2 s2eec000 Toeo O
AU A
1OHM/18lN
Twelfth Symposium
NAVAL
HYDRODYNAMICS
Boundary Layer Stability and Transition
Ship Boundary Layers and
Propeller Hull Interaction
Cavitation
Geophysical Fluid Dynamics
sponsored by the
OFFICE OF NAVAL RESEARCH
the
DAVID W. TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER
and the
NAVAL STUDIES BOARD
of the
NATIONAL RESEARCH COUNCIL
MARINE
BIOLOGICAL
LABORATORY
LIBRARY
WOODS HGLE, MASS.
Vie Tes (CEE
NATIONAL ACADEMY OF SCIENCES
Washington, DC 1979
Partial support for the publication of these
Proceedings was provided by the Office of Naval
Research of the Department of the Navy. The
content does not necessarily reflect the position
or the policy of the Navy, the U.S. Government, or
the National Academy of Sciences and no endorse-
ment should be inferred.
ISBN 0-309-02896-5
Library of Congress Catalog Card No. 79-53803
Available from:
Office of Publications
National Academy of Sciences
2101 Constitution Avenue, N.W.
Washington, D.C. 20418
Printed in the United States of America
PROGRAM COMMITTEE
George F. Carrier, Chairman, Harvard University
William E. Cummins, Vice Chairman, David W. Taylor Naval Shtp
Research and Development Center
Ralph D. Cooper, Office of Naval Research
Stanley W. Doroff, Office of Naval Research
Lee. M. Hunt, Nattonal Research Counctl
Wen Chin Lin, David W. Taylor Naval Shtp Research and
Development Center
Justin H. McCarthy, Jr., David W. Taylor Naval Ship Research
and Development Center
Vincent J. Monacella, David W. Taylor Naval Ship Research
and Development Center
SYMPOSIUM AIDES
Marguerite A. Bass
Yetta S. Hassin
Office of Naval Research
Lavern Powell
David W. Taylor Naval Ship Research and Development Center
Bernice P. Hunt
Joyce L. Wright
Nattonal Academy of Sctences
Grace Masuda
Institute of Medicine
Hope M. Bell
Doris E. Bouadjemi
Beatrice Bretzfield
Mary G. Gordon
Virginia A. Harrison
Debra A. Tidwell
Eva F. Tully
Nattonal Research Counetl
statal,
=
ted
ety
7h Leann
“hin! iy H Terni
Preface
The Twelfth Symposium on Naval Hydrodynamics
was held at Washington, D.C., during the period
5-9 June 1978 under the joint sponsorship of the
Office of Naval Research, the David W. Taylor Naval
Ship Research and Development Center, and the
National Academy of Sciences.
The technical program of the Symposium con-
sisted of eight sessions equally apportioned among
the following four subjects of great current inter-
est in the general field of naval hydrodynamics:
(1) boundary layer stability and transition, (2)
ship boundary layers and propeller hull interaction,
(3) cavitation, and (4) geophysical fluid dynamics.
Tours of the hydrodynamic research facilities of
the David W. Taylor Naval Ship Research and Devel-
opment Center and of Hydronautics, Inc., were also
included in the technical program.
It is interesting to recal that the National
Academy of Sciences was a cosponsor of the First
and Second Symposia in this series which were held
respectively in 1956 and 1958. It is a great plea-
sure to acknowledge once again the invaluable
assistance of the Academy in launching these Sym-
posia and in establishing the high standards of
quality and style for them by which we are guided,
even to this day.
Similarly, the David W. Taylor Naval Ship
Research and Development Center has played an
important role in the series of Symposia on Naval
Hydrodynamics from their very inception. Scien-
tists and engineers from the Center have presented
outstanding scientific papers at each of the Sym-
posia and have, in addition, participated in an
informal manner in the planning of many of the
earlier ones.
For these reasons the Office of Naval Research
is especially pleased and honored at the opportu-
nity presented by the cosponsorship of this Twelfth
Symposium to renew and continue the fruitful col-
laboration with its old scientific allies. We are
deeply grateful for their generous assistance in
the past and present, and look forward with confi-
dence to their continued support in the future.
Of the seemingly endless list of people who
contributed in large and small ways to the planning
and organizing of the Twelfth Symposium the follow-
ing deserve special recognition: Professor George
F. Carrier of Harvard University and the Naval
Studies Board of the National Research Council, who
served as chairman of the Program and Organizing
Committee; Dr. William E. Cummins of the David W.
Taylor Naval Ship Research and Development Center,
who served as vice-chairman of the Committee, and
his colleagues from the .:nter, Dr. Wen Chin Lin,
Mr. Justin H. McCarthy, Jr. and Mr Vincent J.
Monacella, who served on the Committee; Mr. Lee M.
Hunt of the Naval Studies Board, who served on the
Committee and who, with the able assistance of
Miss Virginia A. Harrison, personally carried out
the multitude of detailed arrangements required for
the success of the Symposium; and Dr. Nelson T.
Grisamore of the National Academy of Sciences, who
edited these Proceedings.
A special note of appreciation is extended to
Mr. Phillip Eisenberg, President of Hydronautics,
Inc., for his delightful after-dinner talk at the
Symposium Banquet and for the tour of Hydronautics,
Inc., which he graciously arranged for the partic-
ipants of the Symposium.
To all of these, and many more, the Office of
Naval Research is forever indebted.
Ralph D. Cooper
Office of Naval Research
: il ou , rad y
i i i wana |. Vash i iy
1 J 1 : : 1 i
I _ iV OF y
i it i igi il ‘
Wi we y
1 y \ =! J
I : i ra (ro see t||
i f Bein > Blu,
[ fl i Di Tite 2 it
i ho ee i Uy, ih Aaa seaieet
i i i il ie i L
i _ yee
i ie :
Uy ee fn vy
u H THT Lt ik if 1
hale
ae) Le fant
ipl! ire ie
=i
- yin, a gl tia ‘
11 A Te A ea es a 1 yaa Hatodbtiake
i Ssaihie am
iy
5 be Gia
: ok Lanes th
wih Sa alae
Contents
Preface
Ralph D. Cooper
Introductory Address
Courtland D. Perkins
Introductory Address
Robert K. Geiger
Introductory Address
Myron V. Ricketts
SESSION I: BOUNDARY LAYER STABILITY AND TRANSITION
Stability and Transition Investigations Using the
Navier-Stokes Equations
Hermann F. Fasel
The Physical Processes Causing Breakdown to Turbulence
M. Gaster
The Instability of Oscillatory Boundary Layers
Christian von Kerczek
Heated Boundary Layers
Eli Reshotko
Discussion
SESSION II: BOUNDARY LAYER STABILITY AND TRANSITION
Nonparallel Stability of Two-Dimensional Heated Boundary
Layer Flows
N. M. El-Hady and A. H. Nayfeh
Three-Dimensional Effects in Boundary Layer Stability
Leslie M. Mack
vii
22
25
33
48
53
63
Experiments on Heat-Stabilized Laminar Boundary Layers
in a Tube a,
Steven J. Barker
Some Effects of Several Freestream Factors on Cavitation
Inception on Axisymmetric Bodies 86
Edward M. Gates and Allan J. Acosta
Discussion 109
SESSION III: SHIP BOUNDARY LAYERS AND PROPELLER HULL INTERACTION
Calculation of Thick Boundary Layer and Near Wake of Bodies
of Revolution by a Differential Method ata}
Virenda C. Patel and Yu-Tai Lee
Stern Boundary-Layer Flow on Axisymmetric Bodies 127
Thomas T. Huang, Nicholas Santelli, and Garnell Belt
Theoretical Computation and Model and Full-Scale Correlation
of the Flow at the Stern of a Submerged Body ‘158
A. W. Moore and C. B. Wills
Experimental and Theoretical Investigation of Ship Boundary
Layer and Wake 169
Shuji Hatano, Kazuhiro Mori, and Takio Hotta
A General Method for Calculating Three-Dimensional Laminar
and Turbulent Boundary Layers on Ship Hulls 188
Tuncer Cebeci, K. C. Chang, and Kalle Kaups
Study on the Structure of Ship Vortices Generated by
Full Sterns 209
Hiraku Tanaka and Takayasu Ueda
SESSION IV: SHIP BOUNDARY LAYERS AND PROPELLER HULL INTERACTION
Wake Scale Effects on a Twin-Screw Displacement Ship 225
Arthur M. Reed and William G. Day, Jr.
Influence of Propeller Action on Flow Field Around a Hull 248
Shunichi Ishida
Prediction Of the Velocity Field in Way of the ship Propeller 265
I. A. Titov, A. F. Poostoshniy, and O. P. Orlov :
Recent Theoretical and Experimental Developments in the
Prediction of Propeller Induced Vibration Forces on
Nearby Boundaries 278
Bruce D. Cox, William S. Vorus, John P. Breslin,
and Edwin P. Rood
viii
A Determination of the Free Air Content and Velocity in Front
of the "Sydney-Express" Propeller in Connection with Pressure
Fluctuation Measurements
Andreas P. Keller and Ernst A. Weitendorf
Discussion
SESSION V: CAVITATION
Pressure Fields and Cavitation in Turbulent Shear Flows
Roger E. A. Arndt and William K. George
Secondary Flow Generated Vortex Cavitation
Michael L. Billet
On the Linearized Theory of Hub Cavity with Swirl
G. H. Schmidt and J. A. Sparenberg
Unsteady Cavitation on an Oscillating Hydrofoil
Young T. Shen and Frank B. Peterson
Cavitation on Hydrofoils in Turbulent Shear Flow
Hitoshi Murai, Akio Ihara, and Yasuyuki Tsurumi
Scale Effects on Propeller Cavitation Inception
G. Kuiper
Discussion
SESSION VI: CAVITATION
A Holographic Study of the Influence of Boundary Layer and
Surface Characteristics on Incipient and Developed
Cavitation on Axisymmetric Bodies
J. H. J. van der Meulen
Mechanism and Scaling of Cavitation Erosion
Hiroharu Kato, Toshio Maeda, and Atsushi Magaino
Experimental Investigations of Cavitation Noise
Goran Bark and Willem B. van Berlekom
Cavitation Noise Modelling at Ship Hydrodynamic Laboratories
G. A. Matveyev and A. S. Gorshkoff
Fluid Jets and Fluid Sheets: A Direct Formulation
P. M. Naghdi
Discussion
ix
300
319
327
340
348
362
385
400
426
433
452
470
494
500
516
SESSION VII: GEOPHYSICAL FLUID DYNAMICS
The Boussinesq Regime for waves in a Gradually Varying Channel
John W. Miles
Study on Wind Waves as a Strongly Nonlinear Phenomenon
Yoshiaki Toba
An Interaction Mechanism betwee Large and Small Scales for
Wind-Generated Water Waves
Marten Landahl, Sheila Widnall, and Lennart Hultgren
Preliminary Results of Some Stereophotographic Sorties Flown
Over the Sea Surface
L. H. Holtuijsen
Gerstner Edge Waves in a Stratified Fluid Rotating about a
Vertical Axis
Eric Mollo-Christensen
The Origin of the Oceanic Microstructure
Grigoriy I. Barenblatt and Andrei S. Monin
SESSION VIII: GEOPHYSICAL FLUID DYNAMICS
The Rise of a Strong Inversion Caused by Heating at the Ground
Robert R. Long and Lakshmi H. Kantha
Laboratory Models of Double-Diffusive Processes in the Ocean
J. Stewart Turner
Buoyant Plumes in a Transverse Wind
Chia-Shun Yih; Appendix by J. P. Benqué
Internal Waves
O. M. Phillips
Breaking Internal Waves in Shear Flow
S. A. Thorpe
LIST OF PARTICIPANTS
523
529
541
555
570
574
585
596
607
618
623
629
Introductory Address
Dr. Courtland D. Perkins
President, National Academy of Engineering
On behalf of the National Academy of Engine-
ering and the National Academy of Sciences it is
my distinct pleasure and privilege to welcome you
to our Nation's Capitol, to the home of both Acad-
emies, and to the Twelfth Symposium on Naval Hydro-
dynamics.
We have welcomed the opportunity to join with
the Office of Naval Research and the David W. Taylor
Naval Ship Research and Development Center in organ-
izing and hosting the Twelfth Symposium in this
distinguished series of meetings.
We have, as a matter of fact, a special inter-
est in the continuing success of the series since
we cosponsored the First and Second Symposia with
the Office of Naval Research in 1956 and 1958.
Therefore, it is as gratifying for us as it must
be for the Office of Naval Research to find that
the international community of fluid dynamics and
related specialties continues to find these meetings
a unique forum for the exchange of research results
and the discussion of problem areas of concern to
both military and commercial activities.
The interest and the involvement of the Acad-
emies in naval science and engineering, of course,
has a much longer history. After a careful reading
of the early history of the National Academy of
Sciences, one is persuaded that the Academy would
not have come into being in 1863 had it not been
for the carefully laid plan and persuasive argu-
ments of the Navy's Chief of Navigation, Commodore
Charles Henry Davis. One is further impressed by
the fact that perhaps a quarter of those who signed
the Academy's Charter were affiliated with the Navy
in one way or another. And it is significant that
the first five studies conducted by the fledgling
Academy were requested by the Navy. In case some
of you may be interested, these were:
On Protecting the Bottom of Iron Vessels
On Magnette Deviation tn Iron Ships
On Wind and Current Charts
Sailing Directtons
On the Exploston On the Untted States Steamer
CHENANGO
I don't want to leave you with the impression that
the Academy worked only on naval problems during
the 1863-65 period. We did another study entitled
"On the Question of Tests for the Purity of Whiskey"
--an investigation undoubtedly stimulated by
President Lincoln's remark that he wished he could
supply all his generals with whatever it was that
General Ulysses S. Grant was drinking.
I have taken this short detour through some
early Academy history, not so much to demonstrate
our own long and continuous interest in naval sci-
ence and engineering but to recognize the important
role played by the Navy in supporting science and
engineering throughout its 200-year history. Over
the past 32 years the Office of Naval Research has
continued that tradition by serving as a model for
enlightened government support of basic research.
On a more personal note may I conclude by say-
ing that as a former professor of aeronautical
engineering at Princeton University your technical
program is of special interest to me. Therefore,
I wish you an interesting and productive meeting.
We are pleased that you have chosen to meet at our
institution, and the staff we have assembled to
support you is available to assure that your stay
is a pleasant one.
tee pear Lib
ate
fe Nai
ac
vinenigll
Dd ne te
M ieite 4
(pSaee
a
i! Ne ne
m aa
(a
(ye
ni
Oh ' iw
in
ay ih alk ‘Ani
TT ae viaje hy Nae Fen
Vy, OY
AE y)
MU i
f ]
ee Nowe i
cf Meas
feet
Hint =
Hun
yay ;
i y ’
J nll A
i
f |
‘ Th ’ |
inn i ™ a A i el
( i Lay, i an i ,
Introductory Address
Rear Admiral Robert K. Geiger, USN
Chief, Office of Naval Research
On behalf of the Office of Naval Research I
would like to extend a sincere welcome to all the
participants of the Twelfth Symposium on Naval
Hydrodynamics.
I wish to express my thanks to the National
Academy of sciences for its assistance and role as
a host and cosponsor of the Symposium through its
National Research Council.
Thanks are also due to the third member of the
triumvirate of cosponsors of this, the Twelfth Sym-
posium on Naval Hydrodynamics, namely the David W.
Taylor Naval Ship Research and Development Center,
known more familiarly to most of us old-timers as
the David Taylor Model Basin and often referred to
affectionately as DTMB. This facility has been a
major contributor to the scientific program of each
of the Symposia in this series, as a glance at the
proceedings of any of the Symposia will confirm.
I am happy to say that the present meeting is no
exception and that it is again well represented on
the technical program. However, this is the first
time that it has participated as a cosponsor and I
am especially pleased to acknowledge the invaluable
assistance that our old colleague and ally in the
field of naval hydrodynamics research has rendered
in the organization and management of the present
Symposium.
The first two Symposia of this series were
held in 1956 and 1958 and were also sponsored by
the Office of Naval Research and the National
Academy of Sciences. Many of the guiding princi-
ples that govern the organization of the Symposia
in this series were established in these first
meetings. For example, the selection of a limited
number of central themes of timely naval hydro-
dynamic interest upon which to focus the technical
program of the meeting was introduced in the
Second Symposium.
From the very beginning, the international
aspects of the Symposia were emphasized through the
invitation of speakers from all over the world
wherever outstanding research in naval hydrody-
Namics was going on. Starting with the Third Sym-
posium, the international aspects were strengthened
by locating the meetings outside the United States
and cosponsoring them with relevant organizations
in host countries.
The list of such meetings includes Symposia
held in the Netherlands, Norway, Italy, France, and
England, and we hope to continue this pattern into
the future as long as the series of Symposia con-
tinue to provide a useful forum for the exchange
of valuable information on results of advanced re-
search in the field of naval hydrodynamics.
I am gratified to see so many representatives
of several countries in addition to the United
States, and the number of technical papers pre-
sented by internationally known authorities in
fluid dynamics and related fields.
For the Navy, progress in hydrodynamics re-
search has become increasingly urgent. The Navy
must find ways to discover and correct the problems
that a new design may run into before reaching the
point of full-scale sea trials.
Since the sea is the Navy's business and we
have been involved in it a long time, we are ex-
pected to know it well. Only investigators like
yourselves are aware of how limited is our knowl-
edge of the forces that impact on a buoyant body
propelled through the water. As much as our under-
standing has increased, we know we have much more
to learn. This information can only be obtained
through the arduous bit-by-bit process of basic re-
search, such as you gentlemen pursue.
Today our nation is faced with the dilemma
that we must plan types of ships that are radically
different in design from anything in the past. At
the same time, these ships must be inexpensive to
operate and maintain in addition to satisfying our
traditional standards.
The results of the research that will be re-
ported at this Symposium should help us move toward
that formidable goal. It is clear that all of you
here today are dedicated scientists, so I do not
need to urge you to keep pressing forward in your
search for solutions to the frustrating problems
in hydrodynamics. I would like to stress, however,
that you maintain strong lines of communication so
that as many people as possible can benefit when
you inevitably succeed in your endeavors.
Best wishes for a successful symposium.
Introductory Address
Captain Myron V. Ricketts, USN
Commander, David W. Taylor
Naval Ship Research and Development Center
We at the David W. Taylor Naval Ship Research
and Development Center are both pleased and proud
to join with the Office of Naval Research and the
National Academy of Sciences in sponsoring the
Twelfth Symposiumon Naval Hydrodynamics. While
not a sponsor of the four earlier symposia held in
Washington, the Center was directly and indirectly
involved with all of the previous meetings. Of the
forty-one papers to be presented at the present
Symposium, five are authored by Center researchers,
roughly the same number of papers given by Center
authors at earlier symposia. In addition, much of
the other U.S. research to be presented in papers
to this Symposium was supported by the U.S. Navy's
General Hydrodynamics Research Program which the
Center has administered for nearly thirty years.
It is worthy to note that this year's confer-
ence is directed mainly at the underlying physics
of hydrodynamic processes. The papers are of quite
a fundamental nature, perhaps more so than was true
of many of the earlier symposia. The Symposium
topics are of immense importance to both the mer-
chant ship and naval communities: Boundary Layer
Stabtltty and Transttton because of their relation-
ship to vehicle drag, cavitation inception, and
flow noise; Shtp Boundary Layers and Propeller/
Hull Interactton because a need to accurately pre-
dict vehicle drag, propulsive efficiency, and
vibration; Cavitation, a very major cause of ero-
sion, vibration, and noise; and finally, Geo-
phystcal Flutd Dynamtes which describes the envi-
ronment in which ocean systems must operate. Each
topic area is a subject of current and lively in-
terest and has witnessed remarkable advances over
the past few years.
The very high quality of the research papers
to be presented this week is typical of previous
Naval Hydrodynamics Symposia and has earned for the
series the reputation of being the preeminent inter-
national conferences on ship hydrodynamics. Each
symposium has constituted an exceedingly valuable
open forum which promotes national and international
ties and dialogues between researchers in the field
of hydrodynamics.
I would like to close by saying that my
Center's namesake, Admiral David W. Taylor, the U.S.
pioneer hydrodynamicist and foremost naval archi-
tect, introducer to the U.S. of towing tanks, water
tunnels, transformer of empiricism to scientific
methods, would be very pleased to be associated
with the Twelfth Symposium on Naval Hydrodynamics.
On Wednesday we look forward to welcoming you on a
tour of the hydrodynamic facilities at the Center.
You will see work in progress at our rotating arm
facility, seakeeping basin, towing tanks and turn-
ing basin, and at our largest cavitation tunnel.
Best wishes for a very successful conference.
Pa) : ou i : Toy teat
0 ly { iN i r Ud) had, Lar en
n i 1 i : i : 1 en ee im
on ‘ } enh i y h a i 1 en
: j han Yo iT : , - a i i) Ne
7 ; com, i i 7 TH Dai - un) a
‘s ’ my we Y inet } iy as = 1 - \h ita eat iver
u/ in fet ; ice i Xie ty thi ees Uy fis! | Delf
; i - 7 i i} t ‘a, ae i i Ahh 9 ell \ 1) ee i} thi a, i Jaa fal yey]
' i 10 Oe i at re i eee i 7 " iia, Whey ’ iH
igs verter
Lec!
i Henly!
RE ae
ae eA Fi
‘co
: ieee |
i Lae eee
i
yA Tee ath WER
{ay ie! Be
: Fy
qe ae Ly i
r ut fi fu i
f iy it '
1 ete Th
ee eth
a ja ete uve
i } : p
‘ | Wee
_ eer
f I |
f
i = i
| / l t " 1 ne i
i
j et
! : . 3 ag
i ae I ‘yh ee
; ne
i U 1
1 1 : i i ,
uy
ie! ty
1 \ if 1 /
, ere wu
i f H) I =
1 a 1
Session I
BOUNDARY LAYER STABILITY
AND
TRANSITION
PHILLIP. S. KLEBANOFF
Session Chairman
National Bureau of Standards
Washington, D.C.
eed ih oN
Ae Urea
na 1 at ye
= x x er
‘sl ie aie el
o
ii :
fe hi mn By
qe idl
i
Ly
faly I in
Saimiri
if eed AM
Pathe en
oie
Payee
ii
10 ae Se cc
5
(ee
PAS
moe
i ey My
TIE SALMAN a
Hie ata i
Stability and Transition Investigations
Using the Navier-Stokes Equations
Hermann F. Fasel
Universitat Stuttgart
Stuttgart, Federal Republic of Germany
SUMMARY
With this paper an attempt is made to review the
stability and transition simulations, performed at
the University of Stuttgart, which are based on
finite-difference solutions of the Navier-Stokes
equations. Research in this area has demonstrated
that implicit finite-difference methods for the
solution of the complete Navier-Stokes equations
for unsteady, two-dimensional, incompressible flows
can be successfully applied to investigations of
hydrodynamic stability and to certain aspects of
transition. This approach of numerically solving
the partial differential equations describing the
underlying flow mechanisms promises to be a valuable
aid in transition research. In particular, this
concept may prove to be especially rewarding for
investigations of aspects of stability and transition
which as yet are not feasible with other theoretical
models.
There are two main reasons for the attractiveness
of this approach: Firstly, no assumptions whatsoever
have to be made concerning the basic flow field under
investigation. Thus, for example, all possible
effects resulting from the growth of a boundary layer
in downstream direction can be included in such
investigations. Even strongly converging or diverg-
ing flows, or flows with separation and/or reattach-
ment can be studied. Secondly, no restrictions
have to be made concerning amplitude and form of
the disturbances which are injected into the flow.
Therefore, using larger disturbance amplitudes
certain nonlinear effects of the amplification
process can be readily investigated.
The major aspects of this approach will be dis-
cussed in this paper. Emphasis will be placed not
only on conveying the advantages of such investi-
gations but also on elaborating the difficulties
and shortcomings of such numerical simulations.
Finally, a conjecture concerning the course of
future developments will be attempted.
1. INTRODUCTION
The phenomena occurring in transition from laminar
to turbulent flow have been the subject of inten-
sive research ever since the discovery that these
two entirely different states of flow exist.
From all the research efforts basically only one
universally-accepted theoretical concept evolved,
namely, linear stability theory, verified experi-
mentally by the famous experiments of Schubauer
and Skramstad (1943).
However, experimental evidence has also shown
that linear stability theory is only applicable
for one 'special' transition process, namely,
transition initiated by the presence of very small
disturbances in the flow. In this case a substan-
tial portion of the entire transition process is
indeed well described by this theory, i.e. the
amplification of two-dimensional disturbance waves
(the so-called Tollmien-Schlichting waves) can be
predicted adequately. But even for this special
transition process, triggered by small disturbances,
linear stability theory is inadequate in the
description and investigation of the mechanisms
that follow the growth of Tollmien-Schlichting
waves, and which finally cause the breakdown to
fully turbulent flow. Nevertheless, due to the
relative success of the linear stability theory
and its impressive experimental verification, the
vast majority of theoretical transition investi-
gations were, and still are, based on stability
theory concepts, thus constantly improving and
perfecting this theory.
The inherent shortcomings of this concept
nontheless (such as being applicable only when
transition is initiated by small disturbances, or
that certain assumptions concering the basic and
disturbance flow have to be made to keep the
resulting equations tractible) led to a search for
other means to investigate transition. One of the
more promising concepts that has emerged in recent
10
years is based on direct numerical solutions of the
complete partial differential equations that
describe the flow phenomena arising in the transi-
tion process. This approach became feasible with
the rapid progress in the development of large,
high-speed digital computers.
The main difficulties here arise from the fact
that these flow phenomena can be adequately
represented only when the complete Navier-Stokes
equations (or certain modifications thereof) are
used. Thus, this approach requires the solution
of the Navier-Stokes equations for strongly time-
varying flow fields, due to the highly unsteady
nature of the transition processes. Additionally,
complications increase because the numerical
solutions have to yield reliable results for
relatively high Reynolds numbers (higher than the
critical Reynolds number) to allow onset of
transition. For a numerical solution procedure it
is therefore necessary to allow for adequate
resolution of the large temporal and spatial
gradients resulting from the occurrence of thin
time-varying fluid layers with large gradients
close to solid walls.
The development of finite-difference methods,
which are applicable for such complex, unsteady
flow phenomena as thase occuring in laminar
turbulent transition, is associated with numerous
difficulties which will be elaborated upon in this
Paper. Because of these difficulties relatively
few previous attempts based on such an approach be-
came known. Reasonably successful earlier investiga-
tions of this kind (based also on finite-difference
solutions) are reported, for example, for incom-
pressible flows in a boundary layer [De Santo
and Keller, (1962)], for Poiseuille and plane
Poiseuille flow [Dixon and Hellums (1967), Crowder
and Dalton (1971)] and for a compressible boundary-
layer flow [Nagel (1967)]. These earlier attempts
clearly demonstrated the usefulness and potential
of such investigations. However, due either to
insufficent resolution of the resulting gradients
and/or assumptions made concerning the basic or
disturbance flows, or to shortcomings of the differ-
ence methods used, the results of these calculations
were more of a qualitative nature. Therefore,
relatively little information could be gained
concerning the various phenomena arising in the
laminar-turbulent transition process.
Some years ago, a research effort was initiated
at the University of Stuttgart aiming at the devel-
opment of numerical methods for the solution of the
Navier-Stokes equations which would be applicable
for detailed investigations of various aspects of
Stability and of phenomena occurring in transition.
To date, an effective implicit finite-difference
method has evolved for the calculation of unsteady,
two-dimensional incompressible flows. The ap-
plicability of the numerical method to investigate
stability and two-dimensional transition phenomena
has been demonstrated by realistic simulations of
Tollmien-Schlichting waves. Detailed results of
these calculations are discussed elsewhere [Fasel
(1976)]. With calculations involving large ampli-
tude disturbances [Fasel et al. (1977)] it was addi-
tionally shown that numerical simulations using
the implicit difference method yield results which
enable insight into certain nonlinear mechanisms
of the transition process.
In this paper the major aspects of the numerical
approach using finite-difference methods will be
reviewed and the present state of the developments
discussed. Emphasis will be placed on the advan-
tages of the numerical approach in general and on
directional options chosen for the present method.
Special attention will also be focused on the
difficulties and limitations of such simulations.
2. SELECTION OF THE INTEGRATION DOMAIN
For a numerical solution of the Navier-Stokes equa-
tions using finite-difference techniques a finite
domain in which the equations are being solved has
to be specified. The selection of the integration
domain determines the nature of a physical flow
problem to be simulated, because the boundary con-
ditions required along the boundaries of this domain
determine to a large degree the solution within the
domain. For reasons of simplicity, in the present
studies only rectangular domains of the x,y plane
were considered as depicted schematically in Figures
1 and 2 with the direction of the basic, undisturbed
flow being in the x-direction. Rectangular domains
allow relatively easy application of difference
methods by using simple rectangular meshes. For
example the rectangular domain may be a section of
a boundary-layer flow on a semi-infinite flat plate
(Figure 1) or a section of a flow between two paral-
lel plates (Figure 2).
In selecting the integration domain one has to
consider that boundary conditions must be found for
the 'artificial' boundaries B-C in Figures 1 and 2
and additionally for C-D in Figure 1. These con-
ditions should allow physically meaningful solutions
in the finite domain, i.e. solutions that would be
obtained if the domain were not made finite by
means of these artificial boundaries. Due to the
spatially elliptic (in x,y) character of the Navier-
Stokes equations application of finite-difference
methods requires boundary conditions on all bound-
aries of the x,y domain. Of course, in a mathemat-
ical sense the equations are parabolic because of
the time derivative (See section 3). Selection of
boundary conditions for boundaries representing
solid walls (such as A-B in Figures 1 and 2 and C-D
in Figure 2) generally creates no additional diffi-
culty although consistent implementation in the
numerical scheme is frequently difficult to achieve.
Also, free stream boundaries such as C-D in Figure
1 for the boundary-layer flow can be handled in
satisfactory fashion (see Section 4).
However, the upstream (A-D) and to a larger ex-
tent the downstream (B-C) boundary require special
considerations because the specific treatment of
these boundaries determines the approach to be
taken in a prospective stability and transition
simulation. In selecting the boundary conditions
FIGURE 1.
Integration domain for boundary
layer on flat plate.
FIGURE 2.
Integration domain for plane
Poiseuille flow.
there are basically two different approaches which
lead to entirely different conceptions of the trans-
ition simulation:
1) Use of periodicity conditions at the upstream
(A-D) and downstream (B-C) boundary, i.e.
corresponding disturbance quantities are
equal at the two boundaries for all times.
Here it is assumed that flow phenomena are spatially
periodic in downstream direction where the integra-
tion domain X contains integer multiples of the
spatial wavelength. When the spatial development
is forced to be periodic, the flow responds with a
temporal development. Thus, with this arrangement
the temporal reaction of the flow to an initial
disturbance (at t=0) of the flow field can be
studied. This case corresponds in linear stability
theory to an eigenvalue problem with wave number
a real and frequency 8 complex (6=6,+i6,), i.e.
amplification in time. Figure 3, for example, shows
a typical result of a finite-difference calculation
based on such an approach for a plane Poiseuille
low [Bestek and Fasel (1977) ]. Plotted here is a
time signal for a case which is unstable according
to linear stability theory. The flow is only dis-
turbed once at t=0. After a certain time span,
where considerable reorganization of the disturbance
flow takes place, the disturbances assume a periodic
character with a slight amplification in time-
direction.
The Navier-Stokes calculation for this approach
may be conceived as a means of solving the eigen-
value problem as in linear stability theory, with
a and Reynolds number given and obtaining the fre-
quency 8,, amplification rate 8;, and the amplitude
distribution of the distrubance flow. Of course
these answers could be obtained with considerably
less effort from linear stability analysis. The
advantage of this present approach is, however,
that it can be easily extended to investigations
- FIGURE 3. Temporal development of u'-disturbance at
y/Ay = 3 for initially disturbed flow (small ampli-
tude); spatially periodic case (plane Poiseuille flow).
11
of certain nonlinear effects by merely increasing
the amplitude level of the initial disturbances
[see, for example, George and Hellums (1972)]. An
equivalent study of nonlinear effects formulated
as an eigenvalue problem in a stability theory
analysis would, on the other hand, become consider-
ably more involved.
A major drawback of this first approach is, how-
ever, that it is pratically only applicable for
basic flows that do not vary in downstream direction
(parallel flows), because only then is the period-
icity assumption for the disturbance flow a real-
istic one. Thus, strictly speaking, boundary-layer
flows could not be treated in this manner since they
are basically (although very mildly) non-parallel.
It has been shown that non-parallel effects can
have a strong influence on the stability character-
istics of this flow [Gaster (1974), Saric et al.
(1977) J.
A second, perhaps even more serious disadvantage
of this model is that the disturbance development
in downstream direction cannot be investigated. As
observed in numerous laboratory experiements the
phenomena of transition are not periodic in space
but rather are inherently space dependent. The
disturbance flow may vary rapidly in downstream
direction. This space dependency of the transition
process does not only occur for flows where the
basic flow is already dependent on the downstream
location. It also occurs when the basic flow does
not vary in downstream direction, as was impress-
ively demonstrated experimentally by Nishioka et
al. (1975) for the parabolic profiles of plane
Poiseuille flow between parallel plates. Thus,
this model is not suitable for realistic studies
of transition phenomena.
However, finite difference simulations based on
this approach become considerably less involved
and are less costly in practical execution than for
the second approach discussed subsequently. The
former approach is therefore applicable for funda-
mental investigations of various unresolved ques-
tions in hydrodynamic stability (such as certain
nonlinear effects) or for preliminary studies of
flow simulations based on the approach discussed
below.
2) At the upstream boundary, time-dependent
disturbances are introduced. Use of bound-
ary conditions at the downstream boundary
which allow downstream propagation of the
spatial disturbance waves.
This second approach differs entirely in concept
from the first one. Here, the reaction of the
flow field to the disturbances introduced at the
upstream boundary is of interest, particularly the
spatial developments of the ensuing disturbance
waves. In contrast to the previous approach, this
case corresponds in stability theory to an eigen-
value problem with a complex (a=a,+ia;) and 6 real.
A typical result for a boundary-layer flow of a
calculation based on this concept is shown in Fig-
ure 4. Plotted is the disturbance variable u'
(velocity component in x-direction) versus the down-
stream coordinate x. The downstream development
of the disturbance (in this case amplification) may
be clearly observed. Thus, this approach enables
the calculation of the spatial reaction of the flow
to upstream disturbances, and therefore realistic
simulations of space-dependent transition phenomena
FIGURE 4. Downstream development of u'-disturbance at
y/Ay = 3 for boundary-layer flow disturbed periodically
(small amplitude) at upstream boundary.
as observed in laboratory experiments should be
possible.
For example, realistic numerical simulations of
Tollmien-Schlichting waves (as observed in the
Schubauer and Skramstad experiments) can be per-
formed by using at the upstream boundary A-D per-
iodic disturbances as produced by a vibrating ribbon
in the physical experiments. If the location of
A-D is considered to be somewhat downstream of the
ribbon in the real experiments, eigenfunctions of
linear stability theory may be conveniently used
to disturb the flow in the numerical simulation.
It was shown that the disturbance flow somewhat
downstream of the ribbon is well described by
linear stability theory when amplitudes are small.
The disadvantage of the second approach is that
the development of numerical methods to solve the
resulting mathematical problem is considerably more
difficult than in the first approach. Although in
a strict mathematical sense both problems represent
mixed initial-boundary-value problems, the main
difference between the two concepts is that the
first approach results in a predominantly initial
value problem, where the temporal evolution of an
initially disturbed flow field is calculated.
The second concept leads to a predominantly
boundary-value problem where the spatial reaction
of the flow field (which is also time-dependent,
of course) to disturbances introduced on the left
boundary is to be calculated. In the latter case
difficulties arise from the necessity of finding
adequate downstream boundary conditions which
allow unhindered passage of the disturbance waves
propagating downstream, and properly implementing
them into the numerical method. Since the aim of
this research effort is directed toward realistic
simulations of transition phenomena, emphasis in
the development of finite-difference methods was
placed on methods that were applicable to solving
the mathematical problem resulting from the latter
approach. The remainder of the discussions in this
Paper are therefore also based on this concept.
3. FORMULATIONS OF NAVIER-STOKES EQUATIONS FOR
NUMERICAL METHODS
The Navier-Stokes equations can be cast into various
forms to be used as basis for a finite-difference
method. Each formulation has its inherent advan-
tages and disadvantages. The decision in favour of
a particular formulation has to be governed by the
physical flow problem to be investigated and by the
difference scheme finally used. In most cases, and
also particularly for the present investigations,
such a decision is difficult to make beforehand.
Extensive preliminary numerical experiments are
necessary before a decision can be made in favour
of a particular formulation.
For two-dimensional, incompressible flows the
stream-function-vorticity formulation is most
widely used in numerical fluid dynamics. It is
also a possible choice for the present investiga-
tions. It consists of the vorticity-transport
equation
aw aw dU) 1 3
et NY 1
3 Sipe dy Re 2 (1)
and a Poisson equation for the stream function
Ay = w (2)
~
where A is the Laplace operator, w is defined as
oy = acl ' (3)
TS gee a (4)
With this definition of the stream function the
continuity condition
au ov
ye ee (5)
is satisfied for the continuum equations, however,
not necessarily for the discretized equations. All
variables in Eqs. (1) to (5) are dimensionless;
they are related to their dimensional counterparts,
denoted by bars, as follows
— —_— = oe tu,
alae = 22 = ee = =
mR TREN AUS, Un eae
(pe Oe ee ee | he SE
wa ” 7 {7 UoL ’
where L is a characteristic length, Ug a reference
velocity and Re a Reynolds number (v kinematic vis-
cosity). Thus this formulation represents a system
of two partial differential equations, each of
second order, for the unknown variables w and yw
because u and v in Eq. (1) can be eliminated using
Eq. (4).
A variation of this formulation is the so-called
conservative form for which the vorticity-transport
equation
aw , d(uw) | Oa). Ine (6)
ot ox oy Re
is used instead of Eq. (1). With this formulation
conservation of vorticity is guaranteed for the
continuum equations.
A second formulation of the governing equations
also consists of a vorticity-transport equation (1)
or (6). However, instead of the Poisson equation,
(2), for ~, two Poisson equations for the velocity
components u and v are used
Au = :
¥ (7)
See Dat i'r
Oe ax
which can be derived from the definition of vortic-
ity, (3), using the continuity equation, (5). This
system of partial differential equations for the
w,u,v formulation is of higher order than the w,\W)
system. The higher order allows less restrictive
boundary conditions which is advantageous in appli-
cations to transition simulations as discussed in
Section 4.
A third form of the governing equations is the
so-called primitive variable formulation with the
two momentum equations
du du du dp
—— — Ss ob ——— ee ——
cay ae oinaees: oy x” Re Au
(8)
av dv av dp al
Bc RNs ay OM es 2 St
t roa bod y By ee
(where p = p/pus, with density 0) and a Poisson
equation for the pressure
ise Qe Og oy oe
9
ox oy ax One oY (9)
which is derived from Eq. (8) using the continuity
Eq. (5).
There is also a conservative form of the primi-
tive variable formulation (conserving momentum)
jm . ee) ©) (om). G22 4b.
yt oF qx + dy re ax Re Ae
(10)
ov, Sw , 0) 2 _ 2 , kL _y,
are ox oy hy INS ;
and a Poisson equation in a now different form
32 (u*) a2(uv) a2(v2) aD , Ll
= - oo - 2 - = + A
me ox2 2 ax dy ay2 at Re Dr(tt)
with the so-called dilation term
du Ch
= — + — 2
D x ay (12)
The absence of the dilation terms in a Poisson equa-
tion for the pressure may cause nonlinear numerical
instability, which can be avoided when such terms
are retained (Harlow and Welch, 1965).
Conservative versus Nonconservative Formulation for
Use in Transition Studies
The evaluation of the relative merits of conserva-
tive formulations over non-conservative ones is a
widely investigated subject in numerical fluid
dynamics [Roache (1976), Fasel (1978) ]. Neverthe-
less, satisfactory answers have not yet been found
except for compressible flows for which conserva-
tive formulations are obviously advantageous. One
argument in favour of conservative formulations is
that better accuracy can be obtained. However, for
- incompressible flow problems there are several ex-
amples contradicting this claim. When evaluating
possible advantages of a conservative formulation
13
one has to keep in mind that the respective quanti-
ties (such as vorticity in the w, or w,u,v formu-
lation or momentum for the u,v,p formulation) are
initially only conserved for the continuum equations.
The conservation property may be carried over to
the discretized equations only if certain differ-
ence approximations (in this case, central differ-
ences) are used. For the implementation of the
boundary conditions it is frequently very difficult
or sometimes impossible to employ such difference
approximations required to maintain the conserva-
tion properties for the discretized equations.
For the present investigations, comparison cal-
culations during the early stage of the development
of the numerical method have shown that, for the
W,) or w,u,v systems, almost equivalent accuracy
can be obtained with either formulation. Because
the conservative formulation leads to a somewhat
slower solution algorithm for the solution of the
difference equations, preference was given there-
fore to a non-conservative formulation.
Vorticity Transport (Ww, or W,u,v) versus Primitive
Variable (u,v,p) Formulation
In reviewing literature on numerical simulations
of viscous incompressible flows it is noticeable
that formulations involving a vorticity-transport
equation, rather than the primitive variable form-
ulation, are preferred. The unpopularity of the
u,v,p system is a result of numerable unsuccessful
attempts in applying it to calculations of viscous
incompressible flows. Although a few successful
applications based on the u,v,p system are reported
in more recent literature, there are still serious
arguments against its use for stability and trans-
ition simulations. Difficulties result from prob-
lems associated with the use of a Poisson equation
for the pressure. This equation is often a source
of numerical instabilities, possibly due to difficul-
ties of properly implementing the boundary conditions
for pressure into the numerical scheme. Although
the numerical instabilities could be brought under
control, at least to a degree, (for example by intro-
ducing the dilation terms in Eq. 11) so that solu-
tions could be obtained for steady flow problems,
the inherent inclination of this formulation to
numerical instability still prohibits its use for
transition simulations. Frequently numerical
solutions based on this system are of a slightly
oscillatory nature (although amplitudes are extremely
small) and therefore interaction with oscillations
of the physically meaningful disturbances as oc-—
curring in transition studies cannot be avoided.
For these reasons finite-difference methods de-
vised for investigations of stability and transition
are based on the equations in vorticity transport
form, i.e. either on the w,l) system (Eqs. 1 and 2)
or the w,u,v system (Eqs. 1 and 7). Nevertheless
current efforts are also directed toward develop-
ment of difference methods based on the equations
in primitive-variable formulation. Emphasis is
placed on extreme numerical stability in order to
make this method also applicable for stability and
transition studies. The continuing attraction of
the equations in primitive-variable form results
from the fact that, for the three-dimensional case,
fewer fields of variables have to be stored than
for a vorticity-transport formulation. For the
three-dimensional case, storage requirements are an
14
order of magnitude even more critical than for the
two-dimensional calculations.
Use of Navier-Stokes Equations for the Disturbance
Flow
For stability and transition simulations, the depen-
dent variables, which appear in the different form-
ulations of the Navier-Stokes equations discussed
previously, are those of the total flow, that is,
including both the basic and the disturbance flow.
There is an alternate approach, namely, to decompose
the total flow into the basic flow and a disturbance
flow such that
u=U+tu!’ , v=Vt+v' , p=Ptp' , w=¥+p'. w=lt+w', (13)
where the prime indicates the variables of the dis-
turbance flow and the capital letters denote those
of the basic flow. Substituting relationships (13)
into various forms of the Navier-Stokes equations,
it is possible to rewrite the equations with the
disturbance variables as dependent variables. Sev-
eral terms involving only the basic flow can be
dropped, assuming the basic flow satisfies the
Navier-Stokes equations.
The aspect of directly solving the equations for
the disturbance variables is an attractive one,
since it is the disturbance conditions that are of
interest when performing numerical stability and
transition studies. For this reason this approach
has probably been preferred in earlier attempts.
It also allows for detailed investigations of the
effects of the nonlinear (convective) terms. because,
in a difference method based on this form, the 'lin-
earization' can be conveniently switched on or off.
A careful evaluation of this form of equations,
however, reveals that it also has some major disad-
vantages. The equations in disturbance form contain
several additional terms (involving disturbance
terms with terms of the basic flow) which are not
present in a corresponding formulation for the total
flow. Thus, in finite-difference solutions addi-
tional numerical operations are required. A more
serious disadvantage is that, because of the
additional terms involving the basic flow, the
basic flow quantities have to be kept in fast-
access computer storage to be readily accessible
for the numerical operations in order to avoid ex-
cessive computation times. On the other hand,
using the equations for the total flow the basic
flow quantities are not directly involved in the
solution algorithm. In this case they are only
required for analysis and better respresentation
of the results (for example to determine the dis-
turbance quantities). For this purpose they can
be stored in mass storage of lower speed accessi-
bility.
The availability of sufficient fast-access stor-
age is, even with the latest computer generation,
still a critical limitation for such numerical
investigations of stability and transition. For
large scale simulations involving large numbers of
grid points, use of the disturbance formulation is
prohibitive. For this reason, for the present re-
search effort, use of the equations for the total
flow variables was generally preferred instead of
the disturbance formulation. However, the basic
solution algorithm of the definite-difference method
was developed such that it is applicable with only
minor modifications for either formulation.
4. BOUNDARY AND INITIAL CONDITIONS
The selection of adequate boundary conditions and
the practical implementation into a finite-
difference scheme represents one of the major dif-
ficulties in the development of a finite-difference
model applicable for stability and transition stud-
ies. Difficulties arise from the necessity that
boundary conditions, selected and implemented along
the artificial boundaries (see Section 2) for the
finite integration domain, have to enable solutions
that would be identical to solutions if the govern-
ing equations were solved in the infinite domain.
There is, of course, no way of checking this be-
cause solutions for the infinite domain are not
available. This indicates that, for selecting
boundary conditions, it is necessary to rely on
experience, intuition, and test calculations.
For practical reasons the boundary conditions
at these artificial boundaries have to be such that
physically meaningful results can be obtained with
a relatively small integration domain. The number
of grid points, and therefore computer storage and
amount of numerical operations required for a nu-
merical solution, is directly dependent on the size
of the integration domain. Thus, only with a rela-
tively small domain may the computational costs of
numerical simulations be kept within acceptable
limits. This aspect is of particular importance
during the testing phases of the numerical methods.
There are also other difficulties resulting from
the complicated nature of the governing equations.
For the nonlinear systems of governing equations in
the formulations of Section 3 it is not yet possible
to decide if a given problem consisting of the
governing equations and a set of boundary conditions
is well-posed in the sense of Hadamard (1952). More-
over, it is not obvious whether Hadamard's postulates
for a well-posed problem are adequate to include
physically meaningful solutions only. Additional
difficulties may arise because finite-difference
methods frequently require more boundary conditions
than would be needed for the original differential
formulation if exact solutions were possible
[Richtmyer and Morton (1967) ]. From numerical ex-
perimentation with model equations simpler than the
full Navier-Stokes equations it is known that these
additional 'numerical' boundary conditions are of-
ten a source of numerical instabilities possibly ©
caused by certain inconsistencies. Therefore, one
is confronted with the delicate task of selecting
and implementing the extra conditons (where it is
normally not known a priori which conditions are
the extra ones) in such a way that the numerical
stability of an otherwise stable method would not
be adversely affected.
Initial Conditions
When the simulation of space dependent transition
phenomena is of interest as in the present in-
vestigation the reaction of the flow to disturbances
introduced at the upstream boundary has to be cal-
culated. In this case one may assume an undisturbed
flow as initial condition at t=O enabling the dis-
turbance waves introduced for t>0 to propagate down-
stream into an undisturbed flow field. Denoting
the undisturbed flow field with capital letters the
initial conditions for the w,i) system can be written
as
wW(x,y,0) = 2(x,y) ,
14
W (x,y ,0) = ¥ (x,y) ’ ( )
and for the w,u,v system
w(x,y,0) = 2Q(x,y) ,
u(x,y,0) = U(x,y) , (15)
v(x,y,0) = V(x,y) ,
The undisturbed flow field is obtained by solving
the Navier-Stokes equations for the steady flow.
Of course, for the flow between two parallel plates
the Poiseuille profiles already represent exact
solutions of the Navier-Stokes equations and can
therefore be used directly. For the boundary-layer
flow a solution has to be calculated numerically
by solving the Navier-Stokes equations without the
unsteady termdwft in Eq. (1). The argument could
be raised that in this case Blasius profiles could
be used instead. The differences between the
Blasius solution and a numerical Navier-Stokes sol-
ution are indeed very small. Nevertheless, for
investigations with very small disturbance ampli-
tudes, the differences can be of the same order of
magnitude as the disturbances themselves and there-
fore the transient character of the flow could
become considerably distorted. The boundary condi-
tions used for the calculation of the undisturbed,
basic flow are discussed subsequently in connection
with the conditions used for the calculation of the
unsteady, disturbed flow.
Boundary Conditions
At solid walls (non-permeable, no-slip), such as
boundary A-B of Figure 1 or A-B and C-D of Figure
2, the velocity components vanish
TO , WO - UO 7 WO oc (16)
The vorticity-transport formulations (the u,v,p
formulation will not be discussed further) require
special treatment for the vorticity calculation at
the walls. For the w,i) formulation vorticity can
be calculated from the relationship
2
WO = = (17)
derived from Eq. (2); for the w,u,v formulation
either
Qt) _ OA
a Wisc ime)
derived from Eq. (7b) or
OS = (19)
; resulting from Eq. (3) can be used. Equations (17)-
15
(19) are applicable for the calculation of both the
steady, undisturbed and the unsteady, disturbed flow.
At the upstream boundary A-D the disturbances are
introduced by superimposing onto the profiles of a
basic, undisturbed flow (denoted by subscript B; for
example, Blasius profiles or Poiseuille profiles
could be used for the cases considered in Figures
1 and 2) so-called perturbation functions which are
dependent on y and t only. Thus for the w, formu-
lation we have
w(O0,y,t) = Wply) + Pyly,t) ,
(20)
DOr) = WG) se tera)
and for the w,u,v formulation
w(O,y,t) = SH) Pin Q¥pie)
u(O,y,t) = ugly) + Pyly,t) , (21)
v(O,y,t) = vply) + Pyly,t)
For the calculation of the steady, undisturbed
flow field the perturbation functions in Eqs. (20)
and (21) of course vanish. For simulations of
Tollmien-Schlichting waves, for example, the
perturbation functions are periodic in time where
amplitude distributions (or so-called perturbation
profiles) as obtained from linear stability theory
can be used.
The freestream boundary C-D (Figure 1) for the
boundary-layer flow is an artificial boundary and
requires special considerations as discussed in
Section 2. For both the calculation of the steady
flow and the unsteady, disturbed flow, vorticity is
assumed zero (w'=2=0). For boundary-layer type flows,
vorticity for both basic and disturbance flow (when
disturbances are introduced within the boundary lay-
er) decays rapidly away from the wall and is practi-
cally zero at a distance of two 6 (6 boundary layer
thickness) from the wall.
For the calculation of the steady flow using the
W,u,Vv system suitable conditions for C-D are
U = Ugg (x) (22)
where the freestream velocity Ug (x) may be speci-
fied according to the downstream pressure variation
of the boundary layer flow. A condition for the v
component can be derived from the continuity equa-
tion, ((5))/,,,.using Eq. | (22)
TORU ge LS
dy dx 0 (23)
For the w,i) system a condition equivalent to Eq. (22)
can be used
= = Ugg(x) - (24)
The w',u',v' disturbances decay relatively slowly
in direction normal to the wall. For example, for
Tollmien-Schlichting waves the ' or v' amplitude
at 66*, (for Re*=630, based on displacement thick-
ness 6*) may still be close to 50% of the maximal
amplitude. Therefore Dirichlet conditions (u'=v'=
w'=0) could only be used if the freestream boundary
were very far, for example 506*, from the wall.
16
This would be impractical due to the excessive
amounts of grid points required. On the other hand,
the conditions given below allow a relatively small
integration domain in y-direction. They only postu-
late that the disturbances decay asymtotically in
y-direction. For the w,i formulation such a condi-
tion is
On ae ’
ae - ay : (25)
and for the w,u,v formulation
we att
Onvaa -
(26)
dv! Plea!
dy
where a is the local wave number of the resulting
disturbance waves. Test calculations have shown
that with the conditions (25) or (26), together
with the Dirichlet-type vorticity condition dis-
cussed previously, physically meaningful results
can be obtained when the integration domain in y-
direction includes only two to three boundary-layer
thicknesses.
Selection and implementation of the boundary
conditions at the downstream boundary B-C represents
a very difficult task. These boundary conditions
have to enable propagation of disturbances right
through this boundary, where any effects causing
even the slightest wave reflection have to be
avoided. The conditions found most satisfactory
in this respect are for the w, formulation
Be ae
ro apse
(27)
ay! Oe
Spon ano ata
and for the w,u,v formulation
920! E- Dee '
Roe ans
32u! 2
Dac oteate BOO es te (28)
av! aig Wie
aT atv
Numerical experiments with conditions (27) and (28)
have shown that physically reasonable results are
already possible when, for periodic upstream dis-
turbance input, the length of the integration domain
includes only three to four wavelengths.
For the calculation of the steady flow (for the
boundary-layer flow, for example) boundary. condi-
tions which are compatible with those of the unsteady
calculations are for the w, system
7 = OA SSE SO 4 (29)
and for the w,u,v system
922 92u 92v
ox2 Oke axe Oy og ox2 Y
The boundary conditions Eqs. (27) or Eqs. (28) for
the downstream boundary [also Eqs. (25) and (26)
for the free stream boundary] can be derived assum-
ing neutral, periodic behaviour of the disturbance
flow. However, extensive test calculations have
shown that use of such conditions does not enforce
a strict periodic behaviour of the disturbance flow
near these boundaries. Rather, these conditions
allow damping or amplification of the disturbances
even on these boundaries themselves. These con-
ditions have also proven to be applicable for cal-
culations with periodic disturbance input of large
amplitudes as well as for non-periodic disturbance
input (random disturbances, for example) [see Fasel
et al. (1977) ].
For cases where a is not known a priori it can
be determined interatively. Starting with an ini-
tial guess ao (x) (a is generally a function of x,
of course, although for the derivation of the
boundary conditions it was assumed constant to
arrive at simple relationships) an improved a(x)
can be determined from the resulting disturbance
waves developing in the integration domain. Even
with relatively crude initial guesses ag(x) (for
example ag=0) this interation loop converges
rapidly, and for practical purposes two or three
iterations are sufficient.
There is no formal difference between the bound-
ary conditions (27) and (28) used for the w,W and
w,u,v formulation, respectively. Both sets of con-
ditions specify relationships for the second deriva-
tives in the disturbance variables. Nevertheless a
subtle difference does exist. Condition (27) for
w' implies that (due to the definition of w, Eq. 4b)
for v' a relationship involving the first derivative
is prescribed
—=-a*p'. (31)
This is obviously more restrictive than condition
(28c) where for v' a second derivative is prescribed.
For small periodic disturbances the two sets of
conditions lead to practically the same results,
although the results with the w,W system, together
with conditions (27), exhibit subtle irregularities
near the downstream boundary for the waves propa-
gating through this boundary. The w,u,v system,
together with conditions (28), however become su-
perior to the w, system with conditions (27) when
larger disturbance amplitudes are involved. In this
case, reflection-type phenomena can be observed in
increasing manner at the downstream boundary for the
w,) system. For the investigation of the effects
of a backward-facing step on transition [Fasel et
al. (1977) ] the small vortices traveling downstream
are caught at the downstream boundary when the w,\
system and conditions (27) are used, rendering the
numerical results worthless. Using conditions (28)
with the w,u,v system, on the other hand, allows
smooth passage of these vortices through that bound-
ary. .
For these reasons conditions (28), in connection
with the w,u,v system, have proven to be the best
choice so far in properly treating the downstream
boundary. The relatively small upstream influence
of these conditions can be best demonstrated with
typical results from test calculations. Figure 5
for example, shows a comparison of the disturbance
variable u' for calculations with small periodic
disturbances where first in Eqs. (28) an adequate
value for a (a=35.6, obtained from linear stability
cee oe 40 x/Ax
-0.0004
— a in eq (28) from linear stability theory 'T
—— a=0 in eq (28) y
FIGURE 5. Downstream development of u'-disturbance at
y/Ay = 3 for different boundary conditions at the
downstream boundary (boundary layer on a flat plate).
theory) was used while for the other calculation a
was simply set zero. It is obvious that even with
the poor value for a the upstream influence is re-
stricted to a region of approximately one wavelength,
while the disturbance further upstream is practi-
cally unaffected. This relatively minor upstream
influence can also be observed in Figure 6 where
the amplification curves (for the maximum of u')
are compared for the two cases. The disturbance
amplification further than one wavelength upstream
is practically unaffected by the value used for a
in Eqs. (28).
5. NUMERICAL METHOD
A numerical method for transition studies has to
generally allow for numerical solutions of a
boundary-value problem for the calculation of the
steady flow, i.e. solution of Eqs. (1) and (2) or
Eqs. (1) and (7) (without dw/dt in Eq. 1) with ap-
propriate boundary conditions discussed in Section
4. Further the solution of a mixed initial-boundary-
value problem for the calculation of the unsteady
flow is required, i.e. solution of Eqs. (1) and (2)
or Eqs. (1) and (7) with the boundary conditions for
the unsteady, disturbed flow and initial conditions
discussed in Section 4. The partial differential
equations are of fourth order for the w,) formula-—
tion and of even higher order for the w,u,v-system.
For both formulations the governing equations are
elliptic for the calculation of the steady flow and
parabolic for the unsteady flow. In this paper the
discussion is restricted to application of finite-
difference methods for the solution of the mathe-
matical problems posed.
A difference method for investigations of hydro-
dynamic stability and transition phenomena has to
meet a number of requirements in order to ensure
Lal
Ao
1.5
a ineq.(28) from
linear stability theory
1.0 a=0 in eq (28)
0 10 20 30 40 x/Ax
"FIGURE 6. Amplification curves for maximum of u' for
different boundary conditions at the downstream bound-
ary (boundary layer on flat plate).
17
success. Some of the requirements deemed most
important in this context are as follows:
(i) Stability, convergence
Rigorous mathematical proofs of (numerical) stabil-
ity and convergence for nonlinear problems as dif-
ficult as the one at hand have not been accomplished
as yet. For the present investigation, however,
stability of the numerical method is of fundamental
importance. Numerical instability is frequently
exhibited in form of oscillations which would be
hardly discernible from the physically meaningful
oscillations caused by introduced forced perturba-
tions. Hence, a prospective difference method has
to be highly stable, even for relatively large
Reynolds numbers.
In general, for transition studies of the kind
considered in this paper convergence is also quite
serious. Convergence is not necessarily guaranteed
if for a properly posed problem the numerical scheme
is stable and consistent as is the case for linear
partial differential equations of second order
[Lax's equivalence theorem, see Richtmyer and
Morton (1967)]. However, experimenting first with
small periodic disturbances one can at least empir-
ically check the convergence behaviour of the nu-
merical method by comparing calculations for various
grid sizes with linear-stability-theory results and
experimental measurements. Then for other dis-
turbance inputs, such as large amplitude periodic
disturbances, one hopes that the convergence char-
acteristics do not change significantly.
(11) Accuracy of second order
For these investigations at least second-order ac-
curacy of the numerical method (i.e. the truncation
error of the difference analogue to the governing
equations, initial and boundary conditions at least
of second order) is required to exclude or minimize
undesirable non-physical effects, such as artificial
viscosity, when mesh intervals of practical sizes
are used.
(iii) Realistic resolution of the transient char-
acter of unsteady flow fields
Transition phenomena are of highly unsteady nature,
with the time-dependent behaviour of the flow being
of special interest. Thus, the difference method
has to be such that realistic resolution of the
transient character of such flow fields is possible.
Therefore truly second-order accuracy is also de-
sirable for the time derivative.
(iv) Efficiency with respect to computational
speed and required fast-access storage capacity
Numerical solutions of the complete Navier-Stokes
equations for unsteady flows at high Reynolds
numbers require numerous time-consuming numerical
operations. Therefore computers with large, fast-
access computer storage capacity, reaching even the
limits of modern computer systems, are necessary.
A prospective difference method for transition
simulations has to be extremely efficient, i.e.
maximizing computational speed and minimizing re-
quired computer storage capacity as much as possible,
18
in order to be capable at all of undertaking inves-
tigations of this nature with the computers available
today.
Of the requirements discussed here, numerical
stability is the most stringent one and hence has
to be given most consideration. For this reason
only implicit methods are suitable. Implicit meth-
ods are generally much more stable than their im-
plicit counterparts. For the adequate resolution
of the large gradients, resulting from the strongly
time-dependent flow fields to be investigated, rel-
atively small spatial intervals Ax and Ay are re-
quired. Using explicit methods this could lead to
excessively small time-steps required to maintain
numerical stability. For example, using an explicit
counterpart to the present implicit method, the
time-step, according to a linearized stability anal-
ysis, would have to be more than 100 times smaller
for a practical calculation than when using the
corresponding implicit scheme. To satisfy require-
ment (iv) attention has to be given to making the
implicit difference method extremely efficient and
also to meeting the other requirements discussed
previously.
Experimentation with various implicit difference
schemes suggested that 'fully' implicit schemes are
the most promising for transition studies. 'Fully'
implicit means that all difference approximations
and nodal values for the approximation of governing
equations and boundary conditions are taken at the
most recent time-level. For our fully implicit
method three time-levels are employed to obtain a
truncation error of second order for the time de-
rivative dw/dt in Eq. (1).
For all space derivatives, central difference
approximations with second-order truncation error
are employed. The implementation of the boundary
conditions into the numerical scheme requires
special care so that overall second-order accuracy
can be maintained.
This implicit scheme leads to two systems of
equations for the w,i) formulation and to three
systems of equations for the w,u,v formulation.
These systems of equations can be solved by itera-
tion. Because of the retention of full implicity
the equation system resulting from the vorticity-
transport equation is coupled with the Poisson
equation systems via the nonlinear convection terms.
It is additionally coupled with the systems result-
ing from the Poisson equations via the calculation
of the wall vorticity from Eq. (17) for the w,wW
formulation and from either Eqs. (18) or (19) for
the w,u,v formulation.
A very effective solution algorithm based on
line-iteration has been developed for our method
for this coupled system. It is discussed elsewhere
in more detail [Fasel (1978)]. This solution. algo-
rithm has shown to be equally effective when the
basic equations are transformed to allow for a vari-
able mesh in the physical plane such as, for exam-
ple, to concentrate grid points close to walls where
high gradients are expected. Overrelaxation to
accelerate convergence can be easily implemented as
has been done for several calculations [Fasel et al.
(1977) ]. Another advantage is that the solution
algorithm is readily exchangeable to be applied for
both the governing equations in w,W and w,u,v formu-
lation. This has been successfully exploited in the
investigations of the effects of a backward-facing
step on transition. In this study both formulations
were used in the integration domain; the w, formu-
lation was used in the region containing the corners
of the step which can be treated more conveniently
with this formulation. For the domain bounded by
the downstream boundary the w,u,v formulation was
applied, because it allows use of less restrictive
boundary conditions as discussed in Section 4.
The effectiveness of this solution algorithm can
be best judged by presenting a typical computation
time for a practical calculation. For a periodi-
cally disturbed flow with small disturbance ampli-
tudes, using a 35 x 41 grid and calculating 260
time-steps, the required CPU time on a CDC 6600
is about five minutes, including the calculation
of the steady flow. This is relatively little,
considering that the flow is disturbed at every-
time level and that full implicity is retained in
the numerical method.
6. NUMERICAL RESULTS
The implicity difference method which we have devel-
oped has been subjected to crucial test calcula-
tions to verify its applicability to investigations
of stability and transition. First, the reaction
of the boundary-layer on a flat plate to periodic
disturbances of small amplitudes was investigated
in detail. It was demonstrated that the spatial
propagation of Tollmien-Schlichting waves could be
simulated where comparison of the numerical calcu-
lations with results of linear stability theory and
laboratory measurements showed good agreement. Re-
sults of such calculations for the numerical method
based on the w,u,v formulation are presented and
discussed elsewhere [Fasel (1976) ].
The usefulness of the numerical simulations for
the investigation of two-dimensional, nonlinear
effects was demonstrated by calculating the reaction
of a boundary-layer flow to periodic disturbances of
larger amplitudes. Investigating the propagation
of spatially growing or decaying disturbance waves
in a plane Poiseuille flow (both in the linear and
nonlinear regime) verified that the numerical method
is not limited to boundary-layer flows but rather
that it is equally applicable to other flows of
importance. Finally, numerical investigations of
of transition phenomena in the presence of a two-
dimensional roughness element (backward-facing step)
showed that simulations with this numerical model
allow insight into processes which may possibly be
important for understanding certain transition mech-
anisms. Results of this investigation and of the
investigations mentioned before are discussed in
another paper [Fasel et al. (1977) ].
Because the purpose of this paper is to review
Main aspects of numerical transition simulations,
emphasis here is not on conveying new results or
details of numerical calculations. Rather, results
presented here are intended to be of exemplary
nature and were selected in order to clearly demon-
strate essential aspects of such simulations and
to show what can be expected from such numerical
calculations.
The drawings:in Figures 7 and 8 should facilitate
an evaluation of the potential of such numerical
simulations, and, of course, also point out possible
disadvantages and limitations. Figures 7 and 8
show results for a boundary-layer flow on a flat
plate, disturbed at the upstream boundary with small
periodic disturbances. This case is particularly
Suitable for demonstration purposes. The ensuing
a)
0.0005
IN 44
0.0000 ig
Nj
b)
0.05 AN
v La iY Cc
i
i ly co
Ce ag
‘Ni (C\
0.00 EX /,
—= x/Ay 30 do
FIGURE 7.
b) v', c) w', d) w' (different view).
Tollmien-Schlichting waves that can be studied from
such calculations are thoroughly investigated,
experimentally as well as theoretically, and the
results of these calculations are therefore more
intelligible than those of more complicated phe-
nomena of transition.
For these calculations, based on the w,u,v for-
mulation, the Reynolds number at the upstream bound-
ary is Re*=630. For the periodic disturbance input,
for which perturbation profiles of linear stability
theory gre u ged, the frequency parameter (defined
as F=10 Bo/u, , with disturbance frequency 86) is
We So | Abel Shis case the flow is unstable according
to linear stability theory (the location of the left
boundary corresponds to a point on the neutral curve)
and therefore the disturbances should become ampli-
fied in downstream direction. For the calculations
an egqui-distant grid with 35 points in y-direction
and 41 points in x-direction was used.
In Figures 7 and 8 the function values of the
disturbance flow (obtained by subtracting the
quantities of the basic flow from those of the total
flow) are plotted for all three fields of variables
u',v',w', for which the total flow variables are
directly obtained from the numerical calculations.
To allow simultaneous representation of the func-
- tion values at all grid points a perspective rep-
resentation was chosen where the function values
are plotted versus the downstream coordinate x/Ax
Disturbance variables versus x/Ax and y/Ay (perspective representation) at t/At = 80; a) u',
9)
c) w
i. 0.0015
A
and the coordinate normal to the wall y/Ay. These
perspective representations allow the best possible
qualitative survey of the large amount of data ob-
tained from such calculations.
In Figure 7 the disturbance variables u',v',w'
are plotted for a time instance of t=80At, which
corresponds to a time of two time periods after
initiation of the disturbances at the upstream
boundary. In Figures 7a, 7b, and 7c the view is in
the direction away from the wall, looking slightly
in upstream direction. In Figure 7d the view
is also in the direction away from the wall, look-
ing now, however, downstream. From these figures
the propagation of the disturbance waves into the
undisturbed flow field can be clearly observed.
Figure 8 shows the corresponding drawings for
the three variables u',v',w' at a time instance of
t=250At, that is, more than two time periods after
the disturbance wave reached the downstream bound-
ary. These plots demonstrate that the downstream
boundary conditions work properly. Obviously, the
waves can smoothly pass through this boundary,
causing no noticeable reflections. Even after hun-
dreds of time-steps the flow at and near this
boundary maintains its time-periodic character and
therefore the state of the disturbance flow as rep-
resented in Figure 8 would repeat itself periodi-
cally if the calculations were continued for further
time-steps.
20
a)
0.0000
b)
WS
Wi
p Sr
au
XS
\V7 \
Wey
0.05
FIGURE 8. Disturbance variables versus x/Ax and y/Ay
b) v', c) w', da) w' (different view).
In Figures 7 and 8 the large gradients normal
to the wall of the u' and w' disturbances become
clearly visible (for w' this can be best observed
from Figures 7d and 8d) while v' changes more grad-
ually. The large gradients observable in these re-
sults indicate already the major difficulties and
limitations in numerical simulations of transition
phenomena. In a numerical solution method these
large gradients have to be adequately resolved to
obtain meaningful representation of essential physi-
cal phenomena. For nonlinear disturbance waves re-
sulting from disturbance input with larger amplitudes
[Fasel et al. (1977)] or for other more complicated
transition phenomena the gradients may become even
considerably larger. Using finite-difference meth-
ods of a given accuracy (for example, second order
as for the present method) better resolution can
only be achieved by using additional grid points.
This, however, leads to ever larger equation sys-
tems the sizes of which are limited by computer
storage capacity and computation time.
Some help can be expected from employing vari-
able mesh systems allowing allocation of more grid
points closer to walls, where the gradients are
largest, and using fewer points further away where
gradients are small. This can be best achieved
using coordinate transformations for which test
calculations have shown that sizable savings in
the number of grid points, and also in computation
- 0.0015
c)
Te BN 40
d)
0.0015 4
0.0000
0
(perspective representation) at t/At = 250; a) u',
time, are possible to achieve accuracy comparable
with calculations in an equidistant grid. Addi-
tional improvement may be expected from application
of higher-order accurate difference schemes (higher
than second order) which are presently in the state
of development and about to be used in our numerical
method.
The results shown in Figures 7 and 8 also unveil
the considerable potential and advantages of such
numerical simulations. The finite-difference so-
lutions produce a bulk of data, i.e. the values of
the variables directly involved in the solution
procedure are obtained for all grid points and for
all time-levels that are calculated. The data can
be conveniently stored on mass storage devices,
such as magnetic tape (used for the present calcu-
lations, for example). The data stored can be
processed immediately or at any later data to ob-
tain any specific information desired, or to produce
additional data that might be deemed necessary for
a more detailed evaluation of particular flow phe-
nomena. For example, they can be used to obtain
frequency spectra, Reynolds stresses, energy bal-—
ances, amplitude distributions, or to produce con-
tour plots (equivorticity lines, stream lines) etc.
Another positive side of such numerical simulations
is that if the data would be destroyed or lost, they
could be reproduced identically, which would be
hardly possible in comparable laboratory experiments.
7. CONCLUDING REMARKS
The objective of the present review was to discuss
possible approaches to numerical simulations of sta-
bility and transition based on numerical solutions of
the Navier-Stokes equations using finite-difference
methods. The approach, allowing investigations of
spatially propagating disturbance waves, mainly
elaborated upon in this paper, appears most promis-—
ing for realistic numerical investigations of physi-
cal phenomena occurring in transition. The immense
amount of reproducible data obtained from such cal-
culations allows detailed information of any part
of the flow field which may be helpful to gain in-
sight into essential mechanisms occurring in tran-
sition.
The restriction of the numerical model to two-
dimensional flows has also a positive side. With
this model truly two-dimensional numerical experi-
ments can be performed while in laboratory experi-
ments it is always difficult to completely exclude
unwanted three-dimensional effects. Of course the
later stages of transition are inherently three-
dimensional in nature and therefore for a study of
these later developments a three-dimensional model
would be desirable.
The main difficulties and limitations of such
simulations result from the large gradients which
occur in the transition process. For adequate
resolution of the large gradients which become even
larger for more complicated phenomena, increasing
numbers of grid points are required which may lead
to excessive requirements of computer storage and
computation time.
In spite of these difficulties the number of
numerical simulations of transition, similar to
the approach discussed in this paper, is likely to
increase due to the enormous potential inherent in
such investigations. Emphasis will probably be on
the development of difference methods with higher
accuracy which are applicable for such studies.
Additionally, increasing use of numerical methods
other than finite-difference methods is likely,
such as spectral methods or finite-element methods.
Finally, with continuing progress in the develop-
ment to high-speed digital computers, detailed
quantitative investigations of three-dimensional
transition phenomena will probably become feasible
in the near future.
This research is supported by the Deutsche
Forschungsgemeinschaft, Bonn-Bad Godesberg, con-
tract Ep 5/7.
REFERENCES
Bestek, H., and H. Fasel (1977). Ein numerisches
Verfahren zur Untersuchung angefachter, kleiner
Storungen bei der ebenen Poiseuille-Strémung.
21
Paper presented at the GAMM-Meeting 1977 in
Copenhagen/Denmark. To be published in ZAmMM 58
(1978).
Crowder, H. J., and C. Dalton (1971). On the sta-
bility of Poiseuille flow in a pipe. J. Comp.
Phys. 7, 12.
De Santo, D. F., and H. B. Keller (1962). Numeri-
cal studies of transition from laminar to tur-
bulent flow over a flat plate. J. Soc. Ind.
Appl. Math. 10, 569.
Dixon, T. N., and J. D. Hellums (1967). A study on
stability and incipient turbulence in Poiseuille
and plane-Poiseuille flow by numerical finite-
difference simulation. AIChE J. 13, 866.
Fasel, H. (1976). Investigation of the stability
of boundary layers by a finite-difference model
of the Navier-Stokes equations. J. Fluid Mech.
7a, S85.
Fasel, H., H. Bestek, and R. Schefenacker (1977).
Numerical simulation studies of transition
phenomena in incompressible, two-dimensional
flows. Proc. AGARD Conf. on Laminar-Turbulent
Transition, Lyngby, Denmark 1977. AGARD-CP-224,
Paper No. 14.
Fasel, H. (1978). Recent developments in the
numerical solution of the Navier-Stokes equa-
tions and hydrodynamic stability problems. Proc.
VKI-Lecture Series "Computational Fluid Dynamics",
March 13-17, Brussels, Belgium.
Gaster, M. (1974). On the effects of boundary-
layer growth on flow stability. J. Fluid Mech.
66, 465.
George, W. D., and J. D. Hellums (1972). Hydro-
dynamic stability in plane Poiseuille flow with
finite amplitude disturbances. J. Fluid Mech.
Bil (7) 5
Hadamard, J. (1952). Lectures on Cauchy's Problem
in Linear Partial Differential Equations. Dover.
Harlow, F. H., and J. E. Welch (1965). Numerical
calculation of time-dependent viscous imcom-
pressible flow of fluid with free surface.
Fluids 8, 2182.
Nagel, A. L. (1967). Compressible boundary layer
stability by time-integration of the Navier-
Stokes equations. Boeing Scientific Research
Laboratores,:'Flight Sciences Report No. 119.
Nishioka, M., S. Iida, and Y. Ichikawa (1975). An
experimental investigation of the stability of
plane Poiseuille flow. J. Fluid Mech. 72, 731.
Richtmyer, R. D., and K. W. Morton (1967). Differ-
ence Methods for Initial Value Problems. Inter-
science Publishers, Second Edition, New York.
Roache, P. J. (1976). Computational Fluid Dynamics.
Hermosa Publishers, Albuquerque.
Saric, W. S., and A. H. Nayfeh (1977). Nonparallel
stability of boundary layers with pressure
gradients and suction. Proc. AGARD Conf. on
Laminar-Turbulent Transition, Lyngby, Denmark
1977. AGARD-CP-224, Paper No. 6.
Phys.
The Physical Processes Causing
Breakdown to Turbulence
M. Gaster
National Maritime Institute
Teddington, England
I want to present some recent experimental observa-
tions that provide further insight into the physical
processes that occur in the transition from a lami-
nar to a turbulent boundary layer. We know that
external disturbances, such as free-stream turbu-
lence and sound, excite small pertubations in the
laminar flow, and that under certain conditions
these may develop downstream in the form of growing
wave trains. At low pertubation levels these un-
stable travelling waves are adequately described
by the linearised equations of motion. Measure-
ments on weak artificially excited waves have, by
and large, provided excellent confirmation of linear
theory. Far downstream the amplitudes of the per-
tubation velocities will, however, become too large
for the neglect of the non-linear terms to be valid,
and a non-linear description of the motion is nec-
essary. Even in the relatively simple situation
of the constrained parallel Poiseuille flow, which
has been extensively studied, the non-linear the-
ories so far developed can only weakly describe
non-linear events, and even then the computations
are very involved. These non-linear theoretical
models are nevertheless very helpful in describing
the various interactions between the fundamental,
its harmonics, and the mean flow, but they cannot
go far toward providing a model of the process of
breakdown to turbulence, nor are they intended for
that purposes.
Non-linear analyses have been concerned mostly
with the evolution of purely periodic wave trains.
In the case of linear problems it is quite proper
to consider any disturbance in terms of its Fourier
elements. Knowledge of the behaviour of purely
periodic wave trains enables more complex distur-
bances to be described. Unfortunately this is not
the case when the disturbance is non-linear, and
the welcome simplification obtained by breaking down
a problem into harmonics is no longer valid. When
the initial disturbances arise from natural rather
random stimuli the linear wave train will initially
consist of a band of unstable waves. After some
22
amplification a slowly modulated almost sinusoidal
oscillation will inevitably develop. When the
selective amplification is very large, as is the
case in many boundary layer flows, the modulations
are slow, and it does not seem too much of an
idealisation to treat the non-linear problems
analytically as if it were a purely regular wave
train. It turns out, however, that the degree of
modulation does not have to be large for its in-
fluence on the Reynolds stresses and thus the 'mean
motion' to be very significant. In a typical ex-
periment on a laminar boundary layer over a flat
plate in a low turbulence wind tunnel one finds
that the instability waves are modulated suffi-
ciently to influence the transition process. It
is found, for example, that breakdown to turbulence
occurs violently and in a random manner quite un-
like the type of breakdown that is observed in
controlled periodic wave trains. Measurements on
isolated wave packets also show the effect that
modulation of the wave train has on transition, but
in a more controlled way.
Previously reported measurements [Gaster and
Grand (1975)] on artifically excited wave packets
showed consistent and quite well defined deviations
from the structure predicted by linear theory.
Since the maximum level of the velocity fluctua-
tions measured lay below that for which significant
non-linearity is exhibited by regular periodic wave
trains, the reason for this behaviour was at that
time unclear. In the experiments only one level
of input excitation was used and so there was no
direct way of assessing the importance of the non-
linear terms. These experiments have been repeated
at the National Maritime Institute with various
levels of input excitation and it has now been con-
clusively established that the previously observed
warping of wave fronts and the non-Gaussian char-
acter of some of the hot-wire signal envelopes arose
from non-linearity. This behaviour can best be il-
lustrated by showing a comparison of the hot-wire
signals that arise: (a) with a sinusoidal input,
and (b) a pulsed input. As in the previous series
of experiments the boundary layer flow was excited
by an acoustic device mounted in a recess on the
reverse side of the flat plate. A small hole
through the plate provided the necessary fluid
dynamic coupling at a point on the boundary of the
working face. Figure 1 shows a set of hot-wire
anemometer records taken with the probe mounted
just outside the boundary layer one metre down-
stream of the leading edge. The exciter was driven
sinusoidally at four different amplitude levels
increasing from (i) to (iv). The velocity fluc-
tuations appear to be regular and show no harmonic
or other distortion until the level of turbulence
intensity exceeded 1% peak-to-peak of the free-
stream velocity (see iv). Exciting the flow with
isolated pulses on other hand, produces a some-
what different picture. Figure 2 again contains
four hot-wire records obtained with different
levels of drive applied impulsively. At the lowest
level shown the signal consists of a smooth roughly
Gaussian packet of ripples, but even a small in-
crease in driving amplitude produces a clearly
discernible distortion to this signal. These dis-
tortions are similar to those obtained in the
earlier experiments quoted. As the amplitude is
further increased the signal becomes increasingly
distorted until at some level a secondary burst
of relatively high frequency oscillations appears.
It should be remarked that the amplitude scaling
on both Figures 1 and 2 are identical, showing that
non-linear effects occur at much lower amplitudes
for the impulsively applied disturbance than for
a periodic one. In these particular experiments it
appears that non-linearity becomes apparent in the
hot-wire signal at a peak to peak amplitude of only
1/sth that for a continuous wave train.
The high frequency oscillation appears to be
associated with a steep shear layer that forms
within the velocity profile momentarily as the
wave packets sweep past the measuring station.
These shear layers initially appear on either side
of the centre line, and not surprisingly therefore
the peak levels of the high frequency secondary
oscillation also arise off centre at roughly these
locations. The high frequency waves grow rapidly
with downstream distance, initially developing
exponentially but later the growth levels off. At
that stage the filtered secondary wave packets were
observed to distort in a way reminiscent of the
c— 0,05 Seeso =
FIGURE 1. Hot-wire Signals from Sinusoidal excitation.
23
p————— (0) SESS) ——$_—$$_—
FIGURE 2. Hot-wire Signals from pulsed excitation.
primary wave packet. It was therefore conjectured
that there might be yet a further level of insta-
bility on the secondary wave oscillations when
these became sufficiently large. Just two days
before leaving for this meeting this idea was
tested. Hot-wire signals from appropriate regions
of the flow were filtered to see whether there was
any signal above the frequency of the secondary
oscillations. When the secondary wave amplitude
was large, a burst of high frequency oscillations
could be seen on the oscilloscope. Figure 3 shows
the result of applying a high-pass filter, set to
pass above 2 kHz, to such a hot-wire anemometer
Signal. The time scale of this record is con-
siderably expanded compared with that of Figures 1
and 2, and shows that the oscillation frequency in
the burst was around 5 kHz. The basic primary
wave packet of roughly 150 Hz developed over 1m
before breaking and supporting a secondary burst
of 1 kHz. This secondary instability grew in am-
plitude to levels large enough to indicate the in-
fluence of non-linearities in a distance of roughly
4 cm. The tertiary mode of 5 kHz detected at this
stage seems likely to grow even more rapidly. One
can only presume that further stages in this evo-
lutionary process are inhibited by viscosity.
These experiments on the non-linear wave packet
and its breakdown to turbulence are as yet incom-
plete and it is my purpose here to indicate only the
r———— 04005 Seess, ——_———
FIGURE 3. High frequency burst.
24
most important features of the process. Firstly,
a clear demonstration of the difference between
a purely periodic wave train and a modulated train
as far as the level at which non-linear effects
occur is presented. The local breakdown observed
in the wave packet case is similar to that observed
in the breakdown of the modulated wave trains that
arise from natural random excitations. Secondary
breakdown does of course also occur on large enough
periodic waves, but modulation seems to cause this
phenomenon to take place at somewhat lower levels
of primary disturbance and in a slightly different
form. The artifically driven wave packet embodies
some of the most important features found in natu-
rally occurring waves, and since they can be gen-
erated in a controlled manner the effects can be
quantified. It is essential to understand this
process if one is going to make estimates of where
transition occurs on the basis of the amplitudes
of instability waves calculated from linear theory.
At present, most prediction methods rely solely on
the intensity of the most unstable wave. This is
clearly inadequate as breakdown is also dependent
on the modulation of the wave train, and consequently
the bandwidth of the amplified part of the spectrum
must also be taken into account in some way yet to
be established.
Secondly, the transition from regular waves to
turbulence appears to occur through a cascade pro-
cess. The stresses induced by a modulated wave
train cause steep shear layers to form in the bound-
ary layer. These support instabilities of higher
frequencies and shorter wavelengths than the waves
that caused the distortions, and these grow to large
amplitudes in appropriately shorter distances. This
process must at some stage be tempered by viscosity,
but in these experiments three levels of instability
have been so far detected. The lowest frequency
motion was artifically excited by the input pulse,
while the two successviely higher frequencies were
excited by random turbulence in the flow at the
particular location in space and time where local
instabilities existed. The development of a fine
scale structure is thus a local, almost explosive,
phenomenon. Such a cascade breakdown process pro-
vides the necessary mechanism for the generation of
fine scale motions that arise in a fully turbulent
flow.
REFERENCE
Gaster, M., and I. Grant (1975). An Experimental
Investigation of the Formation and Development
of a Wave Packet in a Laminar Boundary Layer,
Proc Ris) SOC Lond. A\ 34772 53—269%
The Instability of
Oscillatory Boundary Layers
Christian von Kerczek
David W. Taylor Naval Ship Research and Development
Center, Bethesda, Maryland
ABSTRACT
The instability of the two-dimensional flat plate
oscillatory boundary layer induced by a stream with
velocity U, + U; cos wt is considered. The velocity
amplitudes, U, and U;, are constants and U)/U, is
assumed to be small. The instability of this oscil-
latory boundary layer is analyzed by a time-dependent
linear parallel flow instability theory. The change
of the Tollmien-Schlichting growth rates due to the
imposed oscillations are computed to second order in
U|/U,- It is found that for imposed oscillation
frequencies in the range of the Tollmien-Schlichting
frequencies of the underlying Blasius flow, the
boundary layer is stabilized by the oscillations of
the external flow.
1. INTRODUCTION
In this paper, we study the instability of the two-
dimensional oscillatory laminar boundary layer which
forms on a flat plate that is exposed to a stream
with a velocity, U, + U; cos wt, perpendicular to the
plate's leading edge. The velocity amplitudes, U,
and U;, are constants, w is the angular frequency of
the oscillation, and t denotes time. The considera-
tions of the instability of oscillatory flows has
become an important field of research in recent years
and has been reviewed by Davis (1976). The partic-
ular class of problems concerned with the instability
and laminar-turbulent transition of oscillatory bound-
ary layers has been reviewed by Loehrke, Morkovin,
and Fejer (1975). The latter review indicates that
very few studies of instability and transition have
focused directly on the subject of oscillatory bound-
ary layers. Such studies that have concentrated on
oscillatory boundary layers have been mainly experi-
mental investigations which were restricted to low
frequency oscillations compared to the oscillation
frequency of unstable Tollmien-Schlichting waves.
The only analytical work concerning the instability
25
of oscillatory boundary layers has been the quasi-
steady analysis of Obremski and Morkovin (1969) which
was aimed at these low frequency cases.
The study of the instability of oscillatory
boundary layers has technological as well as funda-
mental importance. Examples of a fundamental nature
for which the study of the instability of oscillatory
flows may have relevance are the problems of how
ambient disturbances affect the instability of the
underlying steady boundary layer. Specific examples
might be the effects of ambient acoustic waves or
ambient turbulence on steady boundary layer insta-
bility. The problem of the effects of ambient tur-
bulence on the instability of a steady boundary layer
probably is not completely accessible by the theory
of the instability of oscillatory boundary layers.
However, a sufficiently complex, but organized, am-
bient oscillation may be adequate for duplicating
some aspects of the effects of ambient turbulence
on steady boundary layer instability. We are hope-
ful that this may be the case because of similar
phenomena in the field of nonlinear ordinary dif-
ferential equations. The study of the instability
of forced periodic solutions of nonlinear ordinary
differential equations has furnished a much richer
class of phenomena than the corresponding study of
the instability of only the steady solutions of these
equations [see, for example, Hayashi (1964); in
particular, the results for the forced van der Pol
equation, pp. 286-300].
In the present study, we focus on the very simple
oscillatory boundary layer that was described ear-
lier. The purely oscillatory part of this boundary
layer is approximated by the oscillatory Stokes layer
which has no spatial structure in the plane of the
plate, i.e., it is an exactly parallel flow. Thus,
this model problem may be too simple to reveal any
particularly important features of realistic ambient
disturbances. However, the model problem is a good
starting point and serves as a basis on which to
develop the appropriate methods of analysis for the
26
instability of oscillatory flows. We will be con-
cerned mainly with moderate and high frequency os-
cillations comparable to the oscillation frequencies
of unstable Tollmien-Schlichting waves. Thus, a
direct comparison of our results with the low-
frequency experimental results cited by Loerke,
Morkovin, and Fejer (1975) will not be possible.
The method used here for analyzing the instability
of the oscillatory boundary layer is a combination
numerical and perturbation method [Yakubovich and
Starzhinskii (1975)]. In this method, the changes
in the amplification rates of the free disturbances
of the underlying steady boundary layer are computed
as perturbation series in the amplitude parameter,
U,/Up, for any positive value of the frequency, w.
Certain resonant and combination frequencies are of
particular interest. The numerical method used here
to evaluate the perturbation series allows the ef-
ficient and easy generation of many terms of the
series.
The plan of this paper is as follows: In Sec-
tion 2, we formulate the basic flow whose instability
is to be examined along with the associated theory
instability problem. Section 3 outlines the solu-
tion method. Section 4 discussed the numerical
results. Some concluding remarks concerning the
instability of womewhat more complex oscillatory
boundary layers are contained in Section 5.
2. INSTABILITY THEORY
The basic flow field whose instability is to be in-
vestigated is the oscillatory boundary layer formed
on a flat plate in a unidirectional stream with
speed U, + U, cos wt perpendicular to the leading
edge of the plate and parallel to its plane. Let
the cartesian coordinate frame (x,y,z) be placed
with its origin in the leading edge of the plate,
the x-axis pointing downstream parallel to the plate,
the y-axis perpendicular to the plane of the plate
and the z-axis pointing in the spanwise direction.
For values of the parameter, (ww/Uy) >> 1, the
ratio, 8) = 6/65, of the boundary layer thickness,
6 = Yxv/U,, to the oscillatory Stokes layer thick-
ness, 5, = VY2v/w, is large and the oscillatory
boundary layer resulting for small values of A =
U)/U, can be approximated well [see Ackerberg and
Phillips (1972)] by the sum of the Blasius profile
Up (y) [see Rosenhead (1963), p. 225] and the Stokes
layer profile Ugly, t) [Rosenhead (1963), p. 381].
Let us scale the x- and y-coordinates by the
local value of the displacement thickness
6x = 1.7208 Y¥xV/U (1)
Then the transverse coordinate, n, is defined by
nN = y/éx and x' = x/éx. The time scale is §*/U,
so that dimensionless time is t' = tU,/5* and hence-
forth the primes will be dropped. Then the basic
oscillatory boundary layer profile is given ap-
proximately by
U(n,t) = fp(n) + pre [ite G7 ET 1] (2)
where f£,(n) is the Blasius profile, B = 6x/5o, Q =
wS*/Uo = 282/Rs,, and Rey = U,d*/v.
We shall consider the instability of the basic
flow (2) in a similar manner to the standard two-
dimensional linear instability theory for steady
boundary layers. In particular, the quasi-parallel
temporal instability theory as outlined by Rosen-
head (1963) is followed. The restriction to two-
dimensional disturbances can be justified based on
an extended version of Squires' theorem [see von
Kerczek and Davis (1974)]. The perturbation ve-
locities (u,v) are determined from the stream func—
tion, W(x,n,t) = $(n,t)er*
(3a,b)
: ia
v = Re ae Reiave e
ax
The disturbance equation for the perturbation ve-
locities is then given by
one Stay one oo _ 840
ae EP = Ra 22, + ia (use ane eg (4)
*
where £ = 32/an2 = a2, i = V-1, Re(a) denotes the
real part of a, and a is the wave number of the
sinusoidally varying disturbance in the x-direction.
The boundary conditions are
= OF = (0, ens fy) = © (5a)
an
and
gr = >0O asn>®. (5b)
By analogy with Floquet theory for ordinary dif-
ferential. equations with periodic coefficients
[Coddington and Levinson (1958)], we seek solutions
of (4) and (5) in the form
O= Haney on (6)
where g(n,t) is a periodic function of t with period
27/Q. This is a reasonable choice of solution be-
cause we are mainly interested in the oscillation
induced changes of the principal disturbance mode
of the Blasius flow. The principal disturbance mode
of Blasius flow has multiplicity one.
We shall adopt in this study an absolute defini-
tion of instability which requires that some measure
of the disturbance amplitude becomes infinite as
t >, If the amplitude remains bounded as t >,
then the flow is defined to be stable to infinites-
imal disturbances. However, we must keep in mind
that the local instantaneous amplitude may be im-
portant in this linear theory because a disturbance
may be transiently so large (but bounded) that the
linear instability theory is no longer valid. Fur-
thermore, the instantaneous magnitude as a multiple
of the initial magnitude of the disturbance is an
important quantity for assessing the likelihood of
transition from laminar to turbulent flow. Thus we
shall consider in detail the gross amplification
rate G of a disturbance which we define by
ann
en dt
G= (7)
where em is the total energy of the disturbance de-
fined by
27
efficients, a -a1+++/Ay, are determined by the
boundary conditions once 4 is known. The matrices,
Q,P,J, and V, are the respective representations of
the operators ,£,£7, (fp£-9°fp/dy") and (f-3%df/ay2) ,
together with the boundary conditions (6) in the
space, y, whose basis is the first N Chebyshev
polynomials, T),..., Ty-1 lOrszag (1971)]. The
function, of, is the Stokes layer profile.
1/a
(u2 + v2) axdn (8)
sig
Q =
ae
OSS
co
ae
fe)
Then the relative amplification ratio, ep, /en POs
a disturbance as it grows during the time intérval
from t, to t); can be shown to be
a : =
-(1+ §
a. ao) 5, is ra oe CE esa
aie = TICE) exp 2 A at (9) )
8 ¢ cS Note that the matrices, Q,P, and J, are real constant
matrices and V is real and time periodic and of the
form
ag|2 (1) int (=1) -int
if - =i
u(e) = ff (= + a*|g]* } an (10) VS Ye FETS (15)
°
(1) (-1) ;
where V and V are constant matrices.
and rr = Re(X).
Since the disturbance energy vropogates down the
boundary layer at the group velocity, c. [see
Gaster (1962)] one can compute the aanaelye ampli-
fication ratio, en /ep , by calculating the integral
The matrix, Q, is invertable so that we can
multiply (13) by Ont to get
3 ; ; da iL : Sy =
in the exponential function of Eq. (9) over the as ae (P'+iaJ')atiaAv'a (16)
spatial interval, X) to x], using the transformation, Sy
dx = cgdt. a ,
g where P' = Q Ip etc.; henceforth we shall dispense
with the primes in (16)
The perturbation procedure is most easily and
illuminatingly carried out by transforming (16) so
that the matrix, (PtiaJ)/R , 148 in diagonal form.
That is we will be working directly in the (approx-
imate) eigenspace of the steady Orr-Sommerfeld
equation for Blasius flow. Suppose that the in-
e(Aota1At.--)t (11) vertible matrix, B, transforms (P+iad) /Re into
diagonal form. Then let
3. SOLUTION OF THE DISTRUBANCE EQUATION
Solutions of Eq. (4) in the form (6) can be obtained
as a series in A,
g= (g +g, +- 5.6)
where each term of (11) can be evaluated by solving 5S (17)
appropriate perturbation equations obtained by sub-
stituting (11) into (4) and (5). Such perturbation and substitute (17) into (16) and left-multiply by
equations are basically inhomogeneous unsteady Orr- Bethe
Sommerfeld equations and must be solved numerically.
Our approach is equivalent to this except we reverse ab 2 2s
the procedure by first executing a numerical pro- ae = Db + AEb (18)
cedure which reduces the Eqs. (4) and (5) to a sys-
tem of ordinary differential equations in time. where
These are easily solved by perturbation theory to as
high an order as desired. Salt F
Let us first expand the function ¢ in the Cheby- DSB (pew) 8) 6 Aire +s Ayey | (19a)
shev series o
x 1 (GX) calighs (Si), Sale
OW, = » a (e)0 (y) (12) i} SS gratis}! “Wyss = in) e +E e (19b)
n mea
n=1 and the notation, [an gosor ad, |: stands for a di-
agonal matrix of order n.
where the Day) = cos7!(n cos y), n = 0,1,...are the The problem is now to find solutions of (18) in
Chebyshev polynomials of the first kind and where the form
we have mapped the interval, neo, no), onto ye[-1,1] . = » Ne
Then we use the t-method as described by Orszag b(t) = z(t)e (20)
(1971) to obtain the system of ordinary differential
equations where Z(t+27/2) = z(t). We are mainly interested
in perturbations of magnitude A of the steady flat-
dawn! 7 Spa = plate disturbance mode which becomes unstable far
2 Ge ~ See ane CHLOE: cS) downstream of the leading edge. This mode is as-
8 sociated with one of the eigenvalues of D, say A_,
which for values of x between the two values, a <
where Q,P,J and V are (N-4)x(N-4) term matrices and X], Satisfies ReA, > 0. It is known that Ap is a
a = (a),-.-,a eae The dagger (+) superscript de- simple eignevalue [see Mack (1976)] so that a solu-
notes the transpose of a vector or matrix. The co- tion of the form (20) can be expanded as
28
Z(t) Sze (e)) EVAZ (CE) INWIAN(S) 855, | (Amel)
A= A, + Ao) + AZ Gio) hace (21b)
Substituting these two expansions into (20) and
(18) and equating the terms of equal order in A
yields the set
dz,
Sas (DA oe = 0 (22)
dz, “ *
Sere a Dane = (E-o,1) 2, (23)
dz, m v a
Sra ea (D-ApI)z, = (Gg, We = Rae (24)
ercr
Note that the constant coefficient matrix of
these equations is
1D)) = att = ly gosoniio |
where Y3 = A; - Ano) S pooap NY Sil o de
The only 27/2 periodic solution that is possible
for Eq. (22) when (A5--Ap) FEM roe SW OPI APS oocborel
j *#oe is the solution
B. (9 4) (25)
where 6 is the Kronecker delta and c is an arbi-
trary complex constant. This statement is merely
a restatement of the fact that the eigenfunction
corresponding to the eigenvalue, A,, is the p-th
column of matrix B, i.e., the least stable eigen-
mode of the underlying steady Blasius flow.
Since the solution (20) requires that z(t) be
periodic with period 27/2 in t, we shall need the
inner product <f,g> defined by
= - )
=> = *
SE,GD Soe if ) £9, dt. (26)
where the asterisk superscript denotes the complex
conjugate. We shall also need the adjoint eigen-
function of Eq. (22) that corresponds to the eigen-
value, Yo = 0, and that is 21/2 periodic in t. This
eigenfunction is
+
7. = Cla) 6
Me ( PE (27)
For convenience we normalize Z,, and Vins that
“a5 SS.
o'%o
by setting c=d=1.
The solution of any one of the equations in the
set (23), (24), etc. is obtained from the solution
of the previous member, by the application of the
Fredholm Alternative and the requirement that these
solutions are unique, i.e., they do not contain
multiples of the eigensolution of Eq. (22) and are
27/2 periodic in time. All of the equations of the
set (23), (24) etc. have the form
— - (D-A_I)z. = h(t) (28)
p J
where h(t) is a periodic vector function which has a
Fourier series representation of the form
fA ( t) 2 ne
kK==a0
(29)
where hy are constant vectors and the p-th component,
hoo of hey, is zero. (This property is enforced by
the solution procedures.)
Then, application of the Fredholm Alternative
for solving (28) yields the requirement that
<h(t) -¥,> =0 (30)
Assuming condition (30) to hold (this will be
achieved by properly selecting the 55'S), the
general solution, 24, can be written as
z, = exp [(D-) 1) €] x
t
EB. 6 i exp[-(D-A 1) s]h(s)ds (31)
(0)
Equation (31) is easily evaluated because
Vert
Neonat
exp (PSX t) el ae Waurere al (32)
Equation (32) is the main reason for diagonalizing
the matrix, (1/Rs,) (P+iad) . It makes evaluation
of the exponential matrix and the integral of Eq.
(31) trivial. Thus, by evaluating Eq. (31), and
requiring that z(t) be unique and 21/2 periodic
in t, values for the constant vector, Eq, are ob-
tained which eliminate all the non-27/2 periodic
functions from (31). The result of these calcula-
tions is
co 7
h ;
Z.(t) = oe ee (33)
5} LkQ-Y 9
k=-00
The solution procedure then is to apply Eqs. (30)
and (33) to each of the Eqs. (23), (24) etc. in
sequence starting with (23). These calculations
have been programmed and are quite easily performed.
(Our program does these calculations to the 7th
order term, but more terms can be easily incorpo-
rated.) We are mainly interested in the first two
perturbation terms which result in
o, = 0; 2,(t) = eee Werte (38)
where
E.
=(1 D (35a)
z | )
+
=
zi) = Ra ' (35b)
and
N!
(il) (=) (5) en (GLP)
= IDA Ga a ID a 0 36
%2 , Gas ai 2 ae
jaa
Bo (t) = Fl e28Mt , lO) 4, g(-2)-2i8t 36)
where
N' ar
(a). (1)
Dn 1G
2 £3 74
=1(2) je
nN 2i0-V, , (37a)
N'
(i) = (on) (2) (a)
= = im eG pO aI )
(o) s ‘ vay 3) a) 3)
j=l
+ (37b)
- O58 og /\-¥9) ,
N! +
ak pl) ,C)
=(=2) eo The?
n = j=l . (37c)
~ 212-5
We note that the order A perturbation, 6), of
the eigenvalue _ is zero so that the long-term
effect of a flow? oscillation with amplitude A is
only of order AZ. However, the short-term effect
is still of order A because the eigenfunction,
Z,(t), appears in the term I(t) in the relative
amplification ratio, ep /en,# given by Eq. (9). In
fact, the structure of the Matrix, E, is such that
all values of 6. with odd indices are zero and A
has an expansion in even powers of A about the
simple eigenvalue, ro: This can be surmized easily
from the fact that the phase of the imposed oscil-
latory part of the boundary layer flow should not
play a role in the modifications of the eigenvalue,
X,. Furthermore, note that the solutions, 2) (t)
and Zo(t), exhibit clearly the possible effects, at
second order, of certain resonant couplings. None
of the denominators in (35) and (37) are zero be-
cause y. #£+ ik for any integer values of j or k;
hence these solutions are uniformly valid for any
positive value of the frequency, 2. It is possible,
however, that at resonant frequencies such as at
n=t8m (5), the value of o» will have a relative
Maximum. Of particular importance is that in the
low frequency limit, 2 > 0, the o4's may be singu-
lar. The lower values of 2 will be an important
consideration and will be discussed in detail in the
next section.
4. NUMERICAL RESULTS AND DISCUSSION
‘Before describing the computational results that
have been obtained, we emphasize that in this work
29
the instability of the oscillatory boundary layer
as a whole is being compared to the instability of
the underlying steady Blasius boundary layer. How-
ever, it is easier to describe this comparison in
the terminology of the oscillatory forcing of the
Blasius boundary layer instability. For example,
if the oscillatory boundary layer is less stable
than the steady boundary layer by itself, then we
describe this situation as one in which the imposed
oscillations tend to destabilize the steady flow.
The first set of calculations were made to test
for resonant interactions at second order in A. By
consulting the solutions (35) and (36), it can be
seen that the mean effect of the imposed oscilla-
tions on the eigenvalue, Aon is manifested by the
term, 09. There are two types of resonances pos-
sible. The first type is the "harmonic parametric
resonance" which corresponds to values of 2 given
by Wo/k = 1/2, 1, 2,... where w, is the response
frequency of the disturbance, W, = Q M\p- The
second type of resonance is the "combination reso-
nance" corresponding to values of 2 given bydm
(10ty 5) = 0 (note the denominators of solution 35).
Figure 1 shows the computational results at certain
frequencies 2 in the range, l<w./Q<3. It can be
seen that the imposed oscillations stabilize the
flow. Figure 1 shows that no resonance effects are
predicted at either Wy /2 = 1,2),3, or at Wp /2 = 1417
and 1.74, which correspond to the two possible com-
bination resonances in the frequency range shown.
This lack of resonance effect results mainly be-
cause the external free stream oscillations induce
a significant amount of oscillatory vorticity in-
side the boundary layer only in a region very close
to the wall. This can be seen by examination of
the Stokes layer profile (14) where the exponential
factor has a vertical decay constant, 8, which is
equal to about 5 in the range of frequencies con-
sidered. The main fluctuations of the disturbance
velocity are concentrated at the mean critical layer,
Nc * 0.5 [where no is given by cy = fp(nc) and c,
is the mean phase velocity of the disturbance].
Thus, instead of the Stokes layer interacting
directly with the disturbance of the underlying
steady boundary layer at the level, nc, where most
of the disturbance energy is being produced, it is
confined mainly to the wall region where it cannot
be very effective. Furthermore, the Stokes layer
lacks a spatial structure in the x-direction that
can match in some way the spatial structure of the
9 SpaeatlieeeaGal ae MarR TP al ral) T
-2
w
2
%
ie
3
ac
-4
-6
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
wp /2
FIGURE 1. The growth rate perturbation Re 09 for a =
= 1128 (on the neutral curve) Wp = Wo/Rex
= 0,43 « n0-*.
30
disturbance mode. It is notable that the imposed
oscillations have an increased stabilizing effect
as 2 decreases (Wp /2 increases). This increased
stabilizing effect can be expected for two reasons.
The first reason is found in the solution (35)
which shows that the terms,
(aye
Nast
Oe (+12 alo
may become unbounded if
5!) ay ae
pp pp
remains bounded as 2 > 0, because y, = 0. Secondly,
it can be seen in the Stokes layer profile (14) that
the oscillations of the boundary layer become more
effective in penetrating up to the critical layer
when 2 decreases (i.e., B also decreases since
B = YOR. Ne
*
However, we cannot use the present parallel flow-
model at very low frequencies because in one period,
27/2, of the imposed oscillation, a disturbance
will propogate down.the boundary layer a distance,
6x, that is too large for the parallel flow assump-
tion to hold (i.e., constant boundary layer proper-
ties in the x-direction). For example, the change,
ORs, in the displacement thickness Reynolds number,
R§,, Over the distance, 6x, (near the values of a =
0.15 and R§, = 1200) is given approximately by
SRe = Jeu) . (38)
where N = Wo/Q. Thus, in the range of values of a
and Rsy of our calculations 6Rgx = sO NPSonthakw tor
N = 3, SRsy is nearly 20 percent of the value of
Rgx- Under the circumstance, the parallel flow
approximation is only roughly valid. Nevertheless,
the values of 6Rs, as a fraction of Rg, decrease as
one goes downstream of the neutral curve for fixed
values of the frequency ratio, Wp /Q. Thus, the
parallel flow approximation improves as one follows
a constant frequency disturbance downstream of the
neutral point.
The second set of calculations that were per-
formed was for the amplification of a fixed frequency
disturbance propogating down the oscillatory bound-
ary layer. Two values of Wo/2, equal to 2 and 3,
were chosen for illustration. The disturbance ex-
amined is an unstable Tollmien-Schlichting wave of
constant absolute frequency Oe = Wo/Rg x = 0.43 x 107"
along the constant frequency line a = 0.00133 Réy-
This disturbance first begins to grow in the steady
boundary layer at the values of a = 0.15, Rox =
1128, and ceases to grow at about the values of a =
0.3 and Rs, = 2255-, The disturbance trajectory a
0.00133 Rs* passes nearly through the point in the
a, Rs, plane of maximum rate of amplification.
Figure 2 shows the values of Reo» obtained for
the growing Tollmien-Schlichting wave along the
trajectory, a = 0.00133 Rgx, at the two different
values Wp /2 = 2 and 3. An interesting feature of
the results in Figure 2 is that [Reo | increases
with Rsgx although the quantity 8 also increases
which would seem to indicate further decoupling of
the oscillatory Stokes layer (14) from the distur-
bance oscillations., Presumably, the values of
-14
1000 1200 1400 1600 1800 2000 2200 2400
Rs,
FIGURE 2. Growth rate perturbation Re o2 along a =
0.00133 Rex-
[Reap | decrease as Rox becomes sufficiently large
for then 8 also becomes so large that the Stokes
layer will almost completely disappear. It can be
seen in Figure 2 that the stabilization of the
boundary layer can be substantial for the value of
Wp /2 = 3 and at the larger values of Rox-
Figure 3 shows the values of Red = ReXp + A?Reo5
for the value of A = 0.1 and the three values of
Wp /2 —JO/2 mands se (W5/2 = 0 is equivalent to A =
0). The total effect of the imposed oscillations
with A = 0.1 is not very substantial at the value,
Wp/Q = 2, but at the value of W/L = 3, the sta-
bilization of the flow is significant. We note that
an oscillation amplitude of A = 0.1 is a rather
large value at the frequencies considered here and
would require a large amount of power to achieve
in an experimental test facility such as a wind
tunnel unless the mean flow is very slow.
The rates of amplification shown in Figure 3 can
be summed according to formula (9) to obtain the
relative amplification ratio, ep, /ep,- One can
show that
°
2
x
ae
-2
1000 1200 1400 1600 1800 2000 2200 2400
R5.
FIGURE 3. The amplification rate A, along the trajec-
tory 4 = 0.00133 Ray.
q
Wi
to
where Rg, and Roy) are the values of Rs, at the
locations of the disturbance at the times, t, and
t,, respectively. The value of cy, the group ve-
locity, along the trajectory a = 0.00133 Rs,, was
computed to be about 0.356. We neglected the 0(A2)
modification of c_ due to the imposed oscillations.
This modification of c, is 0(10 °) and thus does
not affect the daiyareal , J, in a substantial way.
From the results shown in Figure 3, one obtains (by
a trapezoidal rule integration), the values of J =x
11,8.7, and 6.3 for w,/Q = 0,2, and 3 respectively.
The integral I(t) of (10) is evaluated by certain
sums and products of the vector components of the
solutions, (35) and (37). We omit the details. The
resulting expression for I(t), to second order in A,
has the following form
I(t) = A] +A ,A2+A (By cosNt+Bysinxt)
(40)
+2 (Cy cos2Nt+Cysin2xNt)
where A)>0,A>2,B,,Bo,C, and Cp are real numbers that
depend on the Reynolds number, Rg,- These coeffi-
cients have been computed along the disturbance
trajectory, a = 0.00133 Rs, and are plotted in
Figure 4. By using the values of Aj,B,, Bg, C, and
Co from Figure 4 (A, = 1.0 by suitable normalization)
in Eq. (40) for the value of A = 0.1 one finds that
I(t)
O.5 <=> &
5 T(t.) 2
at all the values of Reynolds number, Rye for which
the disturbance grows. It is customary to assess
the overall growth of a disturbance by considering
the natural logarithm of the amplification ratio,
ep/eq,- From (9) we have
e
Qn a Qn ze) cr ay
ea oe (Gt)
and one can see that although the term, &n I(t)/I(t,),
contributes an oscillatory factor to &n e Jen, (re-
call that, by following the disturbance down the
plate, t = Rs,) this contribution is minor relative
to the maximum value attained by J. Thus it can be
seen that the major effect of the parallel free
stream oscillations is to reduce the mean growth
rate of the unstable disturbances. This effect is
small for small values of A but can be significantly
large at such large values of A as A > 0.1. We note
that typical free stream turbulence rarely has a
_velocity magnitude as large as 10 percent of the
mean free stream speed.
_ Experimental results on the effects of parallel
free stream oscillation on the instability and
transition of the flat plate boundary layer are re-
31
-10 =e el eee:
1000 1200 1400 1600 1800 2000 2200 2400
i,
FIGURE 4. The coefficients of I(t) along a = 0.00133
Rear W/2 = 3.
viewed by Loehrke, Morkovin and Fejer (1975). How-
ever, we shall not make any comparison with their
experiments because these were for very low frequency
oscillations (w_/Q * 10) for which our parallel
flow instabilitY theory is of doubtful applicability.
An appropriate analytical instability theory for
comparison with these experiments is a quasi-steady
and parallel flow theory [see Obremski and Morkovin
(1969) ].
5. CONCLUDING REMARKS
Our main result is that the parallel free stream
oscillations, which manifest themselves in the
Blasius boundary layer as a Stokes layer, lead to
a mean stabilization of the flow. This stabiliza-
tion is very weak except for oscillation amplitudes
that are at least near 10 percent of the mean free
stream speed. Precise experimental data on the
effects of such oscillations on Blasius boundary
layer instability is not available in the frequency
range considered in this work. However, the results
are in accord with transition data for oscillatory
pipe flows. Sarpkaya (1966) has shown experimen-
tally that transition is delayed substantially when
harmonic axial oscillations are superimposed on
steady pipe flow. Furthermore, von Kerczek and
Davis (1975) have shown that the oscillatory Stokes
layer by itself is very stable, probably at all
Reynolds numbers, so that one might conjecture that
if the Stokes layer begins to dominate the boundary
32
layer (which occurs for low frequencies 2 and large
amplitudes A), then the Blasius boundary layer can
be stabilized by these oscillations. However, the
Stokes layer stability is very sensitive to extra-
neous effects such as streamline curvature. For
instance, experiments show that transition of plane
Stokes layers occurs at Stokes layer Reynolds
numbers, Rs, (where R = ARg,/8) on the order
of 500 [see Li (1954)]. However, if a slight amount
of streamline curvature exists, as would occur in
Stokes layers induced on the bottom of a water chan-
nel supporting free-surface gravity waves [see
Collins (1963)], the transition Reynolds number is
reduced to about 160. Thus, the effect on the in-
stability of the Blasius boundary layer of free
stream oscillations with a spatial structure such
as Up + U, cos (kx-wt) can be expected to be different
from the parallel flow oscillations considered above.
It is well known that ambient turbulence tends
to promote laminar to turbulent transition of the
boundary layer. Thus, if some oscillatory boundary
layer does in fact properly model certain features
of the interaction of the ambient turbulence with
the underlying steady boundary layer then it is to
be expected that such a oscillatory boundary layer
is less stable than the underlying steady boundary
layer. Although the present numerical results show
only a stabilizing effect for the type of oscilla-
tion considered, as inferred above there is reason
to believe that a more complex form of oscillation
of the boundary layer can be destabilizing. The
theory of the instability of forced oscillatory
boundary layers provides an alternative point of
view from that of Rogler and Reshotko (1974) and
Mack (1975) on the role of the interaction of free-
stream disturbances with Tollmien-Schlichting waves.
ACKNOWLEDGEMENT
This work was supported by the Naval Sea Systems
Command.
REFERENCES
Ackerberg, R. C., and J. H. Phillips, (1972). The
Unsteady Laminar Boundary Layer on a Semi-
Infinite Flat Plate Due to Small Fluctuations in
the Magnitude of the Free Stream Velocity. J.
Fluid Mechanics, 51, 137-157.
Coddington, E. A., and N. Levinson, (1955). The
Theory of Ordinary Differential Equations,
McGraw-Hill, New York.
Collins, J. I. (1963). Inception of Turbulence at
the Bed under Periodic Gravity Waves. J. Geo-
physical Res., 68, 6007-6014.
Craik, A. D. D., (1971). Nonlinear Resonant In-
stability in Boundary Layers. J. Fluid Mechanics,
50, PP SI8—413.
Davis, S. H., (1976). The Stability of Time-Periodic
Flows. Annual Review of Fluid Mechanics, 8,
57-74.
Gaster, M., (1963). A Note on the Relation Between
Temporarily-Increasing and Spatially-Increasing
Disturbances in Hydrodynamic Stability. J. Fluid
Mechanics, 14, 222-224.
Hayashi, C., (1964). Nonlinear Oscillations in
Physical Systems, McGraw-Hill Book Co., New
York.
Kerczek, C. von, and S. H. Davis, (1974). Linear
Stability Theory of Oscillatory Stokes Layers,
J. Fluid Mechanics, 62, 753-773.
Li, H., (1954). Tech. Mem. 47, Beach Erosion Board,
U.S. Army Corps of Engineers, Washington, D.C.
Loehrke, R. I., M. V. Morkovin, and A. A. Fejer
(1975). Transition in Nonreversing Oscillating
Boundary Layers, Transactions ASME, J. Fluid
Engineering, 97, 534-549.
Mack, L. M. (1975). Linear Stability Theory and
the Problem of Supersonic Boundary-Layer Trans-
ition, ATAA J., 13, 278-289.
Mack, L. M., (1976). A Numerical Study of the
Temporal Eigenvalue Spectrum of the Blasius
Boundary Layer, J. Fluid Mechanics, 73, 497-520.
Obremski, H. J. and M. V. Morkovin, (1969). Appli-
cation of Quasi-Steady Stability Model to Periodic
Boundary-Layer Flows, AIAA J., 7, 1298-1301.
Orszag, S. A., (1971). Accurate Solution of the
Orr-Sommerfeld Stability Equation, J. Fluid
Mechanics, 50, 689-703.
Rogler, H. L. and E. Reshotko, (1975). Disturbances
in a Boundary Layer Introduced by a Low Intensity
Array of Vortices, SIAM J. Appl. Math., 28, 431-
462.
Rosenhead, L., (1963). Laminar Boundary Layers,
Oxford.
Sarpkaya, T., (1966). Experimental Determination
of the Critical Reynolds Number for Pulsating
Poiseuille Flow, Trans. ASME, J. Basic Engineer-
ing, 88, 589-598.
Yakubovich, V. A. and V. M. Starzhinskii, (1975).
Linear Differential Equations with Periodic Co-
efficients, Translated from the Russian by D.
Lauvish, John Wiley & Sons.
Heated Boundary Layers
Eli Reshotko
Case Western Reserve University
Cleveland, Ohio
ABSTRACT
Heating the walls on which laminar boundary layers
develop in water can delay their transition to
turbulent flow and lead to significant drag reduc-—
tion. This paper describes the work done over the
last several years at Case Western Reserve Univer-
sity in examining the bases and consequences of the
heating phenomenon. Included are theoretical and
experimental studies of the stability of heated wa-
ter boundary layers for both uniform and non-uniform
wall temperature distributions, and experimental
study of the effect of heating on laminar separa-
tion and a quantitative assessment of the prospec-—
tive drag reduction on underwater vehicles.
1. INTRODUCTION
It was noted many years ago in experiments at low
subsonic speeds [Frick and McCullough (1942),
Liepmann and Fila (1947)] that the transition lo-
cation of the flat plate boundary layer in air is
advanced as a result of plate heating. Based on
this observation it had long been suspected that
heating would have the opposite effect in water,
namely that it would delay the onset of transition.
This is because heating in water reduces the vis-
cosity near the wall resulting in a fuller, more
stable velocity profile for a flat plate than the
Blasius profile. Cooling in water (and heating in
air) on the other’ hand tends to give an inflected
velocity profile which is less stable than the
Blasius profile.
These suspicions remained untested until con-
firmed by the analysis of Wazzan, Okamura, and
Smith (1968, 1970). These results triggered a
significant activity in the United States to deter-
mine whether wall heating could realistically be
33
used as a technique for drag reduction. A portion
of this effort was undertaken at Case Western Re-
serve University (CWRU) under the joint auspices
of the Office of Naval Research and the General
Hydrodynamics Research Program of the David W.
Taylor Naval Ship Research and Development Center.
The CWRU effort has been both analytical and
experimental and is ongoing. This paper will re-
view the results to date of the CWRU activity and
indicate current and future directions.
ANALYSIS OF THE STABILITY OF HEATED WATER
BOUNDARY LAYERS
The
ers
analysis of Wazzan et al. (1968, 1970) consid-
the stability characteristics of the boundary
layer to be governed by the disturbance vorticity
equation including consideration of viscosity vari-
ations in the basic flow but ignoring temperature
fluctuations and the coupled viscosity fluctuations.
The disturbance differential equation consists of
the fourth-order Orr-Sommerfeld operator augmented
by some lower order terms and is as follows:
(Gc) (p"=026) - uN = - Slug?” - 2029" + alg)
+ 2u'(6"" = 026)
+ u"(o" + 029) ] (1)
with boundary conditions
(2)
The analysis of Lowell and Reshotko (1974) on the
other hand is based on the following coupled sixth-
order system of vorticity and energy disturbance
equations:
34
(U-c) [(p¢)" - a2 (6o)] - U"(pd) + ifr (U-c)2]'
(od)! |" = 202) 26g)" ] 1 + abe
p
+ 2u!
Nt
+ 1S
R
N
(f=) ee
DIl[F DI[D
I oe
nel]
_
D
eS
(0d)' | ' + a2
|
e
eae es se
ahd ie E 5
) i (3)
= Wath AA Non |. = 1 WP \o aemtiye
EE (Ge)) + (po)T] = GREE {tae + okT']
Die:
- sae } (4)
p
ar |
2 i
N w
LGR DN
aa
3
\
! I
real
K
S}
DI
a
Cy SS CI|R
with boundary conditions
o(0) = ¢'(0) = 1 (0) (0)
(5)
$ (») o'(e) = t(@) 0)
In equations (3) and (4) all properties of the
basic flow are variable. The quantities r, m, and
K are the density, viscosity, and thermal conduc-
tivity fluctuation amplitudes respectively and
the coupling comes about through the viscosity
fluctuations that are directly related to the tem-
perature fluctuations.
Reg “min. crit.
X 1073
—O— Lowell & Reshotko
(1974)
12 —O-Wazzan, Okamura &
Smith (1970)
Ww To. = 60°F
d
10 el) @
dx
MINIMUM CRITICAL REYNOLDS NUMBER
100 200 300
Ura HA)
FIGURE 1. Effect of wall temperature on minimum cri-
tical Reynolds number ,[from Lowell and Reshotko (1974)].
The results of these two analyses for the min-
imum critical Reynolds number with wall heating are
shown in Fig. 1. The curves are very much alike.
Furthermore, the neutral stability characteristics
and the growth rates as calculated in the aforemen-
tioned analyses are sufficiently close so that there
is no important quantitative difference between the
two. The coupling of vorticity and temperature
fluctuations through the viscosity seems therefore
to be rather weak.
As is seen in Figure 1, both sets of calculations
predict significant boundary layer stabilization
(increased minimum critical Reynolds number, de-
creased disturbance amplification rates, etc.) with
moderate heating, but display a maximum and sub-
sequent decrease as the wall to free-stream tem-
perature difference is further increased. The
significant stabilization indicated for overheats
of up to 40°C (70°F) prompted a study of the pos-
sible drag reduction due to heating to see if this
drag reduction technique was in fact worth pur-
suing further.
3. DRAG REDUCTION IN WATER BY HEATING
It is shown in this section that significant reduc—
tions of drag are available to water vehicles with
on-board propulsion system is discharged through
heating the laminar flow portion of the hull. The
analysis is as follows [following Reshotko (1977) |:
For a vehicle with an on-board propulsion system
te E
—_—_
Poy Ses
Cm_)
the friction drag is
= dx +
D q c wdx xJ Coy wdx (6)
where q is the dynamic pressure, cre and Cry are
respectively the laminar and turbulent friction
coefficients, wdx is the area element at length x,
L is the vehicle length, and Xty is the transition
location.
The total drag can be written
D= D,,(D/D,,) (7)
where D/Dp is the ratio of total to friction drag.
For an axisymmetric body this ratio is a function
of the fineness ratio of the configuration.
Hoerner (1958) suggests that
3/2
D/D,, = il) sb 1.5(5) f wr Bao (8)
The drag power can then be written
Pr = Duan u,, (D/D,,) (CEA) (9)
where [Cp A] is the quantity in brackets in equa-
Editon (5) ie
The power available for heating is related to
the thermal efficiency of the power plant as
follows:
p s(22on |g. 2 2 (S=S a) q (10)
where Neff is the effectiveness of transmitting
the reject heat to the water in the desired manner.
If one considers heating only the laminar por-
tion of the hull then the power required to accom-
plish such heating is
x
134
= ar S
Be pu ¢ 4 Che wdx (11)
where c is the specific heat of water, cphe is the
laminar Stanton number for the heated boundary
May ermal ATE =) iti
Applying the available heating power Pa to the
laminar portion of the flow, (Py = Py) + after some
simplification yields
*tr
oe Cha wax
L c
f ce cask f tr ~£Q wdx
x ite ———
qs te eS at 2 2 (12)
ff ee c wdx Ke || a0 ( = in ff
° £2 F th e
The left side is the ratio of overall friction drag
to the laminar friction drag and is configuration
dependent. The right side depends on the dimension-
less ratio CAT/Uay and on the bracketed parameter
in the denominator related to the amount of reject
heat that can be transferred to the boundary layer.
The bracketed parameter in the numerator is a
Reynolds analogy factor which is configuration de-
pendent. In order to close the calculation, a
relation is needed between AT and transition
Reynolds number Re, which is also dependent on
configuration. ee
Example - The Flat Plate
In order to quantitatively evaluate the prospective
drag reduction due to heating, it is necessary to
choose a particular configuration. The flat plate
is chosen because of its great simplicity and be-
cause some information on transition with surface
heating is available. The results should be repre-
sentative of what can be obtained for slender shapes
having pressure gradients that are not too large.
For a flat plate (w = const)
(13)
ff x
L eae
x c dx = 0.074 ( )
fate
a Reu/a Ree a
L x
tr
and by Reynolds analogy
35
(e}
a £2 5 OD (14)
hy 2
Thus for the case of the flat plate, Eq. (12) be-
comes
= x me Fe (15)
1.328 1/2 oe (Dia i
Re DEN ne) Wee
x < th
te
The left side of equation (15) is the ratio of
overall friction drag to laminar friction drag for
a flat plate.
The variation of transition Reynolds number
Rex ~ with overheat AT depends on the choice of
transition criterion. A criterion that has been
shown to give plausible trends is the e? criterion
of Smith and Gamberoni (1956) and Van Ingen (1956).
For low speed flows, these authors correlated tran-
sition Reynolds number over plates, wings, and
bodies with the amplitude ratio using linear sta-
bility theory of the most unstable frequency from
its neutral point to the transition point. They
found that the transition Reynolds number Rex,, as
predicted by assuming an amplification factor of
e? was seldom in error by more than 20%. Wazzan
et al. (1970) have calculated and presented such a
curve for heated flat plates in water a portion of
which is shown in Figure 2. Although not quite
shown on the figure, Rey i reaches a maximum value
of about 260 x 10© at an overheat of about 43°C.
The most recent data of Barker (1978) taken ina
constant-diameter pipe are shown on this figure as
well. Barker obtains a considerable increase of
transition Reynolds number with heating in the
entrance flow boundary layers and his data attests
109
@ Barker (1978)
=
[o)
es}
107
TRANSITION REYNOLDS NUMBER Re x,,
0 10 20 30 40
WALL OVERHEAT, AT, °C
FIGURE 2. Variation of transition Reynolds number for
a flat plate with uniform wall overheat according to
an "eo" transition criterion, T, = 60°F.
36
to the reasonability of the assumed transition
schedule with overheat.
Drag reduction calculations have been performed
for plate speeds up to 24.4 m/sec (80 fps), for
plate lengths of 3.05 m (10 ft), 15.24 m (50 ft).
30.48 m (100 ft), 152.4 m (500 ft), and 304.8 m
1)
D) all
(OOORBE)), endutonmvaluestof l= Gai Jof 2,5,and
Cren
Since the product DN o¢¢/ Op might be very close
Vote
9.
to unity, one may view the aforementioned values of
A ba
the "efficiency factor" = 1) Nee! as approx-
F th
imately corresponding to n
respectively.
Results are presented in Figure 3 for the case
of an efficiency factor of 5 (n, 0.17). Shown
in Figure 3 are D/D the ENG of the drag with
heating to that WER OuCanenG the reject heat for
drag reduction purposes, the corresponding laminar
fraction of the plate x;,,/L, the wall temperature
rise of the laminar region, and finally the ratio
of the computed drag with heating to that for fully
laminar flow over the entire plate.
Generally speaking the drag reduction becomes
noticable as speeds exceed 10 m/sec (20 knots).
Although the drag ratio is not a strong function
of length, the overheat in the laminar region in-
OSs, Osali7/7 ctarcl ()5ilf0)
th
1.0 L, m (ft)
304.8 (1000)
1.0
creases quite significantly with vehicle length.
For ne, 0.17 (Figure 3), drag reduction of about
60% are atainable for vehicle speeds of 25 m/sec
(v50 knots) but the vehicle is far from full lami-
narization. The variation of drag ratio with Nth
is shown in Figure 4 for selected cases. The lower
the thermal efficiency, the larger the drag reduc-
tion and vice-versa. The indication from the cal-
culations is that full laminarization can be ob-
tained in a number of cases (Figure 4) but only if
Nth gets below about 0.03. Since the e” transition
curve (Figure 2) has a maximum value of Rey, be-
low 3 x 108, vehicles with length Reynolds numbers
above 3 x 108 cannot be completely laminarized.
For a plate of given length at a prescribed
speed, the fuel consumption (proportional to D/nty,
the slope of a line through the origin in Figure
4) increases as ntp is reduced. But it is far below
that of the unheated plate.
Real Configurations
Real vehicle configurations involve additional fac-
tors not considered in this flat-plate calculation.
Favorable pressure gradient, for example, can be
very effective in delaying transition while regions
of adverse gradient are otherwise. Non-uniform
longitudinal heating distributions can result in
a more optimal use of the available heat. Effects
Die
= (= arent =5
Dé th
0.8 152.4 (500) a 08 ny ~ 0.17 L, m (ft)
So oS 3.05 (10)
Ww Pad
O}+ 5
4 = 0.6
0.6 = 0
a
: 30.5 (100) 9 15.2 (50)
SI ©
=
< 15.2 (50) z 4 30.5 (100)
0:4) a 0
g 3.05 (10) 2
fo =
- s
0.2 0.2
50 kts 10 20 30 40 50 kts
0 0
0 5 10 15 20 25 0 5 10 15 20 25
U_, m/sec U_, m/sec
an, L, m (ft)
(o)
Bo 100
oe L,m (ft)
ES
wa 50
z z c
Pac) <q 20 30.5 (100)
co 2
wi wu = 10 15.2 (50)
a <
re a 8 3.05 (10)
35
<
34
1
0 5 10 15 20 25 0 5 10 15 20 25
U_,, m/sec U__,, m/sec
FIGURE 3. Drag reduction by use of reject heat of propulsion system for transition
delay. D
= t = 5, (n m WoIL7)) o
[= Nth ) nese] U th
1.0 Ue L
m/sec (FT/SEC) mM (FT)
15.2 (50) 15,2 (50)
15.2 (50) 3.05 (10)
24.4 (80) 15.2 (50)
24.4 (80) 3.05 (10)
st = 0
D
D
Oo COMPLETE LAMINARIZATION
Drac Ratio,
0 cal 2 oe} 4
PoweR PLanT THERMAL EFFICIENCY, 1.
TH
FIGURE 4. Effect of thermal efficiency of propulsive
power plant on drag reduction.
of surface roughness on transition are possibly
more pronounced for heated surfaces than for un-
heated. These factors are presently being studied
both experimentally and analytically by a number
of investigators for the purpose of obtaining an
objective evaluation of the practical capabilities
of this relatively simple and readily available
means of drag reduction. The related experimental
investigations done at CWRU will be described in
the next two sections.
4. STABILITY EXPERIMENTS IN WATER
The first experimental study of flat plate boundary-
layer stability in air was by Schubauer and Skram-
stad (1948) who used hot wire anemometry to measure
the growth characteristics of sinusoidal velocity
disturbances introduced into the boundary layer by
a vibrating ribbon. Ross et al.(1970) repeated the
Schubauer and Skramstad experiment to obtain data
for comparison with improved numerical solutions
of the Orr-Sommerfeld equation. Similar stability
experiments have been performed in water by Wortmann
(1955) and Nice (1973). The results of these ex-
periments are in agreement with the numerical solu-
tions of the Orr-Sommerfeld equation except near
the minimum critical Reynolds number, where the de-
parture from parallel-flow theory seemingly results
from the breakdown of the parallel flow assumption.
Among the attempts to correct the parallel-flow
formulation, those of Bouthier (1972, 1973) and
Saric and Nayfeh (1975, 1977) using the method of
multiple scales yield numerical results which dis-
play the best agreement with experimental results.
A natural extension of the above work is in the
investigation of factors which can increase bound-
ary layer stability. As indicated earlier, one of
these factors is wall heating in water. The ob-
jective of the experimental work done at CWRU was
to see if the predicted increase in stability due
to heating is in fact realized. To this end the
stability of flat plate boundary layer was investi-
gated on both a heated and unheated plate. For
the heated plate, the case of uniform wall temper-
37
ature may be more interesting from an engineering
viewpoint. For example, since the portion of the
plate upstream of the minimum critical point of
the unheated plate is stable without heating, why
not begin heating at the minimum critical point
and use more advantageously, the power that would
have gone to heating the leading edge region?
To systematize the approach to the problem, two
types of nonuniform wall temperature distributions
were studied: step changes in wall temperature
of magnitude AT occuring at a location xg; and
power law wall temperature distributions of the
form T,,(x)-T,. = AxM for n both positive and nega-
tive. The temperature T. is that of the external
stream. In order to isolate the effect of the
parameters, n and xg, on the boundary layer sta-
bility, one of two quantities must be held fixed -
either the total heating power put into the plate,
Qtotal: or the local wall temperature difference
at some reference location 1. (Xree) To: Since
heat losses from the test plate used in this ex-
periment could not be accurately measured, the
total heating power put into the plate could not
be related to the total convective heat transfer
to the boundary layer. Therefore the wall tem-
perature difference at Xyor¢, Ty (Xref) ~To, was held
constant as n and x were varied, with Xref chosen
in the region in which stability measurements were
performed.
Experiment
The experiment was performed in a low turbulence
water tunnel which has a test section 15.5 in. long,
9 in. wide, and 6 in. high. The free stream tur-
bulence intensity in the test section is 0.1 - 0.2%
for free stream velocities ug Lellpte Sec
The flat aluminum test plate, which is 13.6 in.
long, 9 in. wide, and 0.625 in. thick is suspended
from a frame which fits the top of the test section
as shown in Figure 5. The origin of the coordinate
system is located at the leading edge. The x-
coordinate is the running length measured in the
streamwise direction, y is measured normal to the
surface, and z is the spanwise coordinate measured
from the plate centerline. The rounded leading edge
(1/32 inch radius) is located 0.425 in. below the
top of the test section, thus forming a slot which
spans the top of the test section. The turbulent
wall boundary layer of the water tunnel is removed
by suction through this slot. Suction is adjusted
so as to locate the flow stagnation point at a
stable position just downstream of the leading edge
on the test side of the plate. A laminar boundary
layer then develops along the plate starting from
the stagnation point location.
Plate heating is provided by eleven electric
heating elements positioned as shown in Figure 5.
Plate surface temperature is monitored by eleven
thermistors imbedded in the surface of the plate
along the centerline. However, because of the
large temperature gradients which occur in the
plate, the thermistors do not yield an accurate
indication of the plate surface temperature. The
surface temperature is determined from boundary
layer temperature profiles measured with a hot-
film anemometer operating as a resistance thermom-
eter:
The pressure distribution on the plate surface
in both the spanwise and streamwise direction is
38
test section
plexiglas plate
mounting frame
X (INCHES)
monitored using static pressure taps in conjunction
with a manometer board. Artificial velocity dis-
turbances are introduced into the boundary layer
with a phosphorbronze ribbon 0.001 in. thick and
0.125 in. wide which is stretched across the plate
surface 3.75 inches. behind the leading edge. Ribbon
vibration is achieved by passing a sinusoidal cur-
rent through the ribbon in the z-direction in the
presence of a magnetic field maintained by horseshoe
magnets located on top of the plate.
A traversing mechanism located in the water
tunnel diffuser downstream of the test section is
used to position hot-film anemometer probes in the
x and y direction for boundary layer profile
measurements. The z-position of the probes is
fixed at the plate centerline.
Temperature measurements in the thermal boundary
layer are made with a DISA 55D0O1 anemometer and a
55F19 hot-film boundary layer probe operated in the
constant current mode as a resistance thermometer.
This unit is calibrated against the free stream
temperature measured by thermistors extending in-
to the free stream through the side walls of the
test section. Boundary layer velocity measurements
are made with a DISA hot-film system consisting of
two 55F19 probes, a 11M01 constant temperature
anemometer equipped with a 55M14 temperature com-
pensated bridge, a linearizer, r.m.s. voltmeter,
and d.c. voltmeter. The system is calibrated
against the velocity measured by a pitot-static
tube located in the center of the test section. A
General Radio 1900-A wave analyzer is used to
measure the r.m.s. amplitude of the anemometer
signal resulting from ribbon-generated disturbances
in the boundary layer.
The mean velocity profile is measured at x = 5.5
inches, which is the center of the region in which
disturbance growth rates are measured. This posi-
tion is also the value of x;,¢, the point at which
the local wall temperature is held constant as the
temperature distribution parameters n and x, are
varied. The displacement thickness, 6*, is deter-
mined by plotting the mean profile and using a polar
planimeter to graphically perform the integration
— o
6* = Vx of (1 - 4) an, where n = yVu/vx
Ue Ye
Since the maximum wall temperature difference used
in the present work,is T-T,.. = 8°F, the error in-
upstream Flange
aluminum suction
transition piece ———\ \\
\
\\
ribbon-drive \\
)
0.
FIGURE 5. Test plate installation.
curred by using the incompressible formulas given
here to calculate 6* and n is only about 0.1%. All
experimental results reported below are therefore
based on the incompressibte forms of 6* and n. The
Reynolds number, R,, = u 6*/v, is formed using the
kinematic viscosity evaluated at the free stream
temperature.
For a fixed Reynolds number and ribbon frequency,
the ribbon-generated disturbance amplitude is
measured at five stations spaced 0.25 in. apart
between x = 5 inches and x = 6 inches. In this
region the pressure gradient is small (Falkner-Skan
8 < 0.02) and there is no interaction between the
ribbon-generated disturbance and the natural dis-
turbances present in the boundary layer. The dis-
turbance amplitude recorded at each station is the
peak amplitude, defined as A(x) = [u'(n,x)/Velmax:
found by searching through the boundary layer in
the y-direction. The spatial disturbance growth
rate is then calculated from the slope (aA/ax) |
of a polynomial-curve fit of the A(x) data. By
repeating the above process for several different
frequencies the growth rate vs. frequency charac—
teristics of the boundary layer are determined for
a fixed Reynolds number and temperature distribution.
All stability measurements reported here for
non-uniform wall temperature distributions were
performed near R,, = 800. At Reynolds numbers
higher than 800 the ribbon-generated disturbances
become more difficult to follow since background
noise levels in the boundary layer increase with
Reynolds number. At Reynolds numbers lower than
800 the disturbance growth rates are already small
for uniform wall temperature in the range 3°F Ss Hh
(x)-To £ 8°F, and measurement of the decreased
growth rates resulting from non-uniform wall tem-
perature distribution is subject to large relative
errors.
xD
Results and Discussion
Uniform Wall Temperature Distributions
The Mean Flow - A comparison between heated and
unheated mean velocity profiles measured under
identical flow conditions is shown in Figure 6
together with the calculated unheated profile ob-
tained using Lowell's (1974) program for 8B = -0.0036,
which is the measured 8 for the case shown. For
n > 6 the measured velocity is uniform to within 1%.
The unheated boundary layer thickness for this case
is 6 = 0.066 inches (n=6.3). Note that velocities
measured in the region n< 0.75 are consistently
higher than would be expected from the straight-line
nature of the velocity profile in this region.
These velocities may be subject to wall interference
effects due to the size of the hot-film probe rel-
ative to the boundary layer. At the last measured
point, n=0.5, the prongs of the hot-film probe
touch the wall. The probe prong diameter is 0.010
inches (n=0.95 in the present case), while the
sensing element diameter is 0.003 inches (n=0.29).
The discrepancy shown in Figure 6 between measured
and calculated profiles for Ty-To = 0 may be due to
the integrated effect of the upstream pressure
distribution on the measured profile. Note that
the difference between the heated and unheated
velocity profiles is within experimental error.
The heated profile is slightly fuller than the
unheated profile in agreement with Lowell's numer-
ical solutions of the variable fluid property
boundary layer equations. The calculated ratio of
OP eattea nunheated for this case is 0.968 while
the measured ratio is 0.967.
Mean temperature profiles measured at varying
values of T -T_ and R,, are compared to Lowell's
(1974) solution of the boundary layer energy equa-
tion in Figure 6. Note that the thermal boundary
layer thickness is smaller than the velocity bound-
ary layer thickness by approximately the ratio
Sp/5y = pr-1/3 = 0.54, where the Prandtl number of
water is taken as 6.3 at T, = 75°F. Further de-
tails concerning the mean flow field may be found
in Strazisar (1975).
The Disturbance Flow Field - While the CWRU
Water Tunnel has a relatively low turbulence level
of 0.1% to 0.2%, this is still much higher than
Ross et al. (1970) in air. It has nevertheless
been ascertained by Strazisar (1975) that the pres-
ent ribbon-generated disturbances do not interact
with disturbances of other frequencies present in
the tunnel turbulence and furthermore display the
linearity required in order that the disturbances
be considered "infinitesimal".
The development of ribbon-generated disturbances
just downstream of the ribbon is investigated to
insure that the disturbances develop fully before
—— Lowell's solution
Tw-Ta =0°F 6 =-.0036
—— Lowell's solution
Tw-Tea =5F f=0
39
reaching the station where growth rates are first
measured, namely x = 5 inches. Figure 7 shows
the results for a decaying disturbance with
Wo = We) SOTO, inj. = Gol Avse = SL dmenas, mie
dimensionless frequency Wy is defined wy, = (27f)
v/u,* where £ is the ribbon frequency. (The exper-
imental lower branch neutral point at Rgx = 601 is
at wy, = 150 x 1o-®.) points in the region n < 0.75
are shown as broken symbols due to possible inter-
ference effects because of probe proximity to the
wall. The disturbance amplitude distribution
through the boundary layer attains its final shape
at x = 4.5 inches but the peak amplitude rises be-
tween x = 4.0 and x = 4.5 inches. Downstream of
x = 4.5 inches both the shape and peak amplitude
of the disturbance display expected behavior as
seen by comparison with the calculated eigenfunction
for this frequency and Reynolds number obtained
using Lowell's (1974) program. Since the measured
wavelength of this disturbance is 0.66 inches the
appropriate disturbance eigenfunction is seemingly
established in less than 1-1/2 wave lengths.
A measured disturbance temperature amplitude
distribution is compared with the corresponding
numerical solution in Figure 7. The calculated
distribution is scaled by equalizing the area
under the measured and calculated distributions
in the region 0.75 <n <3. The shape of the
disturbance temperature amplitude distribution is
also found to be virtually independent of the
disturbance frequency at a fixed R5x.
Disturbance Growth Rates - Measured disturbance
growth rates as a function of frequency for uni-
form wall temperature are shown in Figure 8 for
Rg*x = 800. The dimensionless spatial growth rate
where A is the amplitude of the disturbance at the
particular frequency under consideration. The
solid lines in Figure 8 are curves faired through
the measured points. The curve through the cir-
cular symbols is for the unheated plate. It is
evident that with increased heating of the plate,
the growth rates progressively decrease and the
range of disturbance frequencies receiving ampli-
fication is diminished. Similar behavior is in-
dicated at other Reynolds numbers as well.
O Ty-Tao=O'F Reg» =940 DO Tyw—Teo =3.5°F R5+=863
4 Ty-Ta =7.8°F Rs+=909 O Ty-Ta =5.4°F Rs«=910
& Tyla =7.8°F Rs+=909
1.0
0.8
0.6
0.4
0.2
0)
0 1 2 3 :
FIGURE 6. Mean velocity and temperature pro-
n 7 files for uniform wall heating.
40
(%)
u
Ue
DISTURBANCE VELOCITY AMPLITUDE,
Ribbon at x = 3.75 inches —— __ Lowell solution Rs + = 890
Two Tee O°F A = 0.656 inches Tyy—Teo = SF w, X 108 = 81
w, X 10° = 138 © Data for Ty—Ta. = 5.2°F
© x=4.0in. w, X 108 = 83 Rs. = 890
r 6
O x=4.5in.
a x=5.0in.
© x=55in.
——--—Solution by Lowell, Rs » = 601
(x = 5.5 in.) =
f 33.0
+} I
3
he
no
w
te
<
ao
c
E 2.0
a °
w
a
=)
=
<
oc
Ww
a
ri
F 1.0
w
S)
Ww
a
=)
=
=)
a
=
<
0
(0) 1 2, 3}
180° PHASE SHIFT
FIGURE 7. Velocity and temperature fluctua-
tion amplitudes in the boundary layer.
Neutral Stability - For reference, the neutral
stability results for the unheated plate will be
presented first. Neutral points obtained in the
present experiment are plotted together with those
from prior investigations in Figure 9. The solid
line in Figure 9 is the non-parallel flow solution
of Saric and Nayfeh (1975) while the dashed line
is the corresponding parallel flow solution of
the Orr-Sommerfeld equation. Lower branch neutral
points in the region Rgx < 500, Wy < 210 x 107© are
T.. = 75 F
Ug = 4.4 ft/sec
x = 5.5 inches
Uy Veg
OF
3.14
4.97
8.87
©Oavo
DISTURBANCE
FREQUENCY,
w, X 108
-4
SPATIAL DISTURBANCE GROWTH RATE, - ~ X 10°
-6
FIGURE 8. Measured disturbance growth characteristics
for uniform wall temperature distributions, Re xy = 800.
(
,
denoted by bars in the present work because dis-
tinct neutral points could not be identified. Ex-
perimental results indicate that a neutral point
lies somewhere in the barred region at each dis-
turbance frequency considered. The present re-
sults are in agreement with the experimental
results of Ross et al. (1970), Schubauer and
420 Theory
@ Non-parallel flow solution (Ref. 7)
p b --—--— Parallel flow solution (Ref. 12)
360 Data
o OC) Present investigation
OQ Schubauer and Skramstad (Ref. 1)
4 Ross etal. (Ref. 2)
Wortmann (Ref. 3)
300
240
180
120
DIMENSIONLESS FREQUENCY, w, X 10°
60
0 400 600 800 1000 1200
DISPLACEMENT THICKNESS REYNOLDS NUMBER, Res «
FIGURE 9. Neutral stability results for the unheated
plate. (Solid symbols denote lower branch points,
open symbols denote upper branch points).
41
4207
360+
©
cS)
x 3004
3
© >
(=)
= (S)
x. 2404 pa CI
= 2
> g
9 = 180}
2 1807 -
3 3
wy < 120}
{120} =
8 P
2 ®
< 60} fa) 60+
c
=
22) 0 0 + + + +
iS) 0400 ~~ 600 800 0 400 600 800 #1000
1000 1200
DISTANCE DOWN THE PLATE, Rs+ = Vx
a) Theory b) Experiment
Skramstad (1948), and Wortmann (1955). and provide
further verification of the departure from parallel-
flow solutions in the region R,,< 500. The agree-
ment obtained allows one to proceed to the case of
the heated plate with some credibility.
Experimentally determined neutral curves in the
(Wy, Rg*) plane for nominal uniform wall tempera-
ture differences of ky = 0,5,8°F are compared
to the parallel flow results of Lowell (1974) in
Figure 10. The experimental results are curves
faired through the measured neutral points, which
have not been shown for the sake of clarity. Com-
parison between the calculated parallel-flow re-
sults and experiment indicates that the departures
between the two found near (Rg) min.crit for the
unheated case persist in the heated cases. It is
readily seen that with increases in Damo
(RG t3) aden. Grete increases and also the range of
frequencies receiving amplification decreases.
Note that while the theoretical neutral curves
according to Lowell's parallel flow calculation
nest within each other, this does not happen ex-
perimentally until Rsx exceeds 860.
Predicted and measured values of (Rg*) min.crit
are compared in Figure 11. The measured rate of
increase in (R§*)min.crit Compares favorably with
that predicted by Lowell (1974) and by Wazzan et al.
(1970). Over the range of values of Ty-T, covered
by the present work it is conjectured that the
=
fo)
fo)
oO
0 2 4 6 8
WALL TEMPERATURE DIFFERENCE, T\,-T.. (CF)
S
MINIMUM CRITICAL REYNOLDS NUMBER,
(Re.+) MIN. CRIT.
FIGURE 11. Effect of heating on minimum critical
Reynolds number.
DISTANCE DOWN THE PLATE, Rs «+ a/ xX
1200
FIGURE 10.
for uniform wall temperature.
Neutral stability characteristics
ap. S 7525
non-parallel flow nature of the boundary layer
serves to reduce the value of (Rg*)min.crit by
about 120 units from that predicted for parallel
flow. This reduction seems independent of the
level of wall heating. A more complete description
of these results ig given in Strazisar, Reshotko,
and Prahl (1977).
Non-Uniform Wall Temperature Distributions
As indicated earlier, the two types of non-uniform
wall temperature distributions studied are a) the
power-law type in which (Ty-T.) = Ax™ and b) step
changes in wall temperature of magnitude AT = a
T.. occuring at location x,. In the discussions
that follow, n is the exponent of the power-law
wall temperature distribution and s = x,/Xyef is
the fraction of the distance to the measuring
station (x;es = 5.5 inches) at which the step
change in wall temperature is located.
The Mean Flow - Mean velocity profiles for
varying values of n, s and Tw(Xref)-To are compared
to the Blasius profile in Figure 12. The discrep-
x = 5.5 inches
Ue = 4.65 ft/sec
Te, = 75k
U/U,
Blasius
FIGURE 12. Mean velocity profiles for varying wall
temperature distributions.
42
ancy between the unheated profile and the Blasius due to equipment limitations. The thermal boundary
solution may be due to small pressure gradient layer near the leading edge is too thin to make
effects. temperature profile measurements with the hot film
practical. The first indication of the wall tem—
perature is thus provided by the thermistor imbedded
in the plate surface at x = 1.2". The heater
Mean temperature profiles and wall temperature
distributions measured for values of Ty (Xye¢)-To
=5°F are compared to relevant solutions of the
boundary layer equations in Figure 19. These nearest the leading edge is located at 0.71" <x
similar solutions were obtained by Runge-Kutta < 0.96". The actual wall temperature thus rises in
integration of the coupled momentum and energy some unknown manner from Ty-To = 0 at the leading
equations assuming variable viscosity and thermal edge to a value near the desired local wall tem-
conductivity. Their development is not shown here. perature at x - 0.71". These limitations are more
The error bars shown in Figure 13 represent the severe for increasingly negative values of n, which
require large temperature differences near the
leading edge, and may be the cause of the discrep-
ancy between theory and experiment seen in Figure
13 for the attempted n = -0.5 profile.
maximum-measurement error. Agreement between the
measured and predicted profiles is reasonable con-
sidering the fact that the wall temperature cannot
be monitored or maintained near the leading edge
1.0 Ug = 4.7 ft/sec
x (inches)
e 2.0
0.8 B30
& 4.25
©) >
gle 0.6F
Te 4
BIE =
i] i
22 x! =
4
Ir
\
ix
0.2 eG
- 2 4 6
x (inches)
0
0 1 2 3 200 400 600 800
nN
Rs *
1.0 Up = 4.53 ft/sec
x (inches)
©) Jes}
0.8 O 4.25
a 55
° 7.0 _
0.6 a
x ne
'
0.4 ix
S
IF
0.2
x (inches)
0 :
0 1 2 3 200 400 600 800
n Rs
1.0 Ug, = 4.79 ft/sec
x (inches)
0.8+
0.6 a
Se 8
IE
i)
0.4 is
ze
IF
0.2
2 4
x (inches)
FIGURE 13. Mean temperature profiles for 0
power law wall temperature distributions, 0 1 2 3 200 400 600 800
(36) Axn. n Rs«
1.0 iy (Xe ep al an OnE
= 07 5a
US 4.4 ft/sec
x = 5.5 inches
0.8
Theory Experiment:
(eo) s=0.0
ia) s= 0.35
s= 0.68
0.6 4
x ee
uw
0.4 3
=
|
x
=
0.2 E
0
0 1 2 3 200
n
Temperature profiles measured at x;of¢ = 5-5" for
several yalues of S = x,/Xref, with AT = 5°F, are
compared to analytic results in Figure 14. The
actual wall temperature does not undergo a steep
step change due to conduction of heat through the
plate upstream of the first heater used in each
case. As a result there is not a unique value of
X,, the step change "location". For purposes of
comparison solutions were obtained to the constant
property energy equation assuming that the temper-
ature profile developed entirely within the linear
portion of the velocity profile. This is a reason-
able assumption for the Prandtl number of water.
Comparison of the measured profiles with these
approximate step change solutions indicates that
the best agreement between theory and experiment
results when x, is taken as the x-location at
which the wall temperature first begins to rise
above the free stream temperature. The choice of
X, is used in all of the results reported herein.
The agreement between measured and predicted
temperature profiles shown in Figures 13 and 14
_—— Unheated
—N=1.0
1958\ w, x 108
SPATIAL DISTURBANCE GROWTH RATE, - = Xx 10°
°
FIGURE 15. Measured disturbance growth characteristics
for power law wall temperature distributions Ty(x) - To
= Ax”, = 800.
Ree
43
600 800 FIGURE 14. Mean temperature
profiles at x = 5.5" for step
Re* changes in wall temperature.
for Ty(Xref)-To = 5°F is typical of that obtained
at local wall temperature differences of 3°F and
8°F as well.
Disturbance Growth Rates - Disturbance growth
rate characteristics for vaying values of n ata
fixed Reynolds number near R§x = 800 with Ty (Xref)
-T. = 5°, are shown in Figure 15. The unheated
case is included for reference. The curves shown
are faired through the measured (a;,W,) points,
which are not shown for the sake of clarity. For
n = +1.0 the maximum disturbance growth rate is
greater than that for n=0 at a given value of Ty
(Xye¢)-T,,, and the band of amplified disturbance
frequencies moves to a higher frequency range.
Similar results are obtained for Tw (Xref) -To = 3°F
and 5°F,.
Disturbance growth rates vs. frequency for
various values of s, with AT = 5°F are shown in
Figure 16 at a nominal Reynolds number of Rgx = 800.
The unheated case is included for reference and
measured points (a;,W,) are once again not plotted
for the sake of clarity. The case s = 0 corresponds
-68
s
Unheated —<S
'
nN
SPATIAL DISTURBANCE GROWTH RATE, - « X 108
& °
FIGURE 16. Measured growth characteristics for a step
change increase in wall temperature, Rox = 800.
44
x 10°
(- =) max
(a) (b)
FIGURE 17. Maximum growth rates for power law wall
temperature distributions, Ty(x) - To = Axt, Rox = 800.
to uniform wall heating beginning at the leading
edge while the case s = 1 corresponds to a step
change in temperature occurring at the measuring
station x = 5.5 inches. The peak disturbance
growth rate displays a minimum as s increases for
each value of AT considered here. The band of
amplified disturbance frequencies also moves toward
a higher frequency range as s increases.
Disturbance growth rate behavior as a function
of wall temperature distribution is summarized in
Figures 17 and 18, where (-01) max is defined as
the maximum disturbance growth rate for a given
value of Ty(Xye¢)-To at fixed values of n is shown
in Figure 17b. We see that positive exponents can
result in large disturbance growth rates at low
wall heating levels. At higher levels of wall
heating the relative reduction in (-0j)may, between
any two temperature levels is greatest as n de-
creases.
The variation of (-;)max, with s at values of
AT = 3°F and 5°F is shown in Figure 18. The min-
imum in (-Gj)max at each wall heating level occurs
near the minimum critical Reynolds number of the
unheated boundary layer. The measured value of
(Oe) annonce for AT = 0 is 400, which corresponds
to s = 0.25 in Figure 18, while the predicted par-
allel flow value of (Rg*) min.crit = 520 £0, AT) =
O corresponds to s = 0.42.
An attempt was made to use the program of Lowell
and Reshotko (1974) to solve the parallel-flow
spatial stability problem for power law, wall
temperature distributions since the solution scheme
allows the mean flow solution to be read directly
into the coefficient matrix of the disturbance
growth rate at a fixed frequency and Reynolds
x 108
(- “max
0 2 4 6 8
FIGURE 18. Maximum growth rates for step change in-
creases in wall temperature.
number is a minimum for n=O, and increases by
maximum of 12% for values of n in the range
- 1/2 £n‘£1. This behavior, which is not con-
sistent with the experimental results, may be due
to the fact that significant changes in wall tem-
perature and therefore in the velocity and temper-
ature distributions are taking place over one or
two wave-lengths in violation of the parallel-flow
assumptions. It is felt that a proper multiple
scales formulation of the stability problem, which
takes into account the rather rapid variation of
wall temperature with x, is required to properly
assess the present results for power-law and step
function wall temperature variations. The results
for non-uniform wall temperature distributions are
given in more detail in the paper by Strazisar and
Reshotko (1978).
5. EFFECT OF WALL HEATING ON SEPARATION
An underwater vehicle is basically a body of rev-
olution having generally favorable pressure gra-
dients forward of the maximum diameter and adverse
pressure gradients downstream of the maximum
diameter. If laminar flow can be maintained all
the way to the adverse pressure gradient region
then the boundary layer will be very easily
separated unless measures are taken to delay such
separation.
An obvious way to delay separation is by suction.
This however involves the complexities of suction
slots, internal ducting and later discharge of
the flow removed from the vehicle boundary layer.
A "cleaner" possibility for separation delay if it
in fact would work is heating.
Wazzan et al. (1970) showed that heating can
cause a separating profile to fill out significantly.
Figure 19 indicates that for a Falkner-Skan B =
-0.1988, an overheat of 90°F, converts a separating
profile to one having the shape factor of a Blasius
boundary layer. This motivated our proposal to
investigate experimentally the potential effect of
heating on delay of laminar separation. Subsequent
calculations by Aroesty and Berger (1975) using an
Y
>
FIGURE 19. Velocity profiles at various wall tempera-
tures for 8 = -0.1988 [Wazzan et al. (1970)].
20
-01 —04 -0.6
Ger
FIGURE 20. Effect of overheat on Falkner-Skan separa-
tion parameter.
approximate procedure showed that despite the
large changes in profile shown in Figure 19, the
value of 8 at separation did not change very much
with heating (Figure 20). This was confirmed as
also shown on Figure 20 by exact calculations of
Strazisar (1975) using Lowell's (1974) program.
The question of the length retardation of separa-
tion on a real configuration nevertheless remained
an open one.
Experiment
This experiment was also performed in the CWRU Low
Turbulence Water Tunnel described in Section 4
using a specially designed two-dimensional model
having an NACA 635-015 profile. The model (Fig-
ure 21) is designed as part of the upper wall of
the test section of the water tunnel. The boundary
layer developing on the upper wall of the nozzle
is removed through a scoop with the bleed rate
adjusted so that the stagnation streamline is
straight and steady. Rod heaters (Figure 22) are
provided over the length of boundary layer develop-
ment. The tests were conducted at rather low unit
Reynolds numbers so as to promote laminar flow
in the separation region and to minimize the power
needed for large temperature differences. The
electric heaters distributed through the plate
provide wall temperatures of the order of 60°F
a Pressure tap
(taps run down centerline of model)
Suction duct
Mounting frame
Model
FIGURE 21. Model as mounted in water tunnel.
45
Experimental model HEATER X_WNCHES
NACA 639-015 profile
Maximum thickness = 1.175"
jh Oo ral
Maximum output of each
heater is 600 watts
electric
D heater
1" electric
4 ~ heater
ayn
8 D
FIGURE 22. Location of rod heaters in ex perimental
model.
above the free-stream fluid temperature in the
region of separation. Wall temperature distribu-
tions are shown in Figure 23.
The separation behavior was determined by com-
binations of the following indicators: 1) indi-
cation of separation by visual observations of a
dye stream injected along the surface through static
pressure holes, 2) location of separation indicated
by the static pressure distribution along the plate,
and 3) use of hot film anemometry to measure bound-
ary layer velocity profiles.
As with many water flow facilities, results are
dependent on the state of cleanliness of the
experimental equipment. In particular, the veloc-
ity profiles were affected by the condition of
the airfoil surface and the screens in the settling
chamber. Even when the screens and airfoil surface
were relatively clean, there was some scatter in
the level of the boundary layer shape parameter
as evidenced by the results for the unheated air-
foil. The effects of heating on shape factor
displayed consistent trends that were generally
independent of facility condition. The experi-
mental setup, procedures and measurement systems
are described in detail by Timbo and Prahl (1977).
HEATER VOLTAGE =
5.
8
70,
140.
100
mb
FIGURE 23. Surface temperature distributions, To = 70°F.
46
Results
When dye was injected into the boundary layer
through the pressure taps along the centerline of
the airfoil, it usually did not all move directly
upstream on the centerline. Some of the dye moved
initially spanwise and then upstream. Regardless
of the path of the dye, its motion was never steady.
When the airfoil surface was polished and the most
accessible of the screens in the settling chamber
cleaned, the most forward upstream position of the
dye on the unheated airfoil was 4.1 inches (x/L =
0.45). With heating, the patter of upstream dye
flow remained indistinguishable from the unheated
case. Thus for the wall overheats tested, up to
80°F measured at x = 4.01" (x/L = 0.44 in Figure
23), there was no separation delay discernable
using the dye injection method.
In looking at pressure distributions, separation
is identified as the point where the experimental
pressure distribution departs from the theoretical
recompression distribution on the aft portion of
the airfoil. The pressure taps will not indicate
a separated boundary layer unless there is con-
tinuous separation at the tap's position. Thus
unless the upstream motion of the dye is very
steady, which is usually not the case, the position
of separation as determined by the most upstream
penetration of dye is consistently farther upstream
than indicated by pressure distributions.
The separation point by examination of pressure
distributions on unheated airfoil occurs at x*4.9"
(x/L = 9.53). This is close to the location x = 5"
predicted for separation using the Thwaites method.
Heating, as reported by Timko and Prahl (1977),
caused no significant alteration in the pressure
4.0
© -9/28
QO -9/29
3.8 4A -10/3
: B = -0,1988 Vy —10/6
4 Falkner-Skan
Theoretical a Experimental
ge Theoretical
(Wazzan and Gazley, 1977)
3.4
3.2
3.0
2.8
2.6
2.4
2.2
2.0
0 10 20 30 40 50 60 70 80
Vergclies lz
FIGURE 24. Variation,of shape factor H with wall heating.
distributions and so again one cannot point to any
delay of separation by heating from these data.
Since the first two indicators showed negligible
shift of separation with heating, the boundary layer
velocity profiles were measured in some detail at
a point upstream of separation with and without
heating. Figure 24 shows the results for boundary
layer shape factor at a station 3.88" downstream
of the leading edge. Heating causes a reduction
in shape factor from the unheated value. The un-
heated profiles correspond to -0.17 < 8 < -0.15
and the slight reduction in shape factor with
heating is in accordance with expectation from the
similar solutions of Wazzan and Gazley (1977).
Despite these shape factor reductions the profiles
are changing so rapidly with longitudinal distance
(hence the scatter in Figure 24) that the separation
location is hardly affected.
Thus for the amounts of wall heating employed
in this study the separation point does not move
noticeably from its unheated position. This ina
sense confirms the results of Aroesty and Berger
(1975) .and of Strazisar (1975) (Figure 20) which
show the theoretical insensitivity of the value
of 8 at separation to heating.
6. CONCLUDING REMARKS
The studies to date reported herein together with
those of Wazzan et al. (1970, 1977), Barker (1978)
and others are such as to justify further investi-
gation of the various elements of the heating
phenomenon. Among the factors affecting the prac-
tical application of heating is the combined effect
of heating and roughness on stability and transition.
The work of Kosecoff, Ko, and Merkle (1976) suggests
that the roughness effect is due to the instability
of the mean profile as distorted by the roughness.
An alternative view being investigated at CWRU is
that the roughness introduces disturbances into
the boundary layer that may subsequently be ampli-
fied by the Tollmien-Schlichting mechanism. In
this view the wavelength of the roughness is im-
portant as well as its height. An experiment has
been planned that will map out the mean and dis-
turbance flow-fields in the vicinity of roughness
elements so that the relevant mechanism can be
identified. This will provide a fluid mechanic
characterization of roughness and help in further
assessment of the effects of roughness on trans-
ition of heated water boundary layers. With further
attention given also to heat exchanger design pro-
pulsion system, and fabrication techniques, there
are promising prospects for the achievement of
drag reduction by heating in water.
ACKNOWLEDGEMENTS
The author wishes to acknowledge the participation
of the following colleagues in the effort reported
in this paper: Dr. J. M. Prahl, Dr. M. Nice,
Dr. R. L. Lowell, Dr. A. Strazisar, and Mr. M.
Timko. All of us are grateful for the sponsorship
of the work by the Fluid Dynamics Program of the
Office of Naval Research and by the General Hydro-
dynamics Research Program of the David W. Taylor
Naval Ship Research and Development Center.
REFERENCES
Aroesty, J., and S. A. Berger, (1975). Controlling
the Separation of Laminar Boundary Layers in
Water: Heating and Suction. Report R-1789-
ARPA, RAND Corp.
Barker, S. J. (1978). Pipe Flow Experiments at
Large Reynolds Numbers. Proc. 12th Symposium
on Naval Hydrodynamics.
Bouthier, M. (1972). Stabilite lineaire des
ecoulements presques paralleles. J. de Mecani-
que, 11, No. 4, 599-621.
Bouthier, M. (1973). Stabilite lineaire des
ecoulements presques paralleles. II. La couche
limite de Blasium. J. de Mecanique, 12, 75-95.
Frick, C. W., Jr., and G. B. McCullough, (1942).
Tests of a Heated Low Drag Airfoil. NACA ARR.
Hoerner, S. F. (1958). Fluid Dynamic Drag. 2nd.
edition.
Kosecoff, M. A., D. R. S. Ko, and C. L. Merkle,
(1976). An Analytical Study of the Effect of
Surface Roughness on the Stability of a Heated
Water Boundary Layer. Physical Dynamics, Inc.
Report PDT-76-131.
Liepmann, H. W., and G. H. Fila, (1947). Investi-
gations of Effects of Surface Temperature and
Single Roughness Elements on Boundary Layer
Transition. NACA Rept. 890.
Lowell, R. L., and E. Reshotko, (1974). Numerical
Study of the Stability of a Heated Water Bound-
ary Layer. Report FTAS/TR-73-95, Case Western
Reserve University.
Nice, M. L.,(1973). Experimental Study of the
Stability of a Heated Water Boundary Layer.
Ph.D. Dissertation, Case Western Reserve
University. (Also Nice, M. L., and J. M. Prahl.
Report FTAS/TR-73-93, Case Western Reserve
University.
Reshotko, E. (1977). Drag Reduction in Water by
Heating. Proceedings. 2nd International Con-
ference on Drag Reduction. Cambridge, Sept.
BHRA, Paper E2.
Ross, J. A., F. H. Barnes, J. G. Burns, and M. A.
S. Ross, (1970). The Flat Plate Boundary Layer,
Part 3, Comparison of Theory with Experiments.
J. Fluid Mech., 43, 819-832.
Saric, W. S., and A. H. Nayfeh, (1975). Non-
Parallel Stability of Boundary Layer Flows.
Physics of Fluids, 18, 945-950.
Saric, W. S., and A. H. Nayfeh, (1975). Non-
Parallel Stability of Boundary Layers with Pre-
sure Gradients and Suction. Paper No. 6, AGARD
Conference on Laminar-Turbulent Transition,
AGARD CP-224.
47
Schubauer, G. S., and H. K. Skramstad, (1948).
Laminar Boundary Layer Oscillations and Tran-
sition on a Flat Plate. NACA Report 909.
Smith, A. M. O., and N. Gamberoni, (1956). ‘Tran-
sition, Pressure Gradient and Stability Theory.
Report EF 26388, Douglas Aircraft Co.
Strazisar, A. J., (1975). Experimental study of the
Stability of Heated Laminar Boundary Layers in
Water. Ph.D. Dissertation, Case Western Reserve
University. (Also, Strazisar, A. J., J. M. Prahl,
and E. Reshotko. Report FTAS/TR-75-113, Case
Western Reserve University, September 1975).
Strazisar, A., (1975). Private communication.
Strazisar, A. J., E. Reshotko, and J. M. Prahl,
(1977). Experimental study of the stability
of heated laminar boundary layers in water.
J. Fluid Mech., 83, Pt. 2, 225-247.
Strazisar, A. and E. Reshotko, (1977). Stability
of Heated Laminar Boundary Layers in Water.
AGARD Conference on Laminar-Turbulent Transition,
AGARD-CP-224, Paper No. 10.
Strazisar A., and E. Reshotko, (1978). Stability
of Heated Laminar Boundary Layers in Water with
Non-Uniform Surface Temperature. Physics of
TECK, AIk 6
Timko, M., and J. M. Prahl, (1977). Experimental
Study of the Effect of Wall Heating on the
Separation of a Laminar Boundary Layer in Water.
Report FTAS/TR-77-135, Case Western Reserve
University.
Van Ingen, J. L., (1956). A Suggested Semi-
Empirical Method for the Calculation of the
Boundary Layer Transition Region. Report VTH-74,
Dept. of Aero. Eng'g., University of Technology,
Delft.
Wazzan, A. R., T. T. Okamura, and A. M. O. Smith,
(1968). Spatial and Temporal Stability Charts
for the Falkner-Skan Boundary Layer Profiles.
Douglas Aircraft Co, Report No. DAC-6708.
Wazzan, A. R., T. T. Okamura, and A. M. O. Smith,
(1970). The Stability and Transition of Heated
and Cooled Incompressible Laminar Boundary Layers.
Proc each pinteaeieatenrans her COM lat2 arn Gul,
Elsevier, (ed. U. Grigull and E. Hahne).
Wazzan, A. R., and C. Gazley, Jr., (1977). The
Combined Effects of Pressure Gradient and Heat-
ing on the Stability and Transition of Water
Boundary Layers. Proceedings 2nd Int. Conf. on
Drag Reduction, Cambridge, Sept. 1977, Paper E3.
Wortmann, F. X., (1955). Untersuching instabiler
Grenzschictschwingingen in einem Wasserkanan
mit der Tellurmethode. 50 Jahre Grenzschicht-
forschuug. Friedr. Vieweg and Sogn, Braunsch-
weig, (ed. H. Gortler and W. Tollmien).
Discussion
CARL GAZLEY, Jr.
Several of us* at Rand and UCLA have made a
series of computations which serve to illuminate
some of the experiments described by Professor
Reshotko. His experiments with non-uniform wall
temperature distribution indicate the sensitivity
of the boundary-layer stability to the way the
surface temperature changes with distance along the
plate. For the power-law variation, AT = Ax ,
Reshotko's experiments for AT < 8°F appear to
indicate decreased stability and increased ampli-
fication rates as the experiment n decreases toward
zero. Our computations indicate the same trend at
low temperature differences, but also show a
reversal at a temperature difference of about 20°F
with an increasing stability with increasing n
above this AT. In fact, very large increases occur
for a AT above 30°.
Our results were obtained both by exact numer-—
ical techniques based on the Orr-Sommerfeld equa-
tion [Wazzan and Gazley (1978)] and by a modifiac-
tion of the Dunn-Lin approximation [Aroesty et al.
(1978)]. The results for flat-plate flow in terms
of the minimum critical Reynolds number based on
displacement thickness are shown in Figures 1 and
2 for values of n = 1 and 2 as a function of the
local temperature difference. The modified Dunn-
Lin approximation is seen to agree remarkably well
with the exact computations. More extensive
results of that approximation are shown in Figure 3
for values of n ranging from zero to 2. For temp-
erature differences above about 30°F, the advanta-
geous effects of an increasing temperature differ-
ence are seen to be very large.
DL APPROX
Res CRIT
w= MODIFIED DUNN-LINN
APPROXIMATION
4 EXACT COMPUTATIONS
0 10 20 30 40 50 60
LOCAL aT=T. -T., °F
Ww e
FIGURE 1. Variation of critical Reynolds Number with
local temperature difference. Flat plate with linear
increase of temperature difference.
*J. Aroesty, C. Gazley, Jr., G. M. Harpole,
W. S. King, and A, R. Wazzan
DL APPROX
Ry cRIT
eee MODIFIED DUNN-LIN
APPROXIMATION
0 EXACT COMPUTATIONS
0 10 20 30 40 50 60
LOCAL AT =T. -7., oF
WwW e
FIGURE 2. Variation of critical Reynolds Number with
local temperature difference. Flat plate with tempera-
ture difference increasing with the square of distance.
Roocrit
0 10 20 30 40 50 60
LOCAL aT=T -T., °F
WwW e
FIGURE 3. Variation of Critical Reynolds Number with
local temperature difference for several surface-
temperature distributions.
REFERENCES
Wazzan, A. R., and C. Gazley, Jr. (1978). The Com-
bined Effects of Pressure Gradient and Heating on
the Stability and Transition of Water Boundary
Layers. The Rand Corporatton, R-2175-ARPA.
Aroesty, J., et al. (1978). Simple Relations for
the Stability of Heated Laminar Boundary Layers in
Water: Modified Dunn-Lin Method.
49
Author’s Reply
ELI RESHOTKO
Dr. Gazley and his colleagues have long been
interested and active in the topic of heated
boundary layers and his comments on the conse-
quences of power-law temperature distributions
are greatly appreciated.
Let me first restate our experimental results.
Referring to Figure 17 of the paper, our experi-
ments for AT < 8°F appear to indicate decreased
amplification rates as the exponent n decreases
toward zero and in fact for some range of negative
values of n, the disturbances become damped. In
the temperature range AT < 8°F, neither our cal-
culations (cited in the paper) nor Gazley's give
any basis for this experimental result.
Nayfeh and El-Hady (private communication)
have recently pointed out that water boundary
layers with non-isothermal walls cannot have
similar boundary layer solutions because of the
variable properties of water. They show that if
one first calculates the non-similar boundary
layer profiles expected at the measuring station
of the Strazisar-Reshotko experiments and then
analyzes the stability of these profiles, the
resulting growth rates are qualitatively in accord
with the Strazisar-Reshotko results as shown in
the figure below supplied to me by Professor
Nayfeh. Note in the figure that as n decreases,
the growth rates also decrease, and although the
calculated maximum growth rates are not negative
for the non-parallel calculations with n = -0.5,
they are very close to zero. This trend is oppo-
site to what was obtained for the stability of
similar boundary layer mean profiles.
Nayfeh and El-Hady's calculations do not go
beyond AT = 8°F. But I believe that they have
made their point that when studying the stability
of water boundary layers with power-law or other
non-isothermal wall temperature distributions,
one must analyze the stability of the appropriate
non-similar boundary layer profiles in order to
obtain even the correct qualitative trends.
Therefore I believe that the results presented by
Dr. Gazley in his comment must be reexamined.
T T Ti T zi teaa
lO Tw(Xpep) Te = 8 F
| 2 =
| RE, x 800
—_—
a SS /
vA x
We
U, UNHEATED” \
y- n= +1.0
AMPLIFICATION RATE oO/R*10°
80 100 {20 140 |60 180
FREQUENCY F <xI0°
Effect of power law wall heating on stability of non-
similar water boundary layer. ---- parallel, non-
parallel. o = Im (a_ + €a,) where a is the quasi-
parallel amplificatfon rate and ea) is the non-parallel
contribution.
x
if
i
i
i
ii
i
Ne
at oe
i f
(valle
i ae
' 2
i Rtas)
i va f
i f
1 if
ovlichie
i)
ir i!
j f
eer iu
ary . \
i i r
5 Un
t etn
l
be SE
, \
yee : i f
ay i
th M i
j Wty '
fy
i \
i ;
{ f
} j
ti
x
: 7
'
{
»
in
F t
“
iy
Mak Wie
we bipeen ar
Session IT
BOUNDARY LAYER STABILITY
AND
TRANSITION
KARL WIEGHARDT
Session Chairman
University of Hamburg
Hamburg, Federal Republic of Germany
s,
Nonparallel Stability of Two-Dimensional
Heated Boundary Layer Flows
N. M. El-Hady and A. H. Nayfeh
Virginia Polytechnic Institute and State
University, Blacksburg, Virginia
ABSTRACT
The method of multiple scales is used to analyze the
linear-nonparallel stability of two-dimensional
heated liquid boundary layers. Included in the anal-
ysis are disturbances due to velocity, pressure,
temperature, density, and transport properties, as
well as variations of the liquid properties with
temperature. An equation is derived for the modula-
tion of the wave amplitude with streamwise distance.
Although the analysis is applicable to both uniform
and nonuniform wall heating, numerical results are
presented only for the uniform heating case. The
numerical results are in good agreement with the
experimental results of Strazisar, Reshotko, and
Prahl.
1. INTRODUCTION
It is generally accepted that the instability of
small amplitude disturbances in a laminar boundary
layer is an integral part of the transition process.
Significant changes in the boundary layer stability
characteristics can be achieved by utilizing dif-
ferent factors, such as pressure gradients, suction,
injection, compliant boundaries, and heating or
cooling of the boundary layer.
Surface heating in a liquid boundary layer can
be utilized to yield a mean velocity profile which
is more stable than the Blasius profile. The rea-
son is that heat transfer alters the shape of the
boundary-layer temperature profile which in turn
alters the velocity profile through the viscosity-
temperature dependence. The effect of wall heating
on the stability of boundary layers in water was
investigated by Wazzan et al. (1968, 1970). They
included the variation of the viscosity with tem-
perature through the thermal boundary layer. They
obtained a modified Orr-Summerfeld equation. How-
ever, they did not include temperature fluctuations
in the disturbance flowfield. Their results show
53
that while cooling the wall has a destabilizing ef-
fect on the flow, moderate heating has a strong
stabilizing effect. Lowell (1974) reformulated the
problem by adding fluctuations for the temperature,
density, and transport properties. The results of
Lowell did not vary appreciably from those of Wazzan
et al. (1970).
The presently available analyses (Wazzan et al.
and Lowell) for the stability of heated boundary
layers in water are all parallel flow analyses.
results of the parallel stability analyses do not
agree with available experimental results. Strazisar
et al. (1975, 1977) performed experiments on the
stability of boundary layers on both unheated and
uniformly heated flat plates. These experiments
confirmed the increased stability resulting from
wall heating in water. Strazisar and Reshotko (1977)
extended their experiments to cases of nonuniform
surface heating in the form of power-law temperature
distributions; that is, Ty(x) - Te = Ax®. Their
results are given only for a displacement thickness
Reynolds number R* = 800 and indicate that, for a
given level of wall heating, cases with n < O have
the lowest growth rates. Strazisar and Reshotko
(1977) found that applying Lowell's analysis (1974)
to the case of power-law temperature distributions
yielded results that did not agree with the experi-
mental results.
In this paper, we use the method of multiple
scales (1973) to analyze the linear, nonparallel
stability of two-dimensional boundary layers in
water on a flat plate, taking into account uniform
as well as nonuniform wall heating. We include
disturbances in the temperature, density, and trans-
port properties of the liquid in addition to dis-
turbances in the velocities and pressure. However,
we present numerical results only for the case of
uniform wall heating and compare our results with
the experimental data of Strazisar et al. (1975,
1977). When the variation of the temperature,
thermodynamic, and transport properties are ne-
glected, the present solution reduces to those of
The
54
Bouthier (1973), Nayfeh, Saris, and Mook (1974),
Gaster (1974), and Saric and Nayfeh (1975, 1977).
The formulation of the problem and method of
solution is taken in the next section, the solution
of the first-order problem is given in Section 3,
the solution of the second-order problem is given
in Section 4, the mean flow is discussed in Section
5, and the numerical results and their comparison
with the experimental data of Strazisar et al. (1975,
1977) is given in Section 6.
2. PROBLEM FORMULATION AND METHOD OF SOLUTION
The present study is concerned with the two-
dimensional, nonparallel stability of two-dimensional,
viscous, heat conducting liquid boundary layers to
small amplitude disturbances. The analysis takes
into account variations in the fluid properties but
neglects buoyancy, dissipation, and expansion ener-
gies. All fluid properties are assumed to be known
functions of the temperature alone.
Dimensionless quantities are introduced by using
a suitable reference length L* and the freestream
values as reference quantities, where the star
denotes dimensional quantities.
To study the linear stability of a mean boundary-
layer flow, we superpose a small time-dependent dis-
turbance on each mean flow, thermodynamic, and
trasport quantity. Thus, we let
G(x,y,t) = Qo (x,y) + q(x,y,t) (1)
where Qg(x,y) is a mean steady quantity and q(x,y,t)
is an unsteady disturbance quantity. Here, q stands
for the streamwise and transverse velocity compo-
nents u and v, the temperature T, the pressure p,
the density p, the specific heat c,, the viscosity
u, and the thermal conductivity k. Substituting
Eq. (1) into the Navier-Stokes and energy equations,
subtracting the mean quantities and linearizing the
resulting equations in the q's, we obtain the fol-
lowing disturbance equation:
or oe ( + ) + om ( + =
ye v ope We 3s DW ay Pov + PVo) = 0 (2)
Ci se My
+ (32 + = @)
eyo x Lh Ome
oy R ox P
avo , 5 2U0
+ a(x ay FS )|} (4)
or 0 oT 8T9 oT
Pol se ap \b) 5 + Uo 9x oy ur Vo al
c
oT dTo
+ (bog& + 9)/uy 22 + |
@ C) C)
an x 0 °Y
femal O oT oO)
RPr c { ax (xo ose ox )
0
3 oT chy
an, Go, ws
qlee tee) ry
Op ile (ep oo = functions (T) (6)
Here, Cpg is the liquid specific heat at constant
pressure, R = pAUaL*/ug is the Reynolds number and
ies = cheba/KS is the freestream Prandtl number.
Moreover,
(Aer Aap GS 2 (e sdk)
wir wiry
(1 + 2e), 1 = 5 ule-1) (7)
where e is the ratio of the second to the first
viscosity coefficients (e = 0 is the Stokes assump-
tion).
The problem is completed by the specification of
the boundary conditions; they are
u=v=T=0O0=aty=0 (8)
I Aue Se 0) AG “Ny ae (9)
We restrict our analysis to mean flows which are
slightly nonparallel; that is, the transverse ve-
locity component is small compared with the stream-
wise velocity component. This condition demands all
mean flow variables to be weak functions of the
streamwise position.
mathematically by writing the mean flow variables in
the form
Up (x1,¥), Vo + EVo(xX1,Y)
i]
Ug
Po = Pg(x1), To= To(x1-yY)
Po = Po(x1+Y), © = 64 sna)
0 ONS Day, Po Po 1rY
Ho = vo(X1-¥), Ko = Ko(*1,Y) (10)
where x; = ex with € being a small dimensionless
parameter characterizing the nonparallelism of the
mean flow. In what follows, we drop the caret from
Vo-
These assumptions are expressed
To determine an approximate solution to Eqs. (2)
-(10), we use the method of multiple scales [Nayfeh
(1973)] and seek a first-order expansion for the
eight dependent disturbance variables u, v, p, T,
O, C., Hw and kK in the form of a traveling harmonic
wave; that is, we expand each disturbance flow
quantity in the form
q(x ),y,t,) [qj] (x1,y)
+ €q9(x,,y) + -.-Jexp(i0) (11)
where
Oo 5 oe aes
a ae a9 (x1), re Tw (12)
For the case of spatial stability, a is the complex
wavenumber for the quasi-parallel flow problem and
w is the disturbance frequency which is taken to be
real.
Substituting Eqs. (11) and (12) into Eqs. (2)-
(10), transforming the time and the spatial deriv-
atives from t and x to 8 and x), and equating the
coefficients of ©9 and € on both sides, we obtain
problems describing the q; and qo flow quantities.
These problems are referred to as the first- and
second-order problems and they are solved in the
next two sections.
3. THE FIRST-ORDER PROBLEM
Substituting Eqs. (11) and (12) into Eqs. (2)-(10)
and equating the coefficients of e€9 on both sides,
we obtain the following
: a)
L} (U1, ,V,,P1,T1) = iagl(Pouy + (Ug - ag)!
3
+ oy (povi) = 0 (13)
Lo (uj,,V],P1,T))
i duo : aod) dug 3U0
aay a a 2h (eo.
R oy ap Jv *O0P1 R dy \aTo dy
_ 2 ot oun ia avi
Roy oF 1 mw POO By
_ 1 ao, Sin OE Ba
R dT dy dy m O Re = © (2e)
L3(4,,V),P)1,T})
9 Ww 1 2
= = — + —
[ teor0(vs =) R voad fr
_ is ,. 20 i, duo 9U0
R Y oy “1 in 2 aTg dy ql
a7 ig PO Bye (15)
55
Ly (W,,V),P1,T))
=| ipgag( up - —) + ees
PON 0 a9 Bie el
0 2 dKQ OT]
+ Vv, -
Po 1 RPr co ay oy
0
1 a2T] _
RPr c 0 oy2 ae ile)
Po
wy = Ww = wh = Oo ae Wy = @) (17)
Chie wala wy = O as y7o (18)
Equations (13) - (18) constitute an eigenvalue
problem, which is solved numerically. It is
convenient to express it as a set of six first-order
equations by introducing the new variables Zin de-
fined by
is _ uy =
ai = ilo Bl see 0 Z213.= Vie
oT]
rah Sig, BG Sho B18. Fe (19)
Then, Eqs. (13)-(18) can be rewritten in the compact
form
az
Hat
- = 0 = Np aeo pO 20
y a 455215 for i ; (20)
j=
Bi = 4213 = Big =O at y =0 (21)
Zllr 213, 215 10) ES 7 ae (22)
where the a, are the elements of a 6 X 6 variable-
coefficient matrix. The nineteen nonzero elements
of this matrix are listed in Appendix I.
We solve this eigenvalue problem by using SUPORT
[Scott and Watts (1977)]. To set up the numerical
problem, we first replace the boundary conditions
(22) by a new set at y = y where y is a convenient
location outside the boundary layer. Outside the
boundary layer, the mean flow is independent of y
and the coefficients aj; are constants. Hence, the
general solution of Eqs. (20) can be expressed in
the form
6
Ba S 2, A; ,cjexP Oy) fore A S Up Pros GO (23)
where the \j are the eigenvalues of the matrix
[a;;], the Ajj are the corresponding eigenvectors
and the c+ are arbitrary constants. The real parts
of three of the Aj are negative, while the real
parts of the remaining \. are positive. Let us
order these eignevalues so that the real parts of
hy,A2, and 3 are negative. Then, the boundary
condition (22) demands that cy,cs5 and cg are zero.
To set up this condition for SUPORT, we first solve
Eqs. (23) for the Cao Lo) and obtain
6
o5exp (1,y) = Waa, soe a) = Wpepceoc (22)
Aa aba} ahat
56
where the matrix [b,;,] is the inverse of (A; -]-
Setting cy = cs = cg = O in Eq. (24) leads to
= 0 for j = 4,5, and 6 at y = y (25)
where the bis are the elements of a 3 X 6 constant-—
coefficient matrix.
Using Eqs. (25) as the boundary condition at y
= y and guessing a value for ag, we use SUPORT to
integrate Eqs. (20) from y = y to y = O and attempt
to satisfy the boundary conditions (21). If the
guessed value for ag is the correct eigenvalue, the
three boundary conditions will be satisfied. In
general, the guessed value is not the correct value
and the boundary conditions at the wall are not
satisfied. A Newton-Raphson procedure is used to
update the value of ag and the integration is re-
peated until the wall-boundary conditions are satis-
fied to within a prescribed accuracy. This leads
to a value for ag and a further integration of
Eqs. (20) leads to a solution that can be expressed
in the form
23 = A(x1) 6, (x) ,Y) nope SS DBAS oo FO (26)
where A is still an undetermined function at this
level of approximation. It is determined by im-
posing a solvability condition at the next level of
approximation.
4. THE SECOND-ORDER PROBLEM
With the solution of the first-order problem given
in Eq. (26), the second-order problem becomes
oa z
5 = a, .Z,.
y a aes
=, 4 FA for i = 1,2,...,6 (27)
al, (bre x]
22] = 223 = 225 = 0 at y =0 (28)
29112231 225 * O as yoy 2 (29)
where the G. and F. are known functions of the Cie
ag and the mean flow quantities. They are defined
in Appendix II.
Since the homogeneous parts of Eqs. (27)-(29)
are the same as Eqs. (20)-(22) and since the latter
have a nontrivial solution, the inhomogeneous Eqs.
(27)-(29) have a solution if, and only if, a solva-
bility condition is satisfied. In this case the
solvability condition demands the inhomogeneities
to be orthogonal to every solution of the adjoint
homogeneous problem; that is,
ic: Be ax nn dete va fay = 0 (30)
i=1
where the W;(x,,y) are the solutions of the adjoint
homogeneous problem corresponding to the eigenvalue
@g- Thus, they are the solutions of
ow.
i
== + Ena =O) sore al = ilpArooe p@ 31)
yy ayes j , , (
dy
Joa
Wo = Wy = We = O at y =0 (32)
Wo, Wy, We > 0 as yy? (33)
Substituting for the G; and Er from Appendix II
into Eq. (30), we obtain the following equation for
the evolution of the amplitude A:
AAG) aes
az ax} = ia, (x,) (34)
where
oo co
6 6
ia) = - » FW dy > G.W.dy (35)
j=1 J j=1 a)
0 0
The solution of Eq. (34) can be written as
A = Agexplie a (x1) dx] (36)
where Ag is a constant of integration.
To determine a1(x1), we need to evaluate dag/dx,
and the 90; /0x)- To accomplish this, we differen-
tiate Eqs. (20)-(22) with respect to x; and obtain
; — )- : Ge )
zs a.
dy \ dx} sai ij\dx,
Sear Wl, sere at Se LHD paod® (37)
1 al
ae = ae = Fay =O at y=0 (38)
Uist wis OES c
dX] ¥ ax] z ax] “ao Ss er (Se)
The initial conditions for the computational pro-
cedures are chosen to exclude any multiple of the
homogeneous solution. The H; are known functions
of Ci, 4 and the mean flow quantities and their
derivatives; they are given by
6 OB 5
= me) d
Hy » c, oxy an
aaah 9
: 3
CLs
Ce Ds 3 8) oe AS Tp Proce (40)
j=1 ax]
Using the solvability condition of Eqs. (37)-(39),
we find that
6
COM). = = :
an »y HW. dy yy GW, dy (41)
Therefore, to the first approximation
2.) = BN eWET gs) Cesailes [va + e€a,)dx - iwt] + O(e) (42)
Y
where the z, are related to the disturbance variables
by Eq. (19) and the constant Ap is determined from
the initial conditions. It is clear from Eq. (42)
that, in addition to the dependence of the eigen-
solutions on x, the eigenvalue a is modified by
€a,- The present solution reduces to those obtained
by Nayfeh, Saric, and Mook (1974) and Saric and
Nayfeh (1975) for the case of nonheat conducting
flows.
5. THE MEAN FLOW
For flows whose thermodynamic and transport
properties are functions of temperature, the
two-dimensional boundary-layer equations for a
zero-pressure gradient are
a
5 (p*u*) + ma (OA) = 6 (43)
) du* 2) du*
*y* + pxky* raves 44
p*u = ON Se 7 Be Bp ) (44)
oT oT a oT*
*u*C* + o*v*ck =
p*u*c ax* p*v SD dy* ay (K* ay*) (45)
The temperature dependence of p and w) couples the
momentum and the energy equations. Note that buoy-
ancy and viscous dissipation effects are neglected.
Although the stability analysis is applicable to
any wall temperature variations, we present stability
results for the case of constant wall temperature
for which the flow is self similar. Thus, we intro-
duce the transformation.
= =f pdy* (46)
where R, is the freestream x-Reynolds number defined
by
R _ p*U*x* /u* (47)
x See S)
Introducing this transformation in Eqs. (43)-(45)
and solving the continuity equations for v, we trans-
form the original set of partial-differential equa-
tions into the following set of ordinary-differential
equations:
Q) du du
an (pu an) a Arie () (48)
a oT oT
Ls wes = 49
on (pk on) + we eS) on (0) (49)
I) 7/
where
lee
i)
nN
g(n) = 5 Jf puan (50)
0
Note that all fluid properties are made dimension-
less by using their freestream values.
Equations (48)-(50) are supplemented by the fol-
lowing boundary conditions:
u=0, T= aes and g = 0 at n 0 (51)
el Se Ls Cyayel ave = a AL as nto (52)
where the subscript w denotes wall values. Equa-
tions (48)-(52) are numerically integrated by using
Runge-Kutta and Adams-Moulten integration techniques
with the liquid thermodynamic and transport prop-
erties computed at each integration step. All nu-
merical results presented here are for water; the
dependence of its thermodynamic and transport prop-
erties on the temperature is given in Appendix III.
6. ANALYTICAL RESULTS AND COMPARISON WITH
EXPERIMENTS
Although the analysis is applicable to both uniform
and nonuniform wall heating, results are presented
only for the case of uniform wall heating for which
the mean flow is self similar.
The only available experimental results for the
stability of uniformly heated boundary-layer flows
are those of Strazisar et al. (1975, 1977). Using
a water tunnel, they introduced disturbances by
vibrating a ribbon and measured the response by
using a temperature compensated hot-film anemometry.
They used the r.m.s. of the stream-wise component
of the disturbance velocity, u, to calculate the
growth rates. They determined the growth rate as a
function of frequency at different Reynolds numbers.
For a parallel mean flow, a, = 0, dp and A are
constants, and the ¢_ are function of y only. Hence,
one can unambiguously define the growth rate o of
the distrubance as the imaginary part of Op; that
is,
o = - Im(a9) (53)
This definition is equivalent to
Oo = Re @ &nu) = Re (2 Qnv) =
ax ox
3 mene
Rey (ine) eRe (ent) (54)
On the other hand, for a nonparallel mean flow, aj,
7 O, A and dg are functions of x, and the Gnyarce
functions of both x and y. Thus, if one generalizes
(53) to take into account €a), one obtains
6 = - Im(d) + €a)) (55)
which is not equivalent to (54). Moreover, the
quantity a); and hence o depend on the normalization
of the Cy because part of the Tn Can be absorbed in
58
A and a}. If one generalizes the definition (54)
and uses (42), one obtains
)
o = - Im(a9 + €a)) + eRe(— Lnz_) (56)
ox n
Thus, the growth rate in (56) depends on the choice
of S, because the axial and transverse variations
of the Ty, are not the same. Since the Z, are func-
tions of both y and x, one may term a stable flow
unstable or vice versa.
Since there are many possible definitions of the
growth rate in a nonparallel flow, one should be
careful in comparing analytical and experimental
results. Saric and Nayfeh (1975, 1977) found that
the best correlation between the nonparallel theory
and available experimental data for the Blasius flow
is obtained if one uses the definition (55). In
this paper, we compare the definitions (55) and (56)
evaluated at the value n where ¢; is a maximum.
Figure 1 shows the variation of the calculated
disturbance growth rates o/R with frequency FR=W/R
for Twate = 0, 3,5, and 8°F and for the displacement
thickness Reynolds number R* = 800. This range of
Tw-Te is chosen for comparison with the existing
experimental results. The growth rate is calculated
by using the definition (55) and by normalizing 7,
so that ¢)>exp(-agy) as ye~. This figure indicates
that the disturbance growth rate decreases with in-
creasing T,-T,. The maximum growth rate is reduced
by approximately 56% by increasing the wall temper-
ature by 5°F. The maximum growth rate is very small
when the wall temperature is increased by 8°F at
R* = 800. Figure 1 shows that the range of unstable
frequencies decreases with increasing T,-Te.
AMPLIFICATION RATE o/R« 10°
80 100 120 140 160 180
FREQUENCY FRxI0®
FIGURE 1. The variation of the spatial growth rate
with frequency for varying wall temperatures at R* =
S00) =a = Nonparallel, ----- Parallel.
MAXIMUM o/Rx10°
Oo 400 600 800 1000 1200 1400 1600
FIGURE 2. The variation of the maximum growth rate
with streamwise position for varying wall temperatures.
Nonparallel, ----- Parallel.
Figure 2 shows the variation of the maximum
growth rate obtained from our analysis with Ty-Te-
It shows that the maximum growth rate decreases with
increasing wall temperature at all Reynolds numbers.
Figures 1 and 2 show a comparison between the
growth rates based on the parallel, (53), and non-
parallel, (55), stability theories. The nonparallel
maximum growth rates are approximately 30% larger
than the parallel ones. Moreover, the nonparallel
critical Reynolds number is approximately 20% lower
than the parallel one for all the values of T,-T
considered as shown in Figure 2.
Figures 3a-3d show comparisons of the experi-
mental growth rates of Strazisar et al. and the
nonparallel growth rates defined by (53), (55) and
(56) for different values of T,-T, and different
values of R*. These figures show good agreement
between the growth rate defined by (55) and the ex-
perimental results, in contrast with the parallel
theory which underpredicts the experimental results
by large amounts. Moreover, including the distor-
tion of the eigenfunction with streamwise position
in the definition of the growth rate yields a growth
rate that is very close to the parallel one and
hence underpredicts the experimental results by
large amounts.
©
7. CONCLUSION
The method of multiple scales is used to analyze
the linear nonparallel stability of two-dimensional
liquid boundary layers on a flat plate for the cases
of uniform and nonuniform wall heatings. We include
disturbances in the temperature, density, thermo-
dynamic, and transport properties of the liquid in
addition to disturbances in the velocities and
pressure. The growth rates calculated from non-
parallel results without including the distortion
of the eigenfunction with streamwise position are
in good agreement with the experimental results of
Strazisar et al. (1975, 1977). The nonparallel
results show that wall heating in water has a sta-
bilizing effect on the flow; there is a decrease in
the disturbance growth rates, a decrease in the
range of unstable frequencies and an increase in
the critical Reynolds number.
a/R «10°
80 100 120 140
FR «I0°
FIGURE 3a. Comparison of the analytical and the experi-
mental spatial growth rates for various displacement
thickness Reynolds numbers and wall temperatures.
Experiments, Strazisar et al. (1975, 1977), 1) o =
NCA pA) GS SINC, se Sei py SY) Kor Se —adin(ay ay Ger)) sp
e€ 0} 1
G1] 9x)
o/Rx10°
60 80 100 120
FRx10°
FIGURE 3b. Comparison of the analytical and the ex-
perimental spatial growth rates for various displace-
ment thickness Reynolds numbers and wall temperatures.
Experiments, Strazisar et al. (1975, 1977), 1) o =
Im(a49), 2) o = -Im(a9 + €0)), 3) O = -Im(ag + €0}) +
€ a] 41
269) ox)
o/R*« 10°
wl = 5.4 °F
R*=910
80 100 120 140 160
FRx10°
FIGURE 3c. Comparison of the analytical and the ex-
perimental spatial growth rates for various displace-
ment thickness Reynolds numbers and wall temperatures.
Experiments, Strazisar et al. (1975, 1977), 1) 5 =
-Im(ag), 2) Go = -Im(a9 + €01), 3) O = -Im(a9 + €4)) +
a ailGa
80 100 120 140 160
FRx 10°
FIGURE 3d. Comparison of the analytical and the ex-
perimental spatial growth rates for various displace-
ment thickness Reynolds numbers and wall temperatures.
Experiments, Strazisar et al. (1975, 1977), 1) 5 =
-Im(ag), 2) o = -Im(ag + €4)), 3) © = -Im(dp + €0]) +
59
60
ACKNOWLEDGMENT
The authors are indebted to Dr. W. S. Saric for
many valuable comments. This work was supported
by the NASA Langley Research Center Under Grant
No. NSG 1255.
REFERENCES
Bouthier, M. (1973). Stabilité linéaire des
écoulements presque paralléles. J. de Mécanique
IA, WS
Gaster, M. (1974). On the effects of boundary-layer
growth on flow stability. J. Fluid Mech. 66,
465.
Lowell, R. S. (1974). Numerical study of the sta-
bility of a heated water boundary layer, Ph.D.
dissertation, Case Western Reserve University;
also, Dept. Fluid, Thermal, and Aerospace Sci.,
Case Western Reserve Univ., Rep. FTAS/TR-73-93.
Nayfeh, A. H. (1973). Perturbation Methods. Wiley,
New York, Chap. 6.
Nayfeh, A. H., W. S. Saric, and D. T. Mook (1974).
Stability of nonparallel flows. Arch. Mech. 26,
401.
Saric, W. S., and A. H. Nayfeh (1975). Nonparallel
stability of boundary layer flows, Phys. Fluids
18, 945.
APPENDIX I
ai2 =]
= 1Po0%0R _, Ox 2
ari 7 (Ug aie! +0,
= i duo
a = = — —_
ze Uo dy
= OoR dUo .
a =
23 Wo y joo (
— 140R
a —
24 7
2
= _ hao doo
E29 Po To (Uo
fog 2 & Eto ous:
5 Ho dTo oy
431 = = 10
1 po
Ag So =e
2 Po Oy
Saric, W. S., and A. H Nayfeh (1977). Nonparallel
stability of boundary layers with pressure gra-
dients and suction. AGARD Conference Proceedings
No. 224, Laminar-Turbulent Transition, Paper
No. 6.
Scott, M. R., and H. A. Watts (1977). SUPORT-A
computer code for two-point boundary value prob-
lems via orthonormalization. SIAM J. Num. Anal.
14, 40.
Strazisar, A. J., and E. Reshotko (1977). Stability
of heated laminar boundary layers in water with
non-uniform surface temperature, AGARD Conference
No. 224, Laminar-Turbulent Transition, Paper No.
10.
Strazisar, A. J., J. M. Prahl, and E. Reshotko (1975).
Experimental study of stability of heated laminar
boundary layers in water, Dept. Fluid, Thermal
and Aerospace Sci., Case Western Reserve Univ.,
Rep. FTAS/TR-75-113.
Strazisar, A. J., E. Reshotko, and J. M. Prahl (1977).
Experimental study of stability of heated laminar
boundary layers in water. J. Fluid Mech. 83, 225.
Wazzan, A. R., T. T. Okamura, and A. M. O. Smith
(1968). The stability of water flow over heated
and cooled flat plates. J. Heat. Trans. 90, 109.
Wazzan, A. R., DT. TT. Okamura, and! Aj Ma1Os eSmictch!
(1970). The stability and transition of heated
and cooled imcompressible boundary layers, Pro-
ceedings, 4th International Heat Transfer Confer-
ence, ed. Grigall U. and E. Hahne, Amsterdam.
ai nsn= Iolo (2 duo 90)
; R Ho Oy Po oy
_ idovo
\e dio 9Po
if (2-p0 2 (200)24
Ang = See Yc = >
as R oH) QoHoPo dy dy Oo Po ay? Qo oy
= 1QoU0 ] dio Uo if doo Oo (Uy _ Ww )
ne R uo dTo 8Y ~~ Woo ATo dy Lo
eg eee
Bing 2 2 Jaguar Ph (Us - aa
ase = |
salves RPr.c 3° ln.
Ko dy
ae ae 9090 ‘Un < wy rapa Le
APPENDIX II
ei i+ FiA = 0
g Bot ros = - ms Ie s lk,
Gs tee HESAV= Li,
Binge tg een a ae
Ge a FEA = 0
Ge a Fell = = orn I,
where
ats doo GA _ J 20
Ie P0G1 + Uo as “| dx, 3X4 en wv
Xx
day duo
ar Ee (to dx, + Qo ep - Po 9x, 10} zt
1, 2 (FF voce - PoUo)ti + R oy Sg 0 RH
27 2)
(SS wrosto re polar
continued on page 62
61
62
iy, abe (r An eg Soy) e ope (Uo th Vo atest} A
= pogwelts = poVoge> + (Foc - poUo) + : [h sue =
(2a) Ble 2 — (Shey + v Ste Shoe (y Blog 5 Boy Ics
te es ea nae)ss} o> fe ee lapRwe
e € Po
(a eo cy Bee ng My - a Be Pou,
+ Vo lls (RPE oko - 2oUo) a 2 paece
APPENDIX III
The variation of the water thermodynamic and transport properties
with temperature is given by
_ | (T* = 3,9863)2(T* + 288.9414) 374.3
ex = | - “sog9n9.2 (T* + 68.12963) + 0.011445 exp(- =>)
p* iin gm/m’, T* in °C.
1.002 ) _ 1.37023(i* = 20) + 8.36 x 1On(T = 20)2
Tia UWS) SF We
Log(
Me Win Cio, Te Alin 2G
K* = - 9.901090 + 0.1001982T* - 1.873892 x10 “T*?
+ 1.039570 x 10° 7T*?
k* in mwatts cm ?K?, T* in °K
cS = 2c! = OsGsley 2 WO “Te? & 2G WOT Ae
= 2.42139 x 105 °1**
Gs in cal Gn K os Te? a Ol
A discussion of the sources and accuracy of these formulas can be
found in Lowell (1974).
Three-Dimensional Effects in
Boundary Layer Stability
Leslie M. Mack
California Institute of Technology
Pasadena,
SUMMARY
Most work in linearized boundary-layer stability
theory has been carried out either on the basis
of two-dimensional mean flow and plane wave dis-
turbances with the wavenumber in the flow direction,
or, for a more general case, by a transformation
of the equations to two-dimensional form. This
procedure can obscure important physical aspects
of wave propagation in two space dimensions. In
this paper the stability equations are retained
in three-dimensional form throughout. A method
for treating spatially amplifying disturbances with
a complex group velocity is adopted and applied
first to oblique waves in a two-dimensional bound-
ary layer, and then to the two-parameter yawed
Falkner-Skan boundary layers. One parameter is
the spanwise to chordwise velocity. For boundary
layers with small crossflow, the maximum amplifi-
cation rate with respect to frequency is calculated
as a function of flow angle for waves whose normal
is aligned with the flow. Next, the minimum crit-
ical Reynolds number of zero-frequency crossflow
instability is obtained for both large and small
pressure gradients, and finally the instability
properties of two particular boundary layers with
crossflow instability are determined for all un-
stable frequencies.
1. INTRODUCTION
Most work in linearized boundary-layer stability
theory has been restricted to two-dimensional mean
flows, and, for these flows, even further restricted
to plane-wave disturbances with the wave normal in
the flow direction.* The latter restriction is
normally justified by reference to the theorem of
*Such a wave is called two-dimensional because
it has only two disturbance velocity components.
All other plane waves have three velocity components
in any coordinate system, and are called three
dimensional.
California
63
Squire (1933), which states that ina two-dimensional”
incompressible boundary layer, the minimum critical
Reynolds number is given by a two-dimensional wave.
Even though the most unstable wave at a given
Reynolds number is two dimensional in accordance
with the theorem, the most unstable wave of a
particular frequency can well be three dimensional.
Furthermore, the unstable three-dimensional waves
can have phase orientation angles (the angle between
the local freestream direction and the wave normal)
up to almost 80°. Any method for the estimation
of transition that is based on stability theory
must take this large range of unstable three di-
mensional waves into account. For a supersonic
two-dimensional boundary layer, even the most un-
stable plane wave at a given Reynolds number is
three dimensional. The two-dimensional waves be-
come of little importance as the Mach number in-
creases above one until the hypersonic regime is
reached, where a two-dimensional second-mode wave
is the most unstable.
When we turn to three-dimensional boundary layers,
there are no two-dimensional waves, but the trans-
formation of Stuart [Gregory et al. (1955) ] reduces
the three-dimensional temporal stability problem
to a series of two-dimensional problems. That is,
the temporal amplification rate can be obtained by
solving a two-dimensional problem for the boundary-
layer profile in the direction of the wave normal.
This approach was carried through numerically by
Brown (1961) for the rotating disk and a limited
number of swept-wing boundary layers. When the
same approach is applied to the spatial theory,
it leads to complex velocity profiles and loses
much of its utility except as a computational device.
Instead of trying to make a two-dimensional
world out of a three-dimensional world, it might
as well be accepted that boundary-layer instability
is inherently three dimensional, even with two-
dimensional mean flow, and to formulate the insta-
bility problem directly as three dimensional [Mack,
(1977); this paper will be referred to as M77]. A
transformation of the dependent variables reduces
the order of the incompressible eigenvalue problem
64
from sixth to fourth order, but the velocity pro-
files and wave parameters are not transformed.
This approach is equally valid for the temporal
and spatial theories, but for the latter a growth
direction must be assigned before eigenvalues can
be computed. In M77 this direction was taken
equal to the direction of the real part of the
group velocity and numerical results were obtained
for two-dimensional incompressible and compressible
flat-plate boundary layers and for the rotating
disk boundary layer.
In the present paper, a theoretical presentation
is given in Section 2 to justify the use of a
spatial mode whose direction of growth is determined
by the complex group velocity. In Section 3, some
results concerning three-dimensional spatial waves
in the Blasius boundary layer are given as an
example. In Section 4, we adapt the family of
yawed-wedge three-dimensional boundary layers
[Cooke (1950) ] for use in stability calculations.
In Section 5, under Boundary Layers with Small
Crossflow, we consider the effect of the flow angle
(the angle between the local potential-flow direc-
tion and the direction of the pressure gradient) on
the maximum amplification rate for small pressure
gradients. Next, in Section 5 we take up cross-—
flow instability and determine the critical Rey-
nolds number for several combinations of pressure
gradient and flow angle. We then obtain the max-
imum amplification rate and instability boundaries
of all unstable frequencies as a function of the
wavenumber vector for a favorable pressure-gradient
boundary layer which is unstable at low Reynolds
numbers only because of crossflow instability.
Finally, in the last part of Section 5, we repeat
the latter calculation for an adverse pressure-
gradient boundary layer with crossflow instability
at a Reynolds number where the boundary layer is
unstable even without crossflow instability. In
all of the examples, only the amplification rate
is calculated, and on the basis of locally uniform
flow. No results concerning wave amplitude are
given, although in Section 2, we make use of a
simple wave amplitude equation in order to properly
define the spatial amplification rate.
2. THREE DIMENSIONAL STABILITY THEORY
Formulation and Transformations
The linearized, incompressible, parallel-flow,
dimensionless Navier-Stokes equations for the
elementary modes
u(x,y,z,t) f(y)
v(x,y,z,t) a o (y)
w(x,y,Z,t) h(y)
p(x,y,zZ,t) tT (y)
exp[i(ax + Bz - wt) ], (1)
where u,v,w are the velocity fluctuations and p
is the pressure fluctuation, can be reduced to
(M77)
'
Z) = Zor
zZi= [ao + 6. + OR(au Hew w)lZ)
2 a2!
+ (aU' + BW')RZ3 + i(a + B )RZy, (2)
2 2
d Cmts
-i2, -|i(au + aw - 0) +2 */z,,
N
a=
i]
for the determination of the eigenvalues. The
primes refer to differentiation with respect to
y, and the dependent variables are
Z, (y) af(y) + Bhly), Z3(y) = oly),
Z,(y) = my).
There are two additional uncoupled equations for
h(y). In Eqs. (2), a and 8B are the complex wave-
number components in the x and z directions, w is
the complex frequency, U and W are the mean velo-
city components in the x and z directions, and R
is the Reynolds number UpL*/v*, where the velocity
scale U* is the potential velocity and L* is a
suitable length scale. Asterisks refer to dimen-
sional quantities. The modes in Eq. (1) can be
termed plane waves in the x,z plane because of the
phase function, even though there is a modal struc—
ture in the y direction.
The boundary conditions are
Z,(0) =O , 423(0) =0, (3)
Z, (y) a> '@ 5 Z,(y) +0 as y +o.
If we choose x to be the direction of the local
potential flow, then z is the crossflow direction
and
WD) ao Ibe Wiy) > 0 asyreo.
Thus U(y) is the mainflow velocity profile; W(y) is
the crossflow profile.
In the temporal stability theory, a and 8 are
real, and Eqs. (2) can be reduced to two-dimension-
al form in two different ways. The first transfor-
mation is
(4)
When W = 0, this is the transformation of Squire
(1933). It relates the eigenvalues of a three-
dimensional wave of frequency w in a velocity pro-
file (U,W) at Reynolds number R to the eigenvalues
of a two-dimensional wave of frequency w/cosy in
a velocity profile U + W tani at Reynolds number
R cosy, where
y = tan’ (8/a)
is the phase orientation angle.
The second transformation,
2 =)
2 +) 1B) )7 A au = ov + BW,
MN
2
a
RrSCReo;
is that of Stuart [Gregory et al. (1955) ]. It
relates the eigenvalues of a three-dimensional
wave of frequency w in a velocity profile (U,W) at
Reynolds number R to the eigenvalues of a two-
dimensional wave of the same frequency in a veloc-
ity profile U cosy + W sini at the same Reynolds
number. The Squire transformation is most useful
for a two-dimensional boundary layer because the
velocity profile is unchanged. Thus all eigen-
values of three-dimensional waves can be obtained
from known eigenvalues of two-dimensional waves
with no additional calculations. In a three-
dimensional boundary layer, the velocity profile
must change and the Stuart transformation is pre-
ferred because the frequency can remain fixed at
a given Reynolds number as the phase orientation
angle ~ is varied.
Spatial Stability Theory
Statement of the Problem
In the spatial stability theory, a and 8 are com-
plex and w is real. Neither transformation is of
much utility except when
a;/B; = o,/8,- (6)
When (6) is not satisfied,a is complex, and in the
Squire transformation both R and w are also com-
plex as well as U for a three-dimensional boundary
layer. In the Stuart transformation, U is complex
for all boundary layers. With complex quantities,
we might as well deal directly with (2), as these
equations have already been reduced to fourth order
and nothing is to be gained from an additional trans-
formation. There only remains the question, to be
answered later in this Section, of whether any use
can be made of the simplification offered by (6).
It is convenient to define a real wavenumber
vector
eS
k = (a,,B,) 4
and a real spatial amplification rate vector
>
G = (-0,, —By),
ae R
in place of the complex vector K - io. The magni-
tudes of the vectors are k and o, and their di-
rections are given by the two angles
v= tan (8, /a,), p= tan” "(B,/a,).
Equation (6) is now seen to be a statement that
k and 6 are parallel (J = ~). Plane waves with
? A) have been termed inhomogeneous by Landau and
Lifshitz (1960).
The solution of the eigenvalue problem set up
by (2) and (3) gives the complex dispersion relation
o = O(K,0,x;2))-
Even with w, x and z fixed, there remain four real
wave parameters: k, W, o and ~. Only two of
these can be determined in a single eigenvalue
calculation, e.g., k and o with w and wt) specified.
The angle i can be considered an independent
variable on the same basis as the frequency. The
problem is to choose . What we are looking for
is a single spatial mode which serves the same
purpose as a two-dimensional spatial mode in a
two-dimensional boundary layer, where it represents
the wave produced by a stationary harmonic source.
“The amplification rate of this mode is used as a
65
measure of the relative instability of different
velocity profiles, and its amplitude can be applied
to the transition problem.
Introduction of an Amplitude Equation
In order to describe wave propagation in the non-
uniform medium of the boundary layer, equations are
needed for the wave amplitude and the change in the
wavenumber vector in addition to the dispersion
relation. Even though no amplitude calculations
are included in this paper, a consideration of the
amplitude equation will help us select jp.
In a nonuniform medium the elementary modes (1)
are not general enough and must be replaced by
wGesy747e) = AGB) explo (})x]£ (y)expli (a,x
+B 2 - wt) ]. (7)
In this, the exponential amplitude factor has been
written separately in terms of the spatial amplifi-
cation rate o()). This amplification rate is the
magnitude of G(K,P,w,X,zZ) considered as a function
of k,W,w,x,zZ with a fixed value of J. Each j de-
fines a coordinate
x = cos x + siny Z
along which the wave growth is directed.
Nayfeh et al. (1978) have derived an equation
for the amplitude factor A(x,z,t) on the basis of
the multiple scales technique, with A considered
to be a slowly varying function of x,z,t, as are
a,8,w and f(y). In a uniform medium, and with A
independent of time, their equation reduces to
A
@ BE a6 dB = ©, (8)
Be (e) ote Z0z
where C = (Cx,Cz) is the (complex) group velocity.
We may note that (8) is also obtained from
2 >
a (V.c)A> = 0, (9)
which is the energy conservation equation of Whit-
ham's theory (1974). Davey (1972) has applied (9)
to non-conservative wave motion in a two-dimensional
mean flow, and refers to the amplitude function A
as a pseudo amplitude, or the 'dispersive part' of
the amplitude.
Spatial Mode - Real Group Velocity
ss
We restrict ourselves first to the case of C real
and define the orthogonal coordinates
x = cos x + siny Za (10a)
gr gx gr
Zee = -siny,, x + cos, Zy (10b)
where
Ves tan (E/E) o
The angle Ugr defines the direction of the charac—
teristic coordinate xg,, which is identical to a
group velocity trajectory, and A is constant along
each characteristic according to (8).
66
The amplitude portion of (7) is now
a(x,z) = A(z expla ()) x], (11)
and (7) can be interpreted as a certain type of
solution for a uniform medium when A is variable,
provided only that A is constant along a character—-
istic. A knowledge of A along some initial curve
completely specifies a along the characteristics
of A, and the characteristics of A are also the
characteristics of a. Therefore we can write (11)
as 2 is os
A(Xgy) = ag (z exp (Og,Xgr) » (12)
gr gr
where (10) has been used to eliminate x and
Sgr = 5 (p) cos (V-Pgy) « (13)
Consequently, (7) becomes
u(x,¥,Z,t) = ao (Zg,) exp (OgyXgy) f(y) exp
[i (4px + Byz - wt)]. (14)
If an is a constant everywhere, the spatial mode
(14) represents a physical wave in the entire x,z
plane that could be produced by a particular
stationary harmonic line source in a uniform medium.
ifag ag is constant only along a characteristic, we
have a form of ray theory, and (14) in turn applies
only along a characteristic (ray). In other words,
x and z are constrained to follow the characteristic.
The latter viewpoint is more useful for a general,
nonuniform boundary layer, and also applies to a
stationary harmonic line source in a uniform bound-
ary layer when the locus and amplitude. distribution
of the source are arbitrary.
Equation (13) was derived in M77 from a general-
ized Gaster relation between temporal and spatial
amplification rates. Its meaning can best be seen
from Figure 1, where the constant amplitude lines
for the two growth directions Ugr and } are shown.
These lines are normal to the direction of growth,
just as the constant phase lines are normal to the
direction of the wavenumber vector. A certain
growth along Xgr in distance Ax xy requires the
amplification rate along x to be 1/cos (b-bgr)
larger than the amplification rate along Xgr to
yield the same growth along x in the shorter dis-
tance Ax = Axgr cos (¥-Pgr) - It is this relationship
between o() and Sgr that is expressed by (13). For
a fixed orientation of the constant amplitude lines
N
Use LINES OF
#—— CONSTANT
AMPLITUDE
FIGURE 1. Wave growth in direction Xgr as described
by constant amplitude lines normal to Xgr and to X.
normal to Xgrr the growth in different directions
follows the usual vector law with the amplification
rate in direction V1 given by
o(p,) = Go. cos(, - eae ©
We can_use (13) to (a) determine o,, from o(p)
provided is known; (b) determine t if two
neighboring values of o() are known; and (c) answer
the question left open previously of whether we can
make use of the simplification in the spatial theory
afforded by (6). The latter is easily done. With
(6), the transformation (5) applies to spatial waves
and gives
Se o(W) cosw. (15a)
With v =w, (13) relates a() to ~O, by
¥ COM ore
= A
= [o(v)cosw](1 + tany cane) cos eee (15b)
It is evident from this expression that (6) is valid
only for can O (or p = Wg,)- However, o() can
be used to~calculate o_, if Jy, is known, on the
same basis as any other o(i). This procedure is
obviously to be avoided when the direction of k is
perpendicular to that of o .
Spatial Mode - Complex Group Velocity
With a complex dispersion relation, the group veloc-
ity, defined as
oo dw aw
é-G2, gS)
is also complex. For pure temporal or spatial modes,
@ is real only at points of maximum amplification
rate. Consequently, it is important to know how
the complex é affects the preceding analysis. With
a and Ce complex, (8) is no longer hyperbolic, as
pointed Out by Nayfeh et al. (1978). However, it
is still possible to proceed by defining a real
characteristic in the three-dimensional space (xy +
ixj,z). Such a technique was used in a different
context by Garabedian and Lieberstein (1958).
The complex vector group velocity is conveniently
described in terms of a complex magnitude and a
complex angle by writing
G@ S&€ cea, C= € sinh, (17a)
x g Z g
where
2 2
cai +o) (17b)
x Zz
is the complex magnitude, and
VG = Vr + Wi (17c)
is the complex angle.
The complex counterparts of (10) are
*
Il
cosW x + sin) 2z,
g g
N
i]
-sin) x + cos) 2,
g g
With x = x + ix,, and x, required to be real,
x, = tanh . (tan Xi 1Z))
ab gi Cig, 46
and
aw = x % = a= =
Xg = SOS 4 (an BEEN) tanh“ GA 2
= 3e ,
With a real, the analysis for real C applies and
gives for the now complex amplitude along the real
characteristic,
A(xg) = 4 (23,)explo() cos() - Vg)%g]. (18)
This expression differs from (12) in that tg is
complex, has been replaced by z, and z
Xgr g gr
(orthogonal to X'grr see below).
We define
Sab Picet Uae = a= wi)
Soa Be Bey oe tanh veri Bore (19a)
as the characteristic coordinate in the physical
plane to replace Xgr - The angle between Xgr and
Xgr is given by
tan ()gy 2 Vgr) = -tanlgy tanh*}g;. (19b)
We can now write the complex amplitude (18) as
A(X) = Ay (Zgr) exp {ow [cos - Ygr) cosh*Pgi
+i sin( = Ygr) coshi)g; sinhYg; ian}
(20)
The real part of the exponential factor defines the
spatial amplification rate along ee to be
= Sgt De
o = o(v) cos(p - vee cosh Teas (21a)
This expression differs from its real counterpart
(13), aside from the factor cosh? wv gis in that ee
Vgr is the real part of the canoes angle Vg
not the angle formed by the real parts of ce ma
Cz. When fp = Dgr'
2
a cosh Dag 0 (21b)
Q
ll
Q
and, unlike o_, o is not directly calculable as
an eigenvalue?” The imaginary part of the exponen-
tial factor of (20) gives the phase difference be-
tween the elementary mode growing along x and the
spatial mode (20) growing along x' . The phase
difference can be written as JT
= rae mr = Fr = P (22)
Boe - a(W)= o(p) sin(y - Vespe) ENN) SHINY a0
where a is the wavenumber component in the ore
direction. We can now write the complex ¢ counter-
part to the pure spatial mode (14) as
) £(y)
ore fi as ap B. Zz > [ee - a) |x - ut}. (23)
With (23) we have arrived at the spatial mode
that will be used for the numerical calculations
to follow. The amplitude growth is along Xgr with
WE 7,Bpe) = & 9 (2gr) exp (ox" ay
67
magnitude o given by (21b) The eigenvalues are
preferably computed with p = Ygrr but as Ygr is
generally not known in advance, or for computation-
al convenience, they can be computed at a neighbor-
ing ~ and o obtained from (2la). If W is sufficiently
close to Wgr, the phase shift given by (22) is
negligible and the orientation angle i is unaffected
by the transformation.
If Vg were independent of v, both (14) and (23)
would also be expected to be independent of . How-
ever, as v departs from Degen o(~) becomes large and
the evaluation of the complex derivatives in (16)
takes place in a region of the complex a and 6
planes well removed from the points which give v
The same difficulty exists in making comparisons
between temporal and spatial amplification rates.
Although the elementary modes with arbitrary |) are
available for the solution of an initial value
problem by superposition, we give physical signifi-
cance _here only to the special spatial mode with
yp =v All of the other spatial modes, as well
as the2éombined temporal/spatial modes with a,f,w
all complex, do not enter the present analysis
except for computational purposes.
OBLIQUE WAVES IN A TWO-DIMENSIONAL BOUNDARY LAYER
Numerical Example of Transformation Formulas
We shall first discuss the transformations from
three- to two-dimensional form and then the trans-
formation between a spatial mode with arbitrary
growth direction and the mode with growth direction
Vgr- A single numerical example for the Blasius
boundary layer will suffice. We use the conventional
dimensionless frequency parameter F = w*v*/U*T,
and choose the length scale to be L* = (x*V*/U*) 2,
With this choice, the Reynolds number appearing in
(2) sig: Rs (Ut x*/v*)%. The subscript 1 refers to
freestream conditions.
For F = 0.2225 x 107+, R = 1600 and ) = 50°, a
direct calculation of the eigenvalues with (6), i.e.,
~ = 50°, or the, completely equivalent two-dimensional
calculations with either the Squire or Stuart
transformations, gives
ks OIG, a) = A119 x 10%.
Application of the wavenumber transformation rule
in (4) and (5) gives
a = 0.1074, -a, = 2.648 x 10-3
a ab
for the complex wavenumber in the x direction.
Ite Vgr is computed in the neighborhood of wp =
50° from (13) by means of the assumption that o
is independent of ) and with the frequency hela?
constant, we find
p= 9.39°
Were 9
to be an approximate value for the real part of the
complex angle of the group velocity vector. (If
the wavenumber is held constant, gy = 8-80°; a
value closer to the angle formed by the real parts
of Cy and C,.) The eigenvalues of the p - 50° wave
with p = 9.39° are
hk O10, G = Say & lor,
gr
68
and in the x direction
a = 0.1073,-a; = 3.085 x Oss).
The eigenvalues computed with (6) differ from these
values in the fourth decimal place, which means
that as has an unacceptable error of 16.5%, an
error which can also be calculated directly from
(15b). Consequently, this example reiterates that
(6), or the real Squire and Stuart transformations,
can only be used if p = 0 (or p= bgr) -
For the check of che transformation of an ele-
mentary spatial mode with growth direction x to the
'physical' mode with growth direction Xgr, we start
by calculating the eigenvalues as a function of
for 0 <p < 95° and the same F, R and yj as in the
previous example. In addition, we calculate the
complex group velocity by evaluating the complex
derivatives of (2) from central differences for
increments in dy, 8, of +0.001 about the calculated
dy, By at each ~. The real and imaginary parts of
the complex angle Ug are listed in columns 2 and 3
of Table 1. The angle Ygr of the growth direction
x!_, as obtained from (19b), is listed in Column
4. Eigenvalues were computed as a function of jp
with wy = 50° by integrating (2), starting at y/L =
8.0, with a fourth-order Runge-Kutta integration
and 80 equal integration steps. The results are
listed in columns 5 and oF
If (13) with Ugr = = tan”
the o() given in column 6, a nearly constant 6
is obtained out to about = 60°. For v SOO 5 eee
decreases steadily, and at pw = 95° it is 21% lower
than the Sgr for pp = v gr’ Columns 7 and 8 give the
angle and Wace ciber : k for y= = v gr as calculated
from the phase-shift formula (22) of the transform-
ation for complex group velocity. The corresponding
amplification rate, as calculated from (21a), is
listed in column 9. Comparisons of directly com-
puted eigenvalues with these k and o are provided
in the last two columns. Column 10 lists the eigen-
value k computed for the Vgr of column 2 and the wp
of column 7. Column 11 lists the amplification
rate 0 obtained from the eigenvalue Ogy accompany—
ing k and from (21b).
We see that the transformation formulas work
quite well out to J = 60°, where the difference
L(Cen/ Cue) is applied to
between columns 9 and 11 is 0.13%. The change of
the o in colum 9 with jj is only about half of the
change given by the transformation with real group
velocity and the correct jj x given by Cy, and Cz,.
In this particular example, at least, the smallest
change of o with | is found if (13) is used with
v a also computed from (13) on the basis of two
neighboring values of o(\) obtained with the fre-
quency held constant and Sgr assumed to be indepen-
dent of ~. The conclusion to be drawn is that in
order to obtain the desired spatial amplification
rate o as defined by (21b), o(W)may be computed at
some convenient v which can differ from the correct
Vor by as much as 40° or 50°, but should be as close
as possible. Only later, after Vgr and 4 are
know, is o(W) converted to o by the transformation
formulas. Almost any of the methods discussed above
for applying the transformations gives acceptable
numerical accuracy.
Effect of Obliqueness Angle on Instability
The frequency F = 0.2225 x 107* used in the examples
of the previous Section is the most unstable fre-
quency at R = 1600, and the maximum amplification
rate for this frequency occurs for ) = 0°. ‘The
distribution of o with ~ is shown in Figure 2 for
this frequency and F x lot = 0.280, 0.1490 and
0.1008. The latter two frequencies are the most
unstable for y = 60° and 75°, respectively. They
have their peak amplification rates, not for ~ = 0°,
but for y = 34.4° and 61.8°, respectively. These
results demonstrate that although the maximum am-
plification rate at a given Reynolds number with
respect to both frequency and orientation occurs
for a two-dimensional wave, the maximum amplification
rate with respect to orientation of given frequency
occurs for.a three-dimensional wave if the frequency
is less than the most unstable frequency.
The envelope curve formed by the individual
frequency curves is also shown in Figure 2. This
curve gives Om;y,, the maximum amplification rate
with respect to frequency, as a function of i. The
envelope curve emphasizes the wide range of unstable
orientations in a two-dimensional boundary layer.
It can be seen that oy,, is not reduced to one-half
TABLE 1 Numerical check of spatial-mode transformation for complex group
velocity. R = 1600, F = 0.2225 x 107",
w= 50°.
% ik @ & TO?
‘ip k o(t) x 103 v k o x 103
TF p vy. eke F oa Y of C7
7) gr gi (9b)! etga, w= 50° (22) (21a) SX D = There
1 2 3 4 5) 6 U 8 9 10 11
0 9.23 =-4.03 9.18 (o)aleyexs} sical GIS) SS) @)qalleysye) 3.145 0.1669 3.146
9.21 9.21 -—4.02 9.16 0.1669 3.127 50.00 0.1669 Salas} 0.1669 3.143
30.0 Deals AIG, GS), alah 0.1670 3.340 50.02 0.1669 iS leSy/) 0.1669 3.138
60.0 9.08 -4.00 9.04 0.1672 4.928 50.06 0.1670 2}, La OMG OMSL 26
90.0 8.67 =-3.88 8.63 OO) akexsi7/ aS SO) S10) © Oaal7/7/ 2.977 0.1676 3.060
95.0 83365) —SieSuskhiss. 0.1710 46.04 50.66 0.1688 2251/0) (0) WS 27
ENVELOPE (o
max)
% 4
x
in}
3
2
]
0
0 10 20 30 40 50 60 70 80
WY (deg)
FIGURE 2. Amplification rate as function of ~ for four
frequencies. Blasius boundary layer, R = 1600.
of its two-dimensional value until ) has increased
to 60°. With unstable waves for -79° < p < 79°, a
consideration of only the two-dimensional wave gives
an incomplete picture of the instability of the
boundary layer.
THREE-DIMENSTONAL FALKNER-SKAN BOUNDARY LAYERS
In order to study the influence of three dimension-
ality in the mean flow on boundary-layer stability,
it is necessary to have a family of boundary-layers
where the magnitude of the crossflow can be varied
in a systematic manner. The two-parameter yawed-
wedge flows introduced by Cooke (1950) are suitable
for this purpose. One parameter is the usual Falkner-
Skan dimensionless pressure gradient; the other
is the ratio of the spanwise and chordwise velocities.
A combination of the two parameters makes it possible
to simulate simple planar three-dimensional boundary
layers.
The inviscid velocity in the plane of the wedge
and normal to the leading edge is
U* = C#(x*) T)
cy c
where the wedge angle is (1/2) R and § = 2m/(mt1) .
We shall refer to this velocity as the chordwise
velocity. The velocity parallel to the leading
edge, or spanwise velocity is
W* = const.
Sl]
The subscript 1 refers to the local freestream. For
this inviscid flow, the boundary-layer equations
in the x_ direction, as shown by Cooke (1950),
reduce to
2
£” + ££" +8 4) = |S O.
h 2
This equation is the usual Falkner-Skan equation
for a two-dimensional boundary layer, and is inde-
69
pendent of the spanwise flow. The dependent vari-
able f(n) is related to the dimensionless chordwise
velocity by
We S36 _f 2 2°) 5
U m+1
and the independent variable is the similarity
variable
iW) SPN oeers Il a
r c
where xo is measured normal to the leading edge.
Once £(n) is known, the flow in the spanwise di-
. * .
rection Zs is obtained from
where
w*
Si
Both f£'(n) and g(n) are zero at n = O and approach
unity as Tabulated values of g(n) fora
few values of By may be found in Rosenhead (1963,
p. 470).
The final step is to use £'(n) and g(n) to con-
struct the mainflow and crossflow velocity components
needed for the stability equations. A flow geometry
appropriate to a swept back wing is shown in Figure
3. There is no undisturbed freestream for a Falkner-
Skan flow, but such a direction is assumed and a
yaw, or sweep, angle yw is defined with respect to
it. The local freestream, or potential flow, is at
an angle Pp with respect to the undisturbed free-
stream. It is the potential flow that defines the
x,Z coordinates of the stability equations. The
angle of the potential flow with respect to the
chord is
n> ©.
ws
54,
-1
Cesta U* 4
i
and @ is related to We. and i by
UNDISTURBED
y FREESTREAM
FIGURE 3. Diagram of coordinate systems used for
Falkner-Skan-Cooke boundary layers.
70
With the local potential velocity, Up = (uxt + we?)’2,
as the reference velocity, the dimensionless main-
flow and crossflow velocity components are
2 2
f'(n) cos 6 + g(n) sin 6,
U(n) (24a)
W(n) [-£" on) + gin) | cos@ siné . (24b)
These velocity profiles are defined by By, which
fixes £'(n) and g(n), and the angle 6. We note
from (24b) that for a given pressure gradient all
crossflow profiles have the same shape; only the
magnitude of the crossflow velocity changes with
the flow direction. In contrast, according to
(24a), the mainflow profiles change shape as §@ varies.
For 0) = 0, U(n) = £.(n);) for 6 = 90°, UM) = g(n);
for 6= 45°, the two functions make an equal con-
tribution.
When the velocity profiles (24) are used directly
in the stability relations, (2), the velocity and
length scales of the equations must be the same as
in (24). This identifies the velocity scale as U*,
the length scale as P
V*X*/U* (x*) : '
{o cy Cc
ie Ss
and the Reynolds number Were as
R=R/cos® ,
c
where R_ = [vs oxy | 4 is the square root of
c i) Ee re
the Reynolds number along the chord. For positive
pressure gradients (m > 0), 8 = 90° at x = O and
8 > 0° as x > »; for adverse pressure gradients
(m <0), 6) ="90° atix = 0) and! 0) 209 as x =); for
adverse pressure gradients (m < 0), 8 = 0° at x =
0 and 8 + 90° as x > ©. The Reynolds number R, is
zero at x = 0 for all pressure gradients, as is
R with one important exception. The exception is
where m= 1 (8, = 1) is the stagnation-point solution;
here it is the attachment-line solution. In the
vicinity of x = 0, the chordwise velocity is
U* = x* (aU* /dx*) x
cy c cy c x=0
The potential velocity along the attachment line is
eno and the Reynolds number is
R(x=0) = we / |v (a08, /8%2) x20]
a non-zero value.
For the purposes of this paper, 86 may be regarded
as a free parameter, and the velocity profiles (24)
used at any Reynolds number. However, for the flow
over a given wedge, 8 can be set arbitrarily at only
one Reynolds number. If 6,45 is 8 at Ro = (R))
the 6 at any other Ro is given by
a | m/(m+1),
tan® = tan® _. [a 7 |
Cerehaic
ref’
For m << 1, the dependence on R_ is so weak that 6
is constant almost everywhere. “one way of choosing
(Re)ref Within the context of Figure 3 is to make
it the chord Reynolds number where p = 0; i-e.,
the local potential flow is in the direction of the
undisturbed freestream. Then 6 is equal to the
yaw angle j_. mee
Figure 4 shows the crossflow velocity profiles
FIGURE 4. Four crossflow velocity profiles, Falkner-
Skan-Cooke boundary layers. INF, inflection point;
MAX, maximum crossflow; SEP, separation pressure
gradient (fy, = -0.1988377) .
for 6 = 45° and four values of 8}. The inflection
point and point of maximum crossflow velocity (Wmax)
are also noted on the figure. In Figure 5, Wmax for
@ = 45° is given as a function of 8} from near sep-
aration to 8, = 1.0. The crossflow velocity for
any other flow angle is obtained by multiplying the
Wmax of the figure by cos8 sin@. The maximum cross-—
flow velocity of 0.133 is generated by the separa-
tion profiles rather than by the stagnation profiles,
where W = 0.120. However, W varies rapidly
with 8 men the neighborhood of separation, as do
all Behen boundary-layer parameters, and for 8, =
-0.190, W is only 0.102.
The function g(n) is only weakly dependent on
12
ai
:
x
6
)
=
=
4h
|
-0.2 0 0.2 0.4 0.6 0.8 1.0
By,
FIGURE 5. Effect of pressure gradient on maximum cross—
flow, Falkner-Skan-Cooke boundary layers.
al
TABLE 2. Properties of three-dimensional Falkner-Skan-Cooke boundary layers.
Bu 8 Ns ng" be Wiese W finf Ninf
SEP 2.2 8.238 3.495 4.024 0.0102 0.00476 0.487 4.306
5.0 8.236 3.489 4.010 0.0231 0.01077 1.100
10.0 8.229 3.466 3.959 0.0455 0.02123 2.156
40.0 8.095 3.075 3.280 0.1310 0.06214 5.709
45.0 8.058 2.986 3.167 0.1330 0.06339 5.696
50.0 8.017 2.897 3.064 0.1310 0.06274 5.516
-0.10 45.0 6.522 1.985 2.698 0.0349 0.01619 1.498 3213
-0.02 45.0 6.098 1.763 2.609 0.0058 0.00267 0.249 2.940
0.02 45.0 BJoesul 1.682 2.578 -0.0054 -0.00248 =0)232 2.835
0.04 45.0 5.854 1.646 2.564 -0.0104 -0.00480 -0.449 2.787
0.10 45.0 5.646 Ae ‘SISAL 2ro2 9 O02 9) -0.01094 -1.029 2.659
0.20 45.0 5.348 1.424 2.482 -0.0423 -0.01924 milo a3} 2.478
1.0 2.4 3.143 0.6496 2.227 -0.0100 -0.00503 -0.406 1.524
10.0 3.196 0.6603 2.226 -0.0410 -0.02021 -1.669
40.0 3.574 0.8050 2.275 -0.1181 -0.05204 —)o Ie)
45.0 3.621 Oss} Bo shoal —o)paliejal -0.05217 —B)E2Eil
50.0 3.661 OF 87/06) (253325 5-0-8 -0.05081 5 Zeb)
55.0 3.695 0.9024 2.366 -0.1127 -0.04804 —B), SS)
80.0 So7/Sal 1.0153 2.524 -0.0410 -0.01704 AS NSH)
87.6 Jo 72) -0.00416 -0.489
1.0260 2.542 -0.0100
Bre and, unlike f'(n), never has an inflection
point even for an adverse pressure gradient. Indeed
it remains close to the Blasius profile in shape,
as underlined by a shape factor H (ratio of dis-
placement to momentum thickness) that only changes
have its wavenumber vector nearly aligned with the
local potential flow, and we can restrict ourselves
to waves with ~ = 0° for the purpose of determining
the maximum amplification rate. With the temporal
stability theory, this procedure is equivalent to
from 2.703 to 2.539 as, goes from -0.1988377 (sep-
aration) to 1.0 (stagnation). The weak dependence
of g(n) on was first pointed out by Rott and
Crabtree (1952), and made the basis of an approximate
method for calculating boundary layers on yawed
cylinders. For our purposes, it allows some of the
results of the stability calculations to be antici-
pated. For waves with the wavenumber vector aligned
with the local potential flow, we can expect the
amplification rate to vary smoothly from its value
for a two-dimensional Falkner-Skan flow to a value
not too far from Blasius as 6 goes from zero to 90°.
The stability results in the next section will
be presented in terms of the Reynolds number R and
the similarity length scale L*. In order that the
results may be converted to the length scales of
the boundary-layer thickness, displacement thick-
ness and momentum thickness, Table 2 lists the
dimensionless quantities ng = 5/L*, ng* = 6*/L* and
H = ng*/Ng of the mainflow profile for several com-
binations of 6, and 6. Also listed are Wmax, the
average crossflow velocity W = (wan) /ng, the
deflection angle of the streamline at the inflection
point, €jnf, and the location of the inflection
point, Ninf. The quantity ng is defined as the
point where U = 0.999.
studying the two-dimensional instability of the
mainflow profile, but is only approximately so in
the spatial theory _unless Vgr = 0, 7S v 7 is
usually small for y = 0°, even with large cross-
flow, we may also view the = 0° spatial results
as a measure of the instability of the mainflow
profile.
In order to place the three-dimensional effects
in context, it is helpful to first consider a small
deviation in the assumed pressure gradient on the
maximum amplification rate of two-dimensional
Falkner-Skan profiles. Figure 6 shows the maximum
spatial amplification rate (with respect to frequency)
as a function of Reynolds number for Blasius flow
and for By = + 0.02. What is noteworthy about
these results is the magnitude of the shift in Omax
for what are quite small pressure gradients. It
is evident that an experiment intended to measure
amplification rates in a Blasius boundary layer to
within an accuracy of 10% is required to maintain
an exceptional uniformity in the flow.
The effect of the flow angle 8 on the maximum
spatial amplification rate of the waves with i = 0°
is shown in Figure 7 for By, = + 0.02 and two Rey-
nolds numbers. In these calculations, gr and bgi
were both taken equal to zero. The amplification
rate Omax is expressed as a ratio to the Blasius
value (0,)max Shown in Figure 6. It will be re-
called that with 6, = 0, g(n) = £(n), and the
velocity profile remains the Blasius function for
all flow angles. The effect of a non-zero flow
angle withs, # 0 is destabilizing for a favorable
pressure ergatthicrate , and stabilizing for an adverse
pressure gradient. Consequently, it reduces the
pressure-gradient effect shown in Figure 6. The
reason for this result is easy to understand by
reference to (24). We have already pointed out in
Section 4 that the spanwise velocity profile g(n)
STABILITY OF FALKNER-SKAN-COOKE BOUNDARY LAYERS
Boundary Leyers with Small Crossflow
In a two-dimensional boundary layer, the most un-
stable wave is two dimensional. Therefore, we can
expect that in three-dimensional boundary layers
with small crossflow the most unstable wave will
x10?
? max
0 0.5 1.0 1.5 2.0 2.5 3.0
R x 109
FIGURE 6. Effect of small pressure gradients on
the maximum amplification rate with respect to fre-
quency for two-dimensional Falkner-Skan boundary layers.
is always close to the Blasius function. Thus as
the flow angle increases from zero the amplification
rate must change from the two-dimensional Falkner-
Skan value at 6 = 0° to a value not far from Blasius
at 6 = 90°.
As discussed in Section 4, the only physically
meaningful flow with 6 = 90° and a non-zero Reynolds
number is the attachment-line flow (f, = 1.0). For
all other values of 8}, R at this flow angle must be
either zero (8, > 0) or infinite (fy < 0). With By
= 1.0 and R = 1000 (R = 404.2, where Ro is the
momentum-thickness Reynolds number), Omax/ (8b) max
= 0.766. The minimum critical Reynolds number of
this profile is (Rg) jy = 268 (the parallel-flow
Blasius value is 201), yet turbulent bursts have
been observed as low as Rg = 250 for small distur-
bances by Poll (1977).
1,20
Omax/( uy Bynax
9 (deg)
FIGURE 7. Effect of flow angle on the maximum amplifi-
cation rate with respect to frequency of = 0° waves
for two boundary layers with small crossflow at two
Reynolds numbers.
We must still show that the waves with p = 0°
properly represent the maximum instability of three-
dimensional profiles with small crossflow. For this
purpose a calculation was made of 0 as a function
of p for Bh = -0.02, 9 = 45°, R = 1000 and F =
0.4256 x 10-*, the most unstable frequency for ) =
0° at this Reynolds number. It was found that the
crossflow indeed introduces an asymmetry into the
distribution of 0 with W, and the maximum of Oo is
located at = -6.2° rather than at 0°. However,
this maximum value differs from the Omay of Figure
7 by only 0.7%. It was also determined that v r=
-0.04° and Wgi = -0.3° (approximately) for ~ = 0°,
which justifies taking both of these quantities
zero in all of the ~ = O° calculations.
Crossflow Instability
Minimum Critical Reynolds Number of Steady
Disturbances
The instability that is unique to three-dimensional
boundary layers is called crossflow instability.
It was discovered experimentally by Gray (1952) and
later given a detailed theoretical explanation by
Stuart in Gregory et al. (1955). This instability
arises from the inflection point of the crossflow
velocity profile. As explained by Stuart, there
is a particular direction close to the crossflow
direction for which the mean velocity at the in-
flection point of the resultant velocity profile
is zero. Consequently, at sufficiently large
Reynolds numbers unstable steady disturbances exist
which have their constant phase lines nearly aligned
with the potential flow.
Although crossflow instability is by no means
restricted to. steady disturbances, these disturbances
do make a convenient starting point for our investi-
gation. The reason is that a suitable initial
guess for the angle ~, which must be known rather
accurately for the eigenvalue search procedure to
converge, is given by
v= (By/[Bnl) (7/2 - leline)»
where €jnf¢ is the streamline deflection angle listed
in Table 2. It turns out that this value is with-
in a fraction of a degree of the angle of the most
unstable wavenumber. There is no such convenient
rule for the wavenumber itself, but the inverse of
Ning the location of the inflection point in the
similarity coordinate, or better still 0.9/Ning is
usually an adequate enough initial guess to ensure
rapid convergence to an eigenvalue.
As the crossflow is a maximum at 0 = 45° fora
given By, we can expect the crossflow instability
to also be a maximum near this angle. Figure 8
shows the minimum critical Reynolds number Roy at
§ = 45° for the zero-frequency disturbances as a
function of f,. For comparison, Roy of the two-
dimensional Falkner-Skan profiles, as computed by
Wazzan et al. (1968), is also given. For adverse
pressure gradients, the steady disturbances become
unstable at Reynolds numbers well above the Roy of
the two-dimensional profiles. On the contrary, for
Bh > 0.07 the reverse is true, and for most pressure
gradients in this range the steady disturbances
become unstable at much lower Reynolds numbers than
the two-dimensional Roy (for fp, = 1.0, the two-
0.4 b . 1.0
B,
FIGURE 8. Minimum critical Reynolds number as function
of pressure gradient: ——, steady disturbances, Falkner-
Skan-Cooke boundary layers with 6 = 45°; ---, two-
dimensional Falkner-Skan boundary layers [from Wazzan
et al. (1968) ].
cr
107} =
BE SEP
6E
4 | | = |
0 20 40 60 80
9 (deg)
FIGURE 9. Effect of flow angle on minimum critical
Reynolds number of steady disturbances for fy, = 1.0
- and separation boundary layers.
73
TABLE 3. Wave parameters at minimum critical
Reynolds number of steady disturbances.
8 i) R k y (p_)
h cr cr (che! gunen
SEP Ps? 535 0.213 -89.41 0.2
5.0 237 0.213 -88.68 0.4
10.0 iZaL 0.215 -87.44 0.9
40.0 46.5 0.230 -83.54 3.0
45.0 46.7 0.230 —{2}2}55)7/ 3.0
50.0 48.4 OR2 351) —OSmons 3.0
-0.10 45.0 276 0.295 -88.42 0.9
-0.02 45.0 1885 0.310 -89.74 0.2
0.02 45.0 2133 0.322 89.76 —{0)5 AL
0.04 45.0 1129 0.327 89.53 —Ofy2
0.10 45.0 527 0.339 88.93 0),
0.20 45.0 328 0.358 88.12 ok, aL
1.00 Qe 2755 OF553 89.60 -0.3
10.0 671 0.547 88.33 ile ib
40.0 219 0.545 84.88 -3.4
45.0 2ale2) 0.540 84.70 =3}5'5)
50.0 212 0.540 84.70 = 8\5'5)
55.0 218 0.538 84.85 —3i3
80.0 563 0.532 88.00 =1155)
87.6 2325 ORS'S82 I) 5 5.1 O53}
dimensional Rg; is 19,280 compared to Roy = 212 for
zero-frequency crossflow instability).
The distribution of Roy with 6 is shown in Figure
9 for Bh = 1.0 over the complete range of 6, and
for the separation profiles (fp, = -0.1988377) over
the range 0° < 6 < 50°. Near 6 = 0° and 90°, Roy
is very sensitive to 6; near, but not precisely at,
@ = 45° Roy has a minimum. This minimum occurs
close to the maximum of ell Are (cf. Table 2), which,
unlike Wy3., is not symmetrical about 6 = 45°. Table
3 lists the critical wave parameters for a few com-
binations of 8, and 6. The extensive computations
needed to fix these parameters precisely were not
carried out in most cases, and so the values in the
Table are not exact. The listed Ygr was obtained
from (13); Ygi was not calculated.
Boundary Layer with Crossflow Instability Only
As an example of a boundary layer which is unstable
at low Reynolds number only as a result of cross-
flow instability, we select 8}, = 1.0 and @ = 45°,
and present results for the complete range of un-
stable frequencies. Although this pressure gradient
can only occur at an attachment line, Figure 8 leads
us to expect that all profiles with a strong favor-
able pressure gradient will have similar results.
For this type of profile, the minimum critical
Reynolds number of the least stable frequency is
very close to the R of Figure 7. We therefore
choose a Reynolds number well above Rr where the
instability is fully developed.
Figure 10 provides a summary of the stability
characteristics at R= 400. For a given frequency,
the eignevalue o(i) can be computed as a function
of either k or , with the other parameter given as
the second eigenvalue. For strictly crossflow
instability, k is the more suitable independent
variable as i) can have an extremum in the unstable
region. All unstable eigenvalues of a given fre-
quency with a specified increment in k were calcu-
74
lated in a single computer run with Ugr = 0°, and
then corrected to an approximate Ygr (k) obtained
from (13) with constant wavenumber. A least-squares
curve fit to o(k) provided Omay, to maximum spatial
amplification rate with respect to the vector wave-
number, and kmax and Wax, the magnitude and direc-
tion of the wavenumber of Omax-
Figure 10a gives Omax aS a function of the di-
mensionless frequency F, and also shows the portion
of the ~-F plane for which there is instability.
The unstable region is enclosed between the curves
marked and w_. These curves represent either
neutral stability points or extrema of }.
The corresponding wavenumber magnitudes are
shown in Figure 10b. The negative frequencies
signify that with taken to be continuous through
F = 0, the phase velocity changes sign. If we
choose ) so that the wavenumber and phase velocity
are both positive, then it is | that changes sign
at F = 0. Consequently, there are two groups of
positive unstable frequencies with quite different
phase orientations. The first group, which includes
the peak amplification rate, is oriented anywhere
from 5° to 31° (clockwise) from the direction opposite
to the crossflow direction. The second group is
oriented close to the crossflow direction itself.
All of the unstable frequencies have in common
that the direction of growth is within a few degrees
of the potential~flow direction. The angle Ugr of
Umax, aS computed from (13), is negative and has
its largest magnitude of just under 6° near F =
-0.60 x 10-+. Orientations other than Umax can
have growth directions further removed from the
flow direction.
Boundary Layers with both Crossflow and Mainflow
Instability
As an example of a boundary layer which has both
crossflow and mainflow instability at low Reynolds
numbers, we select 8, = -0.10 and 6 = 45°. In con-
trast to the previous case, the steady disturbances
do not become unstable until a Reynolds number, R =
276, where the peak amplification rate is already
7.35 x 10-°. [For B} = -0.10 and @.= 0° omax =
MMO Os! ate kes 22 Om according to Wazzan
et al. (1968)]. The distribution of o with jp is
shown in Figure 11 for F = 2.2 x 10-4, a frequency
close to the most unstable frequency of F = 2.1 x
10- . We see that with a maximum crossflow velocity
of 0.0349 (cf. Table 2), the distribution of o about
w = 0° is markedly asymmetric, and the maximum
amplification rate of 7.31 x 1073 is located at jp =
-29.4° rather than near zero. This asymmetry was
barely perceptible for the small crossflow boundary
layers of Figure 7 where the crossflow is only one-
sixth as large. The o at pp = 0° of Figure 11 (5.82
x 10-3) is close to Omax With respect to frequency
of the » = 0° waves (5.91 x 10°3). Since this value
is 20% below the peak amplification rate, the =
O° waves are no longer adequate to represent the
Maximum instability as with small crossflow boundary
layers. Figure 11 also gives the distribution with
of k and Wgr- The latter quantitiy was obtained
from (13) with constant wavenumber, and we see that
it remains within + 7.5° of the potential-flow
direction throughout the unstable region.
Because R = 276 is the minimum critical Reynolds
number of the steady disturbances, the unstable
region terminates in a neutral stability point at
UNSTABLE
0.2 ki STABLE
I i 1 it
O05) Os SNM ONNZESINSSO
Fx104
0 L er
-2.0 -1.5 -1.0 -0.5
FIGURE 10. Instability properties of 8, = 1.0, 8 = 45°
Falkner-Skan-Cooke boundary layer at R = 400. (a) maxi-
mum amplification rate with respect to Wavenumber and
unstable ~ - F region; (b) unstable k-F region.
F = 0. We are particularly interested here in Rey-
nolds numbers where F = O is also unstable, and as
an example, Figure 12 gives results for all unstable
frequencies at R= 555. Figure 12a shows Omax as
a function of F (here, as in Figure 10, Omax 1s the
maximum with respect to k), as well as the unstable
region of the k-F plane; the unstable region of the
W-F plane appears in Figure 12b. These two unstable
regions are quite different from those of Figure 10
where there is only crossflow instability. The
negative frequencies do resemble those of Figure 10
in that the unstable range of ~ is small, of k is
large, and with defined so that F > 0, the orien-
tations are close to the crossflow direction. How-
ever, for the higher frequencies, which are by far
a x 10°, k x 10
=70 -60 =50 =40 -30) -20) -10 0 10 20 30 40
(deg)
FIGURE 11. Effect of wavenumber angle on 9, k and Vgr
for By, = -0.10, 8 = 45° Falkner-Skan-Cooke boundary
layer at R = 276. F = 2.2 x 107".
x103, k x 10
STABLE
UNSTABLE
-60
-80
-100
-0.2 0 0.2 0.4 0.6 0.8 1,0 1.2 1.4
Fx10
FIGURE 12. Instability properties of Bh = -0.10, 9 =
45° Falkner-Skan-Cooke boundary layer at R = 555.
(a) maximum amplification rate with respect to wave-
number and unstable k-F region; (b) unstable -F region.
the most unstable, the unstable regions of Figure
12 bear more of a resemblance to those of a two-
dimensional boundary layer than to Figure 10. The
main differences from the two-dimensional case are
the asymmetry about = 0° already noted in Figure
11, the one-sidedness) Of Wnax, and, for F < 0.4 x
10-4, the replacement of a lower cutoff frequency
for instability by a rapid shift with decreasing
frequency to waves oriented opposite to the cross-
flow direction and which are unstable down to zero
frequency. The instability shown in Figure 12
represents primarily an evolution of the small cross-
flow boundary layers of Figure 7 to larger cross-
flow. Only the frequencies, say |F| < 0.2 x loser
have to do with the pure crossflow instability of
Figure 10. For frequencies near 0.4 x 1074 ,wy
varies little with k in one part of the unstable
region, as with crossflow instability; in the other
part, as with mainflow instability, the opposite
is true. This behavior becomes more pronounced at
high Reynolds numbers.
CONCLUDING REMARKS
All of the numerical results that have been presented
stem from the viewpoint adopted in Section 2 that
75
useful information concerning three-dimensional
boundary-layer stability can be obtained from par-
ticular pure spatial modes just as with two-
dimensional boundary layers. Arguments were given
to support using the modes whose growth direction
is determined from (17) or, more exactly, from
(19b). A transformation (2la), was derived to
enable the use of waves with an arbitrary growth
direction in calculating eigenvalues. The trans-
formation used in the temporal theory to reduce
the three-dimensional problem to a two-dimensional
probelm in the direction of the wavenumber vector
was shown to apply to spatial modes only when this
direction is close to the correct growth direction,
or the latter is the same as the potential-flow
direction WVgr = 0°).
The waves which have their wavenumber vector
aligned with the local potential flow (i) = 0° when
the x axis of the mean-flow coordinate sytem is
also in the flow direction) always have their growth
direction very close to the potential-flow direction.
If the crossflow is small, the maximum amplification
rate of the ~ = 0° waves is almost identical to the
maximum amplification rate of the three-dimensional
boundary layer. Consequently, if we are only in-
terested in establishing the maximum amplification
rate of a small crossflow boundary layer, it can be
obtained from the mainflow profile alone. We used
this approach to obtain the effect of the flow (yaw)
angle on the instability of the Falkner-Skan-Cooke
yawed-wedge boundary layers for small pressure
gradients, and found that yaw reduces both the
stabilizing effect of a favorable pressure gradient
and the destabilizing effect of an adverse pressure
gradient.
With moderate or large crossflow, crossflow in-
stability, which arises from the inflection point
of the crossflow velocity profile, is present and
can destabilize a boundary layer at low Reynolds
numbers which would otherwise be stable. As befits
the name, the unstable waves have their wavenumber
vectors oriented near the crossflow (or opposite)
direction. Also the instability covers a wide band
of unstable frequencies (including zero) and wave-
numbers. The growth direction of all unstable waves
is still near the potential-flow direction. If the
mainflow profile is also unstable, then the unstable
frequencies near zero act as with pure crossflow
instability and the higher frequencies as with pure
mainflow instability. Intermediate frequencies
have the latter behavior for small wavenumbers, and
the former for large wavenumbers.
The results demonstrate why crossflow is more of
a problem for the maintenance of laminar flow with
strong favorable pressure gradients than with ad-
verse pressure gradients. In the former case, cross-
flow provides a powerful instability mechanism
even when the mainflow profile is stable; in the
latter, the crossflow only increases the amplifi-
cation rate over that of an already unstable main-
flow profile. This increase is about 50% for the
6 = 45° separation boundary layer.
ACKNOWLEDGMENT
This paper represents the results of one phase
of research carried out at the Jet Propulsion Lab-
oratory, California Institute of Technology under
Contract No. NAS7-100 sponsored by the National
Aeronautics and Space Administration. Financial
76
support is gratefully acknowledged from the Tactical
Technology Office, Defense Advanced Research Projects
Agency, and from Langley Research Center.
REFERENCES
Brown, W. B. (1961). A stability criterion for
three-dimensional laminar boundary layers.
Boundary Layer and Flow Control, Vol. 2, G. V.
Lachmann, ed., Pergamon Press, New York, pp.
913-923.
Cooke, J. C. (1950). The boundary layer of a class
of infinite yawed cylinders. Proc. Camb. Phil.
Soc. 46, 645.
Davey, A. (1972). The propagation of a weak non-
linear wave. J. Fluid Mech. 53, 769.
Grabedian, P. R., and H. M. Lieberstein (1958).
On the numerical calculation of detached bow
shock waves in hypersonic flow. J. Aero. Sci.
25), LOSE
Gray, W. E. (1952). The nature of the boundary
layer at the nose of a swept back wing, Unpub-
lished work Min. Aviation, London.
Gregory, N., J. T. Stuart, and W. S. Walker (1955).
On the stability of three-dimensional boundary
layers with application to the flow due toa
rotating disk. Phil. Trans. Roy. Soc. London
A248, 155.
Landau, L. D., and E. M. Lifshitz (1960). Electro-
dynamics of Continuous Media, Pergamon Press,
New York, p. 263.
Mack, L. M. (1977). Transition prediction and
linear stability theory. Laminar-Turbulent
Transition, AGARD Conference Proceedings No. 224,
Pp l= to di=22'5
Nayfeh, A. H., A Padhye, and W. S. Saric (1978).
The relation between temporal and spatial sta-
bility in three-dimensional flows, AIAA Paper, to
be presented.
Poll, D. I. A. (1977). Leading edge transition on
swept wings. Laminar-Turbulent Transition,
AGARD Conference Proceedings No. 224, 21-1 to
Aaloilal.
Rosenhead, L. (1963).
Oxford Univ. Press.
Rott, N., and L. F. Crabtree (1952. Simplified
laminar boundary layer calculations for bodies
of revolution and for yawed wings. J. Aero.
Seri do), 35s}q
Squire, H. B. (1933). On the stability for three-
dimensional disturbances of viscous fluid flow
between parallel walls. Proc. Roy. Soc. London
Al42, 621.
Wazzan, A. R., T. T. Okamura, and A. M. O. Smith
(1968). Spatial and temporal stability charts
for the Falkner-Skan boundary-layer profiles.
McDonnell Douglas Report No. DAC-67086, Long
Beach, Calif.
Whitham, G. B. (1974). Linear and Nonlinear Waves,
Wiley-Interscience, New York.
Laminar Boundary Layers,
Experiments on Heat-Stabilized Laminar
Boundary Layers in a Tube
Steven J. Barker
Poseidon Research* and University
of California at Los Angeles
ABSTRACT
There has been considerable recent interest in the
stabilization of water boundary layers by wall
heating. Calculations based upon linear stability
theory have predicted transition Reynolds numbers
as high as 2 x 108 for a zero pressure gradient
boundary layer over a heated wall. The flow tube
experiment described in this paper was intended to
investigate these predictions. The test boundary
layer develops on the inside of a cylindrical tube,
0.1 m in diameter and 6.1m in length. The dis-
placement thickness is small relative to the tube
radius under nearly all operating conditions. The
tube is heated by electrical heaters on the outside
wall. The location of transition can be determined
by a heat flux measurement, by flush-mounted hot
film probes, or by flow visualization at the tube
exit.
A transition Reynolds number of 107 can be ob-
tained without heat, which shows that free stream
turbulence and other perturbations are well con-
trolled. At 7°C wall overheat, a transition
Reynolds number of 42 x 10° has been obtained,
which is at least as high as the prediction for
that overheat. However, as temperature is further
increased there have been no additional increases
in transition Reynolds number, which is in contra-
diction to the theory.
Possible reasons for the differences between
theory and experiment have also been investigated.
New test section exits have been developed to
determine the effects of downstream boundary con-
ditions upon the flow. An instrumented section
has been used to measure detailed velocity profiles
in the boundary layer, and determine intermittency
as a function of azimuthal angle. From these
measurements we can evaluate the possibility of
*
This work was performed by the Marine Systems Division of
Rockwell International, and Poseidon Research. It was
sponsored by the Defense Advanced Research Projects Agency.
77
buoyancy-generated instabilities in the tube.
Future tests will also investigate the influence
of free stream turbulence, streamwise vorticity
in the boundary layer, and wall temperature vari-
ations.
1. INTRODUCTION
Numerical calculations such as those of Wazzan,
Okamura, and Smith (1968, 1970) have predicted large
increases in the transition Reynolds numbers of
water boundary layers with the addition of wall
heating. The stabilizing mechanism is the decrease
in fluid viscosity near the wall resulting from the
heating. This increases the negative curvature
of the velocity profile, making the flow more stable
to small disturbances. The present study is an
experimental investigation of these predictions,
using the boundary layer developing on the inside
wall of a cylindrical tube. This boundary layer is
thin relative to the tube diameter, so that it
approximates a boundary layer over a flat plate.
The numerical predictions of Wazzan et al. are
based on two-dimensional, linear stability theory.
The mean flow is assumed plane and parallel, and
the superimposed small disturbance is described by
a stream function,
W(x,y,t) = o(y) exp ia(x-ct) (1)
Here $(y) is the disturbance amplitude, a is the
wavenumber and is assumed real, and c is the wave
velocity which may be complex. The imaginary part
of c determines whether the disturbance is tempo-
rally amplified or damped. If we substitute this
stream function into the Navier-Stokes equations
and linearize, taking account of the variation of
viscosity || with distance from the wall y, we find
at nm 2 "
GRe Lh (> Ze) Qa
(WU = e) (OY = a4) = U"> =
m
a 6) + 2ut (on! - a6") + uM(o" + a6)] (2)
78
In this equation, U(x) is the external flow velocity
and Re is the Reynolds number based upon free stream
velocity U,, and boundary layer thickness§5. This
is known as the "modified Orr-Sommerfeld equation,"
the variable viscosity terms.
Wazzan et al. have solved Eq. (2) numerically
for the boundary layer over a heated flat plate,
using velocity profiles generated by the method of
Kaups and Smith (1967). The solutions determine
the critical Reynolds number, which is the lowest
Reynolds number at which any disturbance has a
positive amplification rate. The last step of the
calculation is to relate the critical Reynolds
number to the transition Reynolds number, using the
"e to the ninth" criterion of A. M. O. Smith (1957).
According to this empirical criterion, transition
occurs when the most unstable disturbance has grown
to e? (which is 8,103) times its original amplitude.
The linear theory is used in calculating the growth
of the disturbance to this amplitude.
Strasizar, Prahl, and Reshotko (1975) have
measured growth rates of disturbances generated by
a vibrating ribbon in a heated boundary layer. They
found neutral stability curves and were able to
determine critical Reynolds numbers for wall over-
heats of up to 5°F (2.8°C). They found that in this
range of overheats the critical Reynolds numbers
are in reasonable agreement with the theoretical
predictions. These experiments were performed at
moderate Reynolds numbers and did not yield data on
transition or on stability at higher overheats.
The results of the Wazzan et al. calculations
predict that the transition Reynolds number of a
zero pressure gradient boundary layer should increase
with wall temperature up to about 70°F (39°C) of
overheat if the free stream temperature is 60°F
(16°C). At that overheat, the transition Reynolds
number should be in excess of 2 x 10° (based upon
distance from the leading edge). Thus the experi-
ment designed to investigate these predictions must
be able to generate a very high Reynolds number
boundary layer while maintaining low free stream
disturbance levels. The wall should be very smooth
and its temperature must be precisely controlled.
These are the chief considerations that led to the
experimental geometry described below.
2. EXPERIMENTAL APPARATUS
Configuration
A facility in which water is recirculated through
the test section was not used for two reasons. (1)
Heat is continuously added to the test section so
that a recirculating experiment would require some
sort of heat exchanger. (2) The free stream tur-
bulence level in the test section must be less than
0.05 percent, which has previously been difficult
to achieve in a recirculating water facility. The
experiment must then be of the "blow-down" type,
in which water is removed from one reservoir and
discharged into another. Run times of more than
twenty minutes are desired, which requires large
reservoirs. This led to the selection of the Colo-
rado State University Engineering Research Center
as the site of the experiment. Here the water
supply is Horsetooth Reservoir, which provides
water to the laboratory through a 0.6 m diameter
pipe at a total pressure of 6.8 x 10° N/m? (100 1b/
in.*). The discharge runs into a smaller lake be-
HORSETOOTH
RESERVOIR
FILTRATION
TAN
K
(7-FT DIA) 24 IN. DIA.
24 IN. DIA DISCHARGE
UPSTREAM SETTLING LINE
TUBE CHAMBER
(24-1N. DIA
FLOW TUBE
BALL VALVE
36:1
VIBRATION ORIFICE PLATE
ISOLATION DISCHARGE
SECTION
FIGURE 1. Experimental geometry.
low the laboratory. At the maximum flow rate of
this experiment (200 liters/sec), the run time is
effectively unlimited.
The flow tube apparatus consits of a settling
chamber for turbulence management, a contraction
section, a test section and various types of instru-
mentation described below. A diagram of the experi-
mental geometry is shown in Figure l.
Settling Chamber
The inside diameter of the settling chamber is 0.6
m, the same as that of the supply line from the
reservoir. The test section is 0.102 m in diameter,
so that the contraction ratio is 35:1. The settling
chamber is made up of four separable sections, as
shown in Figure 2. The sections are made of fiber-
glass to avoid heat transfer through the walls, and
their total length is 3.35 m. Each end of each
section is counter-bored to hold a 0.15 m long
aluminum cylinder with a 1.3 cm wall thickness.
Each cylinder will hold one or more turbulence
manipulators, including screens, porous foam, or
honeycomb material. This design allows the settling
chamber to be assembled in different configurations,
so that it can be optimized experimentally.
The details of the design and optimization of
the turbulence management system have been reported
separately [Barker (1978)]. The configuration
shown in Fibure 2 was arrived at after a great
deal of testing. There is a considerable body
of literature on the subject of turbulence
Management, and this provided some guidelines
for the optimization of the present system.
The most detailed recent study is that of Loehrke
and Nagib (1972), who measured mean velocity and
turbulence level downstream of various turbulence
Manipulators. Further recommendations for the
construction of a turbulence management system
have been given by Corrsin (1963), Bradshaw (1965),
and Lumley and McMahon (1967).
At the downstream end of the settling chamber is
an additional 0.30 m long section containing porous
wall boundary layer suction. Hot film anemometer
surveys in the settling chamber have shown that
at test section velocities above 9 m/sec (0.26 m/sec
in the settling chamber) the boundary layer becomes
turbulent before the flow enters the contraction.
A thin turbulent boundary layer entering the strong
favorable pressure gradient of the contraction
79
SECTION
FIGURE 2. Schematic of turbulence manage-
-— TURBULENCE MANAGEMENT SECTION ————~}
VIBRATION. C
}—-—4 FT cot meet IET) Lis Nar |e |_— -CONTRACTION
pa u SECTION
/10 PPI FOAM A Loy
/
SUCTION /
VoSECTIONS Seed
UPSTREAM 34 MESH | 6 MESH
BALL SCREEN \
VALVE \
‘1/8 CELL 1/8 CELL 34 MESH
HONEY COMB HONEYCOMB. (2 SCREENS)
3" THICK 3" THICK
section will tend to "relaminarize," as described
by Launder (1964) and Back et al. (1969). However,
this would leave us with unknown initial conditons
at the entrance to the test section. Therefore we
have added the suction section to completely remove
the turbulent boundary layer. This section has a
0.1m length of porous wall surrounded by an annular
plenum chamber. The suction flow from the plenum
is controlled by a valve and a Venturi meter. At
each test section velocity above 9 m/sec, the
suction flow is adjusted to the minimum value
necessary to remove the turbulent boundary layer
at the contraction entrance.
Contraction and Test Section
The 35:1 contraction was designed by a potential
flow calculation using the method of Chmielewski
(1974). The length.to diameter ratio of the
contraction was chosen by balancing the effect of
relaminarization with that of the Goertler insta-
bility in the concave-curved portion. A careful
study of these two effects led to a length to
diameter ratio of 2.25, which made the contraction
1.37 m long. The contraction was constructed in
two sections: a fiberglass upstream half and an
aluminum downstream half. The joint between the
two sections is in the region of greatest favorable
pressure gradient, and has no measurable step across
Estee
Recent velocity measurements in the test section
(discussed below) have led to the design and con-
struction of a new contraction section to replace
the original one. The new contraction will have
an annular bleed flow surrounding an entrance
section which is all convex. In this way the
concave-curved wall, which can produce Goertler
vortices, will be avoided entirely. Results using
this new contraction will soon be available.
The flow tube test section is 6.4 m in length
and 0.102 m in diameter, with a 2.5 cm wall thickness.
It is made of aluminum, and the inside wall has been
polished to a surface roughness of less than 10-7 m
RMS (4 micro-inches). Surface waviness has been
measured as less than one part per thousand for
wavelengths less than 2 cm. The tube has been
optically aligned on site so that it is straight to
within less than 0.018 cm over its entire length.
The outside wall is covered with electrical band
. heaters, which are connected together in groups
covering about 0.30 m of length. Each heater group
ment system.
is servocontrolled by a system which maintains a
preset temperature on a thermocouple located near
the inside tube wall. In this way the inside wall
temperature can be controlled independently of flow
velocity, and different variations of temperature
along the tube length can be studied.
To avoid tripping the boundary layer, no pene-
trations of the inside wall are allowed except at
the downstream end. The only instrumentation in
the test section is an array of thermocouples within
the wall, spaced along the tube length. At each
location, there is one thermocouple on the outside
surface and one in a small hole drilled to within
0.15 cm of the inside surface. The temperature
difference between the two thermocouples determines
the heat flux through the wall at a particular
location. Since heat flux increases by a factor of
about ten at the transition point, these temperature
measurements should provide a good transition
indicator. A total of 53 thermocouple voltages are
digitized and recorded.
During the earlier experiments, there was a single
hot film anemometer probe at the downstream end of
the test section. This probe was located within
the boundary layer and was used to indicate inter-
mittency only. In the more recent measurements, a
new instrumented section has been developed and
installed on the.downstream end of the test section.
This section is 0.61 m long and its inside diameter
matches that of the test section to within 2 x 107°
m. Two types of measurement can be made in the
instrumented section. Very small Pitot tubes can
be used to traverse the boundary layer and measure
mean velocity profiles, and flush mounted hot films
can determine intermittency at various locations.
Since the boundary layer is typically less than
0.5 cm thick, the Pitot tubes must be very small.
The one being used at present has a cross-section
of 0.013 x 0.076 cm. The smaller dimension is
oriented in the direction perpendicular to the wall.
The tube is traversed from the wall to the free
stream by a micrometer, which can position it with
an uncertainty of +0.002 cm. In addition, the
entire central portion of the tube can be rotated
in the azimuthal direction so that the Pitot tube
can be traversed about the circumference of the
test section. The azimuthal rotation can be per-
formed while the experiment is running.
The hot film anemometers in the instrumented
section are all mounted flush with the wall to avoid
tripping the boundary layer. The Pitot tubes are
removed from the section while hot film measurements
80
are being made. The films are used only to determine
intermittency, hence they are not calibrated. There
are eight hot film locations--two streamwise sep-
arated stations each having four probes at different
azimuthal angles. All eight outputs can be displayed
simultaneously on oscilloscope traces or recorded
on a photographic strip-chart recorder.
A high static pressure must be maintained in the
test section to avoid possible cavitation or out-
gassing from heated walls. Therefore the pressure
loss for controlling the flow velocity is located
at the downstream end of the experiment. Originally,
a set of sharp-edged orifice plates was used on the
end of a 1 m long extension tube added to the test
section. Concern over possible upstream influence
of the disturbances generated at the orifice plate
led to the development of a smooth contraction
section for the downstream end. With the smooth
contraction, it is possible to maintain laminar
flow all the way to the exit of the experiment, and
thus determine transition by flow visualization in
the exit jet. In addition, a "plug nozzle" has
been developed, which consists of a strut-supported
central cone which can be moved in and out of the
end of the test section. This adjustable exit
valve permits us to vary the test section static
pressure independently of flow velocity while main-
taining laminar flow all the way to the exit. With
any of these possible exit conditions, the test
section velocity can be determined from the test
section static pressure and the known discharge
coefficient of the nozzle.
3. RESULTS
Free Stream Turbulence
Mean and fluctuating velocities were measured in
the settling chamber by a cylindrical hot film
anemometer. The probe penetrated the settling
chamber wall 0.1 m downstream of the boundary layer
suction section, and could be traversed from the
wall to the centerline. Mean velocities and tur-
bulence levels were measured at many points, and
turbulence spectra were measured at two or three
points for each flow condition. In addition, a
1.2 m long instrumented straight tube could be
substituted for the 6.4 m test section. This short
tube contained a Pitot tube, accelerometers, and
hot film probes. The unheated transition Reynolds
number was measured in the 1.2 m tube for each
settling chamber configuration. This Reynolds
number varied from 800,000 for the empty settling
chamber with no turbulence manipulators to 5.0 x
10® for the "best" configuration. This configura-
tion (shown in Figure 2) includes one piece of porous
foam, two sections of honeycomb, and four screens.
The last screen is located 0.3 m upstream of the
beginning of the contraction, and has a mesh of 24
per cm. All screens in the settling chamber have
more than 55 percent open area, in accordance with
the findings of Bradshaw (1965).
Detailed results of the velocity measurements
in the settling chamber have been reported separately
[Barker (1978)], and are only summarized here. At
test section velocities less than 9 m/sec, the
settling chamber boundary layer remains laminar and
the only effect of the suction is to make it thinner.
The turbulence level is about 0.07 percent at all
distances from the wall for the configuration of
Figure 2. At higher velocities the turbulence level
near the wall reaches 3 or 4 percent with no suction,
but remains 0.07 percent at distances from the wall
greater than 2 cm. As the suction flow rate is
increased, the mean velocity profile shows thinning
of the boundary layer and the turbulence level near
the wall drops rapidly. At the optimum suction
rate, the highest turbulence level near the wall
in the settling chamber is about 0.4 percent. The
suction has no measurable effect upon the mean
velocity profile or turbulence level more than 2
cm from the wall.
The settling chamber velocity measurements and
the unheated transition Reynolds numbers indicate
that the turbulence management system is performing
well. If the turbulence level reduction through
the contraction is proportional to the square root
of the contraction ratio [Pankhurst and Holder
(1952) ], then the turbulence level in the test
section should be about 0.01 percent. This is
lower than the turbulence level recorded in most
wind tunnels, and certainly lower than any previ-
ously reported water tunnel.
Transition Reynolds Numbers
Figure 3 shows measured transition Reynolds numbers
as a function of wall overheat for the uniform wall
temperature case. The results on the upper curve
were obtained with the smooth, laminar flow nozzle
at the downstream end of the test section, using
flow visualization at the exit to determine tran-
sition. The water temperature was approximately
50°F (10°C) during these tests. Note that the
transition Reynolds number rapidly increases with
wall temperature up to 10°F (6°C) wall overheat,
at which it has reached a value of 42 x 10°®.
This represents a factor of four increase in tran-
sition Reynolds number for a relatively small heat
input. However, above 10°F there are no further
increases in transition Reynolds number, while the
theory predicts that it should increase up to about
60°F (33°C) overheat. Previously published results
[Barker and Jennings (1977) ] have shown that varying
the wall temperature distribution does not change
hh
/
7 SMOOTH
7 NOZZLE
ORIFICE
PLATE
o
1
S
x
Ix
= 7 WAZZAN, ET. AL.
© B= 0.07
ONE EXTENSION
0 5 lo 15 20 25 30
OVERHEAT, AT (°F)
FIGURE 3. Transition Reynolds numbers measured at
exit: one extension tube.
81
this result. In fact, uniform wall temperature has
produced the largest transition Reynolds numbers to
date. The primary difference between the results
shown here and those published previously is that
the present experimental curve reaches the limit
Reynolds number of 42 x 10° at a lower overheat
than before. This change is attributed to the
improvement of the exit conditons with the develop-
ment of the laminar flow nozzle.
All of the data of Figure 3 were taken by main-
taining laminar flow over the full length of the
tube and observing transition at the exit. If the
flow velocity is increased further, so that the
transition region moves upstream in the test
section, the measured transition Reynolds numbers
are much lower. In addition, there is a hysteresis
effect when transition is allowed to move more than
about 1 m upstream from the exit. That is, to
restore fully laminar flow over the full tube length
the velocity must be reduced to a value lower than
that which previously yielded fully laminar flow.
This hysteresis may be a phenomenon which is accen-
tuated by the flow tube geometry. The free stream
in the flow tube is confined by the boundary layer,
so that the boundary layer can influence the free
stream once it becomes turbulent. This free stream
influence could propagate upstream, which has led
to conjecture about disturbances from the test
section exit affecting the transition Reynolds
number.
To test this hypothesis of downstream disturbances
affecting transition Reynolds number, a separate
study has been conducted to determine the dependence
of transition upon the tube exit geometry. As dis- FIGURE 4a. Exit jet from smooth nozzle: laminar
cussed above, there are three types of exit nozzle boundary layer.
available: orifice plates, the smcoth contraction,
and the plug valve. In addition, the length of
unheated tube between the heated test section and
the exit can be varied from zero to 3.7 m in incre-
ments of 1.22 m. For each configuration, transition
can be determined either at the exit itself or at
the end of the heated section. Transition at the
exit is easily determined by flow visualization, as
shown in Figure 4. This photograph of the smooth
exit contraction shows laminar flow (4a) and turbu-
lent flow (4b), both at a length Reynolds number
of approximately 40 x 1O®. ma Figure 4a, note
the glassy region very near the exit, which soon
becomes milky in appearance as the air-water shear
layer undergoes transition. The longitudinal streaks
in Figure 4a are appraently due to Goertler vortices
generated in the concave part of the smooth exit
contraction. They are not seen with the plug valve
exit, which has no concave region.
The data of Figure 3 are for one 1.22 m extension
section on the end of the heated section, followed
by either the smooth contraction or the orifice
plate. Transition is measured at the exit in either
case. Note that the transition Reynolds numbers
with the orifice plate exit are about 20 percent
lower than with the smooth contraction, showing a
definite effect of the exit condition. Figure 5
shows the same comparison with 2.44 m of unheated
extension tube between the test section and exit.
Here we see a much larger difference between results
with the orifice and with the smooth contraction.
The smooth contraction transition Reynolds numbers
are nearly the same as with 1.22 m of extension
tube, while the orifice Reynolds numbers have
dropped almost by a factor of two. Clearly the FIGURE 4b. Exit jet from smooth nozzle: turbulent
effect of the exit condition upon transition Reynolds boundary layer.
SMOOTH
NOZZLE
1076
oseanea
5 = 0.07
Rex,
ORIFICE
PLATE
TWO EXTENSIONS
° 5 Te) 15 20 25 30
OVERHEAT, AT (°F)
FIGURE 5. Transition Reynolds numbers measured at exit:
two extension tubes.
number is far more pronounced here than for the
shorter extension tube length. The most reasonable
explanation of this lies in the fact that in the
second case the boundary layer has passed over a
much longer region of unheated wall, which should
have a destabilizing effect. This less stable
boundary layer is then more sensitive to external
perturbations such as the disturbances created by
the orifice plate exit.
As the extension tube length is increased still
further, the transition Reynolds numbers obtained
with the smooth contraction begin to decrease.
Apparently the destabilizing effect of the long
unheated wall is felt even with the low disturbance
exit condition. These results indicate that, under
some conditions, a moderate length of unheated wall
can be used downstream with no measurable reduction
of transition Reynolds number.
When transition is determined at a distance of
1.4 m upstream of the exit rather than at the exit
nozzle itself, the influence of the exit condition
is greatly diminished. Taking the case of the 2.44
m unheated extension as an example, there is a
factor of 2.3 difference in the maximum transition
Reynolds number obtained with the orifice and with
the smooth contraction when transition is measured
at the exit (Figure 5). However, when transition
is measured 1.4 m upstream of the exit, the corre-
sponding difference is only 15 percent in Reynolds
number. Clearly the disturbances present at the
exit nozzle can affect the transition process if
it occurs near the nozzle, but this influence
diminishes rapidly as transition moves upstream
of the exit. Since the highest transition Reynolds
numbers have consistently been obtained with laminar
flow over the full length of the tube, most future
measurements will be made using one of the two
laminar flow exit conditions.
Although it is difficult to assess uncertainties
in transition Reynolds number in this experiment,
some effort should be made. Results for the highest
transition Reynolds numbers exhibit a large amount
of scatter, but most of this can now be attributed
to variations in the free stream particulate content.
The purity of the water supply varies considerably
with weather conditions at the site, and these
changes in purity have been directly correlated
with changes in transition Reynolds number. Under
the most adverse conditions, this effect has reduced
the maximum transition Reynolds number to less than
15 x 108 (compared with 42 x 10© for "clean" water) .
If we compare results that were obtained during
periods of relatively high water purity, the stan-
dard deviation in transition Reynolds number is
about 10 percent of the mean.
This extreme sensitivity of the results to water
purity was quite unexpected, and an effort has been
made to improve the water quality by filtering
upstream of the settling chamber. Measurements of
the particle concentration spectrum have been made
using a Coulter Counter, and some of the results
are shown in Figure 6. The bands on this figure
indicate the typical ranges of concentration that
are obtained in the present experiments, as well
as in the NSRDC towing basin and the ocean. Note
that the flow tube particle spectrum has a steeper
slope than either the ocean or the tow basin, which
implies that for particle sizes greater than 10 U,
the flow tube water is much cleaner than the other
two. The filtration system presently used in the
flow tube effectively removes all particles larger
than 100 iL.
The reason for the strong sensitivity of results
to relatively minor contamination of the water
supply is not understood at present. The most
likely mechanism seems to be a slight increase in
wall roughness due to the adhesion of particles
to the wall. Whatever the mechanism, this effect
will clearly be of importance in hydrodynamic
applications.
Comparison with Theory
Wazzan et al. (1970) have presented numerically
predicted transition Reynolds numbers for heated
PARTICLE CONCENTRATION (COUntS/,))
FLOW TUBE
STATION |
| 10 100
PARTICLE MEAN DIAMETER (microns)
FIGURE 6. Particle concentration spectra: flow tube,
NSRDC towing basin, and open ocean.
83
10 ft/sec
(3 m/sec)
Ug *
-0,10
20 ft/sec
- 0,08}
40 ft/sec
80 ft/sec
-0,04
-0,02
(6.1 m/sec)
(12.2 m/sec)
(24.4 m/sec)
X (FT)
wall boundary layers with zero pressure gradient.
More recently, similar calculations have been
performed for boundary layers in favorable pressure
gradient flows. Before comparing the flow tube
results with such predictions we should estimate
the favorable pressure gradient produced by the
boundary layer displacement effect in the tube.
The most common way to characterize streamwise
pressure gradient in a boundary layer is by the
similarity parameter 8 [Schlichting (1968)]. For
the general class of wedge flows, the external
velocity U is given by U Cx , and the parameter
8 is then 2m/(m + 1). Both m and & are constants
in any wedge flow, and are equal to zero for the
zero pressure gradient boundary layer. We have
calculated approximate local values of 8 in the
flow tube, using the Blasius growth law for the
boundary layer displacement thickness:
6* = 1.72 (vx/Us) 4 (3)
(The calculation can be iterated to include the
effect of pressure gradient upon 6*, but the differ-
ence is negligible.) The resulting values of 8 as
a function of x at several values of U_ are shown
in Figure 7. 8 is proportional to the square root
of x, and thus has its largest value at the down-
stream end of the tube.
Figures 3 and 5, which show transition Reynolds
numbers versus overheat for the flow tube, also
include the theoretical predictions of Wazzan et al.
(1970) for a 8 of 0.07. This represents an approx-
imate average of 8 in the tube for the velocity
range of interest. (Calculations using exact 8
values from the tube will be done in the near future.)
Note that the experimental results lie near or even
above the 8 = 0.07 prediction for overheats from
zero to 13°F (7°C). At this point the experimental
curve quite suddenly levels out, while the predicted
curve continues to rise at an increasing slope.
The predicted curve reaches its maximum at a Reynolds
number of about 250 x 10° (near 45°C overheat),
while the experiment has never yielded more than
AD 16° ,
There are several possible reasons for the
disagreement between theory and experiment at the
higher overheats. (1) The theory does not account
for the destabilizing effects of density stratifi-
cation, which will become increasingly important as
“overheat is increased. Buoyancy effects may
FIGURE 7.
Uk50
8 versus x for several values of
destabilize the flow in three distinct ways: (a)
the bottom of the tube wall is subject to thermal
convection rolls, similar in form to the Goertler
instability; (b) the side wall boundary layer will
experience a cross-flow due to the rising fluid
near the wall; and (c) the top wall boundary layer
will grow in thickness faster than normal because
of the fluid rising up from the sides. (2) The
theory neglects the effects of temperature and
viscosity fluctuations upon the growth of the
velocity fluctuations. There is evidence that this
is a reasonable approximation. (3) The theory relies
upon the e? transition criterion, which may become
increasingly incorrect at higher overheats. This
criterion has never before been applied to boundary
layers with inhomogeneous physical properties.
There is a large distance between the minimum crit-
ical point in the boundary layer and the predicted
transition point using ey ab is questionable
whether the region of linear growth can extend over
such a large range of Reynolds numbers. (4) Wall
roughness is not accounted for in the theory, and
the importance of roughness will increase with wall
heating (and with increased velocity) due to the
thinning of the boundary layer. Roughnesses that
are insignificant at zero or low overheat may become
important as overheat increases.
Velocity Profile Measurements
In view of the differences between experimental
results and computed transition Reynolds numbers,
measurements have been made of boundary layer
velocity profiles in the flow tube to try to
establish the mechanism of transition. If the
buoyancy effects described above are in fact
significant, they should produce measurable devi-
ations from axisymmetry in the mean velocity profiles.
In addition, they might cause transition to occur
earlier on the top, side, or bottom wall, depending
upon which mechanism is predominant. We therefore,
designed the instrumented section (described above)
to be installed on the downstream end of the 6.1m
test section. This contains Pitot tubes for mean
velocity measurements and flush mounted hot film
probes for intermittency measurements. The instru-
mented section has been very successful in measuring
mean velocity profiles in the flow tube. Figure 8
shows a typical measured profile that has been
84
a
= T T T >I
@ = 90° (FROM TOC)
QT= O°F
VELOCITY RATIO,
u/Up
BLASIUS PROFILE
0.5
0.5 1.0
DISTANCE FROM WALL, y/8
FIGURE 8. Normalized velocity profile with zero over-
heat, compared with Blasius profile.
normalized and plotted with a curve representing
the Blasius profile for a zero pressure gradient
boundary layer. Actually, the agreement shown
here is better than it should be due to the positive
B of the flow tube boundary layer.
The most surprising result that has been obtained
with the instrumented section is the large deviation
from axisymmetry in the profiles, even with no wall
heat. Figure 9 shows a plot of 6* (displacement
thickness), 8 (momentum thickness), and H (shape
factor) versus azimuthal angle for no wall heat
at a free stream velocity of 1.60 m/sec. The
dashed lines indicate the calculated values for
8 = 0 and § = 0.16, which is the value of 8 at the
downstream end of the test section. The variations
in 6* and @ are more than 50 percent, which was
totally unexpected. Figure 10 shows an azimuthal
velocity profile, that is, u versus ¢ at a fixed y.
Here we see that the departure from axisymmetry is
wave-like in nature, and that significant changes
in velocity occur over a 15° change in $.
This behavior suggests that the asymmetries may
be caused by streamwise vortices within the boundary
layer, which would have a cross-stream length scale
on the order of the boundary layer thickness.
Such vortices could be caused by the Goertler
instability in the contraction section, as described
above. To test this hypothesis, a new contraction
section is presently being built which will avoid
a : [ oe ——~ . = 4
Oo 90 180 270 360
AZIMUTHAL ANGLE
FIGURE 9. Displacement thickness, momentum thickness,
and shape factor vs. azimuthal angle for zero overheat,
Us. = 155 cm/sec.
the Goertler instability entirely. This new con-
traction will have a fully convex inlet section
surrounded by an annular bleed flow. All fluid
from the settling chamber boundary layer will be
removed by the bleed flow.
Variations in mean velocity profiles due to
heating have in fact been measured, but they are
small relative to the changes with azimuthal angle
shown in Figures 9 and 10. The shape factor H
tends to decrease with increasing overheat as
expected. However, no firm evidence of buoyancy-
driven instabilities has yet been seen, even at
low flow velocities and high overheats.
150 . I ] |
VELOCITY mt ln I ¢ ll! hy
ly
(cm/sec) nett gl J
50
: ° ——— eee! es en i =i
vs. azimuthal angle, 0 30 60 90 '20 EO Tk dD ee ake
AZIMUTHAL ANGLE
FIGURE 10. Velocity, u 300 330 360
(
¢?, at y = 0.51 cm
4. CONCLUSIONS
The flow tube experiment has already demonstrated
that wall heating can have a significant effect
upon transition Reynolds numbers in water boundary
layers. Although the maximum transition Reynolds
number of 42 x 10° is well below the predicted
maximum, this value has been obtained with only
7°C wall overheat. The unheated transition Reynolds
number of 10’ shows that disturbances are well
controlled in the experiment.
Possible causes for the differences between the
predicted and realized transition Reynolds numbers
at higher overheats are still under investigation.
Preliminary results from the instrumented section
indicate that buoyancy-driven instabilities are
not a Significant factor. However, major deviations
from boundary layer axisymmetry have been observed
even with no wall heat. These perturbations of the
unheated flow could themselves have an effect upon
transition Reynolds numbers. This is particularly
true if the actual disturbances are Goertler vortices,
because these vortices would increase in strength
with increasing flow velocity. Since the transition
length is fixed at the end of the tube in this
experiment, transition Reynolds number will be
directly proportional to velocity. Thus the
Goertler vortices could impose a limit in transition
Reynolds number if they begin to dominate the
transition process above some critical flow velocity.
This hypothesis will be tested by the installation
of the new contraction section, which eliminates
the possibility of Goertler vortex formation.
ACKNOWLEDGMENT
The author wishes to acknowledge the participation
and support of the Marine Systems Division of Rock-
well International, and in particular Mr. Douglas
Gile. In addition, the author acknowledges the
Defense Advanced Research Projects Agency, who
sponsored this research.
85
REFERENCES
Back, L. H., R. F. Cuffel, and P. F. Massier (1969).
AIAA Journal, 7, 4; 730.
Barker, S. J., and C. Jennings (1977). The effect
of wall heating on transition in water boundary
layers. Proc. of NATO-AGARD Symposium on
Laminar-Turbulent Transition, Copenhagen, 19-1.
Barker, S. J. (1978). Turbulence Management in a
High Speed Water Flow Facility. Submitted to
ASME.
Bradshaw, P. (1965). J. Fluid Mech., 22 pt. 4, 679.
Chmielewski, G. E. (1974). J. Aircraft, 11, 8; 435.
Corrsin, S. (1963). Turbulence: Experimental
methods, in Handbuch der Physik, 8, pt. 2, 523.
Kaups, K., and A. M. O. Smith (1967). The laminar
boundary layer in water with variable properties.
Proc. ASME-AIChE Heat Transfer Conf., Seattle,
Wash.
Launder, B. E. (1964). Laminarization of the
Turbulent Boundary Layer by Acceleration, M. I. T.
Gas Turbine Lab Report 77. Cambridge, Mass.
Loehrke, R. I., and H. M. Nagib (1972). Experiments
on Management of Free-Stream Turbulence. NATO-
AGARD Report 598.
Lumley, J. L., and J. F. McMahon (1967). Trans.
ASME, D, 89, 764.
Schlichting, H. (1968).
McGraw-Hill, New York.
Smith, A. M. O. (1957). Transition, pressure gra-
dient, and stability theory, Proc. 9th Int. Con-
gress on Appl. Mech., 4, 234, Brussels.
Spangler, J. G., and C. S. Wells (1968) .
WOUGNaAly Mola Shea a Sr
Strasizar, A., J. M. Prahl, and E. Reshotko (1975).
Experimental Study of Heated Laminar Boundary
Layers in Water, Case Western Reserve Univ.,
Dept. of Fluid, Thermal, and Aerospace Science
Report FT AS/TR-75-113.
Wazzan, A. R., T. T. Okamura, and A. M. O. Smith
(S68) aransi, ASME, eG, 90) jooR
Wazzan, A. R., T. T. Okamura, and A. M. O. Smith
(1970). The stability and transition of heated
and cooled incompressible boundary layers. Proc.
4th Int. Heat Transfer Conf., Paris.
Boundary Layer Theory,
AIAA
Some Effects of Several
Freestream Factors on Cavitation
Inception of Axisymmetric Bodies
Edward M. Gates
University of Alberta
Edmonton, Canada
Allan J. Acosta
California Institute of Technology
Pasadena, California
ABSTRACT
Some of the effects of freestream turbulence and a
dilute polymer solution on the fully wetted flow
and the subsequent cavitation inception has been
investigated for three different bodies. Two of
these bodies possess a laminar separation and one
does not. In the fully wetted investigation the
flow on one of the bodies was found to be insensi-
tive to the present disturbances whereas the other
two were found by comparison to be very sensitive.
Although there is a pronounced "Suppression" of
inception by the polymer, it seems clear that the
effects observed are due primarily to the change
in the real fluid features of the flow past the
bodies themselves and not to an intrinsic cavita-
tion process. There appeared to be no special poly-
mer effect, insofar as cavitation is concerned, on
the body not having a laminar separation, confirm-
ing the results of van der Meulen. Due to practical
limitations the effects of turbulence per se on in-
ception could not be separately evaluated.
The inception index on all bodies was found to
be greatly dependent on the distribution of nuclei
within the water tunnel. For those cases in which
a turbulent transition was established well upstream,
travelling bubbles were a common form of cavitation
observed on all test bodies. The number of these
cavitation events were so few, however, that in one
test facility having a resorber, it was just as
likely for an attached cavity to form as it was to
observe a travelling bubble. In both cases the
inception index was far below the customary minimum
pressure coefficient reference value. Nuclei counts
made with the aid of holograms reveal significantly
fewer microbubbles within the flow of this test
facility than in those not having a resorber.
1. INTRODUCTION
Our understanding of the details of the process of
cavitation inception (and thus our ability to scale
86
laboratory results to prototype conditions) is far
from complete [e.g. Acosta and Parkin (1975), Morgan
and Peterson (1977)]. This lack of understanding
is well illustrated by our ability to do no more
than indicate reasons which are believed to be
responsible for the large variations in the results
of the ITTC comparative test series [Lindgren and
Johnsson (1966), Johnsson (1969)]. These results,
some of which are presented in Figures 1 and 2, did,
however, prompt a considerable amount of effort to
investigate more systematically the factors influ-
encing cavitation inception. In particular there
are three areas in which there have been significant
developments: (i) the influence of viscous effects
on inception, (ii) the discovery that in some situ-
ations the presence of drag-reducing polymers in the
water cause a suppression of the inception index,
and (iii) the development of equipment to accurately
measure freestream nuclei populations.
1.0
oO ® . WITHOUT RESORBER FACILITY
2 OS ** WITH RESORBER FACILITY
<
WwW
a
z
0.8
Zz
9° TOKYO - (JAPAN
a |
Pe OL
1S)
Zz Com
mo! PENN STATE(DTMB-BODY A) Ts _____ |
roy SAFH -(U.S.A.)
E “DTMB 36" BODY B
(U.S.A.)
FE 05 a |
= 2 "*>TMB 36" BODY B -(U.S.A.)
ps NPL 2-
r3) ta! rah STATE (SAFH BODY) x aclen
“10 20 30 40 50 60
TUNNEL VELOCITY-U_ , ft/sec
FIGURE 1. Results of the comparative inception test
on a modified ellipsoidal headform sponsored by the
International Towing Tank Conference, Lindgren and
Johnsson (1966).
5. Caltech
8. SSPA
Viscous Effects
Parkin and Kermeen (1953) appear to be the first
investigators to appreciate the influence of the
boundary layer on the inception process. However,
even though their interpretations of the experi-
mental results were used in many subsequent incep-
tion theories [e.g. van der Walle (1962), Holl and
Kornhauser (1969) to name only two] further experi-
mental investigations of these viscous effects
were carried out only much later.
Among these, Arakeri and Acosta (1963), by using
the schlieren flow visualization technique, were
able to observe cavitation inception within the
structure of the flow. A primary feature of the flow
observed by them was a laminar separation in which
the cavitation was seen to occur first. There was
further some suggestion by them that the laminar-
to-turbulent transition itself may promote cavita-
tion, perhaps through a mechanism similar to that
for inception in turbulent pipe flow [Arndt and
Daily (1969)]. In any case, it should be expected
then, that any factor which could influence the
presence of separation or even transition may also
influence the inception of cavitation. One such
well-known factor is freestream turbulence. [For
recent accounts of these effects on transition see,
e.g., Spangler and Wells (1968), Hall and Gibbings
(1972), and Mack (1977)]. Unfortunately, the mea-
surement of turbulence in water is more difficult
than its aerodynamic counterpart and, until recent
87
FIGURE 2. Photographs of dif-
ferent types of cavitation ob-
served in the ITTC tests,
Lindgren and Johnsson (1966).
times, there has been no great demand for determin-
ing the freestream turbulence in water tunnels. For
reference we tabulate in Table I the turbulence
levels for a few water tunnels for which this inform-
ation is available (12th ITTC Cavitation Committee) .
Polymer Effects
It was inevitable that the much-heralded, drag-
reducing polymer solutions would be the subject
of cavitation experiments also. Very early in the
course of this work Hoyt (1966) and Ellis et al.
(1970) found that the inception index was reduced
by as much as a factor of two for hemisphere-nosed
bodies. There was, furthermore, a pronounced change
in the physical appearance of the cavitation, once
it was well developed, as subsequently illustrated
by the beautiful photographs of Brennan (1970). Two
possible explanations for the cavitation-suppression
effect were then advanced: in the first, it was
speculated that the dynamics of individual bubbles
were changed by the presence of the polymer, and
in the second, it was assumed that the basic viscous
flow about the model was altered by the presence of
the polymer. Ting and Ellis (1974) could find no
difference in the collapse time of spark-generated
bubbles in either water or polymer solutions weak-
ening the idea that the bubble mechanics are impor-
tant for this process. Later, however, Holl and
co-workers (1974) in commenting on experiments
88
TABLE I
Ottawa, Canada 0.75%
Kriloff No. 2
Leningrad, USSR 0.4%
NPL No. 1
Feltham, UK 0.5%
MIT
Massachusetts, USA 0.77%
6'' Tunnel
Minnesota, USA 0.8%
carried out at the Garfield Thomas Water Tunnel
(GTWT) noted that there appeared to be no laminar
separation on a hemisphere nose body when polymer
was added to the water, but no direct flow visual-
ization was done. Later van der Meulen (1976)
verified this speculation with the clever use of
schlieren holography to observe simultaneously the
viscous flow and cavitation inception on a 10 mm
diameter hemisphere nose body. His results showed
clearly that when polymer was injected into the
boundary layer that the laminar separation was re-
moved. van der Meulen suggested that the polymer
removed the separation by causing an early transi-
tion to a turbulent non-separating boundary layer.
He then attributed the suppression effect to the
removal of the large pressure fluctuations associ-
ated with the transition zone of the free shear
layer [Arakeri (1975)].
Freestream Nuclei
It is generally accepted that inception begins at
the nuclei in the liquid and that there are two
sources for these nuclei--the test body surface
and the incoming flow. At one time "surface nuclei"
received considerable attention [e.g. Acosta and
Hamaguchi (1967), Holl and Treaster (1966), Holl
(1968), Peterson (1968) and van der Meulen (1972) ].
While on the one hand it was shown that under certain
circumstances, but not in normal cavitation testing,
surface nuclei could exert a controlling influence
upon inception. It seemed evident on the other hand
from the results of the ITTC tests that freestream
nuclei were the more important. Further, the de-
velopment of the concepts of cavitation event count-
ing [Schiebe (1966) ] in conjunction with Johnson
and Hsieh's (1966) trajectory calculations, the
idea of “cavitation susceptibility" [Schiebe (1972) ]
and the development of equipment to measure free-
stream nuclei populations have led to more interest
in the influence of freestream nuclei versus surface
nuclei. In particular, the experiments of Keller
(1972) have prompted considerable interest in mea-
suring and relating freestream nuclei populations
to inception.
Morgan (1972) has reviewed the various types of
instruments available for measuring freestream
nuclei populations and Peterson et al. (1975) have
made an experimental comparison of three of these,
namely; light scattering, microscopy, and holog-
raphy. At the moment holography seems the best
Turbulence Levels in Some Water Tunnels
ORL
Pennsylvania State, USA 0.8%
HSWT
California Institute of
Technology, USA 0.25%
LIWT
California Institute of
Technology, USA 0.05%
(present work) tor See %
in that no "calibration" is required, a permanent
record is obtained, a large volume is sampled, and,
as Peterson observed, one can determine if the
nuclei are solid particles or micro-bubbles.
There is seen to be ample reason then to pursue
these various freestream factors in inception re-
search. Two are primarily fluid-dynamic in nature
and of these the questions concerning freestream
turbulence levels are of historic interest in fluid
mechanics and naval architecture. The cavitation
nuclei however are directly involved in the cavita-
tion inception process and the recent experimental
progress cited above make one hope for a more quan-
titative predictive ability than in the past inso-
far as inception is concerned. The present work is
in the mainstream of these observations; briefly we
report on observations made in two different flow
facilities having widely different freestream nuclei
distributions on identical bodies. In one of these,
the freestream turbulence level is varied over nearly
a factor of 100 (but not in a condition of cavita-
tion then) and we confirm and extend the observa-
tions of van der Meulen on the polymer effect.
Schlieren photography is extensively used to visu-
alize thermal boundary layers on the test bodies
used and in-line holography is used to determine
nuclei populations in the working section.
Before discussing these effects we should com-
ment briefly on the means used for the determination
of the actual inception observation. A standard
procedure has been to observe the test body under
stroboscopic light and to say that inception occurs
when macroscopic cavities or bubbles become visible
on the model. However, this method is observer-
dependent and the trend now is to use cavitation-
event counters free of human judgment. Ellis et al.
(1970) and Keller (1972) have developed optical
techniques which count interruptions of light beams
which are adjusted to graze the model surface where
inception has been observed to occur. Peterson
(1972), Brockett (1972), and Silberman et al.
(1973) have also determined inception acoustically
by locating a hydrophone inside the test model.
There are problems of identifying the types and
location of the cavitation phenomena occurring with
these “events." Aside from the question of tech-
nique, there is also the question of selecting
appropriate threshold levels at which an event be-
comes countable and also the event rate at which
inception is defined to occur. At present there is
no universal agreement of just what these values
should be.
ELLIPSE
0/6
0/2
a
HEMISPHERE NOSE NSRDC BODY SCHIEBE BODY
Cemin = 9.75
(BRASS) (COPPER) (CRES
FIGURE 3. Definition of test models.
20) EXPERIMENTAL EQUIPMENT AND METHODS
Test Models
Three axisymmetric test models were used in the
present experiments: a brass hemisphere nose, a
copper modified ellipsoidal (or NSRDC) body, anda
stainless steel standard headform from the Schiebe
series with a minimum pressure coefficient (Cpmin)
of -0.75. The hemisphere nose and Schiebe bodies
were fabricated specifically for these tests,
whereas the NSRDC body is the same as that used
by Brockett (1972) and Arakeri (1976). Each body
is 5.08 cm in diameter and has a 0.423 cm diameter
hole at the stagnation point for polymer injection.
No quantitative measure of surface roughness was
made, but each model was highly polished [a highly
polished surface typically has a 0.1 x 10-7m rms
finish, Beckwith and Buck (1961)]. The model ge-
ometries are shown in Figure 3.
The models were supported by a two-bladed sting
in the LIWT and by a three-bladed sting in the
HSWT with the nose of the model being about six
SPARK-GAP LIGHT SOURCE (0.032"DIA.)
SLIT. GATHERING LENS F.L.=3"
REMOVABLE fe ——.——| +] STEADY LIGHT SOURCE
MIRROR |
F.L.=12" 2.5
BIND OWS MAN OR WITH
CORRECTOR LENS)
TUNNEL TEST SECTION
WITH HEATED BODY IN PLACE
FOCUSING LENS
F.L.=7", f25
CUT - OFF PLATE
FILM PLATE
FIGURE 4. Schematic diagram of flow visualization
system.
89
body diameters upstream of the sting in each case.
Misalignment from the geometric tunnel center-line
in both the LTWT and HSWT was measured to be about
ORO Ee
Flow Visualization
Thermal boundary layers in the viscous flow past
the test model were observed by schlieren photog-
raphy. The particular schlieren configuation used
is shown schematically in Figure 4 and is essen-
tially the same as that used by Arakeri (1973).
Also following Arakeri, the prerequisite density
gradient was produced by heating the body with in-
ternal cartridge type electric heaters. An example
schlieren photograph obtained using this system is
presented in Figure 5.
Water Tunnel
The two facilities used in the present experiments
were the High Speed Water Tunnel (HSWT) and the
FIGURE 5. A schlieren photograph of a 5 cm diameter
hemisphere showing laminar separation and turbulent
reattachment at a body Reynolds number of 2.6 * 10°.
The maximum height of the separated region is about
2 mm.
90
MUFFLER
/- TURBULENCE GRID
/ WORKING SECTION
f [ ia" SQUARE x 100" LONG
TUNNEL PRESSURE
CONTROL VALVE—
FIGURE 6. Diagram of the Low Turbulence Water Tunnel
(LTWT) .
Low Turbulence Water Tunnel (LTWT) both at the
California Institute of Technology. Since the
HSWT has been described in detail elsewhere [see
Knapp et al. (1948) or Knapp, Daily, and Hammit
(1970) ], it will only be noted here that one, it
has a resorber and two, the freestream turbulence
level has been measured to be about 0.2 percent by
Professor S. Barker. The LTWT [Vanoni et al.
(1950) ] is also a closed loop recirculating tunnel;
but, as can be seen in Figure 6, it has no resorber.
In this facility the maximum test section velocity
and minimum cavitation number are approximately 8
meters per second and 0.3 respectively. The unique
feature of the LTIWT is that the freestream turbu-
lence level in the test section can be varied from
a very low value (for water tunnels) of 0.05 per-
cent to a high value of 3.6 percent. The low tur-
bulence level is obtained by use of small turning
vanes in each elbow of the circuit, a yery gradual
diffuser (included angle is 3°13'), a nozzle with a
16:1 contraction ratio, and by turbulence. damping
screens and honeycombs in the "stagnation" section
of the tunnel just upstream of the nozzle. The
configuration of screens and honeycombs which pro-
duces the 0.05 percent turbulence level is shown
schematically in Figure 7 (with the exception that
no turbulence generating grid is installed) and is
based upon the results of Loehrke and Nagib's (1972)
report.
By inserting different turbulence generating
grids into the tunnel circuit the turbulent intensity
can be gradually increased from 0.05 to approxi-
mately 3.6 percent. The description of these grids
is as follows:
HONEYCOMB
"7" TRIANGULAR CELLS
TURBULENCE DAMPING SCREENS
0.0075" DIA. WIRE, 22 meshes/lineal inch
TURBULENCE
GENERATING GRID
SECOND HONEYCOMB
1/8 x2 HEXAGONAL CELLS
FIGURE 7. Sketch of LTWT contraction nozzle showing
the turbulence manipulators.
grid No. 1: 12.7mm diameter bars with 50.8mm
mesn
grid No. 2: 6.35mm diameter bars with 25.4mm
mesh
grid No. 3: three 25.4mm diameter horizontal
bars on 76.2mm centers
grid No. 4: 0.635mm diameter fishing line with
19.05mm mesh
Grids 1, 2, and 4 are located at the entrance to
the test section as is shown in Figure 7 (the
distance from these grids to the test model is
approximately 1.2 meters). Grid No. 3 is located
in the "stagnation" section immediately after the
final turbulence damping screen. Grid No. 3 has
this particular configuration because (after much
trial and error) it was found to produce a turbu-
lence level which is close to the levels measured
in a number of other facilities--see Table 1.
A DISA constant temperature anemometer was used
to measure the turbulence levels in the test section.
The probe was a wedge-shaped hot film type and was
firmly mounted on the tunnel center-line at the
model position (1.2 meters from the test section
entrance). The results of these measurements have
been summarized in Figure 8.
Polymer Injection System
The injection approach of introducing the polymer
into the boundary layer versus filling the tunnel
with a polymer solution (polymer ocean) was chosen.
After considering a number of injection configura-
tions [Wu (1971)] it was decided to follow van der
5
GRID #|
e) Oo
O D0 oo g GRID #2
GRID #4
A IS & & B& A
GRID #3
Ga BS 8 g@ Bp
0.5
TURBULENCE LEVEL - u‘/U, percent
0.1
FREESTREAM
e
0.05
® @
O 5 10 i) 20 25
TUNNEE VEEOCIinY SU titZselc
FIGURE 8. Summary of turbulence intensity measurements
in the LTWT.
j-2:! CONTRACTION
INJECTOR ~ foie BRASS
SMOOTHING SECTION
PACKED WITH POROUS FOAM
Luoves FOR
CARTRIDGE HEATERS
FIGURE 9. Cross-section of injector used for these
polymer experiments. The body diameter is 5 cm.
Meulen's (1973) example and inject the polymer into
the boundary layer through a hole at the stagnation
point. To do this an injector was designed to in-
troduce the polymer into the boundary layer without
also introducing disturbances. The injector is
shown schematically in Figure 9 assembled inside
the hemisphere nose body and consists of first a
settling chamber 12.7mm in diameter and 31.75mm long.
This section was packed with porous plastic foam
held in place by a sintered brass disc. The pur-
pose of this section is to disperse the jet enter-
ing the injector and provide a smooth flow into the
9:1 contraction which follows. After the smooth
contraction there is a tube with a length to diam-
eter ratio of 22 and this tube ends at the surface
of the model.
To minimize polymer degradation, the polymer
solutions were "pushed" through the injector from a
reservoir by using compressed air instead of a pump.
A check with a turbulent flow rheometer [the same
one as used by Debrule (1972) ] showed degradation
of the polymer after it passed through the injec-
tion system to be minimal. Preliminary tests were
carried out with water as the injectant to ensure
that the injection process itself was not respon-
sible for any observed changes in the flow. Results
of these tests for the NSRDC body are presented in
Figure 10 and show that even at an injection rate
of three to ten times higher than actually used
with polymer solutions no differences are detectable
from the no-injection case.
Nuclei Counter
Nuclei distributions were deduced from holograms
of the test fluid. The experimental apparatus and
method is much the same as used by Peterson (1972),
Feldberg and Shlemenson (1973) and is described in
detail in Gates and Bacon (1978). Essentially it
is a two-step image forming process. In the first
step, a hologram of a sample volume of the water
in the tunnel test section is recorded on a special
high resolution film by a "holocamera." In the
second step, the developed hologram is reconstructed
producing a three dimensional image of the original
volume which can be probed at the investigator's
leisure. The holocamera and reconstruction system
are shown schematically in Figure 11 and 12 respec-
tively.
91
(b)
FIGURE 10. Schlieren photographs showing the effect
of injecting water on the NSRDC body at a body Reynolds
number of 3.2 x 10°, (a) injection rate = O m&/sec,
(b) 1.8 m&/sec, (c) 3.6 m&/sec, (d) 6.6 m&/sec,
(e) 9.8 m&£/sec. No effect is observed.
SAMPLE VOLUME
TEST SECTION
WINDOWS
BC D
1
FIGURE 11. Diagram of the holocamera; (a) dielectric
mirror, (b) iris, (c) dye-quench cell, (d) ruby-flash
lamp assembly, (e) iris, (f£) dielectric mirror,
(g) beam splitter, (h) neutral density filter,
(1) beam expander lens, (j) 25y pinhole, (k) collimat-
ing lens, (1) front surface mirror, (m) p.i-.n. diode,
(n) film pack.
ics)
No
[icc alee ee : and also to let the freestream bubbles go back into
solution or rise to the high points in the tunnel
TV CAMERA circuit.
The same general test procedure was used in the
HSWT except for small differences in pressure mea-
MICROSCOPE surement. However, desinent cavitation observa-
tions were also made in this facility. All holograms
=== TV MONITOR made in the HSWT were done without the model in
=>==2 RECONSTRUCTED place but at conditions of velocity and pressure at
Ty 7t Wee which inception had been observed to occur.
4. PRESENTATION AND DISCUSSION OF FULLY WETTED
HOLOGRAM ON TRAVELING RESULTS
CARRIAGE
ig} b L aL
— BEAM DIAMETER ~5cm reestream Turbulence Levels
COLLIMATING LENS CSN ACIOAS
The influence of gradually increasing freestream
turbulence level upon the viscous flow about each
test body is illustrated in the sequences of
PIN HOLE
|| ~~ microscore OBJECTIVE schlieren photographs presented in Figures 13
‘| through 15. In each sequence of photographs the
test body is seen in silhouette and the flow is
from right to left. The magnification is such that
the surface length shown in these photographs is
He-Ne GAS LASER (5mw)
FIGURE 12. Arrangements to reconstruct and read the
holograms.
3. GENERAL EXPERIMENTAL PROCEDURES
Before any experiments were carried out, the water
in each facility was de-aerated to reduce the
number of freestream air bubbles produced in the
tunnel circuit. This was of particular importance
in the LIWT which has no resorber. During the
present tests the air content in the LIWT was typ-
ically between 7 - 8ppm whereas in the HSWT it was
between 9 - 10ppm (air content levels were measured
with a van Slyke blood gas analyzer). At these air
contents there were very few macroscopic air bubbles
visible in the flow approaching the model in the
HSWT (as will be seen later). However, in the LTWT
there were always many macroscopic air bubbles
easily visible in the approaching flow.
In a typical cavitation test in the LTWT, the
tunnel velocity and polymer injection rate (if any)
were first adjusted to the desired values. Incep-
tion was then obtained by reducing as rapidly as
possible the tunnel static pressure until the pres-
ence of cavitation was visually observed on the
model under stroboscopic illumination. At the
point of inception, a schlieren photograph, a holo-
gram, the tunnel velocity, and the tunnel static
pressure were recorded simultaneously. Each test
had to take less than forty seconds since by that
time the abundant supply of cavitation bubbles gen-
erated at the pump would reach the test section and (e)
dramatically change the freestream conditions. Af-
ter each test, the tunnel pressure was raised to on the flow past the NSRDC body (the flow is right to
about one atmosphere and the tunnel allowed to cir- left) at a body Reynolds number of 1.6 x 105: (a) u'/v
culate for five minutes. This recess between each = 0.05 percent, (b) 0.65, (c) 1.1, (a) 2.3, (e) 3.6
test was required to let the ruby laser cool down percent.
FIGURE 13. The effect of freestream turbulence level
(d)
FIGURE 14. The effect of freestream turbulence level
on the flow past the hemisphere body at a body Reynolds
number of 2.6 * 10°. (Same turbulence values as in
Figure 13.)
approximately 10mm. As can be seen in the first
photograph of each of Figures 13 and 14, the NSRDC
and the hemisphere nose bodies respectively have a
laminar separation. Transition on these bodies oc-
curs on the resulting free shear layer and the flow
subsequently reattaches as a turbulent boundary
layer. With increasing turbulence intensities the
point of transition on the NSRDC body moved upstream
on the free shear layer. As the position of tran-
sition moved forward, the size of the separation
bubble decreased until finally it disappeared when
the position of transition and separation coincided.
Once the point of transition moved upstream of the
point of separation, no further observations of the
thermal boundary layer could be made with the pres-—
ent schlieren system. Unlike the NSRDC model, the
increasing turbulence level seemed to have no ef-
fect upon the viscous flow about the hemisphere
nose body--as can readily be seen in Figure 14.
This rather surprising result will be returned to
later.
As is shown in the first photograph of Figure 15,
the Schiebe body has no laminar separation and tran-
sition occurs on the model surface rather than on a
free shear layer. With increasing freestream tur-
bulence level two effects were noted; first, as can
be seen in Figure 15, the position of transition
moves substantially upstream and secondly, the ap-
pearance of the disturbance appears to change. This
change is not quite so evident in only a few pic-—
tures, but we believe we observe more-or-less peri-
odic and highly amplified boundary layer waves in
Figure 15a and even b. However, for the higher
93
turbulence levels frequent “bursts" interspersed
with a periodic phenomenon seemed to be more common.
A random collection of schlieren photographs of the
same body (Figure 16) at an intermediate turbulence
level shows these various forms more clearly.
Discussion
To quantify the effects of turbulence level, the
position, length, and maximum height of the separa-
tion bubble were measured for the NSRDC and hemi-
sphere nose bodies. For the Schiebe body, which
has no separation, the position of transition was
recorded--the position of transition being defined
as that point at which the first noticeable dis-
turbance occurs in the laminar boundary layer. These
quantities are defined in Figure 17 and were mea-
sured directly from the negatives of the schlieren
photographs with the aid of a scale or reference
(e)
FIGURE 15. The effect of freestream turbulence level
on the flow past the Schiebe body at a body Reynolds
number of 2.5 x 10°. The turbulence levels are those
in figure 13 and the regions shown are, at arc-length
diameter ratios of (a) 0.82-1.07, (b) 0.76-1.01,
(c) 0.60-0.85, (d) 0.61-0.86, (e) 0.47-0.63.
94
FIGURE 16. Random photographs of the flow past the
Schiebe body at a Reynolds number of 3.4 x 10° with
background turbulence level of 1.1 percent. The
region shown covers the arc length diameter ratio of
0.68 to 0.93.
(a) POSITION OF LAMINAR SEPARATION OR
TRANSITION WHICHEVER IS APPLICABLE
U eet
H
(b) LENGTH AND HEIGHT OF SEPARATED REGION
FIGURE 17. Definition sketch of separation location.
079
HEMISPHERE NOSE BODY
[o)
1
x
[o)
NI
a
ESTIMATED SEPARATION
082 LOCATION (THWAITES METHOD)
NSROC BODY
TURBULENCE LEVEL
0.05%
0.65%
O78} 1.10%
°
8
2.30%
3.60%
POSITION OF SEPARATION/ DIAMETER -(S/D),
15x10° 25x10° 35x10° 45xi0>
BODY REYNOLDS NUMBER- UD /y
FIGURE 18. Observed separation locations as a func-—
tion of turbulence level for two bodies.
negative. Note that the position of transition on
the free shear layer coincides with the definition
of the end of the separated bubble.
Each of these measured quantities was non-
dimensionalized by dividing by the body diameter
and are plotted versus the body Reynolds number
with the freestream turbulence level as a parameter
in Figures 18 through 21. For the NSRDC body, Fig-
ure 19 shows that the size of the separation bubble
decreases with increasing velocity and turbulence
level--the critical Reynolds number being reduced
from a value of greater than 4 x 10° at 0.05 joene=
cent to near 2.5 x 10° at 3.6 percent. As was ex-
pected from the schlieren photographs of the
hemisphere nose (Figure 14), Figure 20 shows the
length of the separation bubble is independent
of turbulence level but decreases with increasing
velocity. Finally, Figure 21 shows that as with
the NSRDC body, the position of transition on the
Schiebe body moves forward with increasing velocity
and turbulence intensity.
The most startling result of the above tests was
the insensitivity of the boundary layer on the
hemisphere nose to the present disturbances imposed
by the freestream turbulence. Hall and Gibbings
TURBULENCE LEVEL
@ 005%
0.65%
1.10%
2.30%
3.60%
o
ie
ce}
0.08
‘s
(e)
5
SEPARATION LENGTH/BODY DIAMETER
°
n
fo}
1.5x10> 2.5x10° 3.5x105 4.5x10°
BODY REYNOLDS NUMBER-UD/v
FIGURE 19. The length of the separated region as a
function of freestream turbulence level for the
NSRDC body.
TURBULENCE LEVEL
0.05%
0.65%
0.09 1.10%
2.30%
3.60%
0.08
0.07
0.06
0.05
SEPARATION LENGTH/ BODY DIAMETER
15x10° 25x105 35x10 45x10°
BODY REYNOLDS NUMBER - UD/v
FIGURE 20. The length of the separated region as a
function of freestream turbulence level for the
hemisphere body.
(1972) have summarized the available experimental
data and semi-empirical correlations at that time
for the combined effects of pressure gradient and
freestream turbulence level upon transition. How-
ever, this correlation does not predict an insensi-
tivity to increasing turbulence levels. No doubt
this discrepancy is related to the question of how
the freestream disturbances are assumed to inter-
act with the boundary layer. For example, van
Driest and Blumer (1963) accounted for the effect
of freestream turbulence by using Taylor's assump-
tion that the unsteady perturbation induced in-
stantaneous variations in the velocity gradient.
But, as just noted, this type of correlation did
not work. Later, Spangler and Wells (1968) demon-
strated that not only the intensity, but also the
energy spectrum and the nature of the disturbance
must be taken into consideration. Reshotko (1976)
and Mack (1977) have re-emphasized Spangler and
Wells' conclusions and pointed out the lack of un-
derstanding of the interaction mechanism between the
freestream disturbance and the boundary layer* is
one of the major obstacles in the consistent predic-—
tion of transition. Thus, although the effect of
freestream turbulence on these bodies cannot be pre-
dicted with confidence, we at least may offer some
speculation based on these ideas to explain the be-
havior on the hemisphere nose body.
It is readily possible using the approximate
method of transition prediction suggested by Jaffe
et al. (1970) in conjunction with the stability
charts for the Falkner-Skan profiles computed by
Wazzan et al. (1968b) to determine the critical
frequency, or most unstable frequency for growth,
for each body at a number of velocities. These
estimates are presented in Table 2. We then esti-
mate with the aid of measured energy spectra of
grid generated turbulence, Tsuji (1956) that there
is approximately sixty times as much energy avail-
able in the freestream at the critical frequency of
the NSRDC body than there is at the critical fre-
quency of the hemisphere nose body. Furthermore,
the distance from the position of neutral stability
to the position of separation is only 0.07 diameters
on the hemisphere nose model whereas on the NSRDC
*This is the concept of boundary layer receptivity devel-
oped by M. V. Morkovin [see the review of Reshotko (1976) ].
95
T ar
a TURBULENCE LEVEL
= @ 0.05 %
SS B® 0.65 %
7p) ai.1% 1
7 © 2.3%
Zz 4 3.6%
[e) o—
= ie
2p) =
E Ss
ra 6 |
[= —~S
Ww a ™:
fo) nia
Zz
©
=
w
Qa
HE +t ee
1x 10° 2x10° 3x105 4x0 ~
BODY REYNOLDS NUMBER-UD/v
FIGURE 21. The location of transition on the Schiebe
body as a function of turbulence level.
body it is 0.40 diameters. Thus on the NSRDC body
not only is there considerable more energy available
at the critical frequency, but there is also more
opportunity for disturbances to grow than for the
hemisphere nose body. This same trend is also found
for the Schiebe body at the low turbulence levels.
The critical frequencies are even less than those
of the NSRDC model (Table 2). There is, therefore,
more energy available at those frequencies than even
on the NSRDC model. Finally, the distance from the
position of neutral stability to transition is be-
tween 0.40 to 0.60 diameters--much the same as for
the NSRDC model.
We find it somewhat reasonable then, in retro-
spect, for the hemisphere body to be found insensi-
tive, in the present experiments, to the freestream
disturbances. Regrettably, the present visual
observations are not sufficiently quantitative to
shed light on this basic problem of boundary layer
receptivity to external disturbances and their sub-
sequent growth into turbulence.
By using an oil film technique, Brockett (1972)
found the NSRDC model to have a critical velocity
of 2.8 meters per second at 20°C and Peterson (1972)
reports 4.2 meters per second at 10°C in the NSRDC
12-inch water tunnel. The same body in the HSWT
was found to have a critical velocity of about 9.2
meters per second and it was observed to be above
7.6 meters per second in the LTWT at 0.05 percent
turbulence level. To reduce the value of the criti-
cal velocity to 4 meters per second in the LIWT re-
quired a 316 percent turbulence level, which is as
can be seen from Table 1 a very high value for a
water tunnel test section. (Initially it was thought
unlikely that the disturbance level in the NSRDC
facility is this high. However, after inspecting a
drawing of the facility [Figure 2.3 pg. 26, Knapp
et al. (1970)] such a high level does not seem so
unlikely.) However, in this as well as in most
water tunnel facilities the energy spectrum is not
known, forestalling therefore a direct comparison
of transition phenomena.
The present observations of transition on the
Schiebe body at the lowest turbulence level are
compared with calculations of Wazzan* and experi-
*Private communication.
96
TABLE II
for Several Bodies
Approximate Critical Boundary Layer Frequencies
Rep Hemisphere Nose NSRDC Schie be
(Hz) (Hz) (Hz)
aEOWaex 10° 1070 670 --
Ze, 0) 2 10° 1800 1060 350
3,33 % 19> 2140 1780 650
Bear a10r 3350 2100 1200
ments of van der Meulen (1976) in Figure 22. There
is good agreement of the experimental results and
also with Wazzan's e? calculations.
Polymer Injection
Observations
The influence of gradually increasing the injection
rate of the polymer solution upon the basic flow
about each test body is illustrated in the sequences
of schlieren photographs in Figures 23 through 25.
In Figure 23(a) the maximum height of the separa-—
tion bubble is 0.5mm and on the hemisphere nose
body in 24(a) the maximum bubble height is 0.25mm.
Unlike the freestream turbulence level, the presence
of polymer in the boundary layer was found to
influence the basic viscous flow on all the test
models. As can be seen in the schlieren photographs,
as the polymer injection rate was increased the
position of transition moved upstream in each case.
For the NSRDC and hemisphere nose models a critical
injection rate was reached at which the positions of
transition and separation coincided and the laminar
separation was eliminated. At injection rates above
this critical value the position of transition ap-
Be © van der MEULEN (1976)
= @ PRESENT OBSERVATIONS
— (0.05 PERCENT
gy 1.47 TURBULENCE LEVEL) al
arid B® CALCULATIONS NO
z HEATING
oO L 4 CALCULATIONS 10°F |
= 1.2 HEATING (WAZZAN &
= GAZLEY, 1978)
ep)
za
CO
= 1.Or ° SA s 4
e 2) i ~~
ve —
© 0.8 2
<a
ie)
F 06
i)
oO
a
— 4 4 ~~ 4 jt —d
1 x 10° 2x 10° 3x10> 4x10°
BODY REYNOLDS NUMBER-UD/y
FIGURE 22. Comparison of transition observations on
the Schiebe body.
(d)
FIGURE 23. Flow past the NSRDC body with injection
of 500 wppm Polyox (WSR 301) at a Reynolds number of
1.6 x 10°: (a) no injection; (b) 0.1 mg/sec, G =
0.5 x 1055; (c)) 0.3) me/sec, G = 15) x) dogo; (a) ons
mi/sec, G = 2.5 x 10-©. G is the dimensionless
polymer injection rate.
peared to move further upstream, but with the limited
resolution of the present schlieren system, these
poisitions could not be accurately determined.
Discussion
It would seem desirable to normalize somehow the
injection rate of polymer fluid. We have chosen
to do this by dividing the mass flux of polymer
FIGURE 24. Flow past the hemisphere body with injec-
tion of 100 wppm Polyox at a Reynolds number of 3.9 x
10°. The dimensionless injection values are: (a) G =
0, no injection, (b) 0.5 x 1076, (c) 1.1 x 1078,
(G) 1.7 & 1078, (@) 2.9 2 10-9.
material by the mass flux of the boundary layer
displacement flow. Although this is an arbitrary
normalization, in the present experiments the dis-
placement effect of the injectant fluid was always
much less than the boundary layer displacement
thickness, 6,. Thus we define a quantity G
za cQ
SS npn ome
wo ts
where c is the polymer concentration (weight basis)
in the injectant Q the volume flow rate of injectant
(basically the same fluid as the test medium) with
D and V. being the body diameter and tunnel ve-
locity respectively. For the NSRDC and hemisphere
models dtg5 was calculated at the position of the
laminar separation whereas for the Schiebe body it
was arbitrarily calculated at S/D = 1.00. The pres-
ent results, so normalized, are presented in Fig-
ures 26, 27, and 28. As with the freestream
turbulence level, no change in the position of
separation on the NSRDC and hemisphere nose models
was observed when polymer was injected into the
boundary layer.
The results of the experiments show the presence
. of very small quantities of Polyox to be destabiliz-
97
ing to the laminar boundary layers on the present
test models. This destabilization effect has been
observed before: in fully developed cavity flows
past spheres and cylinders Brennen (1970) observed
distortions in the cavity surface and separation
line due to the presence of polymer. Brennan at-
tributed the changes in cavity appearance to a
polymer induced instability in the wetted surface
flow on the headform. Sarpkaya (1973, 1974) in-
vestigated the flow of dilute polymer solutions
about cylinders and several airfoils and also ex-
plained his observations by suggesting a polymer
induced instability in the laminar boundary layer.
Some later experiments by Tagori et al. (1974)
support some of Sarpkaya's speculation for one of
the airfoils.
A destabilizing effect is rather contrary to the
general impression obtained from the available lit-
erature on the effects of drag-reducing polymers on
fluid friction [see for example Hoyt (1972)]. We
were unable however, to find in the available lit-
erature any satisfactory explanation of the effect
on transition of the polymer fluids.
FIGURE 25. Flow past the Schiebe body at a Reynolds
number of 4.2 x 10° with injection of 500 wppm Polyox.
The dimensionless injection parameters are, (a) G = 0,
(3) 23 8 1O-, (6) 1.5 = 16-5, @) 2.9 % 20°53, mach
frame is 0.2 body diameters in length and they are
centered at arc length ratios of 0.82, 0.75, 0.6,
0.53, respectively.
ite}
iss)
NO INJECTION
G~0.25 x 107
G~0.50 x 1o7&
G~1.00x107&
018
al4
Pad
010
—— (5)
0.04
SEPARATION LENGTH/BODY DIAMETER
) D A
ix105 2xi0> 3x10° 4xio>
BODY REYNOLDS NUMBER - UD /v
FIGURE 26. The length of the laminar separation as a
function of polyox injection on the NSRDC body.
Comparison of Present Results with those of van
der Meulen
van der Meulen (1976) has studied the influence of
dilute polymer solutions (Polyox, WSR 301) upon the
fully-wetted flow and cavitation inception for a
hemisphere nose body and was the first, to our
knowledge, to observe the Schiebe body (Comin =
-0.75). He also was the first to inject the polymer
solution at the stagnation point. To observe the
flow on the test models, van der Meulen used pulsed
ruby laser holography. However, to make the flow
visible a salt was added to the polymer solution.
In his case the injectant was a 2 percent salt--
500wppm Polyox solution. ‘
On the hemisphere nose body he observed that the
injection of the salt-polymer solution eliminated
the laminar separation and he further speculated
that the polymer caused an early transition to a
turbulent non-separating boundary layer. On the
Schiebe body, which has no laminar separation, the
laminar to turbulent transition point was found to
move upstream of the no-injection position. The
present results for this body are seen to agree
qualitatively with those of van der Meulen (Figure
28), although the deduced injection rates of the
G = 0 (NO INJECTION)
G~0.5x107&
G~1.0x107&
G~1.5x107&
0.10
008
006
004
ace
SEPARATION LENGTH/BODY DIAMETER
\x105 3x105 5x10> 7x105 9x10°
BODY REYNOLDS NUMBER - UD/v
FIGURE 27. The length of the laminar separation as a
function of polyox injection on the hemispehre nose
body .
@G=0, PRESENT RESULTS
~2xlO& PRESENT RESULTS
12 < 7xI10® ” ”
~13x10°°
~I5x10"© van der MEULEN (1976)
~20xI0°° " " "
9°
2)
ARC LENGTH LOCATION OF TRANSITION-(S/D),
fe) c
gS @
1 x 10% 2x10? 3x10° 4x10°
BODY REYNOLDS NUMBER-UD/v
FIGURE 28. The position of transition on the Schiebe
body as a function of polymer injection.
latter are rather larger. Even though freestream
conditions of these two tests may not quite be the
same, it is evident because of the nearly one order
of magnitude change in Reynolds number that the
polymer fluid is the chief agent of boundary layer
instability.
5. EFFECT OF FLOW VISUALIZATION ON TRANSITION
It is now well documented that heating a laminar
water boundary layer, tends to stabilize it [see
for example Wazzan et al. (1968a, 1970)]. This
point was further discussed with reference to the
hemisphere and ITTC test bodies by Arakeri and
Acosta (1973) who concluded that for the separating
flows of these bodies, the effect of heating was
on the order of only a few percent. Since the heat-—
ing rate and velocity ranges are similar in the
present experiments, it is expected that the in-
fluence of heating on the hemisphere and NSRDC
bodies is not Significant. However, there is some
question as to the influence of heating on the non-
separating flow on the Schiebe body. Shown in Fig-
ure 22 are averaged observed values of the position
of transition calculated by Wazzan with and without
wall heating. First, it can be seen that there is
good agreement between Wazzan's calculation for an
unheated boundary layer with e? amplification and
the observed position of transition. However, the
point to be noted is that (with a wall temperature
10°F above the ambient water temperature) these
same calculations predict a 40 percent delay in
transition at Rep = 2.5 x 10°. This would suggest
that wall heating is important although not perhaps
sufficient to alter major trends in the present ex-
periments. There is however the qualification that
the calculation assumes a constant wall temperature
while this is not the actual case.
An attempt to measure the actual wall tempera-
ture was made by installing six thermocouples near
the surface of the model at positions of S/D = 0.4,
0.6, 0.8, 1.0, 1.2, 1.4. The position of neutral
stability on this body is S/D = 0.37 and the average
position of transition varied from S/D = 1.0 to
S/D = 0.8. Since it is the heating in the boundary
layer prior to transition that is of importance,
the values of the wall temperatures at S/D = 0.4,
0.6, 0.8, 1.0 were of the most interest. The total
heat flux was set at 250 watts (about 3W/cm2) at
which the schlieren effect was observable. The
wall temperatures were then measured at increasing
values of velocity. It was found that the maximum
wall temperature between S/D = 0.4 and 1.0 varied
from 3°C to 5°C above the ambient temperature.
However, it must be emphasized that these are very
conservative values since the thermocouples are
actually somewhat below the surface in a region of
a high temperature gradient. When this gradient
is accounted for our estimate of the surface excess
temperature is from 1-3°C, a smaller but not neg-
ligible amount. van der Meulen avoided the tem-
perature effect by injecting a two percent salt
solution. On the whole this method and the present
one agree quite favorably (Figure 22). There is,
however, the possibility of instability via a de-
(c)
FIGURE 29. Schlieren photographs of the Schiebe body
with and without salt water injection. The top photo-
graph of each group is without injection; the bottom
photograph shows the injection of MgSO, solution having
a specific gravity of 1.02. The Reynolds number is
Ue67) <9102in) (a) 26150) x) 102 in (((b)),, and) 3433) <1 dloe
in (c).
99
0.65% TURBULENCE LEVEL
fos.
BAND TYPE
BUBBLE TYPE
LOWEST AIR CONTENT
HIGHEST AIR CONTENT
(BROCKETT 1972)
(eo)
@o
fo)
a
fo)
u
CAVITATION INCEPTION NUMBER - Oj
° fo)
B o
1x10° 2x10° 3x05 4xi05
BODY REYNOLDS NUMBER - UD/V
FIGURE 30. Cavitation inception on the NSRDC body.
stabilizing density gradient. This point was ad-
dressed experimentally and in Figure 29 matched
pairs of schlieren photographs, without and with
salt injection are presented. It was found that
although the appearance of the transition changed
markedly the location of transition did not change
significantly.
6. PRESENTATION OF CAVITATION INCEPTION RESULTS
Freestream Turbulence Level
The data on the influence of freestream turbulence
level upon cavitation inception is limited because
of the low maximum water speed in the LTWT of about
8m/s but more importantly because the turbulence
generating grids located at the entrance to the
test section cavitated themselves before the test
models did. Consequently, only the 0.05 and 0.65
percent turbulence level configurations could be
used. The NSRDC body was the only one to be so in-
vestigated. Some of these inception data are sum-—
marized in Figure 30 where they are compared with
Brockett's (1972) data. Inception on the NSRDC
body was always of the band type which occurred
suddenly without any precursor bubble type cavita-
tion. As can be seen in Figure 30, inception oc-
curred at the same value of the inception index for
both turbulence levels, but as illustrated in Fig-
ure 31 the subsequent developed cavitation was much
less steady at the higher turbulence intensity.
The Effects of Polymer Solutions
Hemisphere Nose Body
The type of cavitation and the value of the incep-
tion index were found to be strongly dependent on
the amount of polymer present in the boundary layer.
For a fixed polymer solution concentration and free-
stream velocity the following changes in inception
were observed to take place: at zero injection
rate, incipient band type cavitation as illustrated
in Figure 32(a) always occurred. At injection rates
less than the critical value (the injection rate
at which the separation would disappear), band type
inception still occurred but as can be seen in Fig-
ure 32(b) the surface of the developed cavitation
100
FIGURE 31. The physical appear-
ance of cavitation on the NSRDC
body at two turbulence levels in
the LIWT. The Reynolds number
is 3.4 x 10°. In (a) the turbu-
lence level is 0.05 percent and
the cavitation index is 0.44.
The turbulence level in the re-
Maining photographs is 0.65 per-
cent and the cavitation index is
about 0.35 for all cases.
FIGURE 32. In these photographs
500 wppm of polyox solution is
injected at the nose of the hemi-
sphere body. The cavitation
index is 0.59, and the Reynolds
number is 6.7 x 10° (HSWT). The
dimensionless injection rate, G,
is zero in (a) 1.9 x 107° in (b),
4.4 x 10-© in (c), and 5.24
1o-© in (d). In many instances
the attached cavitation would
disappear.
has a definite wave structure and the separation
line has become very irregular. Inspection of
Schlieren photographs of the fully wetted flow at
this injection rate showed that the position of
transition on the free shear layer had moved upstream
from the no injection case and that the separation
region was smaller in size. With a further increase
in the injection rate to near critical values, dif-
ferent types of cavitation were observed depending
upon the facility. In the HSWI, band type inception
would occur intermittently in patches with irregular
separation lines and surfaces as is shown in Figure
32(c), (d). At injection rates above the critical
value, the same type of behavior took place, but with
the flow altering between fully wetted and patchy
band type cavitation more rapidly. A decrease in
the cavitation number at this injection rate would
make the cavitation more "violent," but no steady
attached cavitation could be obtained. At these
near-and-above critical injection rates the fully
wetted observations showed the laminar separation
had been eliminated with only an occasional short
reappearance. That is, the flow in the region of
interest was almost always turbulent. If then the
injection rate was suddenly reduced to zero, a large
steady cavity would quickly form on the body.
In the LTWT the same sequence of cavitation
events with increasing injection rates would occur
as in the HSWT. However, near and above critical
injection rates, travelling bubble and band type
cavitation would occur simultaneously, unlike the
HSWT where no bubble type cavitation was observed.
NO INJECTION
G~0,35x 107°
G~ 1.75 x 107
G~ 2.3 xior&
G~ 7.0 x 107
G~9.0 x1lo7®
G~ 14.6 x 107°
G~ 23.6 x 1o7®
fo)
x
o@pvpDegncno
to}
co)
oO
‘7
fo)
a
CAVITATION INCEPTION NUMBER - 0;
°
ce)
4 1 — 4
Ixio> 2 3 4 5 6 ? 8 cS) Ix10®
BODY REYNOLDS NUMBER - UD/V
FIGURE 33. Cavitation inception with polymer
injection on the hemisphere body.
This difference will be discussed later. These
inception data have been summarized in Figure 33.
NSRDC Body
The NSRDC body was tested only in the LTWT and it
too was observed to go through a sequence of cavi-
tation development similar to that of the hemisphere
nose body in the LIWT; namely, that the injection
of polymer at sub-critical rates changed the orig-
inal band type inception to simultaneously occurring
intermittent band and travelling bubble type incep-
tion. At above critical injection rates the inter-
mittency became more rapid but still no steady
attached cavitation could be obtained. Examples
of these types of cavitation are shown in Figure
34. Notice in particular Figure 34(d) where only
one cavitation bubble is visible at a cavitation
number of 0.34. Values of the inception index
101
versus body Reynolds number are presented in Fig-
ure 35.
Schiebe Body
The Schiebe body was tested in both the LIWT and the
HSWT, but the influence of polymer was only studied
in the LTWI. Again as for the hemisphere nose body,
the type of cavitation depended upon the facility.
In the LIWT, travelling bubble type inception always
occurred and the presence of polymer was found to
have no significant effect on either the type of
cavitation or the inception index. Lowering of the
tunnel pressure below the inception value produced
a steady, attached cavity of the type normally as-
sociated with the presence of a laminar separation.
On the other hand, in the HSWT, travelling bubble
type cavitation events were extremely rare. In-
ception occurred with the sudden appearance of an
unsteady attached cavity occasionally preceded by
one or two travelling bubble events. Examples of
these types of cavitation on the Schiebe body are
given in Figure 36 and a summary of the inception
data is given in Figure 37. A unique location of
inception could not be accurately determined in
either facility for this body.
7. DISCUSSION
Freestream Turbulence Level
The main purpose of the investigation of freestream
turbulence level upon cavitation inception was to
determine if it could be a contributing factor to
the differences in cavitation results on identical
bodies tested in different facilities. In particu-
lar, could the differences in cavitation inception
on the same NSRDC test body between the CIT HSWT and
the NSRDC 12-inch tunnel be explained by different
FIGURE 34. The physical appear-
ance of cavitation on the NSRDC
body at a Reynolds number of
3.4 x 10° in the LIWT with (a)
no injection, (b) G = 3.4 x 10°77,
cavitation index = 0.45 [same as
in (a)I, (ce) 3.4 x 1077, cavita-
tion index = 0.34, and (d) 7.1 x
10-© at the same index!
102
——————
© NO INJECTION
@ G~0.35x107°
4 BROCKETT (1972)
NO INJECTION
o
n
oO
on
° °
a B
CAVITATION INCEPTION NUMBER - O;
°
Nn
ix10° 2 3 4
BODY REYNOLDS NUMBER - UD/V
FIGURE 35. Cavitation suppression by polymer injection
on the NSRDC body.
turbulence levels? From the proceeding discussion
of the fully wetted results it appears that the
differences in observed critical Reynolds numbers
are probably due to a higher turbulence level in
the NSRDC facility. It follows then that the dif-
ferences in the type of inception for velocities
less than 30 feet per second can be explained in
terms of the different viscous flows. However, we
FIGURE 36.
: Photographs of cavitation on the (same)
Schiebe body in the LIWT (upper picture) at a cavita-
tion index of 0.52 and in the HSWT at an index of 0.41.
The flow speeds are 7.3 and 14 m/s, respectively.
@ HSWT PRESENT TESTS
@ LTWT a i
4 van der MEULEN (1976)
fe)
(2)
S —:—~C, AT ESTIMATED POSITION
OF TRANSITION
x<
AOS
a
z
204
)
Looe
00.3
WO.
z
9°
to
Axio = 4 5 6 7 8 9 1x10®
BODY REYNOLDS NUMBER-UD/v
FIGURE 37. Cavitation inception on the Schiebe body
in three different facilities.
still need to account for the different freestream
populations of nuclei, the subject of the next
section.
Polymer Injection
Some inception data for the hemisphere nose body
with polymer injections are given in Figure 38 as
a function of the injection rate for two concentra-
tions. The same data have been replotted in Figure
39 against the non-dimensional injection parameter
G. It can readily be seen that the two curves have
collapsed onto one. A similar happy result was
found when the dimensions of the laminar separation
bubble on the hemisphere nose body were plotted
versus the parameter G. These correlations of the
inception index and separation bubble dimensions
with G implies that the polymer "effectiveness" is
proportional only to the amount present within the
boundary layer, here taken to be the displacement
thickness.
For the NSRDC and hemisphere nose bodies it can
be seen that increasing amounts of polymer in the
boundary layer produce an increasing suppression
°
=
© 50 WPPM
@ 500 WPPM
9
oO
03
CAVITATION INCEPTION NUMBER - oj
0 5 10 15
POLYMER INJECTION RATE - ml/sec
FIGURE 38. Cavitation index with polymer injection on
the hemisphere body at a Reynolds number of 7.5 x 10°.
Oo 50 WPPM
@ 500 WPPM
{e)
(>)
° fo)
rs a
CAVITATION INCEPTION NUMBER - Go;
°
ol
(o) Ixl0"& 2 3 4 5 6 7 8x1o7&
FIGURE 39. The data of figure 38 replotted against the
dimensionless injection parameter G.
of cavitation index. There is a limit, however,
beyond which no further increase in cavitation
suppression occurs. In the present experiments on
the hemisphere nose body this limiting value of G
is approximately 7 x 107© which also coincides with
the removal of the laminar separation. These
results and others are summarized in Figure 40
where the maximum percent reduction in cavitation
index has been plotted versus the Reynolds number.
These include the "polymer ocean" results of Baker
et al. (1973), Holl et al. (1974), and Ellis et al.
(1970). However, the information from their re-
ports is limited and all that can be said is that
they give values approximately the same as those
noted in the present case. The agreement is be-
lieved to be reasonably good for experiments of this
type insofar as the maximum effect goes. We presume
that similar effects in "ocean" experiments could
be achieved at much smaller concentrations if the
G parameter has significance.
During their cavitation tests Baker and Holl
noted a change in the appearance of the developed
cavitation. From photographic observations of these
changes they speculated that the cavitation attenu-
ation was due to a "flow reorientation in the region
of the laminar separation bubble." They further
suppose [Arndt et al. (1975)] that the amount of
attenuation might depend on a Deborah number,
TV /S51 where T is the molecular relaxation time
of the molecule, V,, the freestream velocity, and
von der MEULEN(1976)-500 WPPM
PRESENT STUDY - SOOWPPM
20 WPPM
50 WPPM
20 WPPM
BOWPPM
2OWPPM _ HOLL etal (53)
} ELLIS etal (1970) |
8
| BAKER etal (1973)
@eseorpcvreo
Db
fe)
nN
le)
PERCENT REDUCTION IN INCEPTION
CAVITATION NUMBER
°
Ixto® 2 3 4 5 6 7 8 9 Ixio®
BODY REYNOLDS NUMBER - UD/V
FIGURE 40. Maximum cavitation inception index suppres-
. sion by polyox WSR 201 on the hemisphere nose. The
Ellis and Baker results are for polymer "oceans."
103
6, is the boundary layer displacement thickness at
separation. It now seems clearly established in
our opinion, that the overall gross effect caused
by the polymer in the flow about these bodies is a
removal of the laminar separation by stimulation of
transition and that this is indeed the origin of the
flow "reorientation" noted by Baker and Holl. Pre-
sumably, the molecular relaxation time has an im-
portant role in boundary layer stability, but as
yet this appears to be unknown; it may be that the
parameter proposed by Arndt is important for some
laminar flows with separation (as it is indeed for
the flow about a circular cylinder), but we think
not in the context of the present experiments.
Since the suppression of cavitation upon these
bodies is a result of the elimination of the laminar
separation by the polymer it is worthwhile to com-
pare the present results with those in which the
separation is eliminated by another method. Arakeri
and Acosta (1976) carried out a series of tests with
a hemisphere nose body and an ITTC body using bound-
ary layer trips to reduce the critical Reynolds num-
ber in the HSWT. It was, briefly, found that with
the trip present and at velocities above the new
critical velocity, the occurrence of cavitation was
significantly suppressed, and that at higher ve-
locities the tunnel would choke from the model
support before the body could be made to cavitate:
The present polymer tests show a very similar large
effect on inhibiting cavitation but not quite as
dramatic as the tripped tests.
8. FREESTREAM NUCLEI AND CAVITATION INCEPTION
Some Observations in the LTIWT
As will be recalled from the description of the
LTIWT, this facility has no resorber which neces-—
sitated cavitation data acquisition before pump-
generated bubbles entered the test section. Ona
number of occasions the cavitation on the NSRDC and
hemisphere bodies was deliberately maintained and
the pump-generated gas bubbles allowed to pass
through the test section. As the number of free
gas bubbles increased, the initially-occurring band
type cavitation was gradually destroyed and replaced
by travelling bubble type cavitation. An alterna-
tive procedure was to lower tunnel static pressure
so that the cavitation number had a value below
-Cpmin but above the inception value and again
allow the pump-generated bubbles to accumulate.
The body would then eventually cavitate with in-
ception then always being of the travelling bubble
type. Schlieren observations of the basic viscous
flow on the hemisphere nose were made at these
gradually increasing freestream bubble populations
and nuclei populations were measured when band type
inception occurred and when this above deliberately-
promoted bubble type inception occurred. The
schlieren observations show (see Figure 41) that
as the number of freestream nuclei increased, the
laminar separation on the hemisphere nose became
unsteady and was finally greatly diminished if not
eliminated. Thus, in effect, the free-stream bub-
bles serve to trip the boundary layer.
Nuclei populations obtained when band type incep-
tion occurred (0; = 0.44) are shown with distribu-
tions obtained when deliberately promoted travelling
bubble inception occurred (0; = 0.58, 0.73) in
Figure 42. As can be seen in this figure, for
104
FIGURE 41. Flow past the hemi-
sphere nose with many freestream
bubbles showing boundary layer
stimulation.
nuclei with radii less than 100 microns all the dis-
tributions are essentially the same whereas for
nuclei greater than 100 microns radius the bubble-
type inception distributions have many more nuclei
than the band type inception distributions. Thus
it seems possible that in facilities with many
Macroscopic freestream gas bubbles, the normally
occurring laminar separation on some bodies can be
eliminated. The subsequent cavitation index and
form of cavitation should then be controlled by the
nuclei population. 4
If so, the experiments on the NSRDC body at that
facility and those tests on the same body in the
HSWT (PARTICLES)
(.24<o<.72)
io!
10'°
le O =0.44 LTWT (BAND)
© O-=0.44
4 0-058 LTWT(BUBBLE)
10° @ 0-073
O O =066-PETERSON(I972)
BUBBLE TYPE
NUCLE! NUMBER DENSITY DISTRIBUTION FUNCTION (m-4)
lo-& 1075 10-4
NUCLEI RADIUS (m)
PIGURE 42. Nuclei distributions measured by holography
in the LIWT (all microbubbles) and in the HSWT (essen-
tially only solid particles).
LTWT, when bubble type inception was deliberately
promoted, should be very similar. This is, in fact,
the case as the inception numbers are more-or-less
the same. Beyond that, nuclei distributions are
known for the two tests [Peterson (1972) and Fig-
ure 42] so that, following the philosophy of Silber-
man et al. (1974), it is possible to estimate the
number of "cavitable" nuclei per unit volume for
each experimental point. A rough estimate of the
number of travelling cavitation events can be easily
made if we take Johnson and Hsieh's (1966) "capture"
radius of 0.01 body radius to determine the flux
of fluid through the cavitating region. These data,
calculated and measured events are tabulated in
Table 3. Peterson measured the event rate acous-
tically and chose one event/sec as the threshold
level because of the agreement with a “visual" in-
ception estimate. (Only the visual estimate was
made in the LTWT.)
Observation in the HSWT
On the whole, the agreement of observations and
event rates is satisfactory and it seems clear in
this circumstance that viscous effects are not of
primary importance and that travelling bubble cavi-
tation, the type studied by the St. Anthony Falls
group, is the prevalent form. But, on all of the
bodies studied we have seen different forms of
cavitation occur, when separation was not present,
if the number of freestream nuclei is very small,
as it is presumably in the California Institute of
Technology HSWT and other resorber facilities. Then,
even on the Schiebe body we see attached forms of
cavitation at inception (see Figure 36) at very low
inception indices with only rare occurrences of
travelling bubble cavitation [see also Arakeri et al.
(1976) ]. In these circumstances the fluid and the
nuclei that it contains pass through regions of
some tension (up to about 1/2 at m in the HSWT). It
is conceivable then that the substantial pressure
fluctuations in transition regions [Huang and Hannon
(1975) ] can initiate cavitation. This is the ra-
tionale for Arakeri's (1975) inception-transition
pressure coefficient correlation. Values of —Cptr
105
TABLE Ml CAVITATION HVENT RATES
$5 Model oy ee R. Cavitatable Calculated Measured
Facility Mat'l. Nuclei/cm Events/sec Events per sec
(ft/sec) (microns) (est)
NSRDC CU 0.62 29.86 We 0.5 Ong 1.0
CU 0.66 29. 86 15 ie 8 Bie WO)
AU-Plated 0.65 29.86 14 Ze Al 3.8 ee
DELRIN 0.69 29. 86 18 OS 0.9 1.0
DELRIN 0.71 29.86 21 2.4 4.3 0
LTWT CU 0.58 20.15 21 Sc 9) 10.7 --
CU 0.64 20. 10 29 2.0 Z.4 --
CU 0.66 ZO). U5 32 0.9 We il --
CU O73 20.25 58 Ne 7 2.0 --
are also shown for the Schiebe body in Figure 37;
again the correlation is suggestive but not con-
clusive.
Further evidence of the difference between a
resorber facility and a recirculating tunnel is
given by the nuclei distributions of the flow in the
California Institute of Technology HSWT. These data
are averaged in the graph of Figure 42. Following
Peterson (1972) it is possible to distinguish par-
ticulate matter from gaseous microbubbles down to
about 10 micrometers. Thus we identify solid par-
ticulates on the one hand and microbubbles on the
other. All of the nuclei reported in Figure 42
for the LTWT are microbubbles. It is significant
that the HSWT shows a very similar distribution of
solid particulates, but very few microbubbles. In
about ten holograms made of the HSWT flow, within
the various sample volumes that were counted, about
100 particles/cm? on the average were found. How-
ever, of these, less than one on the average was a
microbubble, too few even to hazard a guess as to
the distribution. This finding certainly tends to
explain the experimental trends in this facility if
it is assumed (as appears evident) that the solid
particulates do not act as nucleating sources.
In closing this section we have perhaps come
full circle in inception research to re-emphasize
the important role of the cavitation nuclei. The
influence of laminar inception on cavitation is now
much clearer as are the effects of the processes
that cause stimulation of the boundary layer. If
there are many nuclei present (so that a large ten-
sion on the body does not exist prior to cavitation)
it is likely that travelling bubble cavitation will
predominate, then the notion of a "standard body"
to deduce cavitation susceptibility appears to be
useful. However, with only a few nuclei other more
_complex forms of cavitation are seen at inception.
Comparison of Nuclei Distributions
Data from several other investigations, reduced to
the number density distribution function, N(R), by
the following approximation
number of nuclei per unit
Ry + R, with radii between RQ, and R,
ee ee
(R,- R))
are shown in Figure 43. A tabulation of the mea-
suring techniques and test conditions for each in-
vestigation is given in Table 4. All the data have
approximately the same slope, but the values of the
distribution function can differ by several orders
of magnitude, i.e., although the nuclei population
changed by several orders of magnitude, the dis-
tribution of the nuclei sizes remains constant.
The large differences in populations is undoubtedly
a consequence of the large variation in conditions
which existed in the water when the data was col-
lected and is no doubt one of the contributing fac-
tors to the lack of repeatability seen in cavitation
esitese
A goal of cavitation research is to be able to
predict the inception of cavitation and thus be able
to scale laboratory results to prototype conditions.
It is interesting then to compare nuclei populations
in water tunnels to those in the ocean. Medwin
acoustically measured bubble populations in the
ocean near Monterey, California, and in Figure 43
two of his measured distributions are presented.
The summer distribution agrees reasonably well with
the distributions obtained under cavitating con-
ditions in strongly deaerated water. However, in
the LTWT there are considerably more bubbles than
found in the ocean for radii greater than about 30
106
TABLE IV COMPARISON OF NUCLEI MEASUREMENTS
Investigator
Gavrilov (1970)
Peterson et al (1975)
Arndt & Keller (1976)
Keller & Weitendorf (1976)
Medwin (1977)
Peterson (1974)
U.S. Navy (Naval Ocean
System Center, San Diego,
California. Courtesy
Dr. T. Lang)
Present Tests
1975) (o=0.49,
SCATTERING
Measuring
Technique
Acoustic
Light Scattering
Holography
Microscopy
Light Scattering
Light Scattering
Acoustic
Coulter Counter
Coulter Counter
Holography
Facility
Water Tunnel
ene ID)
Water Tunnel
at NSMB
Water Tunnel
at Hamburg
Model Basin
Monterey Bay,
California
Santa Catalina
Channel
California
San Diego Bay
and offshore
LTIWT
Conditions at time
of Measurement
Standing tap water
At inception on 50 mm diameter
NSRDC body o=0.49
Cavitation tests on a sharp edged
disc. Air contents: 6.3 and
12.5 ppm
Propeller test, gassed water,
Air content: ~30 ppm
Various depths and seasons
Various depths
Various depths
Air content ~ 7ppm, o=0.44
@ PRESENT TESTS LTWT
AIR CONTENT ~ 7 ppm
a =0.44
PETERSON etal
(1975)(0=0.49, ©
HOLOGRAPHY) '
10"!
\-
GAVRILOV (1970) \
1o9 AFTER STANDING
5 HOURS)
NUMBER DENSITY DISTRIBUTION FUNCTION N(R),m“
MEDWIN (1977)
(OCEAN, FEBRUARY)
RNDT & KELLER
976)
AIR CONTENT~ |
12.5 ppm
ot
1 10
RADIUS R (micrometers)
KELLER &
WE!ITENDORF
(1976)
GASSED WATER
AIR CONTENT~
30 ppm
ARNOT &
KELLER (1976)
\
V\meowin (1977)
(OCEAN, AUGUST)
FIGURE 43. Nuclei distributions from various sources.
micrometers.
Further, in the winter the measured
bubble population in the ocean is one order of
magnitude less than in the summer. We see then it
is actually possible for laboratory facilities to
have much higher nuclei populations than actually
occur in the ocean. Medwin concludes interestingly
that the microbubbles had a biological as well as
physical origin because the concentration of bubbles
increased with depth. This observation is perhaps
of importance for the Coulter Counter measurements
of Peterson (1974) and Lang (1977). The particulates
measured there, although thought to be of organic
material, may actually also contain some gas.
Finally, it is amazing to observe the wide range
of applicability of fairly simple power laws for
particulate and microbubble populations.
9. CONCLUSIONS
It is clear that the onset of cavitation and its
physical appearance at this onset can be greatly
affected by freestream turbulence and the presence
of minute amounts of long chain polymer solutes.
The present results support the conclusion that
these effects are indirect insofar as cavitation
goes and that the primary effect is on the viscous
flow past the test body. The polymer solutions in
particular promote an early boundary layer transi-
tion which forestalls the presence of laminar separa-
tion much as does boundary layer stimulation by
freestream turbulence or trips. It follows that
cavitation on bodies not having laminar separation
should not be much affected by freestream turbu-
lence or polymer solutions. This appears to be
the case if the test medium has "many" freestream
nuclei so that travelling bubble cavitation is
predominant. However, if only a few nuclei are
present, attached forms of cavitation occur at
inception even on nonseparating bodies. From
recent nuclei measurements in the ocean it appears
that some test facilities may have too many nuclei
and others possibly too few.
ACKNOWLEDGMENTS
This work was supported by the Department of the
Navy, Office of Naval Research under Contract
NOO014-76-C-0156 (in part) and by the Naval Sea
Systems Command, General Hydromechanics Research
Program, administered by the David W. Taylor Naval
Ship Research and Development Center under Contract
NOO014-75-C-0378. This assistance is gratefully
acknowledged. Special thanks are due to Mrs.
Barbara Hawk for manuscript preparation and to
Joseph Katz and David Faulkner for their painstaking
efforts in hologram analysis. Finally we thank
Professor M. Morkovin for his careful and helpful
review of the manuscript.
REFERENCES
Acosta, A. J., and H. Hamaguchi (1967). Cavitation
inception on the ITTC standard headform. Hydro-
dynamics Laboratory, Report No. E-149.1, Cali-
fornia Institute of Technology.
Acosta, A. J., and B. R. Parkin (1975). Cavitation
inception--A selective review, J. Ship Res., 19,
4; 193-205.
Arakeri, V. H., and A. J. Acosta (1973). Viscous
effects in inception of cavitation on axi-
symmetric bodies. J. Fluids Eng. 95, Ser. 1,
4; 519-528.
Arakeri, V. H. (1975). A note on the transition
observations on an axisymmetric body and some
related fluctuating wall pressure measurements.
Je Hluids Eng. , 97, ‘Sex. 1, 1; 82=s8i7-:
Arakeri, V. H., and A. J. Acosta (1976). Cavita-
tion inception observations on axisymmetric
bodies at supercritical Reynolds numbers. J.
Ship Res., 20, 1; 40-50.
Arndt ReEe Acie .n las Basllet siden Wie sHOUsy) and
C. B. Baker (1975). A note on the inhibition
of cavitation in dilute polymer solutions. App.
Res. Lab. Tech. Memo. 75-285, The Penn. State
Univ.
Arndt, R. E. A. and J. W. Daily (1969). Cavitation
in turbulent boundary layers. Cavitation State
of Knowledge, ASME; 64-86.
Arndt, R. E. A. and A. P. Keller (1976). Free gas
content effects on cavitation inception and noise
in a free shear flow; Two phase flow and cavi-
tation in power generation systems. IAHR Symp.
Grenoble, March 16, 1976.
Baker, C. B., R. E. A. Arndt, and J. W. Holl (1973).
Effect of various concentrations of WSR-301
polyethylene oxide in water upon the cavitation
performance of 1/4-inch and 2-inch hemispherical
nosed bodies. App. Res. Lab. Tech. Memo. 73-257,
The Penn. State Univ.
Beckwith, T. G., and N. L. Buck (1961). Mechanical
Measurements. Addison-Wesley, Reading, Mass.
107
Brennen, C. (1970). Some cavitation experiments
with dilute polymer solutions. J. of Fluid
Mech., 44, part 1; 51-63.
Brockett, T. (1972). Some environmental effects
on headform cavitation inception. NSDRC Rep.
3974.
Cavitation Committee (1969). Summary of replies to
questionnaire on water tunnel methods. 12th ITTC.
Debrule, P. M. (1972). Friction and heat transfer
coefficients in smooth and rough pipes with
dilute polymer solutions. Ph.D. Dissertation,
California Institute of Technology.
Ellis, A. T., J. G. Waugh, and R. Y. Ting (1970).
Cavitation suppression and stress effects in
high-speed flows of water with dilute macro-
molecular additives. J. of Basic Eng., Sept.;
459-466.
Feldberg, L. A., and K. T. Shlemenson (1973). The
holographic study of cavitation nuclei. Dis-
cussion to Proc. IUTAM Symp. on Nonsteady Flow
of Water at High Speeds. Leningrad, USSR
(English version; Moscow 1973); 106-111.
Gates, E. M., and J. Bacon (1978). Determination
of cavitation nuclei distributions by holography.
Wo Glasto RAG, AA, ip Zs,
Gavrilov, L. R. (1970). Free gas content of a
liquid and acoustical techniques for its measure-
ment. Sov. Phys-Acoustics, 15, 3; 285-295.
Hall, D. J., and J. C. Gibbings (1972). Influence
of stream turbulence and pressure gradient upon
boundary layer transition. J. of Mech. Eng.
Sethe da Bp alsvaS iss.
Holl, J. W., and A. L. Treaster (1966). Cavitation
hysteresis. J. of Basic Eng., Trans. ASME,
Ser, ID, GH jp I@SoAily.
Holl, J. W. (1968). Sources of cavitation nuclei.
Pres. at the 15th Amer. Towing Tank Conf.,
Ottawa, Canada.
Holl, J. W., and A. L. Kornhauser (1969). Thermo-
dynamic effects on desinent cavitation on hemi-
spherical nosed bodies in water of temperatures
from 80 degrees F to 260 degrees F. Pres. at
the Joint Conf. of App. Mech. and Fluids Eng.
Divisions, ASME, Northwestern Univ. ASME paper
No. 69-—PE=1-
Holl, J. W., R. E. A. Arndt, M. L. Billet, and
C. B. Baker (1974). Cavitation research at the
Garfield Thomas Water Tunnel. App. Res. Lab.
Tech. Memo. 74-62, The Penn. State Univ.
Hoyt, J. W. (1966). Effects of High-polymer solu-
tions on a cavitating body. Proc. llth ITTC,
Tokyo.
Hoyt, J. W. (1972). The effects of additives on
fluid friction. J. of Basic Eng., June; 258-285.
Huang, T. T., and D. E. Hannan (1975). Pressure
fluctuations in the regions of flow transitions.
DTNSRDC Rep. 4723.
Jaffe, N. A., T. T. Okamura, and A. M. O. Smith
(1970). Determination of spatial amplification
factors and their application to predicting
‘transaid onl vArAA mie, 1S),) 0217) SOL —30/8r
Johnson, Jr., V. E., and T. Hsieh (1966). The
influence of the trajactories of gas nuclei on
cavitation inception. 6th Symp. on Nav. Hydro.
Johnsson, C. A. (1969). Cavitation inception on
headforms, further tests. 12th ITTC, Rome;
BSI 392%
Keller, A. P. (1972). The influence of the cavita-
tion nucleus spectrum on cavitation inception,
investigated with a scattered light counting
method. J. of Basic Eng., Dec.; 917-925.
108
Knapp, R. T., J. Levy, J. P. O'Neill, and F. B.
Brown (1948). The Hydrodynamics Laboratory
of the California Institute of Technology.
Trans. ASME, 70; 437-457.
Knapp, R. T., J. W. Daily, and F. G. Hammitt (1970).
Cavitation. McGraw-Hill, New York.
Lang, T. G., (1977). Naval Ocean Systems, Private
Comm.
Lindgren, H., and C. A. Johnsson (1966). Cavitation
inception on head forms ITTC comparative experi-
ments. Pub. of the Swedish State Shipbuilding
Exper. Tank, No. 58. (Also presented at the
llth ITTC Tokyo, 1966).
Loehrke, R. I., and H. M. Nagib (1972). Experiments
on management of freestream turbulence. AGARD
Rep. No. 598.
Mack, L. M. (1977). Transition and laminar in-
Stabwlattye die ee ae PUD 7715).
Medwin, H. (1970). In situ acoustic measurements
of bubble populations in coastal ocean waters.
J. Geophys. Res. 75, 599-611.
Medwin, H. (1977). In situ acoustic measurements
of microbubbles at sea. J. Geophys. Res. 82,
Gp SPARS
Morgan, W. B. (1972). Air content and nuclei mea-
surements. 13th ITTC, Rep. of Cavitation Comm.
Morgan, W. B., and F. B. Peterson (1977). Cavita-
tion inception. 18th Amer. Tow. Tank Conf.,
U.S. Naval Acad., Annapolis, Md.
Parkin, B. R., and R. W. Kermeen (1953). Incipient
cavitation and boundary layer interaction on a
streamlined body. Hydro. Lab. Rep. E-35.2,
California Institute of Technology.
Peterson, F. B. (1968). Cavitation originating at
liquid-solid interfaces. NSDRC Rep. 2799.
Peterson, F. B. (1972). Hydrodynamic cavitation
and some considerations of the infltience of free
gas content. 9th Symp. on Naval Hydro.
Peterson, F. B., F. Danel, A. Keller, and Y. Lecoffe
(1975). Determination of bubble and particulate
spectra and number density in a water tunnel
with three optical techniques. 14th ITTC.
Peterson, L. L., (1974). The propagation of sun-
light and the size distribution of suspended
particles in a municipally polluted ocean water.
Ph.D. Thesis, California Institute of Technology.
Reshotko, E. (1976). Boundary-layer stability and
transition. Ann. Rev. of Fluid Mech., 8; 311-349.
Sarpkaya, T., P. G. Rainey, and R. E. Kell (1973).
Flow of dilute polymer (Polyox WSR-301) solu-
tions about circular cylinders. J. of Fluid
Mech., 57; 177-208.
Sarpkaya, T. (1974). On the performance of hydro-
foils in dilute Polyox solutions. Int'l. Conf.
on Drag-Reduction, Churchill Coll., Cambridge,
England, 4-6 Sept.
Schiebe, F. R. (1966). Cavitation occurrence
counting--a new technique in inception research.
Cavitation Forum, ASME, N.Y., 8-9.
Schiebe, F. R. (1972). Measurement of the cavita-
tion susceptibility of water using standard
bodies. St. Anthony Falls Hyd. Lab. Project
Rep. No. 118, Univ. of Minn.
Silberman, E., F. Schiebe, and E. Mrosla (1973).
The use of standard bodies to measure the
cavitation strength of water. St. Anthony
Falls Hyd. Lab., Rep. No. 141, Univ. of Minn.
Spangler, J. G., and C. S. Wells, Jr. (1968). Ef-
fects of freestream disturbances on boundary
layer transition. AIAA J., 6; 543-545.
Tagori, T., K. Masunaga, H. Okamoto, and M. Suzuki
(1974). Visualization of flow of dilute polymer
solutions around two-dimensional hydrofoils.
2nd Symp. of Flow Visualization, Univ. of
Tokyo, July; 83-88.
Ting, R. Y., and A. T. Ellis (1974). Bubble growth
in dilute polymer solutions. Physics of Fluids,
ili VAG VA GDS
Tsuji, H. (1956). Experimental studies on the
spectrum of isotropic turbulence behind two
grids. J. of Phys. Soc. of Japan, 11, 10;
1096-1104.
van der Meulen, J. H. J. (1972). Cavitation on
hemispherical nosed teflon bodies. Int'l. Ship-
building Prog., 19, 218; 333-341.
van der Meulen, J. H. J. (1973). Cavitation sup-
pression by polymer injection. ASME Cavitation
and Polyphase Flow Forum, N.Y.; 48.
van der Meulen, J. H. J. (1976). A holographic
study of cavitation on axisymmetric bodies and
the influence of polymer additives. Doctoral
Thesis, Netherlands Ship Model Basin.
van der Walle, F. (1962). On the growth of nuclei
and the related scaling factors in cavitation
inception. 4th Symp. on Naval Hydro.
van Driest, E. R., and C. B. Blumer (1963). Bound-
ary layer transition: freestream turbulence
and pressure gradient effects. AIAA J., 1, 6;
1303-1306.
Vanoni, V. A., E. Hsu, and R. W. Davies (1950).
Dynamics of particulate matter in fluid sus-
pensions. Hydro. Lab. Rep. No. 7l.la, California
Institute of Technology.
Wazzan, A. R., T. T. Okamura, and A. M. O. Smith
(1968a). The stability of water flow over
heated and cooled flat plates. J. of Heat
Transfer, Feb.; 109-114.
Wazzan, A. R., T. T. Okamura, and A. M. O. Smith
(1968b). Spatial and temporal stability charts
for the Falkner-Skan boundary layer profiles.
D.A.C. rep. 67086, McDonnell Douglas Corp.
Wazzan, A. R., G. Keltner, T. T. Okamura, and A.
M. O. Smith (1972). Spatial stability of stag-
nation water boundary layer with heat transfer.
Physics of Fluids, 15, 12; 2114-2118.
Wazzan, A. R., C. Gazley, Jr. (1978). The combined
effects of pressure gradient and heating on the
stability and transition of water boundary
layers. Rand Rept. R-2175-ARPA.
Wu, J. (1971). Some techniques of ejecting additive
solutions for drag reduction. Hydronautics
Technical Rept. 7101-1.
Additional Reference*
van Ingen, J. L. (1977). "Transition, pressure
gradient, suction, separation and stability
theory," paper 20 AGARD proc. CP-224.
*
Suggested by Professor M. V. Morkovin
Discussion
M. A. WEISSMAN
My question was "What is your definition of
growth rate?" This is quite a crucial point, for
in comparing theory to experiment, we must make
sure that we are comparing like to like.
The meaning of growth rate for nonparallel
flow is not obvious. Let us consider El—Hady and
Nayfeh's lowest order solution (Eq. 42):
A = iN feGe ry)explif (a + ca )dx - iwt] (1)
1 0 1 0 1
The downstream growth of the magnitude of this
function is not purely contained in the expotential
factor. The change in the eigenfunction, Tt, with
x, also contributes to "growth." In fact, a com-
miete definition of growth would be
1 a|z,|
er el ex
which reduces to
G= @) + eG, + Tel Oe, (2)
109
using (1), where it is understood that Gy and G4
are the negative and imaginary parts of o, and a,.
[Bouthier (1972), Gaster (1974), and Eagles and
Weissman (1975)].
Equation 2 shows that the growth rate is
actually a function of y. (It is also a function
of the flow quantity under consideration, see the
above mentioned references.) However, if we agree
to measure the growth rate at a particular
y-position and if the eigenfunction is normalized
at that position (so that 3|c|/ax = 0 at that
position), then the influence of the changing
eigenfunction on growth rate will disappear (for
this particular definition of growth rate). The
poit is that a, is not uniquely defined; it depends
on the normalization used for Z. [This can also
be seen from examination of the equation defining
Oye Eq. 35]. The authors have neglected to explain
what their normalization was.
REFERENCES
Eagles, P. M., and M. A. Weissman (1975). On the
Stability of Slowly Varying Flow. J. Flutd. Mech.
69, 241-262.
110
Author’s Reply
ALI H. NAYFEH
The growth rate in a parallel flow can be
unambiguously defined, but it cannot be unambig-
uously defined in a nonparallel flow. Because the
eigenfunctions are functions of y as well as x,
Saric and Nayfeh (1977) note that stable flows may
be termed unstable and vice versa. Saric and
Nayfeh (1977) discussed in great detail the differ-
ent possible definitions of the growth rate and
compared these definitions with all available exper-
imental data for the Blasius flow. They found that
all the experimental data (neutral curves or growth
rates) obtained at the values of n for which |u|
has a maxima can be correlated with the nonparallel
results if the growth rate is defined as in (55).
For the heated liquid problem, we arrived at the
same conclusion. Including the distortion of the
eigenfunction with the streamwise position, the
definition, (56), underpredicts the growth by
large amounts.
Discussion
G. CHAHINE and D. H. FRUMAN
The question of whether polymer solutions
affect cavitation inception through changes of the
flow structure or through the inhibition of bubble
growth has been the subject of much controversy.
In this excellent paper the authors seem to adhere
to the first school of thought and disregard the
second. We think that there is ample evidence of
the profound flow changes introduced by the ejected
polymers to support, at least partially, their
contention. However, evidence also exists showing
that the onset of acoustically generated cavitation
is delayed by the presence of minute amounts of
polymers and asbestos fibers [Hoyt (1977)]. also,
in investigating the behavior of spark-generated
bubbles in the vicinity of a solid wall, the dis-
cussers have observed significant changes being
promoted by the presence of the polymers.
Figure 1 shows the geometric dimensions that
have been considered in the analysis of the bubble
behavior. The displacement of point A, where the
re-entering jet originates, divided by the maximum
lateral dimension of the bubble, Rumax, is plotted
in Figure 2 as a function of the dimensionless time
parameter, t/tp, and the parameter, n, which is the
ratio between Rumax and £, the distance between the
center of the spherical initial bubble and the wall.
As shown, the polymer solution has a retarding
A
FIGURE 1
lo}
o oO
Y—e1 1
Se
Le
oA
{eo
lee
a
1 4 1 4 Hono n
effect on the re-entering jet. This effect in-
creases with increasing n [Chahine and Fruman
(1979)]. Together with results shown in Hoyt,
our data further confirm that, in the absence of
flow, bubble behavior is affected by the intrinsic
properties of dilute polymer solutions.
4
4
4
4
Ra ]
Bs A= A 1
1s rae ‘A
er,
Le mR 1
Ma fo) © SS
~ Wi
RUN | ) | LIQUID
6 PY59|0.50| DISTILLED WATER eo \a 4
; 4 PY 71/0.56) POLYOX 250 PPMW Ge Wye
3-4 @ IBY, GBH 139) ies eee ae: aah 4
r © PY641/1. 25] DISTILLED WATER 7 1
IP t/Te.
AO nn Fi CA ere eer rer ell eee a ; SES |
ie) 0,5 1 1,5 2 2,5
FIGURE 2
REFERENCES
Hoyt, J. W., (1977).
tions and Fiber Suspensions.
phase Flow Forum, ASME.
Chahine, G. L., and D. H. Fruman (1979). Dilute
Polymer Solution Effects on Bubble Growth and Col-
lapse. Physics of Flutds.
Cavitation in Polymer Solu-
Cavitatton and Poly-
112
Authors’ Reply
EDWARD M. GATES and ALLAN J. ACOSTA
Messrs. Chahine and Fruman have raised the
question of the relative importance of polymer-
induced changes in bubble growth versus induced
changes in the flow structure with regard to the
suppression of cavitation. Although both experi-
mental [Ellis and Ting (1970); Chahine and Fruman!
(1979)] and theoretical [Street (1968); Fogler and
Goddard (1970)] work demonstrate that in "no-flow"
situations the growth and collapse rates in polymer
solutions are different than those in pure water,
the magnitude and sense (Street predicts an in-
crease in bubble growth rate) of the changes are
open to question. On the other hand, the results
of Hoyt (1976), Brennen (1970), van der Meulen
(1976), and the present work show drag-reducing
polymers have a very dramatic effect upon the flow
structure in jets and axisymmetric bodies. The
authors believe that in the present work the influ-
ence of these profound flow alterations predominate
over any influence of modified bubble dynamics as
nicely shown by them as evidenced by the following
observations:
First, it was observed in the LIWT that cavi-
tation inception on the non-separating Schiebe body
was not influenced by viscous considerations and
was of the travelling bubble type. In this situa-
tion we would expect that if the polymer effect
upon bubble dynamics was significant, it should be
well illustrated under these circumstances. How-
ever, we (like van der Meulen) observed no change
in either the cavitation index or the appearance
of the cavitation at inception. Second, on the
hemisphere nose and NSRDC bodies a similarly large
suppression of the inception index was obtained by
Arakeri and Acosta (1976) through the elimination
of the laminar separation by a mechanical boundary
layer trip - a situation for which there is no
change of bubble dynamics.
From these observations we infer that the in-
fluence of the polymer on cavitation inception is
dominated by changes in the flow structure rather
than modified bubble dynamics. However, in "non-
flow" sitations it must be assumed that modified
bubble dynamics are responsible for the observed
changes and the work of Messrs. Chahine and Fruman
is a useful addition to this area of study.
REFERENCES
1 See Reference from Chahine and Fruman discussion.
Street, J. R., (1968). The Rheology of Phase
Growth in Elastic Liquids. Trans. Soc. Rheol.,
12, ps LOS.
Fogler, H. S., and J. D. Goddard (1970). Collapse
of Spherical Cavities in Viscoelastic Fluids.
Phystes of Flutds, 13, (5), pp. 1135-1141.
Hoyt, J. W., (1976). Effect of Polymer Additives
on Jet Cavitation. Journal of Fluids Engineering,
Trans. ASME, March, pp. 106-112
Session IIT
SHIP BOUNDARY LAYERS
AND
PROPELLER HULL INTERACTION
PETER N. JOUBERT
Session Chairman
University of Melbourne
Melbourne, Australia
wh
A raae
Calculation of Thick Boundary Layer and
Near Wake of Bodies of
Revolution by a Differential Method
Wa Go
Iowa City, Iowa
ABSTRACT
The differential equations of the thick axisymmetric
turbulent boundary layer and wake are solved using
a finite-difference method. The equations include
longitudinal and transverse surface curvature terms
as well as the static-pressure variation across the
boundary layer and wake. Closure of the mean-flow
equations is affected by a rate equation for the
Reynolds stress deduced from the turbulent kinetic-
energy equation. The results of the method are
compared with the two sets of data obtained at the
Towa Institute of Hydraulic Research from experi-
ments in the tail region of a modified spheroid
and low-drag body of revolution, and also with the
predictions of a simple integral approach proposed
earlier. It is shown that the differential approach
is superior, provided due account is taken of the
normal pressure variation and the direct influence
of the extra rates of strain, associated with the
longitudinal and transverse surface curvatures, on
the length scale of the turbulence.
1. INTRODUCTION
In the absence of flow separation, the boundary
layer on a pointed-tailed body of revolution con-
tinues to grow in thickness up to the tail. Over
the rear quarter of the length of a typical body,
the boundary layer thickness becomes large enough
to invalidate the assumptions of conventional thin
boundary-layer theory. The measurements of Patel,
Nakayama, and Damian (1974) on a modified spheroid
as well as those of Patel and Lee (1977) on a low-
drag body indicate that the breakdown of thin bound-
ary layer approximations is manifested by several
concurrent flow features, namely (a) the boundary
layer thickness is no longer small compared with
the local transverse and longitudinal radii of sur-
- face curvature, (b) the velocity component normal
to the wall is not small, (c) the pressure is not
115
Patel and Y. T. Lee
The University of Iowa
constant across the boundary layer, and (d) the
pressure distribution on the body surface does not
conform with that predicted by potential flow theory,
as a consequence of the interaction between the
thick boundary layer and the external inviscid flow.
These features have been recognized in the develop-
ment of the simple integral method of Patel (1974)
for the calculation of a thick axisymmetric bound-
ary layer, and later on, in the formulation of the
interaction scheme of Nakayama, Patel, and Landweber
(1976a,b) which attempted to couple the boundary
layer, the near wake and the external inviscid flow
by means of successive iterations. Although the
overall iteration scheme proved to be quite success-~
ful, the treatment of the boundary layer using the
integral method, and particularly its extension to
calculate the near wake, required many assumptions
which remain untested. The purpose of the present
work was therefore to develop a more rational pro-
cedure in which the differential equations of the
thick boundary layer and the near wake are solved
by means of a numerical method, since it appeared
that such a procedure would provide not only a
more reliable vehicle for the extension of the
boundary layer solution into the wake, but also
yield the detailed information on the velocity
profiles required for the interaction calculations.
This paper describes the new differential method
and evaluates its performance relative to the inte-
gral method as well as the available experimental
information.
2. DIFFERENTIAL EQUATIONS AND TURBULENCE MODEL
In the (x,y,¢) coordinate system shown in Figure 1,
x and y are distances measured along and normal to
the body surface, respectively, and ¢ is the azi-
muthal angle. As shown by Patel (1973) and Nakayama,
Patel, and Landweber (1976b), the momentum equa-
tions of a thick axisymmetric turbulent boundary
layer may be written
116
FIGURE 1. Coordinate system and notation.
hyrt
We Oe US pee i) Se Or (ee
hy ox oy hy phy ox rhy oy p
(1)
U_ ov OV Kees} 1 op
— —+V — - — UL +=—- 2 =0 (2)
hy, ox oy hy po oy
and the continuity equation is
a a ¥
53 (Ur) + ay (xh,V) = 0 (3)
U and V are the components of mean velocity in the
x and y directions, respectively; h, = 1 + ky, Kk
being the longitudinal surface curvature; T = —puv
+ u dU/dy, where p is density, wu is viscosity and
-puv is the Reynolds stress; r = r, + y cos 8 is the
radial distance measured from the body axis, 8
being the angle between the tangent to the surface
and the axis of the body; and p is the static pres-
sure. These equations resulted from order of mag-
nitude considerations and an examination of the
data from the modified spheroid experiments of
Patel, Nakayama, and Damian (1974). Specifically,
from Eq. (2) we note that the static pressure varies
across the boundary layer and that the gradient of
the pressure in the direction normal to the surface
is associated primarily with the curvature of the
mean streamlines.
Equations (1), (2), and (3) also apply to the
wake, with k = 0 and 0 = O (i-.e., r = y)- In place
of the no-slip boundary conditions on the body sur-
face, however, the conditions on the wake center-
line are dU/dy = O and t = O.
If the Reynolds stress is determined by a one-
equation model using the turbulent kinetic-energy
equation, as proposed by Bradshaw, Ferriss, and
Atwell (1967), then the appropriate closure equa-
tion for the flow outside the viscous sublayer and
the blending zone is
1 U oT chs ou
Daa) Naan ox a= ; {2 7 “)
- T 3/2
Hid T max AT,
+ = — = =
Pighiet, Aaa/ow mp he Tw 2 e
aj p
where a, is a constant (=0.15), G is a diffusion
function and 2 is a length-scale function identified
with the usual mixing length. G and % are assumed
to be universal functions of y/é, where 6 is the
boundary layer thickness. The particular forms of
these functions proposed by Bradshaw et al. (1967)
for a thin boundary layer have gained wide accep-
tance and have proved adequate for the prediction
of a variety of boundary layers developing under
the influence of different pressure gradients and
upstream history. In the adoption of this closure
model for the treatment of thick boundary layers
and wakes, however, it is necessary to consider the
influence of transverse and longitudinal surface
curvatures on the turbulence.
Figure 2 shows the conventional transverse and
longitudinal curvature parameters for the modified
spheroid and low-drag body [Patel and Lee (1977) ].
The ratio of the boundary-layer thickness to the
transverse radius of curvature, 6/ro, is seen to be
more than twice as large in the latter case as in
the former. In both cases, however, 6/ro is less
than 0.4 up to X/L = 0.75, so that the boundary
layers may be regarded as thin up to that station.
Over the rear one-quarter of the body length, the
influence of transverse curvature would prevail
not only through the geometrical terms in the mo-
mentum and continuity equations but also through
any direct effect on the turbulence. The precise
nature of the latter is not known at the present
time since the turbulence is also affected by the
longitudinal curvature of the streamlines associated
with the curvature of the surface as well as the
curvature induced by the rapid thickening of the
boundary layer over the tail.
The longitudinal surface curvature parameter ké
is seen to be quite different for the two bodies.
In the case of the modified spheroid, the curvature
is convex up to X/L = 0.933 and zero thereafter,
while that of the low-drag body is initially convex
and becomes concave for X/L > 0.772. Several
recent studies with nominally two-dimensional thin
© Modified Spheroid
& Low-Orag Body
Ly
0.4 Os 0.6 0.7 0.8 0.9 10
X/L
FIGURE 2. Ratios of boundary-layer thickness to the
longitudinal and transverse radii of surface curvature.
turbulent boundary layers [Bradshaw (1969, 1973),
So and Mellor (1972, 1973, 1975), Meroney and
Bradshaw (1975); Ramaprian and Shivaprasad (1977);
Shivaprasad and Ramaprian (1977)] have indicated
that even mild (ké~0.01) longitudinal surface
curvature exerts a dramatic influence on the turbu-
lence structure. In particular, it is noted that
quantities such as the mixing length 2, the struc-
ture parameter a, = -uw/q2 and the shear-stress
correlation coefficient uv/(Vva- vv“) are influenced
markedly, and experiments indicate that convex
streamline curvature leads to a reduction in these,
whereas concave curvature has an opposite effect.
The turbulence measurements on the modified spheroid
and the low-drag body appear to confirm these ob-
servatons although the relative influence of longi-
tudinal streamline curvature and transverse surface
curvature could not be separated readily.
Bradshaw (1973) has argued that whenever a thin
turbulent shear layer experiences an extra rate of
strain, i.e., in addition to the usual dU/dy, the
response of the turbulence parameters is an order
of magnitude greater than one would expect from
an observation of the appropriate extra terms in
the mean-flow equations of momentum and continuity.
For THIN shear layers and SMALL extra rates of
strain he proposed a simple linear correction for
the length scale of the turbulence, viz.
Be a
= U/oy
(5)
where £, is the length scale with the usual rate
of strain, dU/dy, & is the length scale with the
extra rate of strain, e, and a is a constant of
the order of 10. For the axisymmetric boundary
layer being considered here, there are two extra
rates of strain:
Pcs! (6)
ns TP icy
due to the longitudinal curvature, and
fy oe es ot a)
ies
due to the convergence or divergence of the stream-
lines (in planes parallel to the surface) associated
with the changes in the transverse curvature. The
former is a shearing strain while the latter is a
plain strain, and it is not certain whether the
two effects can be added simply in using Eq. (5)
as recommended by Bradshaw (1973). If this is the
case, however, we would expect a greater reduction
in £ in the tail region of the modified spheroid,
where kK is positive and dr,/dx is negative, than
on the low-drag body, where kK becomes negative and
would therefore tend to offset the influence of
the negative dr,/dx. Although the available data
appear to bear this out to some extent, a direct
comparison between Eqs. (5), (6), and (7) and the
data was not attempted, especially in view of
Bradshaw's [Bradshaw and Unsworth (1976) ] assertion
that Eq. (5) should be used in conjunction with a
simple rate equation which accounts for the up-
stream extra rate-of-strain history. He proposes
ae
Me eft :
eS & SS S=
Lo dU/dy So
-and
117
ie yh Sa ese (9)
dx eff 106
where e is the actual rate of strain, eef¢ is its
effective value and 106 represents the "lag length"
over which the boundary layer responds to a change
ine. In order to determine the merit of this
proposal, it is of course necessary to incorporate
it in an actual calculation and make a comparison
between the predictons and measurement. Such an
attempt has been made here.
The functions £2, and G used in the present study
are shown in Figure 3. For the wake calculation,
the linear variation of 2, in the wall region is
replaced by the constant value of 0.09, as shown
by the dotted line in the figure. The local dis-
tribution of the length scale, %, is thus given by
Eqs. (6) through (9) while the diffusion function,
G, and the structure parameter, a ,, retain their
thin-boundary-layer values.
3. SOLUTION OF THE DIFFERENTIAL EQUATIONS
A numerical method available for the solution of
equations corresponding to (1), (3), and (4) for
a thin two-dimensional boundary layer was modified
to introduce the longitudinal- and transverse-
curvature terms. Instead of incorporating the y-
momentum, Eq. (2), into the solution procedure,
however, changes were made such that a prescribed
variation, across the boundary layer, of the pres-
sure gradient dp/dx could be used. This implies
that the pressure field is known a priori. The
solution of Eqs. (1), (3), and (4) together with
Eqs. (6), (7), (8), and (9) can then be obtained
through step-by-step integration by marching down-
stream from some initial station where the velocity
and shear-stress profiles are prescribed. A
staggered mesh, explicit numerical scheme, similar
to that used by Nash (1969), was used to integrate
the equations in the domain between the first mesh
point away from the surface (or the wake center-
line) to some distance, typically 1.25 6, outside
the boundary layer and the wake. The fifteen mesh
points across the boundary layer are distributed
Lap 1
Fi LOF aT
75 f, (Wake)
C0) Sg RS ee
6 08 5|
(G)y=8
06
04
0.2
FIGURE 3. Distributions of empirical functions, 29
and G.
118
non-uniformly to provide a greater concentration
near the wall and the wake centerline. Instead of
carrying out the integration of the equations up
to the wall, i.e., through the viscous sublayer
and the blending zone, the numerical solution at
the first mesh point, located in the fully turbulent
part of the boundary layer, is matched to the wall
using the law of the wall. In the extension of the
method to the wake, the matching between the first
mesh point and the wake centerline is accomplished
by using the conditions 3U/dy = O and t = O on the
centerline. The main differences between the
boundary layer and wake calculation procedures are
therefore the treatment of the flow between the
first mesh point and the wall or the wake center-
line, and the change in 2, at the tail. Note that
the local value of £2 in the boundary layer as well
as the wake is different from 2, due to the lag,
Eq. (8). The length scale recovers the reference
distribution % 4 asymptotically in the far wake.
Since the near wake data from the low-drag body
indicated that most of the adjustment from the
boundary layer to the far wake is accomplished over
roughly five initial wake thicknesses, the lag
length for the wake calculation was taken to be
5 6, rather than 10 6 used for the boundary-layer
calculation on the basis of Bradshaw's (1973) sug-
gestion. Since the extra rates of strain vanish
at the tail (k = 0, dr j/dx = 0), the length scale
approaches the £2, distribution at about five wake
radii downstream of the tail.
Preliminary calculations performed with the dif-
ferential method described above quickly indicated
that the extra rates of strain in both experiments
were much larger than those examined by Bradshaw
(1973) in support of the linear length-scale
correction formula of Eq. (8). In fact, the use
of the linear formula led to a rapid decrease in 2
and indicated almost total destruction of the
Reynolds stress across the boundary layer in the
tail region and the near wake. In view of this,
recourse was made to a non-linear correction formula
in the form
ae
L eff,- 1
— = 1- —
Qo { DU/ay? (8a)
which reduces to the linear one, Eq. (8), for
small extra rates of strain. Equations (1), (3),
and (4), together with (6), (7), (8a), and (9),
were then solved with the following inputs:
A: the measured wall pressure distribution Cow
(i-e., no normal pressure variation) and
L(y/8) = &o(y/8)
B: the measured Cpy with %£(y/s) corrected for
only the longitudinal curvature (e = eg)
C: the measured Cpy with 2(y/6) corrected for
only the streamline convergence (e = e;)
D: as above, but with e = Cn sr Gs
E: using e = e€, + ey in Eqs. (8a) and (9), and
a variable dp/dx across the boundary layer
evaluated by assuming a linear variation in
p from y = 0 to y = 6 and using the measured
values of Cpw, Cpg and 6.
Thus, case A corresponds to an axisymmetric bound-
ary layer with thin, two-dimensional boundary-layer
physics. The other cases enable the evaluation of
the relative influence of the extra rates of strain
as well as the static pressure variation through
the boundary layer. The calculations were started
with the velocity and shear-stress profiles mea-
sured at X/L = 0.662 on the modified spheroid and
at X/L = 0.601 on the low-drag body.
4. COMPARISONS WITH EXPERIMENT
The major results of the calculations are summarized
in Figure 4(a-k) for the low-drag body and in
Figure 5(a-h) for the modified spheroid. However,
in the latter case the calculations are restricted
to the boundary layer since detailed measurements
were not made in the wake. Both figures contain
comparisons between the experimental and calculated
velocity, shear-stress, and mixing-length profiles
at a few representative axial stations as well as
the development of the integral parameters, 62, Ad,
H, H, and Cg, with axial distance. These parameters
are defined by
Seis “en sae PE, BeBe GO)
0 U OU
6 6
A, = fi (ay = =) ray, Ao = ie cz (il > =) xash7,
U
6 6 6
H = Aj/Ad2 (11)
and
tw
Ce = (12)
4pU>
6
Where Us is the velocity component at the edge of
the boundary layer and wake (y = 6), tangent to the
body surface for the boundary layer and parallel
fe) %,
a ~ UV y2
e Yu,
~--= (A) Cg¢¢ =O.
—— a ((E) e-e,+ ey,
. op
Linear Vax
Vy
io) 0.0005 0.0010 0.0015 0.0020
-uv
/y2
FIGURE 4(a). Comparison of measurements with the solu-
tion of the differential equations, low-drag body. Ve-
locity and shear stress profiles at X/L = 0.920.
[ U a, U u— T T T T T aT I
fo) 7Uo fo) %,
0.1000} a la 4 ~< Yo
72 f 4 TUN y2
V, 0.0750
O'My. Ot |
sae A |
0.0875 + (A)@ eff = 0 7777 (A) eee =O
—-—(B)e=e, 0.0625 4 —— (=) GOO, 4
— = 0) 6 8 ; ap
0.0750+ t 4 Linear 7 9x
—-—(D) e=e,+e;
0.0500 a
=e,+ op,
Me (E)e ey 4, Linear 9P/,,
ie 4 Wy
0.0375 4
0.0250 a}
0.0125 4
4
1 1
(0) . 1.0 1.2
di Q V,
Yu5» “Up
L 4 n 4 4
fo) 0.0005 0.0010 0.0015 0.0020
TUNZ 2
t tl fe it i FIGURE 4(c). Velocity and shear stress profiles at
(0) 0.0005 0.0010 0.0015 0.0020 X/L = 1.000.
- Uy,
u 72
FIGURE 4(b). Velocity and shear stress profiles at
X/L = 0.960.
T T maT T T T
T T T T T Q
°
Oo Vy Us
= 0.0750 a uv y2 |
0.0750 4 T UV 2 a °
°
nj
e Wy e 70,
°
—--— (A) Cg¢¢ = 0 fe)
=-~~ (A) egg = 0 ° 0.0625 ct ° |
0.06254, ° . (E) e=eyt &, 0
Trimm Say ESO 2 Linear 9P,
tines® aP/ax g inear 7 ax
eg 0.0500
(o) 4
° yy
yy FS | 7
J 0.0375 7
0.0250 |
|
0.0125 4
(e) JJ]
“ 12
a 10 1.2
Uc" /u n 4 4 4
< : 1 : 4 0.0005 00010 00015 00020
ie) 0.0005 0.0010 0.0015 0.0020 -uy, -
oy, U
7u3 ‘
FIGURE 4(d). Velocity and shear stress profiles at FIGURE 4(e). Velocity and shear stress profiles at
X7/L = 1.06. X/L, = 1.20.
ELS)
120
= T eal T T Sa
Q
° /U,
00750 4 TONE ]
Vv
O 7Uo
———— (A) Cpg¢= 0 °
0.0625 Gi ;
—— (E) e= e+ e,,
Linear 9P/9,
0.0500 4
MA
0.0375 4
0.0250 4
0.0125 4
fo} 1 di Sl
a OR, 08 1.0 12
uy 0
L 4 ——i 4 4
{e) 0.0005 0.0010 0.0015 0.0020
—uy,
7u2
FIGURE 4(f) .
Velocity and shear stress profiles at
X/L = 2.472.
=r T T T T = T T te
0.125+ 4
Asymp. Value
I © Experimental 8 2
o.100- = \ === (A) e44=0 2 |
. toy, —-— (D) e=ey +e, -
X\ —— (e)e=ey +e, Linear 9M, '
, 0.075+ J
0
Ke
8 fe}
i x
0.050; al
0,025+ 4
|
|
°
Co}
05 are
FIGURE 4(h). Boundary layer and wake thickness.
| ——4 : : ro i
es Asymp. Value
2i2 hasan ye 4
} Experimental
aroma
|
| SIM Gy 3O x
eff
ech —-— (dD) e=ep+e, - 1
—— (E)e-e, +e, |
18+ Linear 9P/, |
HLH
16+ 4
14+ |
+
ler tq
8
10+ J = —) n 1 n 1 sf.
OS NOG IO; 7s OG NLO!S 10 i 12 3 147 25
FIGURE 4(3)
Shape parameters.
X/L
ol © 0.601 |
vy 0.920
a 0.960
x 1.000
Colculation, e-ey+e,, Lineor my,
0.10
0,08
0.06
0.04 F
0.02
FIGURE 4(g) .
[ T =a T ——T, To
0.16 Le 5
IV 2 106
o 120 |
O.14F x 247
Calculation, e = ey +e.
oz} * Linear 9%, J
Mixing length profiles.
T T T T T T T T fH
Asymp. Value
L a 8,
eS 2 } Experimental 7
QO! Bs
== (A) €g¢¢=0 x
—-— (D) e=-ey +e -
20F a © ( Nee 4
3, o —— (Eye=ey+e,, \
TS Linear P73,
10
USE
es
Lz
G10)
lo
if °
o.5-
G2
005
FIGURE 4(1i).
Planar and axisymmetric momentum deficits.
Ty T T T
L i x,Q Preston Tubes |
OES (1.651 mm, 0.711 mm)
6 fe} Clauser Plot
g —=-—= (A) Gg4¢= 0
0.004 + Tp Gee Sy >I
(E)e= ept+e,,
Linear OM
0.003 + DE al]
Cr
0.002 + 4
Oooltf- =|
0 N avs a 1 i
0.5 0.6 0.7 0.8 0.9 ie)
“i
FIGURE 4(k). Wall shear stress.
y
A
0.0250 +
0.0125
10) 0.2 0.4 0.6 0.8 1.0 1.2
Q V,
Aso Ms
\ 1 fl L j
ie) 0.0005 0.0010 0.0015 0.0020
- UV,
40
FIGURE 5. Comparison of measurements with the solution
° My,
a —UuV 2
“U2
e Yul,
a UN Gas = ©)
° °
—— (£) e=eytey,
Linear 9P/9, i
e °
e °
of the differential equations, modified spheroid.
(a) Velocity and shear stress profiles at X/L = 0.930.
0.0750-
0.0625+
0.0500
T T >is T aa T
fo) Hy, |
a -UVs 2 °
Vv °
e 70, °
--- (A) eef¢ = 0 ° 4
— (E) e=e, +e, 6
Linear 9P/5,
|
'
1
I
0.0375 | J
y
AL
0.0250 + 4
0.0125 4
4
{e) 1.0 1.2
—aee |
(0) 0.0005 0.0010 0.0015 0.0020
TON
FIGURE 5(c). Velocity and shear stress profiles at
X/L = 0.990.
T T Q “1 T = T 3
0.0750+ oO, |
oO
a -UV, 2
°
Vv
e My,
OCGIs | | ae (Ale eff = O 4
Eee Ee
4 A (B) ey
—--—(C)e=e,
ie)
X/L = 0.960.
a
0.0250 e
0.0125 4
at
ie) 1.2
lo) 0.0005 0.0010 0.0015 0.0020
- Uy,
/u2
FIGURE 5(b). Velocity and shear stress profiles at
0.12
0.10
0.08
0,06
0.04
0.02
XP
0.622
0.960
0.990
Calculation, e=@¢+@,, Linear om,
xogqgoao
0.930 4
FIGURE 5(d).
Mixing length profiles.
121
122
= T p= T ao
0100+ ——== (A) Cee = 0
—-— (D)e=ep +e,
(E)e=e 9 +ey,
=== : ap
— Linear “'/
0.075 oe Ox
TosL
8
7
t o.0s0b
0.025-
i lL te
0.5 06
FIGURE 5(e). Boundary layer thickness.
A H
H
Bal © 5
Ses WW) Gage O
<a> (0) e= ep +e,
2.07 (eye =Op tes, Linear OM, 7
Or 4
AH is
16) =|
Lae 4
ea =
1.0 l | n i a
0.5 0.6 0.7 08 0.9 10
X71.
FIGURE 5(g). Shape parameter.
to the axis for the wake. In the interest of clar-
ity, the results of all the calculations (cases A
through E) are shown only at one axial station
(Figure 4b and 5b), those at other stations being
qualitatively similar.
Considering the most detailed figures, 4b and
5b, first, it is clear that the predictions are
rather poor when the length scale, &, is assumed
to be the same as that in a thin boundary layer
(case A). This is particularly evident in the pre-
diction of the shear-stress profiles across the
boundary layer and the near wake. Incorporation
of the correction to & to account for the extra
rate of strain due to longitudinal curvature (case
B) leads to a marginal improvement in the case of
the low-drag body and a dramatic improvement for
the modified spheroid. This is to be expected in
view of the grossly different surface curvature
histories of the two bodies as noted earlier
(se = Sa ina T To
a 8)
(le AN
720) )—, 2 =|
SA) Cort =O
et am ORC CIE fe)
(E)e=ep+e,,
= 1S) = £ 4
8 Linear Op
L
(x10)
Qo lO
ue
(x10)
OS) |r
(o) rt
O5 0.6
FIGURE 5(f). Planar and axisymmetric momentum deficits.
T T lin erarlies
0.005 a Preston Tube
| O° Clouser Plot
Sa == WN) Gaggs ©
—-— (D)e=e, +e,
O00)
(Eve=e) +e),
Linear oy,
0.003
Cr
HNO |
0.001
(e)
o5 06 0.7 08 0.9 1.0
ATi
FIGURE 5(h). Wall shear stress.
(Figure 2). Nevertheless, it is clear that this
correction by itself is not sufficient to account
for the differences between the data and the calcu-
lations with thin boundary-layer turbulence models
(case A). The application of the correction for
the extra rate of strain due to the transverse
curvature (case C) appears to account for a major
portion of these differences for both bodies. The
influence of transverse curvature is in fact seen
to be somewhat larger for the low-drag body as
would be expected from the fact that 6/ro is greater
in that case (Figure 2). The simple addition of
the effects of the two rates of strain (case D)
leads to a significant improvement in the prediction
of both the velocity profiles and the shear stress
profiles. The incorporation of a variable pressure
gradient across the boundary layer (case E), which
is an attempt to account for the normal pressure
gradients, appears to make a significant improve-
ment in the prediction of the velocity profile in
the case of the modified spheroid, but its influence
is small, and confined to the outer part of the
boundary layer, in the case of the low-drag body
Examination of the velocity and shear-stress
profiles at several axial stations shown in Figures
4a-f and 5a-c suggests that the incorporation of
the non-linear length-scale correction of Eq. (8a),
the associated rate Eq. (9) and the static-pressure
variation in the equations of the thick boundary
layer, which already include the direct longitudinal
and transverse curvature terms, leads to satis-
factory overall agreement with the data for both
bodies. It is particularly noteworthy that the
velocity and shear stress distributions in the
far wake (X/L = 2.472) of the low-drag body are
predicted with good accuracy. The level of
agreement can obviously be improved further by
appropriate modifications in the empirical functions
in the turbulent kinetic-energy equation and changes
in the lag-length used in the length-scale equation.
The predictions of the shear stress profiles are
consistent with those of the mixing-length distri-
butions shown in Figures 4g and 5e insofar as lower
shear stresses correspond to an over correction in
the mixing length. These comparisons provide
further insight into the manner in which the length
scale must be modified to improve the correlation
between the calculation method and experiment. It
is apparent that the consistent discrepancy between
the calculated and measured velocity and shear-
stress profiles near the outer edge of the boundary
layer and wake stems from a poor representation of
the length scale distribution.
It is interesting to note that, for both bodies
the calculation precedure predicts normal components
of mean velocity which are of the same order of
Magnitude as those measured. The relatively close
agreement between the predictions and experiment
for both components of velocity is perhaps a good
indication of the axial symmetry achieved in the
experiments. The large values of the normal veloc-
ity and the influence of static pressure variation
noted above would appear to indicate that incorpora-
tion of the y-momentum equation in the calculation
procedure would be worthwhile. Note that this has
been avoided in the present calculations by using
the measured pressure distributions at the surface
and the outer edge of the boundary layer.
Finally, the comparisons made in Figures 4 (i-k)
and 5 (e-h) with respect to the integral parameters
show several interesting and consistent features.
It is observed that the prediction of the physical
thickness of the boundary layer and the wake is
insensitive to the changes in 2 as well as the in-
clusion of static pressure variation. The under
estimation of the thickness is associated with the
discrepancy, noted earlier, in the velocity profile
near the outer edge of the boundary layer and wake.
The planar momentum thickness 69 and the momentum-
deficit area A» are also insensitive to changes in
2. The variation of static pressure across the
boundary layer appears to make a small but notice-
able contribution to the development of A» in both
cases. However, it is not large enough to account
for the differences between the calculations and
experiment. The predictions of the shape parameters,
H and H, presented in Figures 4j and 5g, appear-to
be satisfactory, especially in view of the rather
_large scale of the plots. Nevertheless, there is
a systematic difference between the data and the
123
calculation in the tail region and wake of the low-
drag body. As indicated earlier, this can be im-
proved by modifications in the empirical functions
and the lag length. The predictions of the wall
shear stress, shown in Figures 4k and 5h, indicate
that the present method gives acceptable results
for both bodies.
5. COMPARISONS WITH THE INTEGRAL APPROACH
An integral method for the calculation of a thick
axisymmetric boundary layer was described by Patel
(1974) and its extension to the wake was proposed
by Nakayama, Patel, and Landweber (1976b). A few
possible improvements in this method were examined
recently relative to the description of the velocity
profiles in the near wake and these are discussed
by Patel and Lee (1977). The most recent version
of this method has been used here to calculate the
development of the boundary layer and the wake of
the low-drag body in order to assess its performance
relative to the experimental data (which were not
available at the time the method and its extension
were proposed) and the more elaborate differential
method.
The results of the calculations are shown in
Figure 6. It is seen that the performance of the
integral method is comparable with that of the
differential method (compare Figures 4h-k with
6a-d) with respect to the prediction of the bound-
ary layer up to the tail. The prediction of the
near wake is, however, distinctly inferior to that
of the differential method, particular with respect
to the physical thickness 6 and momentum deficit
area Ay. The main conclusion to emerge from these
calculations is that the integral method is capable
of giving a good overall description of the flow
features with considerably less computing effort.
The differential approach is to be preferred, how-
ever, Since it affords the opportunity for further
refinement and gives greater details which may be
necessary for many applications. A more thorough
discussion of the integral method and its short-
comings is given in Patel and Lee (1977).
6. CONCLUSIONS
From the present solutions of the differential
equations, using the (one-equation) turbulent
kinetic-energy model of Bradshaw, Ferriss, and
Atwell (1967), it is clear that methods developed
for thin shear layers cannot be relied upon to pre-
dict the behavior of the thick boundary layer and
wake of a body of revolution. Although these cal-
culations have demonstrated that the boundary-layer
calculation can be readily extended to the wake
and that a fairly satisfactory prediction procedure
can be developed by incorporating ad hoc corrections
to the model for the extra rates of strain, along
the lines recommended by Bradshaw (1973), it is
indeed surprising that such modifications, proposed
originally for small extra rates of strain and thin
shear layers, work so well for the two bodies which
are substantially different in shape. In keeping
with recent trends in the formulation of turbulence
models, one inquires whether thick axisymmetric
boundary layers and near wakes ought to be treated
by the so-called two-equation models. From the
yapid changes in the mixing-length indicated by
124
0.125 4
ze ——_ Integral Method x
0.100 IN ° Experiment 8 7
0.075 + \ x |
oy, ;
ty
AL
ol
0.025 +
FIGURE 6. Comparison of ex- ©)
periments with the solution of 6 |
the integral equations, low- 05
drag body. (a) Boundary layer
and wake thickness-
ere T aa T T =I T Saloon eal
Asymp. Value
Integral Method x
2.5 A Experiment 8.
° Experiment Qo
FIGURE 6(b). Planar and axisymmetric momen-
tum deficits.
22+ ———_ Integral Method x
4 Experiment H
fo} Experiment H
2.0F
18-
H,H
16
ise oy 6)
x
Lab ° =
a
°
Lo 1 i 1 1 1 1 th 1 ~~ +}
05 06 0.7 0.8 0.9 1.0 il 12 13 1.4 2.5
FIGURE 6(c). Shape parameter. ae
Ir Tigra pian bz ll ir
0.005 | x4 Preston Tubes =|
(1.651 mm, 0.711 mm)
6 ° Clauser Plot
Q a Integral
0.004 |- s mm Method =|
°
g
0,003 - 4
0.002 - 4
0.00] | =|
(@) Jt ! 1 i 1
0.5 06 0.7 08 09 1.0
x
ae
FIGURE 6(d). Wall snear stress.
the data, this would appear to be desirable since
it would provide an extra equation for the length-
scale of the turbulence in addition to that for
its intensity. This would also enable the incorpo-
ration of the variations in the structure parameter,
a,, observed in the experiments. However, the
recent work of Launder, Priddin and Sharma (1977)
and Chambers and Wilcox (1977) indicates that even
two-equation models, at least of the type available
at the present time, require further modifications
to account for the extra rates of strain stemming
from such effects as streamline curvature, stream-
line convergence, and rotation, two of which are
present in the case examined here.
In addition to the problem of turbulence models,
the thick boundary layer and the near wake contain
the complication of normal pressure gradients. The
available data show that there exist substantial
variations of static pressure across the boundary
layer. The calculations presented here as well
as those performed with the integral method by Patel
and Lee (1977), suggest that the influence of the
normal pressure gradients on the development of
the boundary layer and the near wake is not negli-
gible although it is masked by the rather major
effects of the transverse and longitudinal surface
curvatures on the turbulence. If normal pressure
variations are to be taken into account in a method
based on the differential equations, it is neces-
sary to include the y-momentum equation in the
solution procedure and regard the pressure as an
additional unknown. This is perhaps best accom-
plished by means of an iterative scheme such as
that proposed by Nakayama, Patel, and Landweber
(1976a,b), although other possibilities can be ex-
plored. In view of the success of the present dif-
ferential method, it is proposed to incorporate
the present method in this iterative scheme, in
place of the integral method, to study the viscous-
inviscid interaction in the tail region in greater
detail. é
The representative calculations presented in
Section 5 demonstrate the overall reliability of
~the simple integral method of Patel (1974) for the
125
prediction of the thick boundary layer. Its ex-
tension to the wake is not altogether satisfactory
and this is attributed largely to the lack of a
systematic procedure for the description of the
velocity profiles in the near wake. This method
is ideally suited, however, for rapid calculations
to determine the state of the boundary layer in the
tail region for certain applications.
ACKNOWLEDGMENTS
This research was carried out under the sponsor-
ship of the Naval Sea Systems Command, General
Hydro-Mechanics Research Program, Sub-project
SRO23 O01 O01, administered by the David W. Taylor
Naval Ship Research and Development Center, Contract
NO0014-75-C-0273. The authors acknowledge the
assistance of Professor B. R. Ramaprian through
several stimulating discussions on the influence
of longitudinal surface curvature on turbulent
boundary layers.
REFERENCES
Bradshaw, P. (1969). The Analogy Between Stream-
line Curvature and Buoyancy in Turbulent Shear
WAV Va | Wie saul) We=yelalg Ska, 7/76
Bradshaw, P. (1973). Effects of Streamline Curva-
ture on Turbulent Flow. AGARDograph No. 169.
Bradshaw, P., and K. Unsworth (1976). Computation
of Complex Turbulent Flows, in Reviews of Viscous
Flows. Proceedings of the Lockheed-Georgia
Company Symposium, 448.
Bradshaw, P., D. H. Ferriss, and N. P. Atwell
(1967). Calculation of Boundary Layer Develop-
ment Using the Turbulent Energy Equation. J.
Fluid Mech. 28, 593.
Chambers), Ls 5Le and) DriGe Witlicox (S77) — Grittical
Examination of Two-Equation Turbulence Closure
Models for Boundary Layers. AIAA Journal 15,
6; 821.
Launder, B. E., C. H. Priddin, and B. I. Sharma
(1977). The Calculation of Turbulent Boundary
Layers on Spinning and Curved Surfaces. ASME,
J. Fluids Engineering 99, No. 1, 231. See also
discussions by P. Bradshaw and G. Mellor, (1977),
ASME, J. Fluids Engineering 99, 2; 435.
Meroney, R. N., and P. Bradshaw (1975). Turbulent
Boundary Layer Growth Over Longitudinally Curved
Surfaces. AIAA Journal 13, 1448.
Nakayama, A., V. C. Patel, and L. Landweber (1976a).
Flow Interaction Near the Tail of a Body of
Revolution, Part I: Flow Exterior to Boundary
Layer and Wake. ASME, J. Fluids Engineering 98,
Sie
Nakayama, A., V. C. Patel, and L. Landweber (1976b) .
Flow Interaction Near the Tail of a Body of
Revolution, Part II: Iterative Solution for
Flow Within and Exterior to Boundary Layer and
Wake. ASME, J. Fluids Engineering 98, 538.
Nash, J. F. (1969). The Calculation of Three-
Dimensional Turbulent Boundary Layers in Incom-
pressible Flow. J. Fluid Mech. 37, 625.
Patel, V. C. (1973). On the Equations of a Thick
Axisymmetric Turbulent Boundary Layer. Towa
Institute of Hydraulic Research, ITHR Report
No. 143.
Patel, V. C. (1974). A Simple Integral Method
for the Calculation of Thick Axisymmetric Tur-
126
bulent Boundary Layers. The Aeronautical
Quarterly 25, 47.
Patel, V. C., and Y. T. Lee (1977). Thick Axisym-
metric Turbulent Boundary Layer and Near Wake of
a Low Drag Body of Revolution. Iowa Institute
of Hydraulic Research, IIHR Report No. 210.
Patel, V. C., A. Nakayama, and R. Damian (1974).
Measurements in the Thick Axisymmetric Turbulent
Boundary Layer Near the Tail of a Body of
Revolution. J. Fluid Mech. 63, 345.
Ramaprian, B. R., and B. G. Shivaprasad (1977).
Mean Flow Measurements in Turbulent Boundary
Layers Along Mildly-Curved Surfaces. AIAA
Journal 15, 189.
Shivaprasad, B. G., and B. R. Ramaprian (1977).
Some Effects of Longitudinal Wall-Curvature
on Turbulent Boundary Layers. Proceedings of
Symposium on Turbulent Shear Flows, Penn State
University, 9.21.
So, R. M. C., and G. L. Mellor (1972). An Experi-
mental Investigation of Turbulent Boundary Layers
Along Curved Surfaces. NASA-CR-1940.
So, R- M. C., and G. L. Mellor (1974). Experiment
on Convex Curvature Effects in Turbulent Bound-
ary Layers. J. Fluid Mech. 60, 43.
So, R. M. C., and G. L. Mellor (1975). Experiment
on Turbulent Boundary Layers on Concave Wall.
The Aeronautical Quarterly 26, 35.
Stern Boundary-Layer Flow on
Axisymmetric Bodies
TA DavHuang, N. Santelilkl, wand! Go Belt
David W. Taylor Naval Ship Research and Development
Center, Bethesda, Maryland
ABSTRACT
Measurements of static pressure distributions, mean
velocity profiles, and distributions of turbulence
intensities and Reynolds stress were made across the
stern boundary-layers on two axisymmetric bodies.
In order to avoid tunnel blockage, the entire after-
body was placed in the open-jet test section of the
DTNSRDC Anechoic Wind Tunnel. The numerical itera-
tion scheme which uses the boundary layer and open
wake displacement body is found to model satisfac-
torily the interaction between the thick stern bound-
ary layer and the external potential flow. The
measured static pressure distributions across the
entire stern boundary layer and the near wake are
predicted well by potential flow computations for
the displacement bodies. The measured distributions
of mean velocity and eddy viscosity over the stern,
except in the tail region (X/L > 0.90), are also
well-predicted when the Douglas CS differential
boundary-layer method is used in conjunction with
the inviscid pressure distribution on the displace-
ment body. However, the measured distributions of
turbulence intensity, eddy viscosity, and mixing-
length parameters in the tail region are found to
be much smaller than those of a thin boundary layer.
An approximate similarity characteristic for the
thick axisymmetric stern boundary layer is obtained
when the mixing-length parameters in the tail region
are normalized by the square-root of the boundary-
layer cross-sectional area instead of the boundary-
layer thickness.
1. INTRODUCTION
Many single-screw ship propellers operate inside of
thick stern boundary layers. An accurate prediction
of velocity inflow to the propeller is essential to
meet the ever-increasing demand for improving pro-
peller performance. Huang et al. (1976) used a
- Laser Doppler Velocimeter (LDV) to measure the ve-
127
locity profiles on axisymmetric models with and
without a propeller in operation. The measured
difference between these velocity profiles has
provided the necessary clues to formulate an inviscid
interaction theory for propellers and thick boundary
layers. An iterative scheme was employed to compute
the velocity profiles of the thick axisymmetric
boundary layer. In this approach, the initial
boundary-layer computation proceeds making use of
the potential-flow pressure distribution on the body
[Hess and Smith (1966)]. The flow calculations are
then repeated for a modified body and wake geometry,
by adding the computed local displacement thickness
as suggested by Preston (1945) and Lighthill (1958).
Potential-flow methods are then used to compute the
pressure distribution around the modified body and
the boundary-layer calculations are repeated using
the new pressure distribution. The basic iterative
scheme is continued until the pressure distributions
on the body from two successive approximations agree
to within a given error criterion (1 percent).
The Douglas CS differential boundary-layer method
[Cebeci and Smith (1974)], modified to properly ac-
acount for the effects of transverse curvature, was
used to calculate the boundary-layer over the axi-
symmetric body. The integral wake relations given
by Granville (1958) were used to calculate the dis-
placement thickness in the wake. In the stern/
near-wake region (0.95 £ x/L £ 1.05), where X is the
axial distance from the nose and L is the total
length, a fifth-degree polynomial was used, with
the constants determined by the condition that the
thickness, slope, and curvature be equal to those
calculated by the boundary-layer method at X/L =
0.95 and by the integral wake relations at X/L =
1.05. Comparison with experimental results of Huang
et al. (1976) show that the potential-flow/boundary-
layer interaction computer program predicts accurate
values of pressure, shear stress, and velocity pro-
files over the forward 90 percent of the bodies,
where the boundary layers are thin compared with
the radii of the bodies. Over the last 10 percent
128
of body length, the measured shear stress and ve-
locity profiles became smaller than those predicted
by the theory. These differences are more notice-
able over the last 5 percent of the body length
where the boundary-layer thicknesses are greater
than the radii of the bodies, especially for fuller
sterns.
In order to examine the thick stern boundary-
layer properties in detail, it is necessary to
measure the distributions of static pressure, tur-
bulence intensities and Reynolds stress across the
thick stern boundary layer. The magnitudes of the
eddy viscosity and the mixing-length parameter were
determined and compared with those obtained for
thin boundary layers. It is found that the eddy
viscosity and the mixing length for thick boundary
layers are smaller than those of thin boundary
layers. An improvement to the Douglas CS differ-
ential method can be made by modifying the mixing-
length model in the tail region. The distributions
of measured static pressure, which were found to be
nonuniform across the thick stern boundary layers
and near wake, can be approximated very well by
potential flow computations for the displacement
bodies. The gross curvature effects of the mean
streamlines on the static pressure distributions
outside the displacement surface are represented
very well by those of the potential-flow stream-
TABLE 1 - Offsets for Model 1
K/L Y/t Y/R X/L
0.0000 9.0000 0.0000 - 2684
0050 -0100 2193 28S)
o0se -0142 3118 - 2883
0149 -0175 3835 - 2982
0199 0202 4441 3082
+0249 -0227 4975 .3181
0208 .0248 5454 ~ 3280
0348 0268 5891 - 3380
0398 -C287 6291 ~3479
9447 -0303 6659 “3979
0497 OSS) 7000 . 3678
05417 0333 7315 SON AMaT|
6596 -0347 7607 . 3877
0646 -0359 7877 .3976
0596 0379 8126 .4076
0746 -0381 8355 -4175
0795 -0390 8567 +4274
0845 -0399 8760 - 4374
0895 -0407 8936 -4473
og44 -0414 9097 -4573
oo94 0421 9241 »4672
-1044 .0427 9371 -4771
1093 0432 9466 -4871
1143 -0437 9587 -4970
1H}93 -0441 9676 -5070
1243 .0444 9752 -5169
1292 .0447 9816 -5268
1342 -0450 9869 “9268
11392 -0452 9912 ~5467
144) -0453 9445 ~5567
1491 -0454 9969 -5666
1541 -90455 9986 .9765
1590 -0448 9836 ~ 5865
1640 .0456 1.0000 -5964
1690 .0456 1.0000 .6064
1740 .0456 1.0000 -6188
1789 .0456 1.0000 .6264
1839 .0456 1.0000 .6378
1889 .0456 1.0000 -6454
1938 .0456 1.0000 - 6567
1988 -0456 1.0000 .6681
2087 -0456 1.0000 ~6757
2187 -0456 1.00C0 -G871
2286 .0456 1.0000 -6984
2356 -0456 1.0000 7060
2485 .0456 1.0000 ~7174
2584 -0456 1.0000 -7250
lines of the fictitious displacement body. Thus,
the nonuniform static pressure distributions across
the thick stern boundary layer can be interpreted
mainly as an inviscid phenomenon and can be assumed
to have little effect on the stern boundary-layer
development.
Two axisymmetric bodies without flow separation,
Afterbodies 1 and 2 of Huang et al. (1976), were
chosen for this investigation. Their geometric
simplicity offers considerable experimental and
computational convenience in treating fundamental
aspects of thick stern boundary layers. Afterbody
1 is a fine convex stern while Afterbody 2 is a
full convex stern.
In the following discussion, the experimental
techniques and geometries of the model are given
in detail. The measurements of mean velocities,
turbulence intensities, and Reynolds stresses were
analyzed to obtain eddy viscosity and mixing length.
The application of the present results to improve-
ment of the accuracy of boundary-layer computations
over the entire stern is outlined.
2. WIND TUNNEL AND MODELS
The experimental investigation was conducted in the
wind tunnel of the DTNSRDC anechoic flow facility.
NA Y/R X/L Y/L Y/R
-0456 1.0000 - 7363 -0427 -9382
-0456 1.0000 -7477 +0421 -9235
0456 1.0000 .7553 -0416 SAY
-0456 1.0000 - 7666 -0408 -8951
0455 1.0000 - 7780 -0399 -8755
0456 1.0000 - 7856 -0392 -8615
0956 1.0000 ESTO: -0382 - 8387
0456 1.0000 -8045 -0375 -8225
0456 1.0000 -8159 -0363 - 7963
0456 1.0000 -8273 0350 SASS)
0456 1.0000 -8349 -0341 - 7477
-0456 1.0000 -B8462 -0326 si NSS)
0456 1.0000 -8576 -0310 - 6807
0456 1.0000 - 8652 -0299 - 6560
0456 1.0000 -8765 -0281 - 6167
0456 1.0000 8841 .0268 -5889
0456 1.0000 -8955 -0248 -5445
9455 1.6000 +9069 +0226 -4970
0456 1.0000 -9144 -0211 - 4633
0456 1.0000 -9245 -0189 ~4147
0456 1.0000 +9344 -0166 3636
0456 1.0000 -9443 -0140 . 3078
0456 1.0000 ~9513 -0122 - 2673
0456 1.0000 9563 -0108 - 2380
0456 1.0000 -9612 -0095 - 2080
0456 1.0000 -9642 .0087 - 1900
0456 1.0000 -9662 -0081 -1778
0456 1.0000 - 9682 -0076 - 1669
0456 1.0000 9692 -0074 -1613
0456 1.0000 -9702 -0072 6 US
0456 1.0000 -9722 -0068 1489
0456 1.0000 -9732 -0066 ~1453
0456 1.0000 OO -0063 - 1393
0456 1.0000 +9771 -0062 1364
0456 1.0000 OPA -0059 +1293
0456 9999) -9811 -0056 -1222
0455 SON, -9831 -0053 +1165
0455 9988 -9851 -0050 1107
0455 9977 -9871 -C048 -1051
0453 9952 -9881 -C046 -1018
0452 g915 «9901 -0043 -0951
0450 9883 +9920 -0040 - 0880
0448 9R23 -9940 -0036 -0782
0444 9748 -9960 -0028 -0625
0441 9690 -9980 -0019 -0413
0437 9588 1.0000 0.0000 9.0000
0433 SSH
The wind tunnel has a 2.44 m by 2.44 m closed-jet
test section, followed by a 7.16 m by 7.16 m open-
jet test section. The length of the open-jet sec-
tion is 6.40 m. The maximum air speed which can
be achieved is 61 m/sec; in the present experiments,
the velocity of the wind tunnel was held constant
at 30.48 m/sec. The measured ambient turbulence
level in the open-jet test section without the model
in place was 0.1 percent. Integration of the mea-
sured noise spectrum levels in the open-jet test
section, over the frequency range of 0 to 10,000 Hz,
indicated that the typical background acoustic
noise at 30.48 m/sec was around 93 db re 0.0002
dyn/cm2. These levels of ambient turbulence and
acoustic noise were considered low enough so as not
to unfavorably affect the measurements of boundary-
layer characteristics.
Two axisymmetric convex afterbodies without
stern separation were used for the present experi-
mental investigation. Their afterbody length/
diameter ratios (La/D) were 4.308 and 2.247. The
detailed offsets for Models 1 and 2 are given in
Tables 1 and 2. Each afterbody was connected to a
parallel middle body of length Ly and an existing
streamlined forebody with a bow-entrance length
diameter ratio (L,/D) of 1.82. The total length
of each model (L) is fixed at a constant value of
3.066 m. The diameter of the parallel middle body
(emp) is 27.94 cm. The common forebody and a
portion of the parallel were constructed of wood.
TABLE 2 - Offsets for Model 2
X/L Y/L Y/R x/L Y/L
0.0000 0.0090 0.0000 . 2684 -0456
.9050 -0100 .2193 . 2783 .0456
.0099 .0142 .3118 .2883 .0456
.0149 0175 . 3836 . 2982 -0456
0199 .0202 4443 . 3082 0456
.0249 .0227 -4975 23181 .0456
-0298 .0249 5455 .3280 -0456
-0348 .0268 -5891 . 3380 .0456
-0398 -0287 -6291 .3479 -0456
.0447 .0303 -6659 .3579 .0456
.0497 -0319 .7000 .3678 .0456
.0547 .0333 .7316 SCUY -0456
.0595 0347 .7606 .3877 .0456
-9646 0359 7877 .3976 -0456
.0696 0370 -8126 .4076 0456
0746 -0381 8357 .4175 -0456
.0795 .0390 .8566 .4274 -0456
.0845 0399 .8761 .4374 -0456
.0895 -0407 .8937 .4473 - 0456
.0944 .0414 .9097 4573 -0456
.0994 -0421 .9241 4672 0456
.1044 .0427 .9372 4774 -0456
-1093 .0432 -9487 .4871 -0456
.1143 .0437 -9588 .4970 -0456
-1193 .0441 -9677 .5C070 .0456
.1243 .0444 -9751 .5169 0456
.1292 -0447 9817 5268 -0456
.1342 -0450 . 9869 .5368 -0456
.1392 .0452 .9913 .5467 0456
1441 0453 .9945 5567 -0456
-1491 .0454 -9969 .5666 -0456
21541 .0455 - 9987 5765 .0456
.1590 .0455 -9996 .5865 -0456
-1640 -0456 1.0000 5964 .0456
.1690 .0456 1.00069 6064 .0456
.1740 -0456 1.0000 6188 -0456
.1789 .0456 1.0000 .6264 -0456
.1839 .0456 1.0000 .6378 -0456
.1889 .0456 1.0000 6454 -0456
-1938 0456 1.0000 6567 -0456
-1988 .0456 1.0000 .6681 0456
. 2087 .0456 1.0000 6757 -0456
.2187 -0456 1.0000 .6871 .0456
.2286 .0456 1.0000 -6984 .0456
. 2386 -0456 1.0000 .7060 -0456
.2485 9456 7%.0000 7174 .0456
- 2584 -045€ 1.0000 .7250 -0456
AAAS)
The afterbody and the remaining portions of the
parallel middle body were constructed of molded
fiberglass; specified profile tolerances were held
to less than +0.4 mm, all imperfections were re-
moved, meridians were faired, and the fiberglass
was polished to a 0.64-micron rms surface finish.
The tail ends of the afterbody were shaped to ac-
commodate the hub of an existing propeller. This
modification caused a considerable change of body
curvature in the region of X/L 2 0.96. However,
as will be seen later, the thicknesses of the
boundary layer in this region are much larger than
the local radii of the body. This deficiency does
not cause serious degradation of boundary-layer
flow at that point.
The model was supported by two streamlined struts
separated by roughly one-third of the model length.
The upstream strut had a 15 cm chord and the down-
stream strut a 3 cm chord. The disturbances gener-
ated by the supporting struts were within the region
below the horizontal centerplane. Prior to the
experiments, pressure taps and Preston tubes were
used to check the axisymmetric characteristics of
the stern flow at X/L 0.90, 0.95, and 0.98. The
circumferential variations of pressure and surface
shear stress on the upper half of the two after-
bodies at these three locations were within two
percent. All the final measurements were made in
each body's vertical centerplane along the upper
meridian where there was little extraneous effect
Y/R X/L Y/L Y/R
1.0000 .7363 -0456 1.0000
1.0000 .7477 -0456 1.0000
1.0000 -7553 -0456 1.0000
1.0000 . 7666 -0456 1.0000
1.0000 .7780 0456 1.0000
1.0000 .7856 -0456 1.0000
1.0000 .7952 0456 1.0000
1.0000 - 8050 .0455 .9996
1.0000 -8147 .0454 .9959
14.0000 .8245 -0450 -9871
1.0000 .8342 -0443 .9723
1.0000 .8459 .0431 .9452
1.0000 . 8556 -0417 £9153
1.0000 .8654 .0400 .8789
1.0000 -8751 .0381 . 8364
1.0000 .8849 .0359 .7881
1.0000 -8946 .0335 -7349
1.0000 .9044 .0309 .6775
1.0000 -9141 .0281 .6162
1.0000 .9239 .0251 .5514
1.0000 .9336 .0220 . 4840
1.0000 .9453 .0182 .3993
1.0000 .9512 sO1G2" a Ses
1.0000 .9570 .0142 Sit
1.0000 .9609 .0128 . 2808
1.0000 - 9648 .0114 .2501
1.0000 .9662 .0109 . 2383
1.0000 . 9682 .0101 .2221
1.0000 .9692 .0098 .2145
1.0000 .9702 -0094 .2055
1.0000 .9722 .0087 .1901
1.0000 .9732 .0083 .1818
1.0000 .9751 .0075 .1654
1.0000 £9771 .0068 .1490
1.0000 .9791 .0060 -1309
1.0000 9811 .0056 i222
1.0000 .9831 -0053 -1165
1.0000 .9351 -0050 -1108
1.0000 -9871 -0048 .1052
1.0000 .9881 .0046 .1019
1.0000 -9901 .0043 .0951
1.0000 +9920 .0040 .0879
1.0000 .9940 .0036 .0781
1.0000 -9960 .0029 . 0626
1.0000 .9980 .0019 .0412
1.0000 1.0000 0.0000 0.00060
1.0000
130
from the supporting strut. One half of the model
length protruded from the closed-jet working sec-
tion of the wind tunnel into the open-jet test
section. The ambient static pressure coefficients
across and along the entire open-jet chamber (7.16
x 7.16 x 6.4 m) were found to vary less than 0.3
percent of dynamic pressure. The tunnel blockage
and the longitudinal pressure gradient along the
tunnel length were almost completely removed by
testing the afterbody in the open-jet test section.
The location of boundary-layer transition from
laminar to turbulent flow was artifically induced
by a 0.61 mm diameter trip wire located at X/L =
0.05. When the flow was probed with a hot-wire,
the trip wire was found to effectively stimulate
the flow at a location 1 cm downstream from the
wire. As a result of the parasitic drag of the
wire, the boundary layer can be theoretically con-
sidered to become turbulent at a virtual origin
upstream of the trip wire. This virtual origin for
the turbulent flow is defined such that the sum of
the laminar frictional drag from the body nose to
the trip wire, the parasitic drag of the trip wire,
and the turbulent frictional drag after the trip
wire equals the sum of the laminar frictional drag
from the nose to the virtual origin and the turbu-
lent frictional drag from the virtual origin to
the after end of the model [McCarthy et al. (1976)].
The location of the virtual origin on the forebody
with a 0.61 mm trip wire at X/L = 0.05 was found
to be at X/L = 0.015 for a length Reynolds number
of 5.9 x 10©. The location of transition in the
mathematical model for the present boundary-layer
calculation is specified at this virtual origin.
The length Reynolds number based on the distance
from the trip wire to the end of the parallel middle
body is larger than for 4 x 10© for the two after-
bodies. It can be assumed that a fully established
axisymmetric turbulent boundary layer exists at the
beginning of the afterbody and that the trip wire
has no peculiar effect on the boundary-layer char-
acteristics of the stern.
3. INSTRUMENTATION
A 1.83-cm Preston tube was taped to the stern at
successively further aft locations in order to
measure the shear stress distribution along the
upper meridian of each stern. The Preston tube
used was calibrated in a 2.54-cm water pipe flow
facility described by Huang and von Kerczek (1972).
Pressure taps (0.8 mm diameter) were used to mea-
sure steady pressures at the same locations as the
Preston tubes. The taps were connected by "Tygon"
plastic tubes to a scanning valve located inside
the model. The output tube from the scanning valve
was run from the model through the supporting strut
to a precision pressure transducer located on the
quiescent floor of the open-jet chamber. The pres-
sure transducer was a Validyne Model DP 15-560 de-
signed for measuring low pressure up to + 1.4 x 10"
dyn/cm? (0.2 psi). The zero-drift, linearity, and
hysteresis of this transducer system were carefully
checked and the overall accuracy was found to be
within 0.5 percent of the dynamic pressure.
A Prandtl type pitot-static pressure probe of
3.125-mm diameter with four equally spaced holes
located at three diameters aft of the nose was used
to measure static pressure across the boundary
layer. The yaw sensitivity of the static pressure
probe was examined by yawing the probe in the free-
stream. It was found that the measured static pres-
sure was insensitive to the probe angle up to 5°
yaw. The response of measured static pressure to
probe angle was nearly a cosine function of yaw
angle for yaw angles less than 15°. The static
pressure probe was aligned parallel to the model
axis for all of the static pressure measurements.
The local angles between the resultant velocity of
the boundary-layer flow and probe axis were found
to be less than 15° (5° for most cases). The maxi-
mum static pressure coefficient in the boundary
layer was less than 0.2. Thus, the error in the
measured static pressure caused by not aligning the
probe with the local flow was less than 0.8 percent
of the dynamic pressure.
The mean axial and radial velocity components
and the Reynolds stress were measured by a TSI, Inc.
Model 1241 "X" wire. The probe elements were 0.05
mm in diameter with a sensing length of 1.0 mn.
The spacing between the two cross elements is 1.0
mm. A two-channel TSI Model 1050-1 hot-wire ane-
mometer and linearizer were used. The "X" wire,
together with temperature compensated probes, were
calibrated at the factory and supplied with their
individual linearization polynomial coefficients.
This eliminated the time-consuming linearization
process. The frequency response of the anemometer
system claimed by the manufacturer is dc to 200 kHz.
Calibration of the "X" wire was made before and after
each set of measurements. It was found that this
hot-wire anemometer system had a 40.5 percent ac-
curacy (40.15 m/s accuracy at the free stream ve-
locity of 30.5 m/sec) during the entire experiment.
The accuracy of cross-flow velocity measurements
by the cross wire was estimated by yawing the cross-—
wire in the free stream. It was found that the ac-
curacy of the measured cross-flow velocities was
about one percent of the free stream velocity.
The linearized signals were fed into a Time/Data
Model 1923-C Real-Time analyzer. Both channels of
analog signal were digitized at a rate of 80 points
per second for ten seconds. These data were imme-
diametely analyzed by a computer code to obtain the
individual components of mean velocity, turbulence
fluctuation, and Reynolds stress on a real time
basis.
A traversing system enclosed in a 15 cm chord,
streamlined strut was used to support both the
static pressure probe and the cross-wire probe. The
traversing system was mounted either on an I-beam
along the axis of the lower floor of the open-jet
chamber or on the ceiling of the closed-jet section.
The combination of these two mounting arrangements
allowed the measurements to be made at any axial
location along the stern and up to 50 percent of
the body length downstream from the aft end of the
body. Positioning of the traversing system was
achieved by manual adjustment in the axial direction
and by remote control in the radial direction. The
total radial traverse of the probe was 25 cm. The
radial position of the probe was monitored by a
potentiometer to with a +0.01 mm accuracy.
4. COMPARISON OF EXPERIMENTAL AND THEORETICAL
RESULTS
In the following, the experimental results for the
thick stern boundary layers are presented and com-
pared with theoretical results. The theories used
in the comparison are the Douglas CS differential
boundary-layer method in conjunction with the dis-
placement body concept. The iteration procedures
for numerical computation are given by Huang et al.
(1976). In this investigation, the displacement
body concept for solving the interaction between
the thick stern boundary layer and potential flow
will be examined and an eddy-viscosity model will
be evaluated.
Measured and Computed Pressure and Shear Stress
Distributions
Significant improvement in the accuracy of measur-
ing surface pressure and shear stress have been made
by using a precision pressure transducer. The
present results are more reliable than the earlier
results of Huang et al. (1976), although the dif-
ferences are small.
The measured and computed values of the pressure
coefficient, Cp = 2(p - Pa) /OUSs. are compared in
Figure 1 for Afterbody 1 and in Figure 2, for After-
body 2; p is the local static pressure, p is the
mass density of the fluid, U, is the free-stream
velocity and po is the ambient pressure (the qui-
escent chamber static pressure of the open-jet sec-
tion). The pressure coefficients computed on the
displacement body were carried radially back to the
hull surface and the radial distribution of pres-
sure at a given axial station was assumed to be a
constant between the hull surface and the fictitious
displacement surface. The maximum error in the
static pressure associated with this assumption is
less than two percent of the dynamic pressure (next
section). The agreement between theory and measure-
ment is excellent for both afterbodies. The results
0.15
Ry = 6.6 x 10°
THEORY
MEASUREMENT
0.10 °
0.65 0.70 0.75 - 0.80
131
suggest that the displacement body concept as used
by Huang et al. (1976) permits accurate computation
of the pressure distribution on the stern.
The measured and computed distributions of local
shear stress, C;, are compared in Figure 3. The
agreement between theory and measurement is
also very good for both afterbodies except for
x/L > 0.95 where the measured values of C, are lower
than the computed values.
Measured and Computed Static Pressure Distribution
The measured and computed static pressure coeffi-
cients for Afterbody 1 are compared in Figure 4 at
various locations across the stern boundary layer
and in Figure 5 for the near wake. Figures 6 and 7
show the comparisons for Afterbody 2. The off-body
option of the Douglas potential-flow computer code
was used to compute the static pressure distribu-
tions off the displacement body. As can be seen in
Figures 4 through 7, the computed static pressure
distributions across the entire stern boundary layer
and near wake mostly agree well with the measured
static pressure distributions. The discrepancy
between the measured and computed values of C, is
in general less than 0.01 which is about the accuracy
of the measurement.
As will be seen later, both displacement bodies
are convex from the parallel middle body up to X/L
= 0.91 and become concave downstream from X/L >
0.91. However, the actual afterbodies are convex
all the way up to X/L = 0.96. The measured values
of C, shown in Figures 4 through 7 increase with
radial distance for X/L < 0.91, indicating that
the mean streamlines are convex; and measured values
of Cp decrease with radial distance for X/L > 0.91,
0.85 0.90 0.95 1.0
X/L
FIGURE 1. Computed and measured stern pressure distribution on afterbody 1.
0.20
= 68 x 10°
THEORY
0.10 MEASUREMENT
p
0.0
—0.10
—0.20
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0
X/L
FIGURE 2. Computed and measured stern pressure distribution on afterbody 2.
0.003
MEASUREMENT
AFTERBODY 1
Ry = 6.6 x 10°
0.002 THEORY
MEASUREMENT
AFTERBODY 2
R, = 6.8 x 10°
THEORY
0.6 0.7 0.8 0.9 1.0
mputed and measured shear stress distribution on afterbodies 1 and 2.
FIGURE 4.
X/L = 0.755
THEORY
X/L = 0.914 DISPLACEMENT
SURFACE
MEASUREMENT
ON THE BODY
Toa ed:
MEASUREMENTS THEORY
v
Computed and measured static pressure distributions across stern boundary layer of afterbody 1.
Tages aes ae | ea
MEASUREMENT THEORY
O
X/L = 1.000
MEASUREMENT
THEORY
DISPLACEMENT SURFACE
FIGURE 5. Computed and measured
static pressure distributions
across near wake of afterbody 1.
134
3.0
° | X/L_ MEASUREMENT THEORY
by 0.846 ° —___
. 0.934 © —_-—
L, | 0.970 4a
in 0.977 Do soa
O 1.000 O
° TENS
y\
2.0 oe)
fo) ANS)
\\
‘ ro
X/L = 0.846 9° sa
r ° \
"max O OY AN
\O
oO WO
ON
QD
ato KS MEASUREMENT RIN AGO
THEORY Ww. ‘Od
DISPLACEMENT XY rat H
X/L = 0.934
SURFACE QUT AN WY
OY
MEASUREMENT 3
CNiecopy oo as eNE
emo
Q, MT?
SON
i]
—0.20 —0.10 0.10 0.20
c
p
FIGURE 6. Computed and measured static pressure distributions across stern boundary layer of afterbody 2.
indicating that the mean streamlines are concave.
Thus, the curvatures of the mean streamlines are
more closely related to the curvatures of the dis-
placement body than the actual body. The close
agreement between the computed and measured static
pressure distributions again supports the displace-
ment body concept for computing the potential flow
outside of the displacement surface.
Yo is the body radius; x is the axial distance; u
and vy, are the mean velocity components respectively
parallel to and normal to the meridan of the body
(s and n directions); v is the kinematic viscosity
of the fluid; Tul is the Reynolds stress; and u!
and vy, are the velocity fluctuations in the s and n
directions respectively. The Douglas CS method as-
sumes that the Reynolds stress depends upon the local
flow parameters only, e.g.,
5. MEASURED AND COMPUTED MEAN VELOCITY PROFILE Sal gpa for! oS nis ne
=ulv' = 5 = (3)
The incompressible steady continuity and momentum Se 6 2 HON iy SS ©
equations for thin axisymmetric turbulent boundary i.
layers are . du.
where ¢, = 22 = gn (eddy viscosity in the inner
d(ru,)/ds + d(xrv,)/dn = 0 (1) fo) region)
and co
€. = 0.0168 y (Ul=u))idn = 00168 UR TOA,
u_du_/3s + v du /dn ° G35 if @ SQ er tere p
Sis Ties,
(2) : é . .
= -dp/pds + d[r(vdu /on)=u'v']/ron (eddy viscosity in the outer region),
s sn r
2 = 0-4 xine) {2 = exp = = in (yy
u,(s,0) = v(s,0) = 0 at n = 0 OS a e
where (mixing-length parameter in the inner
region) ,
r(s,n) = xr _(s,n) + n cosa
fe} Tw —5 :
A = 26 Lira , (Van Driest's damping
a= tan”* (dr, /ax)
factor),
135
6 Us where u,/U, and V1,/U, are computed by the CS method.
oF =f (1 - —)dn, [displacement thickness The potential-flow pressure is assumed to be con-
fo} e (planar definition) ] stant between the body surface and the displacement
; surface and is equal to the pressure Pq computed on
y 2 Ant & BLE (2) Tees (Ghisetisteney the displacement body. The value of U, used in Eqs.
E18 6 Recto) (4) and (5) is equal to vl - Pq and U, is assumed
,
to be parallel to the body surface.
& = 8995 , (boundary-layer thickness), The displacement-body concept can be used to
improve the computed values of u, and We outside
ae (wall sheamistress)), of the displacement Surface of thick boundary layers,
e.g.,
Ue is the potential-flow velocity used in the
boundary-layer calculations, and at y,, ej is equal u(r) u, (n) U_cos(8-a)
to €,. A computer code to solve for the values U ~ U_cos(@-a) U Cese!
us/Ue and vp/Ue has been developed by Cebeci and © P °
Smith (1975) using Keller's numerical box scheme. : (6)
The velocity components measured in the present v,,(n) Smee) ;
investigation are u, and v,, the components in the ~ U sin(6—a) U Sei
axial and the radial directions of the axisymmetric P ©)
body. The computed values of uy and v; are given
5 v_(n) u_(n) U_cos (6-a)
Vi 1g s p
SSS 58 ee a SS “Sting
a, (x) us (n) U, v,,(n) U UR t eeS(O=@) Us
U = U v cosa - yp sina, (4)
fe) e fe} e fe} vy, (™) UL oaeWre)
+ 7 cosa (7)
v(x) a, (n) UL v (n) U umn (=e) U, ,
7 = a — sina + i cosa, (5)
fo) e fo) e [o) where the variation of the inviscid static pressure,
MEASUREMENT THEORY
X/L = 1.000
MEASUREMENT
THEORY
DISPLACEMENT
SURFACE
0 0.05 c 0.10 0.15 0.20
p
PIGURE 7. Computed and measured static pressure distributions across near wake of afterbody 2.
MEASUREMENT
MEASUREMENT
|
X/L
(A) 0.755
(B) 0.846
(C) 0.914
(D) 0.934
X/L
(E) 0.964
(F) 0.977
(G) 1.000
THEORY
(A) X/L = 0.755
me ee )=—DIRECT
IMPROVED
ea
THEORY
O
ed mean
axial
a olig
(E) X/L = 0.964
(F) X/L = 0.977
pas = 0 0.2
U U
o oO
easier OLRECT
— — — — —— So mproveod
and radial velocity
(D) X/L = 0.934
(G) X/L = 1.00
0.4 0.6 0.8 1.0
distributions across stern boundary
C,,(r), across the thick boundary layer is expressed
in terms of the inviscid resultant velocity U
Name) S valve Cp(r)Jand @ is the angle between the
inviscid resultant velocity U, and body axis (0 is
positive when Up is directed away from the axis).
In the first improvement the values u,g/(U,,cos(0-a) )
are taken as the computed values of f' = us/U, in
the CS method with Ue equal to the inviscid resul-
tant velocity on the displacement body. At the edge
boundary layer, the value of ug(U,cos(8-a)) is equal
to 1.0. The value of v,/(U.sin(68-a))is also equal
to 1.0 since the boundary-layer-induced normal ve-
locity is assumed to be equal to the inviscid normal
velocity of the displacement body at that point
(Lighthill (1958)]. The theoretical proof for an
axisymmetric body has not been worked out in the
literature and will not be given here. However,
the validity of the assumption will be borne out
by the experimental measurements of Vy: Therefore,
Eqs. (6) and (7) reduce to the proper limit at the
edge of the boundary layer, e.g.,
u(r=6)) U
TE Kane: yg cost: (8)
fo) °
v.(r=5)) U
uU = T sin6é, (9)
fe}
which are the inviscid axial and radial velocity
components of the displacement body, where Us =
1.4
1.2
1.0
0.8
0.4
0.2
AL ey7)
Vlas Coe (CTO Vo Outside of the boundary layer,
Eqs. (B) and X9) are also valid so long as the
local inviscid values of U, and 6 for the displace-
ment body are used. The improved values in Eqs.
(6) and (7) account for the variation of the in-
viscid static pressure and potential-flow vector
across the thick boundary layer and make appropriate
use of the results of the CS method. As already
noted, the variation of static pressure computed
across the boundary layer outside of the displace-
ment surface agrees quite well with the experimental
results.
Figure 8 shows the comparison of the mean axial
and radial velocity profiles at several axial sta-
tions on Afterbody 1, and Figure 9 shows the mea-
sured axial velocity profiles across the near wake
of Afterbody 1. The theoretical results at X/L =
1.00 were calculated at X/L = 0.998. Figures 10
and 11 show comparisons of the measured and computed
velocity profiles for Afterbody 2. The mean axial
and radial velocity components uy, and Vy were mea-
sured by a cross-wire probe and the experimental
accuracy of measurements of u,,/Ug and Vr/U5 were
respectively about 0.5 percent and 1.0 percent.
As shown in Figures 8 and 10, the theoretically
computed velocities, which account for the variation
of static pressure distribution across the thick
boundary layer, agree better with the measured axial
and radial profiles outside of the displacement sur-
face. These results suggest that a simple improve-
ment of the existing boundary-layer computation
method can be made for the thick stern boundary
FIGURE 9. Measured mean axial velocity distributions
across near wake of afterbody 1.
138
0.8
0.6
I~)— —9- — — 0
A
(C) X/L = 0.970
(B) X/L = 0.934
(D) X/L = 0.977
>
0.2
t)
uy Mp o x
a ee — (LDV)
X/L Uy U, ra uy
(A) 0.840 © QO 0.962 THEORY
(B) 0.934 Q 0.484 oO uy v
(C0970 A JA 0206 uU ~
° °
(D) 0.977 O o.149 (0) — = — - — DIRECT
oem ocean Sao IMPROVED
FIGURE 10. Measured and computed mean axial and radial velocity distributions across stern boundary
layer of afterbody 2. 3
layer by means of the displacement body concept.
However, it is important to point out that the
measured axial velocity profiles in the inner region
are in general smaller than the theoretical values.
The eddy viscosity model plays an important role in
this region; therefore, it is essential to examine
the eddy viscosity model used for computing the
thick stern boundary layer. Figures 8 and 10 also
show the comparison of the axial velocities mea-
sured by the cross-wire and by LDV (Huang et al.,
1976). The agreement is very good inside the bound-
ary layer. However, due to the artifical seeding
of oil mist required for the LDV, the axial veloc-—
ities near the edge of the boundary layer measured
by LDV are smaller than that by the cross-wire.
6. COMPARISON OF MEASURED AND COMPUTED INTEGRAL
PARAMETERS
The integral parameters are derived from the mea-
sured velocity distribution. The two-dimensional
displacement thickness is defined as
= u (x)
x (10)
where 6, is the boundary thickness measured radially
normal to the body axis and Ux(r) is the value of
the axial component of inviscid flow velocity com-
puted about the displacement body. The value of
,
U,,(r) is computed by the potential-flow method ex-
cept inside of the displacement surface where it is
assumed that UL (x) = UL, (XQ) with re) being the radius
of the displacement surface. The boundary-layer
thickness 6, is defined at the radial position where
the measured value of uy (r) equals) (01995 3U57\(@>) eae
is difficult to obtain 6, precisely since the ac-
curacy of the Uy/U, measurement is only about 0.005.
Nevertheless, the overall accuracy of the values of
6, estimated in the present investigation is about
10 percent.
A measure of the mass-flux deficit in the thick
axisymmetric boundary layer is defined by
r_ +6 ry +6%
(0) "32 u(r) OMG
= i oS rdr = rdr abil
i U_(z) ff are
xr x ie
fo)
where r_ is the local body radius and 6* is the
axisymmetric displacement thickness. Thus, the
axisymmetric displacement thickness becomes
Ge a 1 O% Yr =
2 z= (2) +2 (12)
12 r 2
max max max
where Ypax is the maximum radius of the body.
The displacement body in the present investiga-
tion is defined by rq = 5# + xr, rather than the
planar definition, rg 6* + ro. Similarily, a
measure of the momenta lax deficit is defined by
FIGURE 11.
across near wake of afterbody 2.
Measured mean axial velocity distributions
5
r +6
fo)
x u(r) 7 u(r)
Do A saeray| cee ee
x x
The measured and computed values of 6* and 6, are
shown in Figure 12 for Afterbody 1 and in Figure
13 for Afterbody 2. The measured values of d¢ and
6, for X/L > 0.90 are slightly larger than the
computed values for both bodies.
The transverse curvatures of the boundary-layer
flow with respect to the body radius, (ro + Oe) Pes
and (ro + 5,)/ro, are also shown in Figures 12 and
13. A drastic increase of the values of (ro + Ss) AZ)
and (rg + 6;)/Yg occurs at X/L = 0.9, indicating the
important effect of transverse curvature on the
stern. The longitudinal curvature of the body is
denoted by K, = (d°r,/dx*) [1 + (dro/dx*)]~3/% and
the longitudinal curvature of the displacement body
is denoted by Kg = (d2rg/dx2) [1 + (drg/dx) 2173/2.
A positive sign for K, or Kg indicates concave sur-
pacer The values of nae and Ka' max are shown
in Figures 12 and 13. There is a significant dif-
ference between Ky and Ka in the thick boundary
layer region. In each case, the curvature of the
displacement body is convex up to X/L = 0.92, then
changes to concave and remains concave throughout
the entire thick boundary-layer and near-wake region.
The curvature of the body surface is convex up to
X/L = 0.96. As already shown in Figures 4 and 6,
the measured distributions of static pressure and
139
hence the curvatures of the mean streamlines are
much more closely related to the displacement body
than to the actual body. The magnitudes of the
maximum concave and convex radii of curvature of
the displacement bodies are estimated to be 8 max
and 30 Yrmax for Afterbody 1 (Figure 12) and 7 rmax
and 8 Yrmax for Afterbody 2 (Figure 13), respectively.
The magnitudes of the radii of curvature of the
mean streamlines outside of the displacement body
are expected to be larger than 10 rmax.-
7. MEASURED TURBULENCE CHARACTERISTICS
The cross-wire probe was used to measure the tur-
bulence characteristics in the thick boundary layer.
The measured Reynolds stresses and the measured
mean velocity profiles were used to obtain eddy
viscosity and mixing length.
Measured Reynolds Stresses
The turbulence characteristics in the thick boundary
layer can be represented by the distributions of
Reynolds stresses, namely, -u'v', u'2, v'2, and
w'2, where u', v', and w' are the turbulence fluc-
tuations in the axial, radial, and azimuthal direc-—
tions, respectively. Figures 14 and 15 show the
measured distribution of Reynolds stress =v’ /U,2
and three components of turbulence intensity at
several axial locations along the two afterbodies.
In general, for a given location, the intensity of
the axial turbulence-velocity component has the
highest value and the intensity of the radial com-
ponent has the smallest value. The degree of
anisotropy decreases as the stern boundary layer be-
comes thicker. Furthermore, the increased boundary-
layer thickness is accompanied by a reduction of
turbulence intensities and a more uniform distribu-
tion of turbulence intensities in the inner region.
The variation along the body of the radial location
of the maximum values of the measured Reynolds stress
=u 9/057 layer is small. The spatial resolution of
the cross-wire probe is not fine enough to measure
the Reynolds stress distributions in the inner re-
gion when the boundary layer is thin. As the stern
boundary layer increases in thickness, the location
of maximum Reynolds stress moves away from the wall
(Figures 14 and 15). The values of Reynolds stress
=u'v" decrease quickly from the maximum value to
zero at the edge of the boundary layer. As shown
in Figures 14 and 15, the shape of the Reynolds
stress distribution curves in the outer region is
quite similar for all the thick boundary layers.
It is interesting to note that the shapes of the
Reynolds stress distributions in the inner regions
are different from those measured in the wake at
X/L = 1.057 and 1.182 (Figures 14 and 15); this is
a typical characteristic of a developing wake
([Chevray (1968)]. The Reynolds stresses experience
a drastic reduction in magnitude near the edge of
the boundary layer.
A turbulence structure parameter defined by aj
= =u'v'/aq2, where q?2 = uj," + v,'2 + w'2, is of
interest. The measured distributions of a, are
shown in Figure 16. Most thin boundary layer data
show that a, is almost constant (a, = 0.15) between
0.05 and 0.86. The present thick stern axisymmetric
data shown in Figure 16 indicate that a, is almost
constant up to 0.6 6,, and the magnitudes of aj
140
x
3
aE
eS
=
RE
ae
x
3
RE
~o
0.75
FIGURE 12.
0.75
0.80
AFTERBODY 1
0.80
Transverse
AFTERBODY 2
0.80
0.85
and longitudinal curvature parameters for a
Qj TRANSVERSE CURVATURE
0.85
1a] irvature parameter:
~~
anes:
—— —
———_ LONGITUDINAL CURVATURE
r
o max
0.90 0.95 1.00 1.05 1.10
EDGE OF BOUNDARY LAYER AND NEAR
THEORY
MEASUREMENT
An De mS
DISPLACEMENT SURFACE
Se Se SS ON
THEORY MEASUREMENT
0.90 0.95 1.00 1.05 1.10
xX/L
(r+ 5 )/r
oO r oO Pa
>
EDGE OF BOUNDARY LAYER AND NEAR WAKE
THEORY MEASUREMENT
Foi Ses
THEORY
DISPLACEMENT SURFACE
ed, er
MEASUREMENT
0.90 0.95 1.00 1.05
1.20
1.20
FIGURE 14.
(A) X/L = 0.755
(C) X/L = 0.934
(E) X/L = 1.0076 (F) X/L = 1.057
0.02 0.04
V w7/u, > v v7/u, . IV w'7/U, 3
Measured distributions of Reynolds stresses for afterbody 1.
(B) X/L = 0.846
0.06
<u'v/U
o
(G) X/L = 1.182
141
(B) = 0.934
T ] Useerorrl Lapp leita amin cel firstly
(A) X/L = 0.840
(D) X/L = 0.977
(C) X/L = 0.970
0.10
e)
0.04 0.06
Vo?u,. Voru,. Vw?su,, -uvsu,?
molds stresses for afterbody 2
0.08
it) 0.02
Measurea distributions of Rey
a
decrease toward the edge of the boundary layer. The
values of a; also decrease in the inner region of
wake at X/L = 1.057 and 1.182 of Afterbody 1. It
should be pointed out that the measured values of
q? contain the free-stream turbulence fluctuation,
no attempt having been made to remove the free-
stream turbulence fluctuation from the measured
values of q*. The measured reduction of a) near
the outer edge of the boundary layer is in part
caused by the larger contribution of the free-stream
turbulence to q2 than to -u'v". Nevertheless, the
measured values of the turbulence structure param-
eter a, are quite constant across the inner portion
of the boundary layer where the effect of free-
stream turbulence is small.
Eddy Viscosity and Mixing Length
The measured distributions of shear stress =u'Tv™
and mean velocity gradient, du,/dr, were used to
calculate the variations of eddy viscosity and
mixing length across the thick stern boundary layers
according to the following definitions
Tg ou (14)
SU = ea
or
and
ou du,
ee 2. =e mS
u'v Q les | ay (15)
The experimentally-determined distributions of
eddy viscosity, €/Ugd5p*1 are shown in Figure 17 for
Afterbody 1 and in Figure 18 for Afterbody 2, where
Us is the potential-flow velocity at the edge of
the boundary layer and 6,* is the displacement
thickness (based on the planar definition, Eq. 10).
Figures 19 and 20 show the experimentally-determined
distributions of mixing length 2/6, for the after-
bedies, where 6, is the boundary-layer thickness
measured normal to the body axis. As shown in
Figures 19 and 20, the measured distributions of
0.018
0.016
0.014
THIN
0.012 BOUNDARY
143
eddy viscosity agree resonably well with the eddy-
viscosity model of Cebeci and Smith (1974, Eq. 3)
when the boundary layers are thin. However, as the
stern boundary layer thickens, the measured values
of e/Us Sp* in the thick stern boundary layers are
only about 1/6 of the values for thin boundary
layers given by the Cebeci and Smith model (1974).
The measured distributions of mixing length shown
in Figures 19 and 20 also agree quite well with the
thin boundary layer results of Bradshaw, Ferriss,
and Atwell (1967). Again as the boundary thickens,
the measured values of 2/6, reduce drastically.
The values of 2/5, in the thick stern boundary
layers are only about 1/3 of those of the thin
boundary layers. Similar reductions of eddy vis-
cosity and mixing length in thick stern boundary
layers were also measured by Patel et al. (1974,
USN) 3
As the axisymmetric boundary layer thickens in
the stern region, the boundary layer thickness Sy
and the displacement thickness Sp* increase dras-
tically. However, the values of eddy viscosity and
mixing length do not have enough time to respond to
this change. Therefore, neither the eddy viscosity
model of Cebeci and Smith (1974), nor the mixing
length results of Bradshaw, Ferriss, and Atwell
(1967) can be applied to the thick stern boundary
layer.
8. TURBULENCE MODELS
In most works, the basic assumption made in the
differential methods for calculating turbulent
boundary layers is that the mixing length or eddy
viscosity is uniquely related to the mean velocity
gradient and the boundary-layer thickness parameter
at a given location. So long as the boundary layer
is thin and the change in boundary-layer properties
due to the pressure gradient is gradual, this simple
assumption is know to be satisfactory [see e.g.,
Cebeci and Smith (1974)]. When the past history of
boundary layer characteristics is important, Brad-
FIGURE 17. Measured distributions
of eddy viscosity for afterbody 1.
144
shaw et al. (1967) argue that the turbulence energy
equation can be used to model the memory effect.
In order to determine the rate of change of tur-
bulent intensity along a mean streamline, three
assumptions have to be made: namely, that turbu-
lence intensity is directly proportional to the
local Reynolds stress, aj =u /q2 0.15; that
= =u'v'/
the dissipation rate is determined by the local
Reynolds stress and a length scale depending on
n/é; and the energy diffusion is directly pro-
portional to the local Reynolds stress with a fac-
tor depending on the mixing value of Reynolds stress.
On the basis of thin boundary-layer data two em-
pirical functions for the last two assumptions were
proposed by Bradshaw et al. (1967). The first as-
sumption, 2/6 £,; (n/é), was found not to be
0.018
0.016
0.014
0.012
0.010
U;5
6 0.008
0.006
0.004
0.002
Measured distribu-
FIGURE 18.
0.2
0.10
0.08
0.06
0.04
0.02
Lstribu-
applicable to the present thick axisymmetric stern
boundary layers. The deviation of the apparent
mixing length along the curved boundary from that
of a thin flat boundary was also noted and dis-
cussed by Bradshaw (1969). A simple linear cor-
rection to the length scale of the turbulence by
the extra rate of strain was made by Bradshaw (1973).
The extension of this concept has just been made for
the thick axisymmetric boundary layer by Patel et
Gilly (ALS) 7S)
It is important to note that the boundary-layer
thickness of a typical axisymmetric body increases
drastically at the stern. Most of the rapid change
takes place within a streamwise distance of a few
boundary-layer thicknesses. Most of the empirical
functions for solving the turbulence energy equa-
Ole! OSS ae ee
= 0.0168
THIN Usd,
BOUNDARY
LAYER
O € 0.0168
fe} = 6
Usdp 1 +55 ee)
5,
X/L
0.4
THIN BOUNDARY
LAYER 2
© 0.755
O 0846
©} 0.934
A 0.964
QO 1.0076
© 1.057
QO 1.182
0.10
0.08
0.06
0.04
tion will undergo rapid changes in basic forms. The
one known for certain is the empirical function for
mixing length. Therefore, it may be difficult to
compute the rate of change of the turbulence energy
or the extra rate of strain in the region.
Fortunately the present measured distributions
of Reynolds stresses shown in Figures 14 and 15 are
quite similar in the outer region and differences
appear in the inner region where the turbulence is
reduced in intensity and more homogeneous. In such
an axisymmetric flow configuration, the character-
istic length scale is more closely related to the
entire turbulence annulus between the body surface
and the edge of the boundary layer rather than the
radial distance between the two. Therefore, we
propose that the mixing length of an axisymmetric
turbu-length boundary layer is proportional to the
square root of this area when the thickness in-
creases drastically at the stern:
Qo We, + S.) 2 - ie
In order to examine this simple hypothesis, the
present measured values of 2/V(rg + 6,) 2 = ro?
together with the data of Patel et al. (1974, 1977)
are shown in Figure 21. The solid line is the best
fit of the present data. The present values of 2
are slightly greater than those for Patel's modified
spheroid (1974) and are slightly lower than those
for Patel's low-drag body (1977). The data in
Figure 21 support this simple hypothesis although
the data are quite scattered due to large varia-
tions of stern configurations and Reynolds number,
and probable measuring errors.
The existing thin turbulent boundary-layer dif-
ferential methods can be applied to the forward
portion of the axisymmetric body up to the station
where the boundary layer thickness increases to
about 20 percent of the body radius. Further down-
THIN BOUNDARY
LAYER
145
FIGURE 20. Measured distribu-
tions of mixing length for
afterbody 2.
stream, the apparent mixing length of the thick
axisymmetric stern boundary layer (2) can be roughly
approximated by the mixing length for a thin flat
boundary layer (2,) by
Ghee 3.336 (to)
ag
which is the solid line of Figure 21. At the aft
end of the stern Yo is zero and the value of 2/2,
is 1/3.33. This simple approximation of the mixing
length for thick axisymmetric stern turbulent bound-
ary layers can be incorporated into most existing
differential methods. As noted earlier, the mea-
sured axial velocities inside the thick boundary
layer (especially in the inner region) are smaller
than the computed values (Figures 8 and 10). The
present CS method overestimates the magnitude of
eddy viscosity (Eq. 3) for the thick stern boundary
layer. While the mixing length approximations ob-
tained in the present investigation can be incorpo-
rated into the CS method to predict more accurately
the thick stern boundary-layer velocities, further
refinement of the theoretical methods is desirable.
9. CONCLUSIONS
In this paper, we have described recent experimental
investigations of the thick turbulent boundary lay-
ers on two axisymmetric sterns without shoulder flow
separation. A comprehensive set of boundary layer
measurements, including mean and turbulence veloc-—
ity profiles and static pressure distributions, are
presented. Two major conclusions can be drawn:
The Lighthill/Preston displacement body concept
has been proven experimentally to be an efficient
and accurate tool for treating the viscid and in-
146
(A) PRESENT AFTERBODIES 2
——
© 0840
0 084 Cj 0.934
© 0934 [0970
Q 0964 © 0977
QO 1.0076
0.05
(B) MODIFIED SPHEROID
PATEL ET AL (1974)
(C) LOW-DRAG BODY
PATEL ET AL (1977)
y/5
viscid stern flow interation on axisymmetric bodies.
The measured static pressure distributions on the
body and across the entire thick boundary layer and
wake were predicted by the displacement body method
to an accuracy within one percent of dynamic pres-
sure. Theoretical predictions of the measured
axial and radial velocity profiles outside the dis-
placement surface were improved significantly when
the variations of the static pressure and radial
velocity of the displacement body were incorporated
into the computation.
Neither the measured values of eddy viscosity
nor mixing length were found to be proportional to
the local displacement thickness or the local
boundary-layer thickness of the thick axisymmetric
boundary layer. As the boundary layer thickens
rapidly at the stern, the turbulence characteristics
in the outer region remain quite similar but the
turbulence reduces its intensity and becomes more
uniformly distributed in the inner region. The
measured mixing length of the thick axisymmetric
stern boundary layer was found to be proportional
to the square root of the area of the turbulent
annulus between the body surface and the edge of
boundary layer. This simple similarity hypothesis
can be incorporated into existing differential
boundary-layer computation methods.
ACKNOWLEDGMENT
The work reported herein was funded under the
David W. Taylor Naval Ship R&D Center's Independent
Research Program, Program Element Number 61152N,
Project Number ZR 000 O1.
REFERENCES
Bradshaw, P., D. H. Ferriss, and N. P. Atwell
(1967). Calculation of boundary layer develop-
ment using the turbulent energy equation. J.
Fluid Mech., 28, 593-616.
Bradshaw, P. (1969). The analogy between streamline
curvature and buoyancy in turbulent shear flow.
J. Fluid Mech., 36, 177-191.
Bradshaw, P. (1973). Effects of streamline curvature
on turbulent flow, AGARDograph No. 169.
Cebeci, T., and A. M. O. Smith (1974). Analysis of
turbulent boundary layers. Academic Press, New
York.
Chevray, R. (1968). The turbulent wake of a body
of revolution, ASME. J. of Basic Engineering,
90, 275-284.
Granville, P.S. (1953). The calculation of the
viscous drag of bodies of revolution. David
Taylor Model Basin Report 849.
Hess, J. L., and A. M. O. Smith (1966). Calculation
of potential flow about arbitrary bodies.
in Aeronautical Sciences, Vol. 8, Pergamon Press,
New York, Chapter 1.
Progress
147
Huang, T. T., and C. H. von Kerczek (1972). Shear
stress and pressure distribution on a surface
ship model: theory and experiment. Ninth ONR
Symposium on Naval Hydrodynamics, Paris; avail-
able in U.S. Government Printing Office, ACR-203-
Viole loos —20Or
Huang, T. T. et al. (1976). Propeller/stern/
boundary-layer interaction on axisymmetric bodies:
theory and experiment. David Taylor Naval Ship
Research and Development Center Report 76-0113.
Lighthill, M. J. (1958). On displacement thickness.
J. Fluid Mech., 4, 383-392.
McCarthy, J. H., J. L. Power, and T. T. Huang (1976).
The roles of transition, laminar separation, and
turbulence stimulation in the analysis of axi-
symmetric body drag. Eleventh ONR Symposium on
Naval Hydrodynamics, London; Published by
Mechanical Engineering Publications Ltd.,
London and New York.
Patel, V. C., A. Nakayama, and R. Damian (1974).
Measurements in the thick axisymmetric turbu-
lent boundary layer near the tail of a body of
revolution. J. Fluid Mech., 63, 345-362.
Patel, V. C., Y. T. Lee, and O. Guven (1977). Mea-
surements in the thick axisymmetric turbulent
boundary layer and the near wake of a low-drag
body of revolution. Symposium on Turbulent Shear
Stress, Pennsylvania State University, University
PEWS IDs Do A998) 5 B3S\0
Patel, V. C. and Y. T. Lee (1978). Calculation
of thick boundary layer and near wake of bodies
of revolution by a differential method. ONR
Twelfth Symposium on Naval Hydrodynamics, (This
Volume, Section III).
Preston, J. H. (1945). The effect of the boundary
layer and wake on the flow past a symmetrical
aerofoil at zero incidence; Part I, the veloc-
ity distribution at the edge of, and outside the
boundary layer and wake. ARC R&M 2107.
APPENDIX
The raw data and derived results of the present
experiments are tabulated in the following so that
they can be used independently by other investi-
gators. Table 3 shows the measured pressure and
shear stress coefficients on Afterbodies 1 and 2.
Tables 4 and 5 provide the measured static pres-
sure coefficients across the stern boundary layers
and near wakes of Afterbodies 1 and 2, respectively.
Tables 6 and 7 contain the values of measured mean
axial and radial velocities, three components of
turbulence fluctuations, and Reynolds stresses
across the boundary layer and near wake of After-
bodies 1 and 2, respectively. The experimentally
derived data on eddy viscosity, mixing length,
planar and axisymmetric displacement thickness, and
boundary layer thickness are also given.
148
TABLE 3 - Measured Pressure and Shear Stress Coefficients on
Afterbodies 1 and 2
6
AFTERBODY 1, RX = 6.6 x 10 AFTERBODY 2, Ry = 6.8 x
oe oe
a c c a P c
L max p T L max p
0.7060 0.9690 -0.062 = 0.6000 1.0000 -0.013
0.7455 0.9267 -0.064 0.00281 0.7000 1.0000 -0.024
0.7952 0.8423 -0.050 0.00265 0.7455 1.0000 -0.035
0.8449 0.7192 -0.024 0.00248 0.7952 1.0000 -0.106
0.8946 0.5480 +0.018 0.00213 0.8449 0.9476 -0.160
0.9145 0.4633 +0.050 0.00185 0.8946 0.7349 -0.010
0.9344 0.3636 +0.074 0.00163 0.9145 0.6137 +0.053
0.9543 0.2396 +0.112 0.00130 0.9344 0.4834 +0.090
0.9642 0.1900 +0.133 0.00115 0.9543 0.3317 +0.170
0.9741 0.1394 +0.135 0.00104 0.9642 0.2547 +0.183
1.0000 0.0000 +0.116 - 0.9741 0.1740 +0.198
1.0000 0.0000 +0.185
TABLE 4 - Measured Static Pressure Coefficients Across Stern Boundary
Layer and Near Wake of Afterbody 1
x/L = 0.7553 x/L = 0.9144 x/L = 0.9344
é r-r) = rr) a Tr
"max Tmax C Tmax vnax Cp Tax "ax Cp
0.9127 0 -0.0560 0.4633 0) 0.050 0.3636 0 0.0740
0.9345 0.0218 -0.0530 0.4997 0.0364 0.0604 0.4214 0.0578 0.0821
1.0283 0.1156 -0.0510 0.5181 0.0548 0.0604 0.4981 0.1345 0.0791
1.1298 0.2171 -0.0500 0.5508 0.0875 0.0587 0.6102 0.2466 0.0674
1.2392 0.3265 -0.0480 0.5892 0.1259 0.0570 0.7253 0.3617 0.0682
1.4736 0.5609 -0.0434 0.6687 0.2054 0.0546 0.8375 0.4739 0.0624
1.6767 0.7640 -0.0380 0.7397 0.2764 0.0514 0.9497 0.5861 0.0593
1.8720 0.9593 -0.0350 0.8519 0.3886 0.0492 1.0989 0.7353 0.0518
2.0908 1.1781 -0.0302 0.9670 0.5037 0.0464 1.2509 0.8873 0.0471
2.2861 1.371 -0.0287 1.0835 0.6202 0.0434 1.4000 1.0364 0.0421
2.4736 1.5609 -0.0270 1.1957 0.7324 0.0400 1.5563 1.1927 0.0368
2.8798 1.9671 -0.0226 1.3093 0.8460 0.0370 1.7054 1.3418 0.0333
3.2783 2.3438 -0.0214 1.4386 0.9753 0.0338 1.8546 1.4910 0.0305
3.7079 2.7952 -0.0197 1.5906 1.1273 0.0301 2.0066 1.6430 0.0264
4.3251 3.4124 -0.0158 1.7426 1.2792 0.0269 2.0904 1.7268 0.0256
WASOSI W4298F 1OROZ4 1S 5022 i74m2; O09 010235,
2.0451 1.5418 0.0221 3.3580 2.9943 0.0114
2.1971 1.7338 0.0206 3.6960 3.3324 0.0099
2.3491 1.8858 9.0183 4.0355 3.6719 0.0079
PBHRRWNHRPRPRPRrFRrRRrFPODOCOCCCOCCO0O
10
SiOLOlIOLO LOLOrOrO1Oore
[o)
i=)
iy)
(>)
S
BPRWNWRPRPRrRrRrrROOCOOOCOCCoO
SOLO OLOLOROROL OOF OLOROLOEOVOL OTS:
TABLE 4 (Continued)
Sel, S On CyAl x/L = 0.9830 x/L = 0.8462
ag ey r 16 * me
= Tax C Tax rnax CS = Yr =
max max max
0.1364 0 0.135 0.1164 0 Ost 7/155 0 -0
0.1600 0.0236 0.1441 0.1278 0.0114 0.1250 0.7450 0.0295 -0
0.2324 0.0960 0.1400 0.1647 0.0483 0.1249 0.7620 0.0466 -0
0.3447 0.2083 0.1293 0.2031 0.0867 0.1241 0.7791 0.0636 -0
0.4569 0.3205 0.1172 0.2471 0.1307 0.1224 0.8018 0.0864 -0
0.5748 0.4384 0.1058 0.2826 0.1662 0.1211 0.8217 0.1063 -9
0.6827 0.5464 0.0950 0.3153 0.1989 0.1189 0.8572 0.1418 -0
0.7949 0.6586 0.0856 0.3892 0.2728 0.1142 0.8572 0.1801 -0
0.9057 0.7693 0.0781 0.4659 0.3495 0.1090 0.9297 0.2142 -0
1.0251 0.8887 0.0705 0.5412 0.4248 0.1036 0.9694 0.2540 -0
1.1373 1.0009 0.0628 0.6136 0.4972 0.0974 1.0092 0.2938 -0
1.2466 1.1102 0.0574 0.7684 0.6520 0.0858 1.0873 0.3719 -0
1.3645 1.2281 0.0536 0.8437 0.7273 0.0817 1.1598 0.4443 -0
1.4796 1.3432 0.0490 0.9162 0.7998 0.0774 1.2322 0.5168 -0
1.5918 1.4555 0.0450 0.9914 0.8750 0.0735 1.3061 0.5906 -0
1.7395 1.6031 0.0406 1.1463 1.0299 0.0650 1.3785 0.6631 -0
1.8901 1.7537 0.0374 1.2968 1.1804 0.0555 1.7000 0.9945 -0
2.0137 1.8773 0.0350 1.4431 1.3267 0.0505
1.5966 1.4802 0.0441
1.7343 1.6179 0.0409
x/L = 1.00 x/L = 1.0076 sh = i057 Sih & Waiey
16 r a6 ag
ie C r C r C Cc
max p max 2) max p max p
0.061 0.1295 0 0.0995 0 0.0418 0 0.0181
0.098 0.1209 0.099 0.0987 0.057 0.0413 0.036 0.0174
Qobye OWtO O,292- ©0868 O,152° O,040 —~O,07%41 @.0nz2
0.249 0.1022 0.327 0.0897 0.247 0.0426 0.149 0.0178
0.361 0.0940 0.453 0.0815 0.344 0.0424 0.224 0.0189
0.477 0.0865 0.585 0.0740 0.433 0.0415 0.303 0.0194
5537 W075 Os720, OL0677% O.527 > O.0400) 0.637 O.Olee
0.702 0.0716 0.855 0.0604 0.616 0.0396 0.561 0.0180
0.814 0.0656 0.980 0.0553 0.831 0.0364 0.716 0.0176
0.928 0.0604 1.116 0.0503 1.000 0.0353 0.864 0.0170
TOSS OO 527. 2417 WN OMO46S el l6 ON = ON0333s Og. OL Ol70
W273) 100464515382) 1004267 91,342) 040316" ol sll6s) © O-ol70
Loos Oo0205 2.5 O.0K7 MoS ©0298) Tale —@,oeH
1.724 0.0344 1.724 0.0344 1.724 0.0249 1.464 0.0160
1.900 0.0230 1.621 0.0150
2.240 0.0195 1.756 0.0140
2.580 0.0168
2.750 0.0150
149
150
TABLE 5
x/L
HIH
NNN FPRP RRP RRP PRR O
max
-9855
.0536
.1417
.2141
.2937
- 3406
-4173
.4968
.6374
.9357
.2085
-4926
. 8860
NNFRrRrROOOCOOCOCOC oO
RPreroooocna0qcco0co
Measured Static Pressure Coefficients Across Stern Boundary
Layer and Near Wake of Afterbody 2
0.8400
1e=ae
NrRPrRPOODOOCOCOCOCOCOCCO
-0
-0
-0
-0
=0).
-0.
=10)6
-0.
-0.
-0.
=<
=.
-0.
CHS) Cy(ey(o) Koyo (Se) ey (yoyo yie) (SiS)
SY LS iSyeC ft SS) (S) (Sy-o(e) Sy )(S)
x/L
NrrFrFODOCOCCOCCCOCCOCCOO
Sila) (ele) (sriere) (2) ee) S) (Ss)
SOLO sO LOFOLOLOROLOL ORO LOLS
0.9336
SLOZOLOLOLOLOLO LOFOlO.OLOLOr©)
SOLO ,OLOLOLOLO VOLO LO ONOFOVOrOoro
x/L = 0.9702
r ETO
Ts (¢,
max max p
0.2419 0.0364 0.1884
0.2930 0.0875 0.1854
0.3754 0.1699 0.1740
0.4379 0.2324 0.1654
0.5061 0.3006 0.1563
0.5686 0.3631 0.1482
0.6623 0.4568 0.1342
0.7262 0.5207 0.1252
0.8541 0.6486 0.1110
0.9961 0.7906 0.0957
1.1467 0.9412 0.0804
1.4393 1.2338 0.0604
eV: Wo Sle Me@Ak3e}
2.0941 1.8886 0.0340
2.4038 2.19835 0.0247
IP SO} 7/ x/il) =) Lls2
r
G 10 C
p max Pp
0.0471 0 0.0004
0.0467 0.0426 0.0067
0.0492 0.2102 0.0095
0.0503 0.3722 0.0099
0.0447 0.5313 0.0090
0.0484 0.6889 0.0120
0.0450 0.8523 0.0179
0.0493 1.1250 0.0191
0.0462 1.7159 0.0172
0.0439
0.0368
0.0340
0.0260
0.0234
0.0232
0.0198
0.0181
TABLE 6 - Measured Mean and Turbulence Velocity Characteristics for Afterbody 1
x/L = 0.755, r_/r = 0.9127 tan a = -0.0671
O° max
es) uy vy Jar2 Jae wi2 pee uty! TR €
max UG Us U5 U Y we, oF O. Uso
0.013 O67 O05 O07) O.0883 O47 On08 Ooi O.084
0.034. 0.719 -0.018 0.069 0.039 0.049 0.146 0.168 0.121 0.0080
0.08: On Or O.0683 Os083 OLOlG ONS OnIG2 Ole WrOlAo
0.074 0.820 -0.019 0.064 0.036 0.043 0.123 0.170 0.264 0.0152
0.02) OG O01 O08 OsOGS ~O.042 “OLUIO O.NG2 ORIG Oily
0.110 0.881 -0.019 0.057 0.032 0.039 0.0969 0.167 0.393 0.0149
0.129 0.908 -0.019 0.052 0.030 0.036 0.0809 0.165 0.460 0.0145
OMmi1Go! 051) 08020) \0s038) 0N024) 0,052) 00589) sOnuzey) On6045) o,00911
0.203 0.983 -0.020 0.027 0.017 0.018 0.0159 0.118 0.725 0.0072
0.241 1.003 -0.019 0.012 0.010 0.011 0.0022 0.061 0.861 0.00125
0.280 1.015 -0.017 0.007 0.006 0.006 0.0004 0.024 1.00 0.00017
0.361. 1.020 -0.016 0.003 0.003 0.003 0.00005 0.019 1.29
O.“5 ICO O02 0,002 0,002 002 =
0.6% 1.000 0.008 ©0022 0,008 O00 <
1,768 1000 -W.008 O.00F O.002 002 =
&* $* 8
2 = 0.0426, = = 00444 Mi ="0R280
max max max
x/L = 0.846, r /r_4 = 0.7155, tan a -0.1343
aT uy vy fie Sie (a2 Aopen -uly! TEXO €
r U_ Um 5 Y % wa aS Sy Uso"
max eo} (e)
O05 O05 Os073 O.067 Oc0k3 ~ WL0AS Oss Oe) OIA
0.0280 0.660 -0.082 0.066 0.035 0.042 0.110 0.150 0.0800 0.00536
0.0843 O78 0.080) OGG O.058 0.050 O.108 O16? O.i153° O,00oo
0,070 O77 0,08 0,089 @.05 O.088 OOP Osis O.al7 O,0128
QnOgS4y) ONS06) 08093), 005400) 0505595 0F036) 0809555. Ohl 79m MOnzelN ss ononad
01700 O40 O,098 O02 O.052 O05 O00 O.i67 0.283 O06
O15, 0.664 0.08 @,052 GO.08l § O.055 O.082 OniG7 O6e05 —W.0140
0.1600 0.890 -0.093 0.050 0.031 0.035 0.0815 0.174 0.457 0.0142
0,105 O98 0,093 O.042 ©0285 O08 O.0888 0.145 —G.557 —W.0nas
02168 O98. 40,098 6.055 0.023 0.025 0.0204 O.129 O.G18 O.00me
0.2250 0.665 =.0S 0.029 0,02, @.025 O03 O07 O.6720 Ooms?
0.2560 0.073 0.08 0,026 0.018 ©0283 0118 O.094 O.75l O,005i8
0.2770 OOS, 0,087 0.099 0,014 O07 ©0025 0.07 O.791 O,0May
0.3341 1.003 -0.084 0.005 0.005 0.006 0.00045 0.046 0.955 0.00053
0.6040 1.002 -0.063 0.003 0.002 0.002 0.00002
1,203 1,000 O05 0,002 0.002 @.002 ©
DMOONSHNI O00) 20024) ONOO2MNONOO2NNOROO2M =n0
6* 6x 6
- = 0.0489, = 5 0,082, 22 2 O88
SOLO FOLOL.O OOO lo:
SISGLEKOIOVOLOLONORoFOto lS:
oO
“I
co
CriOlrOlOoLOoxOvORONCIe
SS OLOLOLOLOLOhOL OLOrOlO)
151
TABLE 6 - Continued
x/L = 0.934, Bfte oe
r-Y, be Wes ut
U_ U_ Un U
max fo) fo) fo)
0.0127 0.425 -0.096 0.055 0.
0.0511 0.541 -0.104 0.049 QO.
0.0909 0.613 -0.105 0.047 0.
0.1704 0.727 -0.103 0.047 0.
0.2360 0.805 -0.097 0.040 0.
0.3309 0.884 -0.091 0.038 0.
0.4105 0.931 -0.087 0.026 0.
0.5284 0.964 -0.076 0.007 QO.
0.6477 0.974 -0.066 0.003 0.
0.8366 0.982 -0.055 0.003 0.
1.1093 0.989 -0.044 0.002 0.
1.4470 0.993 -0.034 0.002 0.
1.7926 0.997 -0.028 0.002 0.
2.1803 1.000 -0.024 0.002 0.
x/L = 0.964, A ee = 0.190, tan a
ae wi ‘e a2
max US Ue U5
0.0145 0.294 -0.085 0.045 0
0.0700 0.428 -0.093 0.045 0
0.1168 0.509 -0.086 0.046 0
0.1751 0.586 -0.083 0.045 0
0.2589 0.682 -0.075 0.044 0
0.3469 0.768 -0.069 0.042 (0)
0.4748 0.868 -0.061 0.035 0)
0.6069 0.936 -0.055 0.021 QO.
0.7361 0.964 -0.047 0.004 0
0.8654 0.978 -0.041 0.003 0
1.0671 0.988 -0.035 0.002 0
1.2674 0.994 -0.029 0.002 0
1.4719 0.997 -0.026 0.002 0)
1.6793 0.991 -0.022 0.002 0
1.881 0.996 -0.020 0.002 0
2.1893 0.998 -0.018 0.002 0
6 *
= 0.364, tan a -0.2440
ee wi Loge saa es €
fo) Yo U5 a* Yr Usep
034 0.039 0.104 0.182 0.023
033 0.038 0.0759 0.154 0.0929 Q.00302 0
032 0.037 0.0740 0.161 0.165 0.00424 0
032 0.056 0.0700 0.154 0.310 0.00517 0
028 0.034 0.0556 0.157 0.429 0.00513 0
025 0.028 0.0300 0.105 0.602 0.00413 0.
018 0.021 0.0139 0.096 0.746 0.00287 0
006 0.007 0.00088 0.066 0.961 0.000465 0O
003 0.003 0.00002 0.003 1.178 0.0000277 0
002 0.002 0.00003
002 0.002
002 0.002
002 0.002
002 0.002
on Gs 6
aE = O.L064, |= 0.0251 = 0S
r r r
max max max
= =0)..2770
ait) wy! Tyat Saniitand r-r
See SS
- 0 fe) (e) a0 6 P
028 0.034 0.0767 0.193 0.0186
.028 0.033 0.0615 0.158 0.0897 0.00176
032 0.035 0.0654 ORSa 0.150 0.00232
032 0.036 0.0651 0.150 0.224 0.00286
.030 0.036 0.0605 0.146 0.332 0.00336
.028 0.036 0.0528 ON LS7 0.445 0.00323
024 0.032 0.0354 0.125 0.609 0.00299
014 0.017 0.00968 0.106 0.778 0.00145
004 0.004 0.00090 0.005 0.944 0.000304
003 0.003 0.00004 0.014 Valog
002 0.002 0.00002: 0.016 1.368
002 0.002 0.00003
002 0.002 0.00002
002 0.002
002 0.002
002 0.002
a oF
— = 0.2343, — = 0.78
max max
[o-)
R[x
.0160
0212
.0262
.0308
.0328
0371
0349
0237
oooooocooco
BDO DOO oe Oe)
153
TABLE 6 - (Continued)
x/L, = 1.0076, soa ie = 0.
r oe Vy a? ne) wie 100 uy 7% € Q
Tax UG os Uy 5 UN US ae Sy U5°F 6.
0 0.368 -0.059 0.044 0.029 0.034 0.0251 0.162 0
0.061 0.385 -0.053 0.039 0.027 0.031 0.0385 0.154 0.067 0.00219 0.0296
0.101 0.426 -0.047 0.037 0.027 0.030 0.0495 0.165 Onna 0.00209 0.0248
0.139 0.462 -0.043 0.038 0.027 0.032 0.0524 0.164 ORS S: 0.00227 0.0262
0.213 OF 535 -0.037 0.039 0.027 0.033 0.0580 0.174 0.234 0.00248 0.0272
0.288 0.607 -0.020 0.041 0.028 0.034 0.0625 0.173 0.316 0.00292 0.0309
0.365 0.670 -0.010 0.042 0.029 0.035 0.0610 0.159 0.401 0.00285 0.0305
ORS 77, 0.781 -0.005 0.042 0.029 0.034 0.0530 0.141 0.524 0.00246 0.0282
0.589 0.871 -0.005 0.036 0.028 0.032 0.0380 0.130 0.647 0.00239 0.0325
0.702 0.929 -0.004 0.033 0.023 0.025 0.0106 0.057 OL 772 0.00105 0.0275
0.811 0.961 -0.004 0.016 0.012 0.013 0.0020 0.035 0.891 0.00038 0.0208
0.923 0.977 -0.004 0.004 0.003 0.003 0.0009 0.027 1.014 0.000207 0.0182
1.040 0.993 -0.004 0.002 0.002 0.002 0.0002
Om oF o
= = 0.243, = = 10.350, = 0.91
max max max
XY / ily = SOS, ie_//ze =Hi0)
o’ ma
oe x ie uate) vie wi2 penne -u'y! ae e 2
Tax aD M5 Uy os i we in oF Us°p Ps
0 0.519 -0.049 0.036 0.026 0.034 0.0117 0.038 0 = -
0.034 0.538 -0.043 0.035 0.026 0.032 0.0235 0.080 0.037 0.00236 0.0293
ORAS 2. 0.609 -0.024 0.037 0.025 0.030 0.0416 0.144 0.165 0.00351 0.0328
0.232 0.667 -0.015 0.038 0.024 0.029 0.0493 0.137 0.252 0.00396 0.0340
5 alal OLw22 -0.010 0.040 0.028 0.031 0.0527 0.158 0.338 0.00438 0.0364
0.388 0.774 -0.006 0.040 0.026 0.031 0.0505 0.156 0.422 0.00420 0.0357
0.460 0.824 -0.004 0.037 0.026 0.031 0.0460 0.152 0.500 0.00394 0.0350
0.573 0.894 -0.004 0.037 0.028 0.030 0.0400 0.132 0.622 0.00357 0.0364
0.614 0.916 -0.004 0.034 0.021 0.027 0.0353 0.235 0.667 0.00336 0.0370
0.693 0.952 -0.004 0.025 0.017 0.023 0.0186 0.129 0.753 0.00255 0.0357
0.770 0.981 -0.003 0.018 0.011 0.015 0.00569 0.098 0.837 0.00104 0.0341
0.847 0.989 -0.003 0.004 0.003 0.003 0.00033 0.051 0.921 0.00029 0.0305
0.923 0.991 -0.003 0.003 0.002 0.002 0.00010 0.024 1.003 0.000323 =
1077, 0.991 -0.002 0.002 0.002 0.002 0.00007 = Naka -
1276 0.993 -0.002 0.002 0.002 0.002 0.00005 = -
1.459 0.994 -0.002 0.002 0.002 0.002 0
1.697 0.998 -0.002 0.002 0.002 0.002 0
Oe é* §
aE Ost =) = oLeszi == = pep
r r r
max max max
154
TABLE 6 - (Continued)
X/il= ale BD lee
SS) [It IOs Cy ie) (s} SS) (S)'(S)
x/L =
>
~—
wo
SLOT OVOLOLOtO Oro eOLO1Oro©
= 0
ax
ux ‘y \ ul2 yi2 Ww 2 1002u -u'v'
im Tah Vi) Ti ae Taare 2
U, U, U5 US U5 UZ q
632) | -OMO1ON) 10-0400 w OK 034) On034 0 0
644 08008)" 0040) 08054) ONossi On0055. Os 01s8
G0 <O.005 O00 OL O.025 O.0I40 0.055
63. 0,005 OxMO- Wes OOH D063 W207)
Te 0,004 O05 C.050 0,052 O,0403 O.ilA
ks O08 O06 O025 @.050 O00 G.lBR
BS 7 eAOO03) 10032 = 0025 te OMOZSHEONOS40N) NOnI46
RO) =O, Ose “G.0Rl ~~ Os0e5 Os0R90 ~~ Opis
7 O02 WelRs Onc” O,02, 0.0200 050
oo . EOS 'OL0O “C.0d) O00 0083 O,025
995 -0.002 0.003 0.002 0.002 0.00006 0.035
5 <0 O007 , O00 O.00R O.onm =
GOSH a = OF0 01 ONOOZMENONCOZMMONOO? 0 u
6* 6*
— = 0.1543, — = 0.2832, — = 0.95
r ae) ay
max max max
0.914, r/r.. = 0.9145 x/L = 0.977; r,/r,,, = 0.1364
aes Be 3 ee ace uy : ts
Tax (e} OY, Tax UR Us
0.0042 0.448 0.085 0.0073 0.318 0.033
0.0136 0.482 0.091 0.0128 0.334 0.039
ONS W520 © W 20M 0.0183 0.345 0.041
0.0280 0.552 0.096 0.0238 0.360 0.046
0.0392 0.574 0.104 0.0349 0.381 0.042
0.0500 0.598 0.107 0.0459 0.400 0.047
0.0700 0.639 0.112 0.0624 0.428 0.055
0.0907 0.675 0.108 0.0845 0.459 0.045
0.1100 0.706 0.111 0.1011 0.483 0.049
Ost s07/0 nO 7542 OOS 0.1294 0.524 0.049
DIS - Oona Wanws Oot 0.855 ~O.085
0.1772 0.796 0.099 0.1845 0.595 0.045
0.2058 0.833 0.096 0.2176 0.634 0.044
0.2344 0.864 0.093 0.2563 0.676 0.040
0.2687 0.894 0.091 0.2901 0.713 0.036
0.2980 0.919 0.089 0.3328 0.748 0.033
0.3324 0.941 0.088 0.3839 0.796 0.035
0.3782 0.964 0.085 0.4287 0.837 0.033
0.4067 0.979 0.082 0.4949 0.884 0.030
0.4475 0.984 0.078 0.5563 0.915 0.028
0.4876 0.987 0.074 016225) 08947. 0.029
0.5276 0.990 0.069 0.6894 0.960 0.025
0.5677 0.992 0.067 0.7501 0.962 0.020
0.6142 0.992 0.063 0.8170 0.976 0.019
0.6599 0.992 0.059 0.9494 0.982 0.016
0.7229 0.993 0.054 1.0218 0.984 0.013
0.7915 0.994 0.052 Ino) Woes — OsonA
0.8609 0.995 0.047 1.2487 0.989 0.011
0.9417 0.995 0.043 1.3253 0.991 0.009
1.0275 0.998 0.041 1.3984 0.991 0.009
1.1369 1.000 0.032 1.4756 0.993 0.007
iezo7iee wIeOOIy 104052 1.5474 0.994 0.006
1.4522 1.003) 0.027: 1.6253 0.994 0.006
1.6474 1.004 0.024 1.7026 0.995 0.005
1.8770 1.004 0.005 1.7743 0.995 0.005
6+ 6* 6*
= 0.0772, = = 0.0875 — = 0.1866, = = 0.2469
max max max
§
= 0.54 tana = -0.2094 = = 0.80 tana = -0.1036
max
LS) (SS) (SSS) O) S) Sys) ©)
Ww
wo
N
x/L = 1.000, De / oe
SEOLOKOKOLOLORO YO)
(=)
fo)
os
|
Oo
u
x
re eee eee oo ooo ooo ooo oo ooo loko nholonolic)
Oe Go oO
uw
ise)
NO
Ww
ito}
ss
ale
-90
fon
fos)
oo
he ee eee ee ee eo oN ooo ooo oo Molo nooo Noho—moio})
ax
TABLE 7 - Measured Mean
x/L = 0.840, r/r =
Cpa a
¥ U U
max ie) 10)
0.018 0.703 -0.091
0.058 0.8444 -0.109
0.0935 0.9096 -0.106
0.1304 0.956 -0.112
0.1645 0.9911 -0.115
0.2380 1.037 -0.110
0.3094 1.042 -0.103
0.4500 1.038 -0.089
0.6659 1.028 -0.073
1.1602 1.013 -0.051
1.7668 1.004 -0.038
6*
up
r
max
x/L = 0.9336, Tatas =
r-r u Vv
° RE aac
Tr U U
max ° fo)
0.0102 0.3206 -0.088
0.02 0.402 -0.091
0.055 0.546 -0.115
0.079 0.598 -0.117
0.112 0.652 -0.119
0.140 0.6955 -0.122
0.201 0.776 -0.121
0.228 0.8063 -0.119
0.300 0.878 -0.115
0.330 0.903 -0.113
0.375 0.930 -0.110
0.427 0.9524 -0.104
0.483 0.970 -0.094
0.555 0.984 -0.090
0.660 0.995 -0.085
0.9273 1.000 -0.070
1.340 1.000 -0.052
and Turbulence Velocity Characteristics for Afterbody 2
0.9618 tana = -0.1047
ane yi2 wi2 10024 -u'y! io
US Yo Uo Uo a Sy
0.0723 0.0364 0.0387 0.1366 0.169 0.064
0.0676 0.0328 0.0368 0.1234 0.176 0.204
0.0600 0.0320 0.0340 0.0950 0.164 0.328
Os, | OL0233- Wn0rOS | O07 Oda” O.718e
0.0402 | 0.0213 0.0240 0.0411 0.155 0.577
OLOLOL) (OX0091N) HOn00929) OF 00S35 eu On las moreso
O00 W005 O.00s j= 1.086
0.0025 0.002 0.002 2
0.002 0.002 0.002 =
0.002 0.002 0.002 =
0.002 0.002 0.002 u
6x 6
OOE%, =— > O08, => 3 OAs
max max
0.4839 tan a = -0.3216
ate yt fyr2 -u'v'-u'yv! BE
Uae fh URm Uae ome aq tenes
[e) (0) e} fe) Le
0/0569)" 0.030" 104035810093. On173. OxOg2
O06) O04 O04 @L095 Ol) OsOss7
0.0541 0.0343 0.038 0.0990 0.178 9.098
0.052 0.032 0.038 0.0801 0.154 0.141
0.053 0.031 0.037 0.0824 0.160 0.20
0.049 0.030 0.036 0.0785 0.170 0.250
0.049 0.030 0.033 0.0759 0.173 0.359
QOASH wn ON O27HMOROSONNOFOSI751 NO; LO SHONd Or
0.037 0.026 0.030 0.0495 0.165 0.539
O02 0.023 0,025. 0.0632 Msl5SS OLS
0.029 0.019 0.022 0.0167 0.099 0.670
0.018 0.014 0.015 0.0078 0.105 0.763
0.011 0.010 0.000 0.0027 0.084 0.863
0.004 0.002 0.003 0.0007 0.014 0.991
0.002 O.004 Oc) = 1.179
0.002 0.002 0.002 0 1.656
0.002 0.002 0.002 2.393
é* 6* 8
— = 0.1126, = = 0.1296, = = 0.560
max max max
oo0o0o00 00000000
OOOO OOOO) OLS)
0.05 0.0180
0.0697 0.0250
0.0780 0.0280
0.0809 0.0291
0.0623 0.0224
La L
6 / ae,
r (reo iste
0205 0.0124
0260 0.0157
0322 0.0195
0336 0.0203
0420 0.0254
0401 0.0243
0451 0.0273
0454 0.0275
0442 0.0268
0421 0.0255
0355 0.0215
0299 0.0181
155
156
TABLE 7 - (Continued)
x/L = 0
Sf S) Syeoy SS) SS) (Sy (S) ) (>)
DS) [3 (SESS) SOOO SOQ) S| ©
SEORLOFOROLOLOLOLO LOLOL OlOr©
SIOLOLOLOROZOLOLOLOnOFOre:
sGl7Ad)5) Se //ae
o’ ma
x
0.
OLOLO LOLOL OVOLOTOLOLOVORSre
SLOLOLOLSLOVOLOLOLOLOR OLS:
tan a -0.4077
yi2 w!
wv. U
oO fo)
0.0244 0.241
0.0304 0.030
0.0333 0.0352
0.0324 0.0359
0.0314 0.0355
0.0290 0.0354
0.0271 0.0315
0.0232 0.0290
0.0168 0.0219
0.007 0.007
0.002 0.002
0.002 0.002
0.002 0.002
0.002 0.002
6*
a 0.239
max
tan a = -0.3901
ui2 , Tanta
uv. Ua
ie} ie}
0.0175 0.0230
0.0289 . 0.0319
0.0301 0.0344
0.0314 0.0354
0.0317 0.0364
0.0287 0.0339
0.0219 0.0250
0.012 0.014
0.003 0.003
0.002 0.002
0.002 0.002
0.002 0.002
0.002 0.002
6*
oo = 0.233,
max
-u'v' -ulv!
100
uz q?
fe}
0.0531 0.185
0.0756 0.177
0.0850 0.164
0.0872 0.171
0.0800 0.165
0.0666 0.152
0.0545 0.156
0.0420 0.149
0.0199 0.126
0.00173 0.107
6*
5 eS 0229);
max
-u'v' -u'v!
100—
ug ae
OROZ19 me Omazil
0.0544 0.152
0.0692 0.161
0.0788 0.161
0.0770 0.156
0.0625 0.159
0.0355 0.157
0.00868 0.124
0.000417 0,033
6*
= 0294),
Ty
SO*OVOROROLOLOLOROI©:
Lis) eyo) S) SY OC)
Ww
>
i)
SOLO .OLOLOLOROIO
SEOEOLOLOVORO!S
.00166
.00209
00244
-00278
-00287
-00289
-00243
.00154
.000339
.78
Q L
o. 22
r J (x, +6,) 2x2
.0182
-0215
-0248
.9295
.0334
.0372
.0357
-0326
.0250
-0206
.0236
-0260
-0306
.0356
.0367
.0300
0201
OLOsOLOLOVOLOLO OS)
STOLOTOIOs OGIO: OS:
.0147
.0174
.0201
.0238
.0270
.0301
.0229
.0264
.0202
157
TABLE 7 - (Concluded)
x/L = 1.000 x/L = 1.057 x/L = 1.182
Yr ox xr uy r “x
r Um 2 U rr. U
max fe} max fe) max (e)
0.089 0.3460 0.000 0.4336 0.000 0.5760
0.102 0.3605 0.026 0.4440 0.040 0.5850
O25 Oo5705. Onl0R WpcW2 0.085 0.5942
0.135 0.3755 0.216 0.5623 0.124 0.6006
ONG) ‘OokES ~ Ooe08 O.6729 0.203 0.6216
0.204 OcI@ O462 O.7729 0.281 0.6764
0.238 0.4520 0.560 0.8488 0.398 0.7715
0.279 0.4975 0.675 0.9088 0.519 0.8538
0.307 0.5300 0.786 0.9595 0.637 0.9211
OokSh) 0.5083 lalAl W,OR 0.752 0.9581
0.393 0.6475 1.659 0.9856 0.904 0.9670
0.434 0.7020 2.151 0.9893 1.172 0.9780
0.474 0.7485 1.644 0.9887
0,36 Oocass 50 2.174 0.9988
0.646 0.8850
0.722 0.9325 ra Dacehs 8s
0.877. 0.9700 — = 0.1880
22 SZO 6* max
1.692 0.9970 Toe :
A007 10S Bae bra eee
2.265 1.0025 re
8
$* a= 100 Srila a5
ep = "0/2336 Tmax Pen eh
26 max
max
6*
BS 0,870
a6)
max
8
TF = 0.96
Theoretical Computation and Model and
Full-Scale Correlation of the Flow at the
Stern of a Submerged Body
A. W. Moore
Admiralty Marine Technology Establishment
Teddington, England
Go 136° Watilalss
Admiralty Marine Technology Establishment
Haslar, England
©British Crown Copyright 1979.
ABSTRACT
This paper describes an empirical method devised
for modifying measurements made at a propeller
position at the rear of unpowered bodies such that
the flow at the same position on a full-scale self-
propelled body may be predicted. ms
A boundary layer calculation procedure for esti-
mating boundary-layer velocity profiles at the
tail region of a body of revolution is discussed,
and the inclusion of a simple representation of a
propeller is described. Comparisons between
velocities measured at Reynolds numbers of order
10® and calculated velocities show reasonable
correlation both for unpowered and for powered
bodies of revolution. It is shown how the results
of boundary-layer velocity calculations are used
to derive a method for modifying flow measurements
at model scale to represent full-scale flow over
the propeller disc area. Comparisons are made
between predictions based on this method and
measurements on powered and unpowered bodies at
high and low Reynolds numbers.
1. INTRODUCTION
For many applications a self-propelled marine
vehicle has a propeller fitted at the rear of the
body where it gains in propulsive performance and
in cavitation performance by operating in the
relatively slow moving fluid in the hull boundary
layer. It follows that a fundamental requirement
for propeller design is a knowledge of the boundary
layer flow at the propeller position. This infor-
mation is not usually known since there are no
theoretical methods presently available for calcu-
lating the boundary flow at the rear of a powered
asymmetric body with appendages. An estimate of
the required flow field can be obtained from
measurements at model scale but as the Reynolds
number based on model length is considerably lower
158
than the full-scale value, it is necessary to make
some modification to the measurements to simulate
the effect of a thinner boundary layer at full
scale. If the flow field is measured on an un-
powered model, as is often the case, further
modification is required to allow for flow acceler-
ation due to the propeller.
This paper describes an approximate method
which has been developed for estimating corrections
required to flow measurements on unpowered bodies.
A boundary layer calculation procedure is briefly
outlined and then compared with data from tests on
axisymmetric bodies at low Reynolds numbers and
non axisymmetric bodies at both low and high
Reynolds numbers.
2. BOUNDARY LAYER CALCULATION
The method is based on the work of Myring (1973)
and only a brief outline is presented herein. An
iterative scheme is adopted in which a boundary
layer calculation is done for a given pressure
distribution over the body and a potential flow
calculation is done to calculate the pressure
distribution over the body with boundary layer dis-
placement thickness added. In the boundary layer
calculation procedure, an integral method is used
in which the laminar flow region is calculated
using the method of Luxton and Young (1962) and the
turbulent flow is calculated using a method similar
to that due to Head (1960). The transition point
must be specified and it is assumed that momentum
area and a shape parameter are continuous at
transition.
An important feature in Myring's method is his
treatment of the turbulent boundary layer in the
region of the tail. The usual boundary layer
assumptions become invalid in this region where the
ratio of boundary layer thickness to body radius
tends to infinity so Myring defines a momentum
area and a displacement area which overcomes the
problem and which reduce respectively to body
radius times momentum thickness and body radius
times displacement thickness far from the tail
where boundary layer thickness is small. A con-
ventional momentum integral equation is derived in
terms of the defined parameters and this is solved
using an empirical relationship for skin friction
coefficient which assumes that wall shear-stress
does not change sign. Therefore the method is only
applicable to bodies on which the boundary layer
remains attached. It is also assumed that the
variation of static pressure across the boundary
layer is negligible. This latter assumption has
been found to be incorrect for bodies with blunt
tails (i.e., cone angles greater than 30°) and an
empirical modification has been made based on the
work of Patel (1974) who developed independently
a method which is similar to Myring's but which
recognises the importance of static pressure varia-
tion. The modification introduced in the present
method is that the predicted velocity distribution
along the body is changed empirically in the tail
region, the change being related to differences
between measured and predicted velocity distribu-
tions at the rear of a given body with a blunt
stern. It has been found that this modification
results in improved correlation between measured
and predicted boundary layer velocity profiles.
A simple actuator disc representation of a
propeller has now been included in the potential
flow part of the calculation in order to give a
first approximation to the acceleration effects
on the flow caused by the action of the propeller.
PREDICTED
x MEASURED
0-7
LOCAL VELOCITY
MODEL VELOCITY
se)
Ca)
aT
° f 2 3 4
DISTANCE FROM HULL (%)
BODY LENGTH
FIGURE 1. Measured and predicted velocities 0.96L from
the bow of a body of revolution with tail cone angle of
AS
159
— — — PRESENT THEORY
NO MODIFICATION.
PRESENT THEORY WITH
EMPIRICAL MODIFICATION TO
ALLOW FOR STATIC PRESSURE
GRADIENT ACROSS THE THICK
BOUNDARY LAYER.
x MEASUREMENTS (PATEL, 1973)
LOCAL VELOCITY
MODEL VELOCITY
°
a)
nl 1 n
ie) !
en
4
2 3
DISTANCE FROM HULL (70)
BODY LENGTH
FIGURE 2. Measured and predicted velocities 0.96L from
the bow of a body of revolution with blunt stern [Patel
(1973) ].
3. RESULTS AT LOW REYNOLDS NUMBER ON AXISYMMETRIC
BODIES
Comparison between Predicted and Measured Results
The main interest in the present work is in the
prediction of boundary layer velocity profiles in
the tail region of a body and the results presented
in this section relate to model measurements under
conditions giving a Reynolds number based on model
length from 1 x 10© to 6 x 10°.
The velocity measurements shown in Figure 1 were
made at a station 0.96 L from the bow of a body of
revolution of length L and having a relatively fine
stern (cone angle 26°). The measurements are of
total velocity whereas the calculation method gives
values of velocity component parallel to the hull.
The theoretical curve in Figure 1 is obtained by
applying a small correction to the calculated
velocities to allow for the difference between
local flow angle and hull angle. It can be seen
that the resulting predicted curve gives values to
within 4% of the measured velocities. Detailed
measurements at the rear of a body of revolution
having a blunt stern have been reported by Patel
et al. (1973) and results for a station 0.96 L
from the bow are shown in Figure 2. The broken
line is the theoretical boundary layer profile
predicted from Myring's method with no allowance
for static pressure variation across the boundary
layer. This curve is significantly different from
the measured velocities which are more than 10%
less than predicted values in the inner part of
the boundary layer. Correlation between measured
and predicted results is improved when the empirical
modification allowing for static pressure variation
160
has been made and the resulting curve is seen in Velocity profiles close to the propeller plane
Figure 2 to be in better agreement with the measure- with and without propeller operating have been
ments. reported by Huang (1976). A typical example is
Theoretical results obtained with the simple shown in Figure 4 where it can be seen that the
representation of a propeller included in the Myring theoretical prediction for the unpowered
method indicate that the propeller can produce body gives velocities which tend to be too low.
large local changes in the boundary layer flow. An Nevertheless the discrepancy is less than 4% of the
example for which measurements are also available measured values.
is shown in Figure 3 where results are presented
for various stations along a body of revolution
having a fine tail and contra-rotating propellers. Reliability of Harmonic Analyses of Measured Flow
The measurements are of total velocity and were Fields
made with rakes of probes fixed to the body, the
rake at the forward propeller plane being removed The comparisons between theoretical prediction and
when the propeller was fitted. The velocity pro- measurement indicate that the calculation method
files on the unpowered body are well predicted gives a good approximation to velocity profiles
except close to the tail where boundary layer measured on powered and unpowered bodies of
separation appears to be present: it is noted revolution. The method is not expected to pre-
earlier that the calculation procedure will not dict velocities to better than 4% in absolute
predict separation. The changes produced by the terms but this is satisfactory for the purpose of
propellers are in surprisingly good agreement with deriving a simple means for modifying model measure-
predicted changes considering that an actuator ments to represent full-scale values. It is re-
disc representation of the propellers has been quired to obtain a representative flow field over
adopted. The velocities in a region close to the the propeller disc area and an essential starting
hull are under-predicted at the two rearmost point is to have reliable model data not only in
stations and at the station very close to the tail the sense that velocities can be measured accurately
the velocities near to the edge of the boundary at a given point, but also that, if a Fourier anal-
layer are also underpredicted. Apart from these ysis is made of the velocities measured during one
discrepancies the effect of the propeller is well revolution at a given radius, then a good approxi-
represented. mation to the magnitudes of wake harmonics is
1-0 ve) + OF O
PREDICTIONS
WITHOUT PROPELLERS
— — WITH PROPELLERS
MEASUREMENTS
+ WITHOUT PROPELLERS
O WITH PROPELLERS
/ = 0-763
/, = 0.621
LOCAL VELOCITY
MODEL VELOCITY
X/1 = 0.898
PLANE OF FORWARD PROPELLER
ie) | 2 3 4 O ! 2 3 4
DISTANCE FROM HUB (*/o)
BODY LENGTH
FIGURE 3. Velocity predictions and measurements on a torpedo-like body.
>|>
EVE
V/V
roy Ke)
4/4
Ww] Ww
>\|>
aia
<|w PRESENT
o}a
o|O O-4 THEORY EXPT.
4/|=
+ NO PROPELLER
0-3 —_—-— ® WITH PROPELLER
0-2
O:-1
fe} 1 (— n J
° I 2 3 4
DISTANCE FROM HUB (°%o)
BODY LENGTH
FIGURE 4. Velocity profiles immediately ahead of the
propeller DTNSRDC body 5225-1 [Huang (1976) ].
obtained. This information is relevant to the
estimation of unsteady forces generated by a pro-
peller. Some tests have been made in a wind tunnel
to assess the reliability of model measurements at
a typical propeller position on a three dimensional
body. Inflow non-uniformity was introduced by
fitting four struts to the body and the velocity
field was measured by a single traversable pitot-
static prove with head 1.5 mm in diameter. Measure-
ments at a given radius were made on different runs
with incremental steps of 1°, 2°, and 3° in the
circumferential position of the probe and 10 repeat
runs were made with 3° incremental steps. A Fourier
analysis of each set of results was made and the
harmonic spectra are summarised in Figure 5. It
can be seen that the standard deviations in the
magnitudes of wake harmonics are quite small show-
ing that misleading information concerning the
relative magnitudes of different wake harmonics
would not be obtained on any one run. The differ-
ences in magnitudes from the runs with 1°, 2°, and 3°
steps in probe position are also quite small in
general although a few wake harmonics, such as ll,
do show significant changes. No consistent trend
is observed in comparing amplitudes at low harmonic
number but at harmonic numbers greater than 25 the
amplitudes obtained from the run with 1° steps tend
to be higher than those from other runs, the impli-
cation being that choosing a coarser step size has
resulted in a small loss in accuracy.
The amplitudes of wake harmonics at harmonic
numbers greater than 20 are small (less than 0.005
times tunnel speed) except for harmonic numbers
which are multiples of 4. These higher values are
161
associated with the wakes from the four struts
which each produce a 'trough' in the measured flow
field. The high harmonic amplitudes at high har-
monic numbers implies a possible inaccuracy in
results from a Fourier analysis based on the finite
number of measured points. This was investigated
theoretically by assuming an idealised wake defect
giving a triangular waveform as indicated in
Figure 6. The number of wake defects and wake
width could be varied and for each assumed flow
field an exact Fourier analysis was obtained ana-
lytically and the results were compared with similar
analyses determined numerically with the waveform
described at discrete points as specified in the
measurements. Figure 7 shows results obtained with
4 narrow wake defects and 120 points specifying the
velocity profile. Two wake widths are considered;
when maximum wake width is 9° harmonics above 20
are in reasonable agreement with the exact solution
although harmonics below 20 are too low; when wake
width is reduced to 44° the amplitude of harmonics
from the exact solution falls slowly with increas-
ing harmonic number whereas the amplitudes deter-
mined numerically show no reduction in amplitude.
In this case, where points are specified every 3°
and the width of each wake defect is only 4%°,
‘aliasing' in the numerical results is not un-
expected. Such pitfalls in numerical analysis are
well known and Manley (1945) shows that erroneous
values in analyses of the type described above
might be expected at harmonic numbers given by
(N-jK) where N is the number of specified points,
K the number of wake defects and j is an integer.
A parametric study for triangular waveforms in
0:04 _
§ 1° steps (1 RUN)
2° STEPS (I RUN)
3° STEPS (MEAN OF I! RUNS)
STANDARD DEVIATION
— ce
0 SSS 22
N
\
ry \
w \)
w \
a \
uw \
4 0-02 N
Ww
z \
z
> |
- \
=
Ww \
> o0-Ol N
e \
a \
a \
Bs \
S \
© N
$ ©
> 3
«
<
=x
FIGURE 5. Harmonic analysis of different measurements
of a non-uniform flow field.
162
AMPLITUDE
e 277 277
kK
FIGURE 6. Theoretical representation of wake defects
in the flow field.
which N, K, and wake width were varied showed that,
in general, errors did not become significant until
wake width was less than twice the angular spacing
of the specified points, i.e., 720°/N.
4. USE OF THE PREDICTION METHOD IN PROPELLER DESIGN
A knowledge of the flow in the region of a propeller
is required first, in order to design it, and
second, to estimate its performance characteristics.
The former requires an estimate of the unpowered
mean velocity through the propeller position to-
gether with the radial variation of mean circum-
ferential velocity. Of the latter, the prediction
of unsteady propeller forces in particular also
requires the detail wake structure at the propeller
position in the powered condition.
The theoretical boundary layer prediction method
outlined in Section 2 cannot be used directly to
predict the above wake information for practical
vehicle configurations because of limitations such
as its restriction to unappended bodies of revolu-
tion. However, it can be employed indirectly by
using the method to predict the changes from model
testing conditions to full-scale vehicle conditions
and then applying these scale effects to available
model data.
The procedure adopted for the predictions dis-
cussed in the following section was to replace the
non symmetric, appended vehicle by an equivalent
body of revolution. Powered and unpowered boundary
layer predictions were then carried out and, by
assuming a simple power law for the boundary layer
velocity profile, the mean circumferential veloci-
ties were determined for the equivalent model and
full-scale bodies. In this way it was possible to
estimate at any position the scale effect upon the
unpowered wakes, the propeller induction effects,
and any combination of the two. These effects were
then applied to all the measured unpowered model
data to give predictions of both model and full-
scale powered wakes for comparison with measured
data.
5. COMPARISON BETWEEN PREDICTED AND MEASURED RESULTS
ON NON-SYMMETRIC BODIES
As part of a programme to investigate the effects
of scaling and propejler induction on wakes, experi-
ments have been carried out on two practical vehicle
forms covering a range of Reynolds numbers, based
on body length, from approximately 1 x 10” to
6 x 108. The two vehicles concerned were propelled
by a single centre line propeller and were fitted
with a set of cruciform after-control surfaces just
ahead of the propeller. The afterbody form was
axisymmetric in both cases, one vehicle having a
fine stern (vehicle A) and the other a blunt stern
(vehicle B).
The low Reynolds number data were obtained in the
ship tanks at AMTE (Haslar) using small conventional
pitot static tubes. For body A, measurements were
made at a position 26 percent of the local control
surface chord aft of the control surface trailing
edge. This corresponded to 28 percent of the
propeller diameter forward of the propeller. The
measurements were made at 2° intervals over an angle
of approximately 90° centred on one control surface,
and at radial distances from the body surface of
12.5 percent and 25 percent of the propeller radius.
For body B, data were obtained 24 percent of the
local control surface chord aft of the control
surface trailing edge, corresponding to 22 percent
of the propeller diameter forward of the propeller.
In this case 6 pitot static tubes were used cover-
ing a range of radial distances from the hull of
12.5 percent to 65 percent of the propeller radius.
The high Reynolds number data were obtained from
trials carried out at sea on vehicle A using 5 con-
°
(@) N=120, K=4, WAKE WIDTH 9.
“10
X NUMERICAL
EXACT
AMPLITUDE
°
ur
4 6 12 16 20 24 28 32 36 40 44 48 52 56 60 64
HARMONIC NUMBER
1 °o
(®) N=120, K=4, WAKE WIDTH 4/2.
10
fo)
a
KIX OMX YK OX EN:
AMPLITUDE
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64
HARMONIC NUMBER
FIGURE 7. Comparison between an exact Fourier analysis
of the theoretical velocity profile and a numerical
analysis of the same profile specified at a discrete
number of points.
ventional pitot static tubes at each of the above
radial positions.
The high Reynolds number measurements could only
be carried out at self-propulsion conditions.
However, the model experiments in the ship tank
were run over a range of propulsion conditions,
the model speed, propeller rpm, and resistance being
recorded.
Analysis of Experimental Data
The low Reynolds number results for vehicle A are
presented in Figures 8 and 9 while the equivalent
high Reynolds number trail data is given in
Figure 10. For body B the available data is
restricted to that obtained in the low Reynolds
number ship tank tests and the results are pre-
sented in Figures 11 to 13. For the sake of
brevity the velocity profiles given in Figures 11
to 13 have been limited to those for alternate
measurement radii.
It can be-shown that the propeller diffusion
ratio, defined as the ratio of the mean velocity
through the propeller to the unpowered mean wake
velocity through the propeller position, can be
obtained from the propeller thrust or hull resis-
tance together with the mean volumetric wake and
thrust deduction. Thus, using the model powered
and unpowered resistance measurements and values
DIFFUSION RATIO
1-308 ————— SELF PROPULSION.
1:226 —— — —
1-130 -—-—-
0-8 |:000— --—— UNPOWERED
0-7
0-6
os
LOCAL VELOCITY
MODEL VELOCITY
o4
CONTROL
SURFACE
50 40 30 20 10 ° 10 20 30
ANGULAR POSITION (°)
DIFFUSION RATIO
1-308 ——— SELF PROPULSION
1-226—-———
1-130— -—
1!- OOO— --—UNPOWERED
0-6
LOCAL VELOCITY
MODEL VELOCITY
Soa CONTROL
SURFACE
50 40 30 20 10 fo) 10 20 30
ANGULAR POSITION (°)
163
of wake and thrust deduction obtained from previous
model tests, the propeller diffusion ratio has been
calculated for the model propulsion conditions
pertaining during the experiments. Similar calcu-
lations have been carried out for the sea trial
conditions using data obtained from previous pro-
pulsion trials. The results of these analyses are
given on Figures 8 to 13, and also in Table 1 which
summarises the experimental and trial conditions.
The velocities just ahead of the propeller have
been averaged at each radius to give the variation
of the mean circumferential velocities with diffu-
sion ratio presented in Figures 14 and 15. [In an
attempt to quantify the secondary flow component in
the above velocity profiles the ratio of the mean
peak velocity to the mean minimum velocity has been
evaluated and plotted in Figures 16 and 17. The
normal parameter used to specify the velocity defect,
namely the ratio of the minimum velocity in the
"trough' to the mean velocity at the edge of the
"trough' is given in Figures 16 and 18. No values
are given for the inner radius on body B because,
as can be seen from Figure 11, the wake defect is
not clearly defined at this position. The latter
parameter is also compared in Figure 19 with an
empirical relationship based on two-dimensional
data [e.g., Raj (1973)].
The results of using the Myring based boundary
layer prediction method as described in Section 4
for the powered model and trial conditions are
also plotted in Figures 14 to 18.
FIGURE 8. Vehicle A model velocity profiles
at position 12.5 percent of propeller radius
from the hull.
EYe)
FIGURE 9. Vehicle A model velocity profiles
at position 25 percent of propeller radius
from the hull.
164
DIFFUSION RATIO 1-160 SELF PROPULSION.
o-8
a °fo PROPELLER RADIUS
z FROM HULL
ares °.
>|Vo ." 1
clo O-7 25 fo
oa
O}>
Ww] yy O-6
WE
ajo
ol
ro) o5
as 4 CONTROL
=) SURFACE
re
FIGURE 10. Vehicle A full-scale velocity ° 1 1 | —1__ 1 ———1_____1_____
profiles 12.5 percent and 25 percent of 50 40 30 20 10 ° 10 20 30 40 50
propeller radius from the hull. ANGULAR POSITION (°)
DIFFUSION RATIO.
1-233 SELF PROPULSION
IRe=———
Lett
O7 | -OO0O0— --—_UNPOWERED
ai
ole
glo
= a)
ry od
>|>
a
2|8
ro} fe)
a)\=
é CONTROL
SURFACE
FIGURE 11. Vehicle B model velocity pro-
files at position 12.5 percent of propeller 50 40 30 20 ite} ° 1o 20 30 40 50
radius from the hull. ANGULAR POSITION (°)
DIFFUSION RATIO
1-233————_SELF PROPULSION
OS 6 Ss
Dols <1
0-8
>
Ele
oe
Slo
gla
SI>
4
dir
ule
o|O
par) > 3
CONTROL
SURFACE
FIGURE 12. Vehicle B model velocity pro-
files at position 33.5 percent of propeller 50 40 30 20 !o ° !0 20 30 che) we)
radius from the hull. ANGULAR POSITION (°)
Discussion of Measured Data
It is clear from Figure 16 that the relative magni-
tude of the velocity defect at the two radii con-
sidered on vehicle A is virtually unaffected by the
propeller, and is subject to only a very small
scale effect. The latter gives rise to a reduction
in the depth of the velocity defect between the
model and full-scale equivalent to an increase of
between 1 percent and 3 percent in the ratio of
minimum velocity in the 'trough' to the mean veloc-—
ity at the edge of the 'trough'. It can be seen
from Figure 19 that for vehicle A the actual values
of the velocity defect are considerably lower than
predicted by the empirical relationship derived
from two-dimensional test results. This is not
DIFFUSION RATIO
1-233 ————SELF_ PROPULSION
1-175 -—— —
1th) ——-—
1-0 |-OOO— --—UNPOWERED
0-9
>|>
Ele
1S) ©
alg o-8
ry
wis
LA i
2 Fr 0-7
19,
Sls
4 CONTROL
SURFACE
50 40 30 20 Te} ° 10 20 30
ANGULAR POSITION (°)
surprising since the two radii concerned are close
to the hull and the velocity defect is developing
in a complex three-dimensional flow field influenced
by the secondary flow and this is possibly leading
to a more rapid mixing of the flow. The results
obtained over a much larger distance from the hull
on vehicle B support the above hypothesis since it
can be seen from Figures 18 and 19 that as the
distance from the hull increases the magnitude of
the wake defect increases and approaches the two-
dimensional value. The model results for vehicle
B shown in Figure 18 tend, in general, to indicate
a small increase in the depth of the wake defect
as the propeller diffusion ratio increases. The
maximum value of this increase in the wake defect,
between the model self-propelled and unpowered
condition, is only of the order of 3 percent. This
change is somewhat surprising since the propeller
produces a favourable pressure gradient aft of the
control surfaces, and on the evidence of two-
dimensional data this would be expected to reduce
the wake defect.
® FULL SCALE MEASUREMENTS.
+ MODEL EXPERIMENTS.
FULL SCALE PREDICTIONS
0-8
‘alte Jo PROPELLER 2 e
olo RADIUS FROM // - ea
° S HULL ae a
J|4 07 a
wis
>
al4 MOpeL. Ma
< 3 o6 REDICTIONS
Fe
z|\z
5|4
2\5
o|% 0-4
= = FULL SCALE MODEL SELF
w = encore PROPULSION PROPULSION
2
1-O | 1-2 193 1-4 rs
PROPELLER DIFFUSION RATIO.
FIGURE 14. Mean circumferential velocity in the measur-—
~ ing plane for vehicle A.
165
FIGURE 13. Vehicle B model velocity pro-
50 files at position 54.5 percent of propeller
radius from the hull.
In contrast to the velocity defect the secondary
flow can be seen from Figures 16 and 17 to be
significantly reduced by the presence of the
propeller, this reduction becoming larger as the
diffusion ratio increases. Additionally, at equal
propeller diffusion ratio, the full-scale secondary
flow is significantly less than measured on the
model. From the data obtained on vehicle A
(Figures 14 and 16) it can be seen that, comparing
the results at model and full-scale self-propulsion
conditions, the magnitude of the secondary flow and
the mean circumferential velocity at the two radii
considered agree to within 2 percent and 3 percent
respectively. Although comparison between the
velocity profiles is difficult because of the non-
symmetry of the trial data, Figures 1 to 3 indicate
that at these conditions there is also reasonable
agreement between the velocity profiles. These
results indicate a possible condition for similar
1-0
°lo PROPELLER
> 0-9 | RADIUS Bra
S FROM HULL Ie aug
> 9 = =
fe) = =
Ela 0-8 | 65°0+- 4 +
OU} w A 7
o> ee ap 7
- +
Ny 54°54 +7
S|5 t
Oo O-'7 ee
|= 2 =
<jw 44-04 on esa
z if + se
. =
ial f° 6 y ES
| 2 2 a
| O a ny Oe
s|= 33-547 we
=) ° as
rs} 3 0-5 ee a
4 w a + a
(5)
ar 23-O+ a oe
z $ 0-4 Z
| _ © +MODEL EXPERIMENTS
= 7 ———PREDICTIONS
Osa iene
; MODEL SELF
UNPOWERED PROPULSION
1-0 t+ 1-2 1-3 1-4
PROPELLER DIFFUSION RATIO
FIGURE 15. Mean circumferential velocity in the mea-
suring plane for vehicle B.
166
@®FULL SCALE MEASUREMENTS
+MODEL EXPERIMENTS
°/o PROPELLER RADIUS
FROM HULL.
MODEL PREDICTIONS
1-3
MAXIMUM SECONDARY FLOW VELOCITY
MINIMUM SECONDARY FLOW VELOCITY
.
° FULL SCALE
25 /o __ PREDICTIONS.
— — —,MODEL PREDICTIONS
5
FULL SCALE
UNPOWERED PROPULSION
lo Il 1-2 1-3
PROPELLER DIFFUSION RATIO
MODEL SELF
PROPULSION
1-4
MINIMUM DEFECT VELOCITY
VELOCITY AT DEFECT EDGE
FIGURE 16. Relative magnitude of the secondary flow
and velocity defect for vehicle A.
inflow to the propeller at model and full-scale;
however, it should not be regarded as a general
conclusion on the basis of this one experiment.
Additionally although the propeller inflow may be
similar at self-propulsion the propeller thrust
loading, as indicated by the diffusion ratio, will
be different.
Comparison between the wake defect and secondary
flow model measurements for the two vehicles
(Figures 16 to 18) show generally similar magnitudes
for the former, but a much larger secondary flow in
the case of the body with the fuller afterbody.
The latter effect can also be seen in the velocity
profiles given in Figures 8 and 1l.
Comparison between Predicted and Measured Results
It can be seen from Figure 14 that the mean circum-
ferential velocity predictions for the powered model
of vehicle A are always higher than measured. The
maximum differences occur at model self—propulsion
conditions and are 7 percent and 4 percent for the
positions 12.5 percent and 25 percent of the pro-
peller radius from the hull respectively. Both
the measured data and the predicted velocities can
be seen to vary linearly with propeller diffusion
ratio. For the blunter stern, Figure 15 indicates
that for radial positions between 23 percent and 44
percent of the propeller radius from the hull the
predictions of mean circumferential velocity are
generally in good agreement with the measured data.
For the two outer radii the predictions tend to
be high as in the case for body A, the maximum
errors at model self-propulsion being of the order
of 4 percent. However, for the innermost radial
position, the powered predictions are up to 14 per-
cent below the measured values. t is apparent
for Figure 15 that, in contrast to the other radii,
the model results for this position are not linear
with propeller diffusion ratio because of the low
velocity obtained in the unpowered condition. Since
the measured data was linear at a similar radial
position for body A this suggests that the poor
powered prediction of velocity for body B is due to
the low unpowered velocity measurement which is
used as the datum for the prediction. This low
measured velocity may be the result of flow separa-
tion on the vehicle with the blunt afterbody which
is suppressed by the favourable pressure gradient
produced when the propeller is operating.
Comparison between the full-scale and predicted
mean circumferential velocities in Figure 14 show
the latter to be less accurate than for the model
case, the predicted values being 15 percent and 9
percent high for the inner and outer positions
respectively. However, correlation of propulsion
data from sea trials and model experiments on
vehicle A suggest an equivalent full-scale hull
Reynolds number of one-tenth of the true value and
TABLE 1 Experimental and Trial Conditions
Hull
Reynolds Diffusion
Vehicle Conditions Number Ratio Remarks
A Model 13 xe 1Od mesos Self propulsion
1.226
1.130
A Trial 5.5 x 108 1.160 Self propulsion
B Model ToD 3 AO? “a Dsis) Self propulsion
1.175
abSatalat
MAXIMUM SECONDARY FLOW VELOCITY
MINIMUM SECONDARY FLOW VELOCITY
FIGURE 17.
+ MODEL EXPERIMENTS
— — —PREDICTIONS
2-0
°lo PROPELLER RADIUS
FROM HULL. 4.9.4 2-309
i)
4-192 PREDICTION FOR
12°5°lo RADIUS.
23-0 12-5
ee) Ua 1-2 1-3 1-4
PROPELLER DIFFUSION RATIO.
Relative magnitude of the secondary flow
for vehicle B.
MINIMUM DEFECT VELOCITY
FIGURE 18.
+ MODEL EXPERIMENTS
——-—PREDICTIONS
fo PROPELLER RADIUS
FROM HULL.
23 -O4+——__+—__ + |
33-5 4+——=_ 5§ —+—
44-O0+—=
54°5 See ln
O-8 — —
a arr
6510 cay
VELOCITY AT DEFECT EDGE
9
)
MODEL SELF
UNPOWERED PROPULSION
10 Vl 1-2 1-3
PROPELLER DIFFUSION RATIO
Relative magnitude of the velocity defect
for vehicle B.
167
MODEL DATA I VEHICLE A
MODEL DATA 2 VEHICLE B
1-0
fo PROPELLER RADIUS
FROM HULL
0-9| 33-5 g1 125 & 25-0
wi
Zlo
sige eee EMPIRICAL VALUES
8 of ‘OR APPENDAGE
an
rare GEOMETRY.
Ww O-7
air
o]o
4
-
w)< 0-6}
>
=
3|5
P 3 ro) O-5
z aa)
z|5
0-4{
al sae ie 1 = SS
° O02 O04 06 O8 10
DISTANCE FROM CONTROL TRAILING EDGE
LOCAL CONTROL SURFACE CHORD
FIGURE 19. Variation of model velocity defect with
distance from the control surface.
if this is used for the predictions the above
differences become + 1 percent and - 2 percent
respectively. Thus the speed trial and full-scale
wake data become compatible and both suggest a
scale effect on the flow velocity for the vehicle
A with the finer stern much smaller than predicted.
This may be due to the fact that the full-scale
vehicle is hydraulically rough at all but the very
lowest speeds while the prediction method assumes
hydraulically smooth conditions.
The process of adding the predicted mean circum-
ferential velocity changes to all measured veloci-
ties are described in Section 4 naturally leads to
a change in the ratios used herein to describe the
relative magnitudes of the velocity defect and
secondary flow. For the velocity defect Figures 16
and 18 show that the predicted magnitude decreases
slightly with increasing diffusion ratio such that
at model self-propulsion the relative magnitudes
are 3 percent higher than measured for body A and
up to 6 percent for body B. The predicted relative
magnitude of the wake defect at the full scale
condition is within 2 percent of that measured,
although as already noted the absolute velocities
are 15 percent and 9 percent higher than measured.
The use of a smaller scaling effect based on the
equivalent Reynolds number discussed above would
slightly reduce the above error in predicted
velocity defect.
The predicted relative magnitude of the secondary
flow can be seen from Figures 16 and 17 to decrease
with increasing diffusion ratio but at a slower
rate than actually measured on the models. Thus,
the propeller is having an influence on the develop-
ment of the secondary flow in addition to the simple
change in relative magnitude arising from the
propeller induced velocity. It is clear that at
model conditions, the difference between the
measured and predicted secondary flow is much
greater for the blunter afterbody form of vehicle
168
B. At model self-propulsion conditions these
differences are up to 5 percent for vehicle A but
60 percent for vehicle B. The secondary flow pre-
diction for the full-scale conditions on body A
given in Figure 16 can again be seen to be higher
than the measured values but only by up to 4 percent
at the two radii considered. In this case the use
of a smaller scaling effect would lead to higher
predicted values such that the differences between
these and the measured values would increase to the
order of 6 percent.
The above results show that the agreement between
the measured and predicted data has been limited and
further work is required before the proposed scaling
method can be regarded as satisfactory. The princi-
pal requirement is for further high Reynolds number
data and it is proposed to obtain this by additional
full-scale trials, together with experiments on
models in a compressed air wind tunnel.
6. CONCLUSIONS
An integral boundary-layer calculation method for
bodies of revolution is shown to give a good pre-
diction of boundary layer velocity profile for
attached flows in the tail region of a body.
Inclusion of a simple actuator disc representa-
tion of a propeller in the calculation method gives
a reasonable first approximation to the effect of a
propeller on the flow.
Comparison between results from Fourier analyses
of measurements from runs repeated a number of
times and of measurements made with different
incremental steps in probe position indicates that
wake harmonics can be determined reliably. from
measurements at model scale.
Fourier analyses of idealised velocity profiles
representing wake defects in an otherwise uniform
flow field have been obtained analytically. Com-
parison between these results and numerical harmonic
analyses of the same profile specified at a dis-
crete number of points shows no significant differ-
ences in the amplitudes of wake harmonics at high
harmonic number provided that the width of the wake
is not too small.
The measurements presented herein indicate that
the velocity defect produced behind a control sur-
face is only slightly affected by either the
presence of a propeller aft of the control surface,
or by the change in Reynolds number from model to
full-scale.
Near the hull, where the flow is influenced by
secondary flow effects, the velocity defect behind
a control surface is much smaller than predicted
from two-dimensional data. For positions outside
the influence of the secondary flow the velocity
defect approaches the two-dimensional value.
The velocity defect is of a similar order of
magnitude for the two bodies examined. However, the
secondary flow effects are significantly larger
for the vehicle with the blunter stern.
The secondary flow produced by the interaction
of a control surface with the hull boundary layer
is reduced significantly by the presence of a
propeller aft of the control surface, and from
model to full-scale conditions. This reduction
increases with increasing propeller diffusion ratio.
By using the unpowered model measurements as
datum it has been possible to predict the model
powered mean circumferential velocities to within
4 percent for radial positions from the hull greater
than 12.5 percent of the propeller radius. At this
radius itself, the predictions are within 7 percent
for the finer stern model and 14 percent for the
fuller stern; however, the latter may be due to
separation effects which are not taken into account
in the prediction method.
Predictions of the mean circumferential velocity
at the full-scale conditions for the vehicle with
the finer stern are high by up to 15 percent. If
the ship prediction is made at a reduced Reynolds
number suggested by speed trial results the pre-
dictions come within 2 percent. Predictions of the
powered velocity defect are wit in © percent for
model conditions and 2 percent for ship conditions,
the latter figure applying to either the true or
reduced full-scale Reynolds number. Predictions of
the model powered secondary flow are within 5 per-—
cent for the body with the finer stern, but up to
60 percent for the fuller form. However, for the
full-scale conditions obtained on the finer stern
the predictions are within 4 percent at the true
Reynolds number, and 6 percent at the reduced value.
A practical method of estimating propeller in-
duction and wake scaling effects has been proposed
and demonstrated to give limited agreement with
model and full-scale data. Further experimental
data are required to refine the method and to this
end high Reynolds number model experiments are
planned to be carried out in a compressed air wind
tunnel, and further full-scale trials scheduled.
REFERENCES
Head, M. R. (1960). Entrainment in the turbulent
boundary layer. British ARC, R & M 3152.
Huang, DT. T., S. Santelia, HH. LT. Wang, and IN-vICE
Groves. (1976). Propeller/stern/boundary-layer
interaction on axisymmetric bodies: theory and
experiment. DINSRDC Rep 76-0113.
Luxton, R. E., and A. D. Young. (1962). Generalised
methods for the calculation of the laminar com-
pressible boundary layer characteristics with
heat transfer and non-uniform pressure distri-
bution. British ARC, R & M 3233.
Manley, R. G. (1945). Waveform analysis. Chapman
and Hall, London.
Myring, D. F. (1973). The profile drag of bodies
of revolution in subsonic axisymmetric flow.
RAE TR 72234. (Unpublished) .
Patel, V. C. (1974). A simple integral method for
the calculation of thick axisymmetric turbulent
boundary layers. Aeronautical Quarterly,
NG Iie ks
Patel, V. C., A. Nakayama, and R. Damian. (1973).
An experimental study of the thick turbulent
boundary layer near the tail of a body of
revolution. TIowa Institute of Hydraulic
Research Report, 142.
Raj, R.-, and B. Lakshminarayana. (1973). Charac—
terisics of the wake behind a cascade of airfoils.
J. Fluid Mechs, 61, Pt. 4.
Experimental and Theoretical Investigation
of Ship Boundary Layer and Wake
Shuji Hatano,
Kazuhiro Mori and Takio Hotta
Hiroshima University, Hiroshima, Japan
ABSTRACT
Characteristics of the boundary layer and wake flow
of ships are investigated experimentally and at-
tempts are made to estimate their velocity distri-
butions.
Boundary layer characteristics, before the onset
of separation, are studied; a three-dimensional
boundary layer calculation is carried out by the
integral method, while examining the boundary layer
assumptions and the validity of auxiliary equations
by direct measurements of velocity and static pres-
sure profiles in boundary layer as well as skin
friction distribution on hull surface.
Assuming that the wake is the domain of influ-
ence of the boundary layer and consists of three
sub-regions, i.e., vorticity diffusion region,
separated retarding region, and viscous sublayer,
different governing equations for each sub-region
are derived by local asymptotic expansions.
Velocity distribution in the vorticity diffusion
region is estimated in two steps: first, vorticity
distribution is found by solving the vorticity
diffusion equation, then velocity distribution is
calculated from the obtained vorticity distribution
by invoking Biot-Savart's law.
Satisfactory agreements are attained between
calculations and measurements both for boundary
layer and wake.
1. INTRODUCTION
Introductory Remarks
The prediction of the viscous flow field around
ship hulls, boundary layer on the hull surface, and
the wake, is one of the most important problems in
ship hydrodynamics. Important design-conditions,
such as estimations of viscous resistance or wake
distribution on a propeller disk, are all closely
connected with this problem. Instabilities of ship
169
Maneuvering and propeller-excited-vibrations are
also presently urgent problems in practice; they
are also fundamentally connected with the viscous
Calculations of a ship boundary layer have been
carried out by many investigators during the last de-
cade; e.g., Uberoi (1969), Gadd (1970), Webster and
Huang (1970), Hatano et al. (1971), Himeno and Tanaka
(1973), and Larsson (1975). They have solved bound-
ary layer equations in integral forms. Cebeci et al.
(1975), as well as Soejima and Yamazaki (1978), has
tried to solve them by the finite-difference method.
Such remarkable progress in ship boundary layer
calculations are mainly due to studies of two-
dimensional boundary layers and to the use of high
speed computers. Though some of them yield good
results, an absence of experimental examination of
boundary layer assumptions or auxiliary equations
can be found when applying them to shiplike bodies.
Experimental examinations are very important because
most of auxiliary equations are derived from two-
dimensional experiments.
On the other hand, as to the ship wake, many
experimental studies have been carried out not only
for ship models but also for full scale ships, e.g.,
Yokoo et al., (1971) and Hoekstra, (1975) mainly
discussed the prediction of full scale wake charac-—
teristics based on model wake survey.
Rational theoretical studies are still more im-
portant. As to theoretical studies of wake, we
must retreat to problems of flow behind rather
simple obstacles like flat-plates, circular cylin-
ders, or bodies of revolution. Even in such cases,
most treatments are based on potential theory such
as free-streamline theory or cavity-flow theory,
reviewed by Wu, (1972). However, because vortici-
ties existing within wakes are mainly generated in
boundary layers of hull surfaces and shed into wakes
viscously and convectively through separations, the
prediction of wake flow should be treated in close
relation to boundary layer flow.
The previous works by Hatano et al.,
(1975, UIT) 5
170
were carried out from this standpoint. But they
are only the beginning of research on ship wakes
and many future problems were pointed out, especially
requirements for further experimental studies.
The present authors are firmly convinced that,
for such viscous flow problems, marriages of experi-
mental and theoretical studies are primarily impor-
tant in order to make further progress. Because
of this, the present paper is divided into two parts;
experimental studies on ship boundary layers and
wakes (Section 2 and 4), and theoretical studies
and numerical calculations (Section 3 and 5).
Coordinate Systems and Models Used
Two coordinate systems are employed throughout the
present paper. One is the right-hand linear coordi-
nate system, O-xyz, whose origin is at midship and
on the waterplane and the oncoming flow, Ug, is in
the x-direction. The other is the streamline coor-
dinate, x]x2x3; the curves of constant x2 coincide
with potential flow streamlines on hull surface and
x3 is normal direction to hull surface (Figure 1).
All quantities are dimensionless by half ship
length 2 (=L/2), ship speed Ug, and fluid density p,
unless specified in another form.
For the present research three ship models,
GBT-125, GBT-30, and MS-02 were used whose body
plans with potential streamlines and principal di-
mensions are shown in Figure 2 and Table l.
GBT-125 and GBT-30 are practical tanker ship
models, similar in geometry to each other. GBT-125
is a double model and was used under submerged con-
ditions for studies of boundary layer flow. MS-0O2,
which was used for the studies of wake flow, has a
rather simple stern form; the framelines are ellip-
tic and given by the equation,
CHlOlae
0.7
xa
Yo = bo | 1- (=) 1 (2)
and bg is the half breadth of the waterplane at
x = 0.4 (S.S.3) and d is the draft. The remainder,
(x < 0.4), has a practical hull form. This is be-
cause the practical stern form produces a very com-
plicated stern flow, e.g., an intensive longitudinal
vortex, not suitable for the present investigations.
Experiments were carried out in the circulating
water channel and the towing tank of Hiroshima
University.
x >a (a=0.4) (1)
Coordinate systems.
TABLE 1 PRINCIPAL DIMENSIONS OF MODELS
GBT-125 GBT-30 MS-02
1.250/™) 3.000'") 3.000!™)
.193 -462 -485
065 o Si 165
Ch .836 .836 .768
NOTATION
L, ship model length and half length
Q
b ship model breadth
Cy, block coefficient of the ship model
d ship model draft
p density
v kinematic coefficient of viscosity
Ve eddy viscosity coefficient
g gravity acceleration
Up velocity of oncoming flow, ship speed
F, Froude number =U0/VgL
Re Reynolds number =UoL/v
€ small parameter for asymptotic ex-
pansions = Rg 8
X,Y,Z orthogonal linear coordinates
X],*%2,%3 orthogonal curvilinear coordinates
E,n,G distances along X],X2,x3 coordinates
hj,ho,h3 corresponding metric coefficients
K,,K2 convergences defined by Kj =
Sel ohoney, eeu il ohy
hyh2dx] 2 hyho 9x9
normalized distances for vorticity
*) diffusion region, separated re-
tarding region, and viscous sub-
layer respectively
q velocity vector
Gy, viscous part of velocity vector
u,v,w velocity components in x,y,z direc-—
tions excluding uniform flow
41/92/43 mean velocity components in x) ,x2,xX3
: directions
fluctuating velocity components in
X1],X2,*3 directions
U, resultant velocity at boundary layer
edge
velocity components at boundary layer
edge in x)],X9,x3 directions
UTITY LUTY 2
~
SIwsaidr
x
WX OUT?
' ' '
G1192793
U,,V1 Wy
RorirYirWi| asymptotic terms of normalized mean
CLOSE velocity for vorticity diffusion
Cae ) region, separated retarding region,
IL Aoo oo
and viscous sublayer region
af, 07 Wi } asymptotic terms of normalized fluc-
(Gil pAeb oo) tuating velocity for separated re-
tarding region
wW vorticity vector
Wye 1 Wy 1 We vorticity components in x,y,z di-
rections
W1,W2,W3 vorticity components in X],X2,X3
directions
asymptotic terms of normalized vor-
ticity for vorticity
diffusion region
p pressure
STREAMLINE NO. 1
GBT-125
& GBI-30 3
STREAMLINE NO.1
MS -02 |
ee ae
Wes
@
99 (@=11 92, 000)
6
* *
617697911
991,912,922
H
pressure far upstream
pressure coefficient =(p-py)/z pu2
al,
0
asymptotic terms of normalized
pressure
boundary layer thickness
three-dimensional boundary layer
thickness parameters defined by
Eqs. (5)),) (19)
shape factor of streamwise velocity
profile =6}/6)
angle between surface streamline
and external streamline direction,
positive in x2 direction
index for power-law velocity profile
parameter for wake part of wall-wake
law in q}, q2 components
coefficients of wall-wake law
wall and wake functions of wall-
wake law defined by Eq. (10)
resultant skin friction
components of skin friction in x}
and x2 directions
friction velocity
entrainment function
parameter for separation
positions of onset of separation
and reattachment
integral region for induced velocity
gradient vector £
symbol of orders f=O(€); lim ¢ =M
(M:constant) eo)
2. EXPERIMENTAL STUDIES ON BOUNDARY LAYER
Kinds of Experiments and Measuring Techniques
In order to examine the boundary layer assumptions
and the validity of semi-empirical equations in
case of ship-like bodies, the following kinds of
experiments were carried out [Hatano et al.,
(1978) ].
7
FIGURE 2. Body plans and potential
flow streamlines of models.
Static pressure measurements on hull surface
Static pressure holes of 0.6mm were arranged on the
hull surface along streamlines and the static pres-
sure was measured by towing ahead and astern.
Static pressure measurements in boundary layer
Static pressure in the boundary layer was measured
by using a static pressure tube. It is 1.2mm in
diameter with two 0.4mm $¢ holes on diametrically
opposite sides. A traverser with a micrometer was
used to move the probe normal to the hull surface.
The preliminary experiments showed that the static
pressure was free from incident flows whose attack
angles were less than 20°.
Velocity measurements in boundary layer
A total head probe, made from hypodermic tubing of
outside diameter 0.28mm and 2.7mm respectively,
was mounted on the traverser. Total pressure was
measured after locating flow directions by yawing
the directionally-sensitive hot film probe. Using
the measured static pressure, velocity was estimated
and decomposed into streamwise and crossflow
components.
Local skin friction measurements
Local skin friction on the hull surface was mea-
sured directly by a floating-element type friction
meter [Hotta, (1975)]. The floating element is 14mm
in diameter with gaps of 0.05mm to the mounting case
and balanced by electromagnetic force.
All experiments described above were carried out
using the GBT-125 under submerged conditions at a
depth of about 6 times the draft of the model. The
Reynolds number was kept constant at 10°.
172
0.4
0.0
0.2
0.0
FIGURE 3. Static pressure distribution on
the hull surface (GBT-125).
Experimental Examinations of Boundary Layer Assump-
tions and Semi-Empirical Equations
Boundary Layer Assumptions
The usual first approximate calculations of the
boundary layer were carried out under the assump-
tion that the static pressure is constant across
boundary layers and is equal to the inviscid flow
pressure. These assumptions are open to experimental
examination when the boundary layer thickness is
not thin, especially in the case of ship-like bodies.
Static pressure distributions on the hull sur-
face along streamline Nos. 5, 7, and 11 are shown
in Figure 3 with calculated potential flow pressures.
Potential flow calculations were carried out by the
well-known surface-source method [Hess and Smith,
(1962) ] representing the hull by 254 x 2 small rec-
tilinear panels. Static pressures while being
towed onward are in good agreement with those calcu-
lated, except near the stern, where pressure has
not recovered and is slightly low. However, towing
astern shows good agreements even near the stern.
This means that displacement effects of the boundary
layer are appreciable near the stern.
Figure 4 shows static pressure profiles in the
boundary layer . It was observed that pressure pro-
files are almost constant across the boundary layer
except for some positions where the pressure is mono-
tonically increasing or decreasing in that normal
direction. The tendencies of increments are signif-
icant at S.S.1% or S.S.1% of streamline No. 11. This
can be referred to the centrifugal force due to the
small radii of curvature of the bilge keel. On the
other hand, a decrease can be found:for all the
streamlines at S.S.% or S.S.4, which may be the
effect of separation. (As described later, flow
*Static pressure on the hull surface does not agree with that
of Figure 3. While the measurements whose results are shown
in Figure 3 were carried out in the towing tank, those shown
in Figure 4 were in the circulating water channel. The
discrepancies are all due to this difference in experimental
conditions; the cross section of the circulating water chan-
nel is restricted to 1200mm
estimated.
‘ 820mm and pressure is under-
)
i
CALCULATED BY POTENTIAL THEORY
° MEASURED BY TOWING AHEAD
+ DO. BY TOWING ASTERN
6
STREAMLINE NO. 5
STREAMLINE NO. 7
STREAMLINE WO. 11
can be assumed to have separated near S.S.4.) There
the concept of boundary layer itself should be dis-
carded.
It can be safely pointed out that the pressure-
Simm) STREAMLINE NO. 5
aw
7)
au
ae
e
Hep
ecoo
o
Pecce
Pee eegaoee 0%
—
t
L
(ae
-0.2 -0.2 -0.1 -0.1 -0.1 0.0 0.0 0.1
e e
(mm) STREAMLINE NO. 9 ° le eo)
60 3 : 2 “4 of
3 le o. e ©
q 8 : 3 © )
1 e
SeSe) e ° 5 e
. re es : ais °
e ° e e e 3
e e
e H ° 5 3 ei
20 ii if ae
x es dae SE ( ne) \G
-0.2 -0.2 -0.1 -0.1 0.0 0.0 0
(mm) e
1 STREAMLINE NO. 11 : 416
e Te ° e
le e 1 e 1
1 18 e ° WO eT
wf = S517 : 2a as
e e e e e 5
e s e e 0 e
tae et Omnia
20 ! ij i ° e
/ \
fee ee ( Ne
0 St
-0 -0.2 -0.2 -0.1 -0.1 0.0 0.0 0.1
FIGURE 4. Static pressure profiles in the boundary
layer (GBT-125).
173
constant assumption can be employed unless the radii If velocity profiles are represented by Eqs. (3)
of curvature are not significantly small; the dis- and (4), the boundary layer thickness-parameters
placement effects are important near the stern and 64, 6,1 and shape factor H are
should be taken into account in higher order calcu-
lations [Hatano and Hotta (1977)]. cot nls bag ae
On = a ff (Uy q,)dt = rik Oley
e
Velocity Profiles 1 5 A
—— - lie = 5
Mi ip t CU = Cele Sempre 2
In order to calculate the boundary layer equations e
by an integral method it is convenient to represent es
velocity profiles by analytical functions which in- m $1 — nt2
clude several parameters. 11 n
The most commonly used formulae are based on a
1/n-power law and on a wall-wake law. The former and Eqs. (3) and (4) can be written in other forms,
has a definite merit of simplicity. The latter, H-1
developed by Coles (1956), has more freedom than = iS H-1 2
70, = I ] (6)
the 1/n-power law and can be expected to represent e 0, ,H(A+1)
velocity profiles more exactly. H-1
Mager's expression is well known as the three- Dr
dimensional velocity profile model based on a 1/n- = Cc H-1 ] ~ & H-1 2
? Cy AU, = reete\3 > Saree [el — ——] .
power law, Mager (1951). He gave the streamwise ee a H (H+1) ony H (H+1) (7)
and crossflow velocity profiles as
> 4 1/n, 3 Tf 944 and 53 are integrated and 8 is determined
en, a 6 (3) from measured velocity profiles, then velocity pro-
files represented by Mager's model can be calculated
17a 2 d with th
ie Gah wt BG from Eqs. (6) and (7) and can be compared wi e
92/0, ras ( 9) ) € ) i) measured profiles.
Figure 5 shows the comparisons of them. It can
where n is a variable parameter. be safely pointed out that Mager's model is employ-
(mm) | |
40 e
1 |
STREAMLINE NO. 5 [
30
552
20 j }
10)
q | H Ge/
| if Ue pS en — Ue
40
STREAMLINE NO. 9
30
2
20 ;
S.S.8
10 |
°
0 i
0.0 0.2 0.0
=o ESTIMATED BY MAGER'S MODEL
@ MEASURED
30
STREAMLINE NO.11
S50”
q2
a= Ne FIGURE 5. Velocity profiles represented by
0.6 0.6 0.6 0.4 1.0 -0.2 0.0 0.00.2 Mager's model.
174
[ 5.5.34
i
40 i STREAML INE
: es
FIGURE 6. Crossflow profiles in
the boundary layer on the AFT hull wu,
surface (GBT-125).
able for the velocity profiles of ship-like bodies
as far as streamwise components.
Figure 6 shows crossflow profiles measured on
aft parts of a model. As easily observed, there
are some profiles which have reverse type (S-shaped)
profiles. For most of remaining parts, the cross-
flow angles are very small and do not show reverse
type profiles. Because Eq. (4) has only one inflec-
tion point, such S-shaped profiles can not be repre-
sented by it.
To represent even reverse crossflows, more gen-
eral polynomial expressions are proposed [e.g.,
Eichelbrenner (1973), Okuno (1977)]. However, they
require additional equations or boundary conditions
and it is reported they do not always yield improve-
ments [Okuno (1977)]. This is because the cross-
flow does not always have such universal profiles
near the stern.
On the other hand, the three-dimensional veloc-
ity profiles based on Coles' wall-wake law can be
represented by
q,/ ec Ce (8)
a/v = Gr fyi hy sins + 958,05) (9)
where
say <u ) = * log, 5 ( =e ) +B, (10)
eZ) = 5 [1-cos( me )] ;
and
Bs 2
u_ = ( Te / p) . (11)
fp eer STREAMLINE
Sine.
J+ J, are variable parameters, for wake parts, u
is the friction velocity, and k, B are constants.
£. given by Eq. (10) is called the wake function.
Figure 7 shows the existence of such parts in case
Ue
of ship-like bodies also. Velocity profiles
deviate from linearity when approaching the outer
edge of the boundary layer. Velocity profiles,
represented by Eqs. (8) and (9), are compared with
measured profiles. Here parameters g, and Jo are
determined by the condition that q equals U, and
dy equals zero at the boundary and u, is determined
by a least-squares fit to the measured profiles.
The values of Clauser, 5.6 and 4.9, were used for
1/K and B respectively. Good reproductions are
examined except crossflow representations.
As to crossflow profiles, the situation is not
much improved from Mager's model; reverse crossflow
observed in experiments can also not be represented
by the wall-wake law. The finite-difference method
may be a possible step toward representation of any
type of velocity profiles.
Local Skin Friction
In the case of turbulent flow, most of the friction
is due to the turbulence (Reynolds' stress). For
this reason it is necessary to introduce additional
equations to determine it in closed form.
Ludwieg and Tillmann's semi-empirical equation
for the skin friction [Ludwieg and Tillmann (1949) ]
is most commonly used; it is
u_9 0.268
11 .
t /pu2 = 0.123x1079-678H ( © —— (12)
WwW) e Vv
Because Eq. (12) is obtained from two-dimensional
experiments, the validity should be examined when
applied to three-dimensional flow.
When Coles' wall-wake law is employed for the
velocity profile, the skin friction can be deter-
—— ESTIMATED BY 10
COLES' WALL-WAKE LAW
@ MEASURED
STREAMLINE NO.5
oe?
10 Sol 1°? 00 voce ah
STREAMLINE
NO.9
FIGURE 7. Velocity profiles represented by Cole's
wall-wake law.
mined from the friction velocity. But it should
also be examined experimentally.
In Figure 8, three kinds of experimental values
of skin friction are compared along streamline
Nos. 9, 11, and 18; directly measured values, those
Tous
+
STREAMLINE
NO.9
eS)
DIRECTLY MEASURED
ESTIMATED FROM COLES’ WALL-WAKE LAW
175
obtained from Ludwieg-Tillmann's formula, Eq. (12),
and those from friction velocity, Eq. (11). For
estimations of the latter two values, measured
velocity profiles are invoked. Calculated results
are also shown here for later discussions.
The values of Ludwieg-Tillmann's formula produce
fairly good agreements with those directly measured,
which implies that Ludwieg-Tillmann's expression
is also good for three-dimensional flow.
Entrainment Equation
In streamline coordinates,
is given by
the continuity equation
3 3 aitabee
dx, (the) + 3x, (12h1) 1 nyh7a 423 =| Oe (13)
Integrating with respect to X51 from zero to 6,
gives
ack
28 (G6) = 262
hj, 9x} 1 hgdx2
5 ee oky 1 9Ue
=F - (6-67) {—— =2 + —— <=} (14)
I
hjh, 3x, Uh, 3x,
where F is the entrainment function given by
a a6 06
b= Wee. v Vee 7 Wal s (15)
Equation (14) is often used as the third (auxiliary)
equation when the boundary layer calculation is
carried out by the integral method. Here, F should
also be given in someway in closed form.
In two-dimensional flow, Eq. (14) is reduced to
F= [U,, (6-63) ]- (16)
ele adh
We chai ©
Head (1960) gave a relationship between F and
(6-67) /01 (=Hg_6%) which was examined by two-
dimensional experiments.
© 00. BY LUDWIEG-TILLMANN'S FORMULA
2.0 88
Comparisons of local
3.0
2.0
=106 OR
3.0 Mo lw —caLcuaTen 9S
STREAMLINE NO.18 v
° Vv °
v
2.0 ° ° <I ° o % 3 3 8 v °
FP 9 8 Te 6 5 4 3 2 1 NLP
riction (GBT-125).
176
Introducing an assumption that the entrainment
equation of three-dimensional flow is related ex-
clusively to the streamwise quantities, Cumpsty and
Head (1967) employed the Head's entrainment function
for three-dimensional boundary layer calculations.
This is of course open to criticism.
In Figure 9, Head's entrainment function and
experimental values, obtained from Eq. (14), invok-
ing measured velocity profiles, are compared. It
can be mentioned that Head's function gives rather
good mean lines both in relation to Hg-3, - H and
F-H5-6] - The values of Hs-§1 are not fairly related
to H in the fore part, where laminar flow may still
exist, and neither F to Hg-6] in the aft part. The
former does not seriously effect F. We should bear
in mind here that the determination of boundary
layer thicknesses is not clear in the three-
dimensional case and accurate estimations of their
derivatives are very difficult.
Himeno and Tanaka (1973) used the moment of
momentum equation as the third equation instead.
In this case, assumptions for the Reynolds" stress
are also required and significant improvements are
not always found.
Summarizing the above discussions it can be
safely concluded that the integral method, where
either Mager's model or the wall-wake law is used
for velocity profile, Ludwieg-Tillmann's equation
for skin friction, and entrainment equation for
auxiliary equation, is expected to yield meaningful
results. Moreover, it can be also pointed out that
improvements can be attained when the second order
approximation for the static pressure is taken into
account near the stern. However, in the region
where reverse crossflows or large crossflow angles
exist, although the boundary layer assumption is
not violated, the integral method is no more
available.
BOUNDARY LAYER CALCULATIONS
According to the preceding conclusions, boundary
layer calculations were carried out by the integral
method and compared with experimental results.
Basic Equations and Auxiliary Equations
The integrated boundary layer equations are given
in streamline coordinates by
FIGURE 9.
function with experiments.
Comparison of Head's entrainment
,
0611 0842 On Oe 2
DE + on + cali ea Oem K; (6)) 890)
=e. f wu 4 (17)
2021 Ue | 011 We
0g on U, on Ue an
(H+1+ 212)
911
- 2K)}89] =T Ue pu2 , (18)
where 9)1, 8,2, 92), and 899 are momentum thickness
parameters defined by
2 yee -
te Pils deh Wisenels. o
6
us 912 = f q2(U;-q))daz ,
2 0 =
We Pan = 4 cin Whissiayiehs p
2 6
WE Opp = f q2(Vi-q2)dz . (19)
The entrainment equation is employed as the third
equation;
aU
al e
Hol i
Us 9&
3E 0 (20)
3
(6-67) - noe = F - (6-6}) (-Kj+
For the function F, the relation of Head is used,
which has already been examined.
If Mager's velocity profiles are employed here,
boundary layer thickness parameters are given using
811, H and B,
851 = 6),E(H)tan® , 695 = 06 1)C(H)tan28 ,
612 = 9110(H)tanB , 65 = 6),D(H)tanB ,
6-6] = 6,,N(H) , (21)
where
CH) = - Gaya
D(H) 16H
~ 3G) Ges) (Gas) 7
]
go eh
STREAML INE
° NO.5
* N0.9
vy NO.11
estes by2it> afi
av) = "Gen Ge) 2
J(H) = E(H) - D(H) ,
Seoul
MED = sey = Big ge (22)
Then Eqs. (17), (18), and (20) are reduced to
simultaneous differential equations in 36) ]/3&,
dH/9§, and dR/9E;
(i=1,2,3) (23)
of OWNM og Chel 2,98
dj = JtanB a J 811 tanBs— J8) sec Bon
611 9Ue 2
+ — ——" =_
(H uN dE + Kj) 6) (1-Ctan“s)
2
ap U
wl Te @ 7%
a2 = Etanf, b2 = E'8))tan8, co = E6,)sec“B,
81, 9U
ta ee ial Anjo ay 2,0H
do gE ant DE Ctan Bo C'6)),tan Bon
- 2C611 tangsec?B5= ~ (1+i+Ctan?B)K 0) 1
2
+
2K)E0),tanB + cee Hf puUctanB 1’
(24)
a3 =N, b3 = N'O)1, c3 = 0,
d3 = ptangee2 + DOqneenee «. Denmsec2 eo
5 an on
n
1 dUe
+ F - N6O))(-Ki+ — —,
Eee NEHGS] a, OG
(The ' means differentiation with respect to H.)
If Ludwieg-Tillmann's skin friction formula,
Eq. (12), is used, all the coefficients of Eq. (24)
are known at earlier & coordinate.
This formulation is the same as that of Cumpsty
and Head (1967).
Numerical Calculations and Discussions
Numerical calculations were carried out for GBT-125
at Re=10°. First, 18 streamlines were traced inter-
polating the 254 x 2 descrete values of velocity,
obtained by the surface source method, and x, coor-
dinates were determined.
The differentials with respect to nm were numeri-
cally determined along the n axis which was defined
by bending short segments orthogonally to the xj
axis. This is the main difference from Cumpsty-
Head's original calculations. For such calculations
as 0Ue/dn, 39911/dn, and so on, the differentials
with respect to n should be carried out as care-
fully as possible. Most numerical errors stem from
‘these terms.
iby 7/
@u/e 102 50
1.2
CALCULATED '
1.0 e MEASURED oe ?
3
0.8 1.0 e e
STREAMLINE NO.5S
FIGURE 10. Comparisons of momentum thickness (GBT-125).
0.5 x 107", 1.4, and 0.0 were used for the
initial values of 8,,, H and 8 at S.S. 94(x=-0.85).
These values were obtained from Buri's two-
dimensional formula assuming the flow is turbulent
just from F.P. (see Figure 1). Fortunately they
do not seriously affect the calculations.
About 200 steps were taken and Eq. (23) is
integrated with respect to € by Lunge-Kutta-Gill's
method (five points for each step).
In Figures 10, 11, and 12, calculated results of
611, H, and § along typical streamline Nos. 5, 9,
and 11 are shown along with experimental results.
The skin friction is shown in Figure 8. Streamline
No. 5 generates a simple, quasi-two-dimensional
curve on the hull surface and it may be expected
the flow can be truly represented by the present
framework. On the other hand streamline No. 11
passes through a region where the boundary layer is
rather thin and also through a bilge corner where
pressure increments were observed.
The experimental values of the streamwise momen-
tum thickness, 6),,, of streamline No. 11 were much
greater than those calculated around S.S.1. This
discrepancy can be related to the fact that S.S.1
of streamline No. 11 corresponds to the position
2.0
ef @ STREAMLINE NO. 5
e
1.5 SS =
© oT Temas eaer eae e
2.0 CALCULATED
@ MEASURED
Aun) NO.9
1.5 OFT OSs es _—
O e ° e ee 5708
ee
2.0 =106
Re=10
NO.11
O a
@
1.5 ee e 5 ° oie ‘
ee °
FIGURE 11. Comparisons of shape factor (GBT-125).
178
ee
CALCULATED
P (degree) @ MEASURED °
10 5
O STREAMLINE NO.5 e
e. e t
3 STREAMLINE NO.9
—————
! !
a) Re=10°
o
STREAMLINE NO. 11
FIGURE 12. Comparisons of crossflow angle (GBT-125).
just behind the bilge keel and the occurrence of
bilge separation can be suspected.
Shape factor H, in every case, does not vary
significantly and agreements between calculations
and measurements are good except near the stern.
There, as shown in comparisons of 8, large cross-
flow angles existed and the present scheme can not
be employed here.
It is interesting that large crossflow angles
can also be observed in experiments near the bow.
They create a suspicion of the occurrence of bow-
bilge separation.
Skin friction Tw, Shows also good agreement.
It is observed that both experimental and calculated
values do not decrease. This suggests three-
dimensional separation differs a little from that
of two-dimensional where skin frictions vanish.
As a whole, it can be safely concluded that,
except near the stern, calculated results show good
agreements with measured as far as integral quanti-
ties like 6); or H. It can be also concluded that
the present scheme, using integrated mementum bound-
ary layer equations as governing equations, can be
appreciated in spite of its brevity.
EXPERIMENTAL STUDIES ON BOUNDARY LAYER SEPARATION
AND WAKE
Kinds of Experiments and Measuring Techniques
The characteristics of separation and separated
flow of ship-like bodies are dim. Experiments may
throw light upon them. In order to discuss the
characteristics of separation and separated flow,
the following experiments were carried out in addi-
tion to the previous experiments. All experiments
were carried out with MS-02 and experiments (c) and
(d) used GBT-30 also. Experiments were executed
at the speeds of Fy=0.1525(Re=2.17x10°) and Fn=0.16
(Rg=2.38*10°) for MS-20 and GBT-30 respectively.
Flow Observations
Planting twin tufts on the hull surface, flow di-
rections near the stern were observed by a submerged
camera; one tuft was just on hull surface and the
other was 22mm off, normal from surface.
Free-surface flow around the ship stern was also
observed in relation to the separated flow by the
aluminium powder method.
Velocity Measurements in Separated Flow Region
Velocity in the separated region was measured using
a hot film anemometer. The probe is a conical type,
2mm in diameter. One horizontal plane of z=-0.02
was covered where framelines are almost vertical.
Because the probe was set parallel to the uniform
flow, the velocity is not quantitatively accurate.
Velocity Measurements in Wake
Two five-hole pitot tubes were used for velocity
measurements in the wake; 8mm-diameter tube for MS-02
and 10mm-diameter tube for GBT-30. For estimations
of vorticity, measurements were carried out on three-
dimensional lattice-points spaced 0.025, 0.015, and
0.015 in x, y, and z directions respectively.
Vorticity Estimations in Wake
The vorticity can be estimated by differentiating
the measured velocity distributions;
ow av du ow av du
‘Om = p = =—= HS a o
x ay dz “y az teen eB 3x dy -
(25)
The differentials were obtained numerically by
three-point approximation.
Discussions on Boundary Layer Separation and Wake
Flow
Boundary Layer Flow near Separation
Figure 13 shows flow directions near the stern of
MS-02 obtained by the twin tufts method.
It was observed that, very near A.P., both tufts
are drooping. This means that the velocity is al-
most dead; in other words, separation has occurred.
On the remaining parts, the outer tufts show
almost the same direction as the calculated poten-
tial flow direction; on the other hand the inner
tufts differ greatly from them and produce large
crossflow angles. A reference to the surface pres-
sure distribution gives a clear explanation that
flow near the hull surface, whose velocity is very
low, cannot make further steps against the pressure
increments and change direction suddenly from the
external streamwise direction toward the low-
pressure regions. Significant occurrences of shear
flow and generation of vortices are assumed which
correspond to beginnings of three-dimensional sepa-
ration.
The above situation can be understood more
clearly from velocity profiles in the boundary layer
near separation. Figure 14 shows the velocity pro-
files of GBT-125 along streamline Nos. 5, 9, and
11. A sudden large crossflow occurs near S.S.%
for all the streamlines and, correspondingly, the
streamwise velocity profile also changes. The
STREAMLINE NO.72 = ——
Ne ate
>
NO.9 noaooe
" OUTER TUFT
Ae sen TUFT
Slmm) trem ine 5 1]
of STREAMLINE NO.5
0 Ape ede
0.4 0.4 0.4 0.4
maximum crossflow velocity amounts to about half
of the streamwise velocity.
Such behaviors of flow near the stern are not in
the category of boundary layer flow, therefore,
boundary layer calculations should be stopped and
another treatment employed.
Criterion for Boundary Layer Separation
It is necessary to introduce some criterion for
boundary layer separation in order to change the
governing equations from boundary layer to some
others.
There are many criteria mainly for two-dimensional
separation [e.g., Chang (1970) ].
A parameter, I',, defined by
Speed (26)
is proposed.
179
FIGURE 13. Flow directions near
stern and isobar lines (MS-O2).
FIGURE 14. Velocity pro-
files in boundary layer
near the stern (GBT-125).
The proposal is based on the experimental facts
that the beginning of three-dimensional boundary
layer separation is closely related to the pressure
gradients, as discussed in the previous section,
and that boundary layer flow, such as with large
momentum thickness and small skin friction, can no
longer exist. Therefore, flows with large values
of IF cannot exist in real flow in the sense of
boundary layer flow. On the other hand, if the
boundary layer assumptions are kept, the calculated
values of Ty can increase without any upper bound.
Figure 15 shows I’, obtained by the boundary
layer calculations and from experiments. The calcu-
lated values get increasingly large approaching
the stern, but experimental values do not and they
seem to have some upper bound.
The value of [, = 20 is reasonable as a criterion
for separation, because, as shown in Figure 14,
large crossflow angles were observed near x=0.9
(S.S.4) and the onset of separation is suspected.
Of course, more experimental data are necessary
for the present discussion and further experimental
180
CALCULATED
Oo — ESTIMATED
FROM MEASURED DATA
SEPARATION
STREAMLINE NO.5
SEPARATION
STREAMLINE NO.9
pe ae
°
SEPARATION
10
STREAMLINE NO.11
0
5.5.2 1s 1 a AP
FIGURE 15. Criterion for separation.
and theoretical studies may give a firmer founda-
tion for the present criterion.
Flow Field after Occurrence of Separation
Once separation has occurred, the flow field differs
greatly from the unseparated boundary layer flow.
The existence of the dead region, pointed out in
the previous section, is one phenomena.
Figure 16 shows velocity profiles, after the
occurrence of separation, measured by the hot film
anemometer. The bars in the figure represent fluc-
tuations in velocity. The region where the velocity
fluctuates so intensively and is very low consists
of a characteristic thin layer, a separated retard-
ing region. It can be definitely distinguished
3 (mm)
100 —t
INTENSIVE
FLUCTUATION
®
1
Soe >
FIGURE 16.
near the separation position
Velocity profiles
(MS-02) .
from the outer part where the flow does not differ
greatly from the unseparated flow, The newly—
generated vortex is confined to this region.
Figure 17 is free-surface flow of MS-O2. It
shows more clearly the existence of the above
mentioned, separated retarding region. The divid-
ing streamline can be observed which coincides with
the border line of the separated retarding region.
In the case of practical ship forms, we have not
enough information as to whether or not such regions
exist. But from the velocity profiles of GBT-125
(Figure 14), their existence can be supposed in
those .cases also.
According to the present experimental studies,
it is implied that any single approximate equation
of the Navier-Stokes equation completely governs
the flow field near the stern.
Eddy Viscosity Coefficient in Wake
In order to predict turbulent terms in the Navier-
Stokes equation, there is a concept of eddy viscos-
ity. It is based on an idea that momentum loss due
to turbulence can be represented by momentum loss
due to friction and the coefficient is constant as
to positions and directions. According to this
assumption, the Navier-Stokes equation is written,
ae 2
q- Vu, - w.Vq = VAY Wr (27)
where v, is the eddy viscosity coefficient.
Equation (27) is a kind of diffusion equation
with vg the diffusivity coefficient. It can be
determined experimentally; substituting the measured
values of velocity and vorticity into Eq. (27)
leaves only Ve as an unknown.
Using experimental data of the GBT-30, covering
1.08<x<1-16, ve is determined by the least-square
method. The estimated values of vg are not unique;
they differ slightly for each direction, 2.7 x 101,
2.4\x 107%, and|1.6 x 10>* fox w,, wy, ands.) the
mean value is 2.2 x iO", and consequently the
equivalent Reynolds number, based on the eddy vis-
cosity, is about 1/300 of the real Reynolds number.
e e @
e
® @ )
@
5 e ©
e
3 : ¢
5:8, 2 1
8 S.S. SS mee 14
FIGURE 17. Free-surface flow near the stern (MS-0O2).
Subdivision of the Flow Field
It has been made clear by experimental studies
that the separated flow has at least two, quite dif-
ferent viscous regions where no single approximate
equation of Navier-Stokes equation seems to be
valid for both. It can be proposed to subdivide
the flow field near the stern into five regions as
shown in Figure 18; potential flow region, boundary
layer region, vorticity diffusion region, separated
retarding region, and viscous sublayer region.
Their characteristics are as follows.
Potential flow region:
The region where the viscous term can be wholly
neglected and only displacement effects should be
taken into account.
Boundary layer region:
The region where the boundary layer assumption
is valid and the backward influence of separation
can be neglected.
Vorticity diffusion region:
The region where the vorticity, which has been
generated in the boundary layer, is diffused con-
vectively and viscously. No vorticity is newly
generated in this region. Because the dividing
streamline is a kind of free-streamline, the pres-
sure on it might be constant.
Separated retarding region:
The region where the velocity is very small
and the turbulence is intensive. Because even a
recirculating flow can be observed, the governing
equation for this region should be an elliptic type.
Viscous sublayer region:
This is the very thin layer region which just
adheres to the hull surface. The molecular viscos-
ity is predominant and the velocity profile should
satisfy the no-slip condition on the hull surface.
CALCULATION OF VELOCITY DISTRIBUTIONS IN THE SHIP'S
WAKE
Approximation of Navier-Stokes Equation by Local
Asymptotic Expansion
In order to get appropriate approximations of the
Navier-Stokes equation for each region, local asymp-
totic expansions of relevant quantities are made,
using small parameter e defined by
3 Ht/8
SS Bes (28)
181
The quantity e€, was first introduced by Stewartson
(1969) and © << 1 in case of a large Reynolds number.
If the x3 coordinate can be assumed to be
linear, i.e.,
h3 = 1, (29)
the continuity equation and Reynolds equations are
written in streamline coordinates as follows.
aq Clee) i 943
I SKS = 0 30
hj,dxj h2dx9 0x3 191 22 te)
Sh Cenk 4. Gea, Sell. _ Oeil fees
hy 0X] ate ho 9x9 139%, 20192 192
eye C) Pp 12 a Ot O25
hy) 0x, ( p q1 ) haox9 q1d2
) 1 ' ' ' 1 1
- ~— q3q] + 2Kyqjq5 + K) (q]2-q9")
8x3
WY) C) dW3
+—J— “(ph =
ho le 202) (31)
Gal Bah 5, Gh en Ee Oe ae ieee
iy On he Sea eee, Pe
aa eae EO atid Wastes on
re oe tee ES tp Cue
2 2
- Ky (q] - 42) + 2K) qa}q49
re | Gra 7 Bee |e (32)
eb eh So, BES os LE heh
hy dx] ho ax9 8X3 3X3
' J ' 1 ' '
= + K +K
hy ox 4193 hy dX> 9293 19193 29293
v a
(hyw = = — (ams)
hyhp | ax en gee Atos
(33)
A. POTENTIAL FLOW REGION
B : BOUNDARY LAYER
C ; VORTICITY DIFFUSION REGION
D ; SEPARATED RETARDING REGION
SEPARATION f : VISCOUS SUBLAYER
POSITION
SHIP HULL
FIGURE 18. Subdivision of separated flow field near
the stern.
182
where W], W2, W3 are the components of vorticity
given by
aq a
yy Se
hodx9 ax3
ed pe ed
oe 3x3 h) 0x, ‘ (34)
ee eee q
3 hy dx, hy dxXo 291 112-
and they satisfy
Vew = 0. (35)
In the reduction of Eqs. (31, 32, 33) from the
Navier-Stokes equation, conventional predictions
for turbulent components are used; the velocity is
assumed to consist of time-averaged terms and
fluctuating terms.
On the other hand, if the constant eddy viscosity
can be assumed, the following equations are derived
directly from Navier-Stokes equation;
—— (y 2-wW2q)) + = (w1q3-0341)
hgdx9 1 dX3
it) Be a2 3 ey 303
=v) Gr 2 yO. S ==)
2 oxo ax3 hj) 0x) hodxo aX3
9 w2 dW
= ——— - - + ¥
Rep ote) orm ) Seaaeprey |e Se!
eee = ))) ar eo -W392)
hj) 0x] W29Q1-W192 aX 3 293-302
i OS 34 wy 93
= y Say ( —)
e né os oats hodx2 hj) dx) dx3
9 dW) dW.
7 Tmo, SIC) SS ners ic OO
(Ky - 2 (wyq3-w3qy) + (Ko- 24 ( )
Le tata le oe 2% HaOxoY was
i pe 1 94 a ew dw
yy eS re a i ee ara np Ls
e E x2 = h2 aR) os che OnE x ma
1 1 2 2
a dw
p= (K 4K = (ears 3
axa MSAD) (Ky 1) Os
= teonealSaS|
hodx9 (38)
where
1 dh 1 9h,
K Sao ak Sf en eee,
1l h2 ox, / Ko9 aD Fis, (39)
1 2
Vorticity Diffusion Region (C-region)
For the vorticity diffusion region, the constant
eddy viscosity is assumed.
Introducing non-dimensional curvilinear small
line segments 3, dn, and dz, we represent the dif-
ferentiations
OAD (40)
Here we assume the derivatives by new variables
Ebay Gull O(a), alaGion
nor
= 5 41
5 0(1) (41)
= 0(1) , a 0(1) '
on
Raat
The origin of the new variables coincides with that
of O-x)x 9x3 but ==0 corresponds to the position
of separation.
We tentatively assume that the asymptotic series
for velocity and vorticity of C-region have the
following forms;
qi /Ug = tig (E02) + eu, (&,n,6) ar eau (Gann) ar G00 G
epi) = mn Spns) 2 Erm (Ecee) 2 aac 5
q3/Ug ee e7wy (E,n,c) ap eS (E,n,c) chiateheta ,
Ml = 2 a, (Esme) +, (E,neE) +
Up/L Ee Ey aa Eo ms cee ,
Sa Aye cca Oe a Le a eect
Taye = cE herteb) +S OL (Ene) ee
a
U/L & (43)
All the quantities appearing in Eqs. (42) and (43)
are assumed to be O(1).
Moreover, we introduce non-dimensional variables
k) ,k2,k11,k22 by
IS, = WEP) op ES SR SS OK o
koo = L*eKo9 p (44)
whose orders are O(1) for all the regions.
Substituting Eqs. (40), (42), and (43) into
Eq. (30) and Eq. (35), we get as leading terms,
55
Oy (45)
3€
du 3a
=e (46)
an dt
and into Eqs. (36, 37, 38) we get,
Ola eats LO (47)
aA ni to? + aF ci to) O 4
a M5 Ot Bye kg (48)
€3 Uh dz2 Epanlel ,
2~.
BOS ENS ROGUE A (49)
e3) Uh Wae2m | OE, td 0” :
In order for the viscous diffusion term to exist,
Ve/UpL should be at least O(e3).
We have obtained four equations, Eqs. (46-49),
for three unknowns, U9 1 Wyys and Orly but it can be
easily shown that one of them is not independent.
Changing variables back into the original ones,
we get, as the governing equations for C-region,
r) C)
hp dx W241) ote dxq (3a) = 0) 9 (50)
a
= es = ©. 5 (51)
3
a oxgccsa) > Ont (52)
The terms of order 0(1/e2) are neglected in the
above equations.
Separated Retarding Region (D-region)
Introducing normalized variables, E, nN, i for the
separated retarding region in the same manner, the
orders of differentiation are assumed,
a _ a ) a Lee
hy dx] L Ede hodxo L forehal
a i @
ox3 Le 30g ’ =)
3 a 3
a O(a) 5 oa O1) ae = OG) (54)
Velocity and pressure are assumed to be expanded
asymptotically,
q1/Ug = € (0, +0} ) + ©? (ap +09) woo 0 ’
qo/Up = E(vitvi) + e2 (oto) +... , (55)
= een ee We nO
q3/Up = €> (wi tw)) + e*(Wotwo) +... ,
2
(p-p,,)/PU, = €P] + Epa + --- , (56)
where Uy, Usr-++ are all time-averaged variables
Ag Lad
and u,, u,,..-. are fluctuating. Here the fluctu-
ating terms of pressure are omitted because they
do not appear in the basic equations.
The vorticity can be also expanded asymptotically,
ol iL OM ah ON
SS ae er a OCI) ig
Ug/L Se he c o
w 3u au
ZUG ay OL 2s ore
U/L €2 ar € 3¢
Ug 9 9n (57)
Under these assumptions, the leading terms of the
continuity equation are written,
Ce foN Oo (58)
9e an ac
“and the governing equations are
183
{a8 a 8 » SB Bae Ac Move tee ae
ar & pases a AWitre oF Vilage i + Wars ar Wome) Uy
ao m2
-koujvj+ mi]
) at 3 U2 Uist
_ test 20,05) + a (alve + 09V})
O patar aint ann a2 12
? eben oP u2W}) -2k9ujV} = ky (ay Savalt )
u
Gages 7 (59)
, ov a Be ng OY
Wilma o0 Wlme oF WA
3 ae ap
8) ae Oe ie SP). 808
+e aa) + (iipe Wiis! Var Weis © eae
+kouy2- eat
1 a 1 @ pSoan fe) 0%) Oo ,.148
SS Sse | een) oP Sela”) oP Satan)
é On E an ae
E
Oane nen AGP ao ssi
+ eae + vow]) + k, (a) Na ) - 2k,0,¥v |
pee a2¥1 + O(e%) (60)
e6 UpgL a2 ‘
3 3
ete 5 OC) =O, (61)
lrg dt
The leading terms of Eqs. (59, 60, 61) yield
p1(&,n,S) = const. (62)
Equation (62) means that the pressure is constant
throughout the present region as far as O(c) is
concerned.
Now the second terms of Eqs. (58, 59, 60) yield
Oe oO BS Big, a Bom
19E 195 38 aE P2 1
a ,n0n8 oO ,nias
= SH) = salen) op (63)
3 87
SN ee
W152 + wigs + Bie = ae t1V¥1)
a on 2 ) SOO
SSO a Wi) = ae) (64)
ee Va ea)
C)
a EHC (65)
Cc
In the above equations, the molecular viscosity
184
disappears but its effects are still existing in-
directly through turbulence.
Equation (65) gives the so-called boundary layer
approximation. But because cross terms of fluctu-
ating components exist in Eqs. (63) and (64), they
do not always yield the same type as boundary layer
equations which can not predict the recirculating
flow observed in experiments.
Viscous Sublayer (E-region)
In the viscous sublayer, E-region, the no-slip con-
dition must be satisfied on the hull surface. Here
the intensity of turbulence may be very small and
all the turbulent terms in the Reynolds equations
vanish infinitesimally.
The following asymptotic expansions are assumed
from the Blasius solution.
2
* *
CHYAUO = SWYy “EWA sP cco py
2
q2/Up = evi a vi ut ROSE (66)
5
q3/Ug = etwt a? & wh Ti ooo p
where the orders of each term are all O(1).
The derivatives are represented by
Cheb te riliert a. 8
web Ps SE inte Fy Goa 2
Gs Bale a8)
axg | betoe 7 sou
and their orders are
3 a a
—— = (0) p == 0 1 = 5
dE (1) a5 (1) ac 0(1) (68)
Substituting the above assumptions into Eqs.
(31, 32, 33), the leading terms are obtained as
follows in original variables;
+ + g3——
Why 8x] qh Fax 139 x
1 98P aq) aS
p hy, dx, ax2 uf ( )
aq 945 Tey)
Wnjax; hoax, * dx,
so ase (70)
P hodxo ax2 i
DITO. 2 (71)
dXx3 :
The continuity equation is
oq) 8a oq3 (5)
+ + =
h) 0x] hodx9 ax3
Here the quantities of O(e%) are omitted.
These are the boundary layer equations themselves.
They must be matched with the solution for the D-
region in quite the same manner as the conventional
method of boundary layer calculation.
The following matching conditions should be
satisfied when governing equations are solved.
(i) for upstream;
Wii Se Sieh) 2 es Se
Ess0 VOCE Ne) > Un, pee wien e) = ve,
Teme) ==
Bee wiGomee) = (73)
whe re Up, Vp, and wp are the velocity components
in the boundary layer in the x),x2,x3 directions
respectively.
(ii) far from the hull surface; the solutions
should be matched to the solution of the A-region,
potential flow.
(iii) between the C- and D-region;
ug(E,n,0) =0, uw (En,0)
lim a Gee 8 Oi ne Se
Pro O1(Esn/o) - S5e Gig (Enc) |=, | . (74)
miGeonpO) = Se Soak 5
vi (E,n,0) = oe vil€n,c) , (75)
wi(&sn,0) = 0, wo(E,n,0)
lim} * ape 2 2 2
Poco | W1 (Erno) tpewi (Erm 2) |z_o (76)
(iv) between the D- and E-region;
MEO ean) o vi
a 8 A ee
En) = pees neo) (78)
wi(En,0) = 0 , wolE,n,0) =
lim She ore ea Ae = By
a Ego) 1S rs wi (Erb) | a6 (79)
The governing equations for the D-region do not
close. Some auxiliary equations are required, but
this problem is left for future work.
Numerical Calculations for the C-Region
To solve the derived equations analytically is al-
most impossible; this is because not only are the
equations non-linear but also the hull surface,
where the boundary conditions are prescribed, is
very complicated in geometry. Instead, they must
be solved numerically. But it may be still more
difficult because the calculation should be carried
out for all the regions at the same time in order
to satisfy the matching conditions. However, this
difficulty can be removed by an iteration method;
the surface consisting of dividing streamlines (DSL)
is given a priori in the beginning as the inter-
mediate region between C- and D-regions where the
matching is carried out. Of course the surface of
DSL can be obtained finally as a solution of the
flow field, but the assumption of DSL makes it pos-
Sible to solve the governing equations in every
region almost independently and it is expected that
repeated iterations may bring forth a reasonable
solution.
The flow in the C-region can be determined by
taking a new streamline coordinate system O-x)X5%34
where the x,-axis coincides with DSL and the x,-
axis is normal to the DSL surface.
By the finite-difference scheme, Eqs. (48)
(49) are transformed into tridiagonal linear equa-
tions for k > 2;
and
wo(i,j,k-1) - 2€(i,j3,k)wo(i,k,k) + wo(i,j,k+t1)
=Ag(i,3,kK) , (80)
W3(2,3,k-1) = 2C(i,j3,k)w3(i,5,k) + w3(i,3,k+1)
= A3(i,3,k) , (81)
where wo(i,j,k) etc. denote those values at x)=xjj,
SSH! and X31
ae Ac? ny,
C(i,j,k) =1 + vbe Gin (Loa pbs) p
Agi(i;a 7k) = = Wo(i=1,5,k=1) +
3 ai Ac? ena Sots
Aids (GALS) phe) || 2b a Con (st ala pis) || Wy (simak 5) petal)
VeAE
Aaa) = = wis\(G—1 a), kK)
: : Ac2 Z : : :
2w3 (i-1,5,k) |1 - —— qj (i-1,3,k) | - w3(i-1,5,k+1),
VeAE
(82)
and AE, At are short segments in the x), x3 directions.
Equations (80) and (81) can be solved by the
forward marching procedure if the velocity profile
of q; is given at the separation position. Here the
value of vorticity at k=l, on DSL, is made equal to
that at k=2.
Once the vorticity distributions are obtained
throughout, the boundary layer and wake, say V,
0.27 AssuMED 0.2 0.2
DSL
UV.W '
‘ a an
0 See
Ol ae ar)
(Geena
-0.2 -
-0.4 2=-0,015 -0.4 -0.4
-0.6
0.2
0
-0.2
o Z=-0.075
-0.4 slave -0.4 LAS, HA
-0.6
2 -0.8 -0.8
185
velocity distributions can be calculated as induced
velocity of vorticity by invoking Biot-Savart's law;
(x )=V 1 ppp W_ Bh ‘ay'az!
eh SOGOU * an Vv ie Y] ested
'
where w, is the mirror image of w' whose components
are Wx, Wy, Wz and
ee (Feo a Ge) ae (aa!)
ee = (EEE )e => Wa) & (a9) = (84)
Because Eq. (83) gives the viscous component of
velocity, the potential component should be added
to qy;.
In the present calculation, DSL is determined
from experiments for the first iteration; it con-
sists of line segments, departing at x5=0.9 and
reattaching at x;=1.1 (see Figure 18). The stream-
wise velocity q,; in Eq. (82) is given by a quadratic
function of ~ which is equal to U; at the outer edge
and to 2/3 U; on DSL. The integral intervals for
x and ¢ are 0.005 and 0.0025 respectively.
In order to obtain the velocity distributions at
x=1.025, the region covering from x=0.8 to x=1.4
is integrated in Eq. (83). Here, 300-times molecular
kinematic viscosity is used as Ve.
The boundary layer and the potential flow calcu-
lations are carried out in the same manner as in
Section 3.
In Figure 19, typical calculated results of the
first iteration for MS-02 are shown compared with
experiments. The ship speed is Fpy=0.1525 and the
corresponding equivalent Reynolds number is about
8700. Here the calculations for the D- and E-regions
have not been carried out; therefore both regions
are excluded from the vorticity-integrating region V.
Satisfactory results are obtained, as far as
C-region is concerned, especially in u and w. The
velocity v is always underestimated, in other words,
overestimated in the negative direction; this may
uo
V4 em =
we —-—--
°
Fp=0.1525 FIGURE 19. Velocity distribu-
( R,#2.17x106 ) tions in wake at (1/8)L AFT from
een A.P. (MS-O2) .
Fy20.1525
-0.05 ( R,#2.17x108 )
---- CALCULATED =}
—— MEASURED
MIDSHIP SECTION Pes
FIGURE 20. Wake distribution at (1/8)L AFT from A.P.
(MS-02) -
be because in the present calculations the potential
components are determined with no attention to dis-
placement effects.
Figure 20 shows the calculated wake distribution
compared with measured. They do not always produce
quantitative agreement with each other, but compli-
mental uses of the present calculations with model-
wake survey may offer a useful method for the
prediction of full scale wake characteristics.
It is expected that much further improvement can
be attained by taking into account the D- and E-
regions.
CONCLUSION
The flow characteristics of boundary layers and
wakes of ship-like bodies are discussed. The fol-
lowing remarks can be mentioned as conclusions;
(i) The pressure-constant assumption of boundary
layer is a good approximation except near the
ship stern or bilge keel where there is a
small radius of curvature. The pressure does
not recover near the stern because of the dis-
placement effects of the boundary layer.
(11) Most commonly used semi-empirical equations
for velocity profiles, skin friction, and
entrainment can be safely employed in case of
ship-like bodies, but the functional expres-
sion for crossflow in boundary layer has a
certain limit for large or reverse crossflows.
(111) The integral method of boundary layer calcu-
lation may be carried out more effectively by
a hybrid use of integral and finite-difference
methods.
(iv) The three-dimensional boundary layer separa-
tion is closely related to pressure distribu-
tion on the hull surface. Its initiation is
referred to the occurrence of large crossflow.
(v) The eddy viscosity coefficient is about 300-
times the molecular one, in the ship's wake.
(vi) The separated flow region has sub-regions
which have different characteristics and no
single approximate equation of Navier-Stokes
equation is valid uniformly for all regions.
(vii) The local asymptotic expansion method is
promising for the separated flow. Further
experimental investigations as to turbulence
are necessary.
ACKNOWLEDGMENT
The assistance of graduate students of the Faculty
of Engineering of Hiroshima University, who partici-
pated in carrying out experiments and numerical
calculations, is cordially appreciated.
REFERENCE
Cebeci, T., K. Kaups, and J. Ramsey (1975). Calcu-
lations of Three-Dimensional Boundary Layer on
Ship Hulls. Proc. of First Intern. Conf. on
Numerical Ship Hydro., 409.
Chang, P. K. (1970). Separation of Flow, Pergamon
Press) Lede Oxford, pp.) 139)
Coles, D. (1956). The law of the wake in the tur-
bulent boundary layer, J. of Fluid Mech. 1, 191.
Cumpsty, N. A., and M. R. Head (1967). The Calcula-
tion of Three-Dimensional Turbulent Boundary
Layers, Part 1: Flow over the Rear of an In-
finite Swept Wing. Aero nautical Quart. 18, 55.
Eichelbrenner, E. A. (1973). Three-Dimensional
Boundary Layers. Annual Review of Fluid Mech. 5,
M. Van Dyke, and W. G. Vincenti, ed., pp. 339-360.
Gadd, G. E. (1970). The Approximate Calculation of
Turbulent Boundary Layer Development on Ship
Hulls. Trans. RINA 113, 59.
Hatano, S., M. Nakato, T. Hotta, and S. Matsui
(1971). Calculation of Ship Frictional Resis-
tance by Three-Dimensional Boundary Layer Theory.
J. of Soc. of Nav. Arch. of Japan 130, 1.
(Selected Papers from J. of Soc. of Nav. Arch.
dik n- WA)
Hatano, S., K. Mori, M. Fukushima, and R. Yamazaki
(1975). Calculation of Velocity Distributions
in Ship Wake. J. of Soc. of Nav. Arch. of Japan
138), 54; Hatano, S=, K. Mori, and T > Suzuka!
(LOM V2ndsRepoxt, Do, 415) LOE
Hatano, S., and T. Hotta (1977). A Second Order
Calculation of Three-Dimensional Turbulent Bound-
ary Layer. Naval Architecture and Ocean Engi-
neering 15, 1.
Hatano, S., K. Mori, and T. Hotta (1978). Experi-
ments of Ship Boundary Layer Flows and Considera-
tions on Assumptions in Boundary Layer Calculation.
Trans. of The West-Japan Soc. of Nav. Arch. 56
(to be published) .
Head, M. R. (1958).
Boundary Layer.
and Mem. 3152.
Hess, J. L., and A. M. O. Smith (1962). Calcula-
tion of Non-Lifting Potential Flow about Arbi-
trary Three-Dimensional Bodies. Rep. No. E. S.
40622 Douglas Aircraft Co., Inc..
Himeno, Y., and I. Tanaka (1973). An Integral
Method for Predicting Behaviors of Three-
Dimensional Turbulent Boundary Layers on Ship
Surfaces. J. of The Kansai Soc. of Nav. Arch.,
Japan 147, 61.
Hoekstra, M. (1975). Prediction of Full Scale Wake
Characteristics Based on Model Wake Survey.
Intern. Shipbuilding Progress 22, 204.
Hotta, T. (1975). A New Skin Friction Meter of
Floating-Element Type and the Measurements of
Local Shear Stress. J. of the Soc. of Nav. Arch.
of Japan 138, 74.
Larsson, L. (1975). A Calculation Method for Three-
Dimensional Turbulent Boundary Layers on Shiplike
Bodies. Proc. of First Intern. Conf. on Numerical
Ship Hydro., 385.
Entrainment in the Turbulent
Aeronautical Res. Council Rep.
Ludwieg, H., and W. Tillmann (1949). Untersuchungen
uber die Wandschubspannung in Turbulenten
Reibungsschichten. Ing. Arch. 17, 288.
Mager, A. (1952). Generalization of boundary layer
momentum integral equations to three-dimensional
flows including those of rotating systems. NACA
Rep. 1067.
Okuno, T. (1977). Distribution of Wall Shear Stress
and Cross Flow in Three-Dimensional Turbulent
Boundary Layer on Ship Hull. Nav. Arch. and
Ocean Eng. 15, 10.
Stewartson, K. (1969). On the flow near the trail-
187
ing edge of a flat plate II. Mathematika 16,
106.
Uberoi, S. B. S. (1969). Viscous Resistance of
Ships and Ship Models. Hydro-og Aerodynamisk
Laboratorium Rep. Hy-13.
Webster, W. C., and T. T. Huang (1970). Study of
the Boundary Layer on Ship Forms. Jour. of
Ship Res. 14. 153.
Yokoo, K., H. Takahashi, M. Nakato, Y. Yamazaki,
H. Tanaka, and T. Ueda (1971). Comparison of
Wake Distributions Between Ship and Models. J.
of Soc. of Nav. Arch. of Japan 130, 41. (Selected
Papers from J. of Soc. of Nav. Arch. 11, 25.)
A General Method for Calculating
Three-Dimensional Laminar and
Turbulent Boundary Layers on Ship Hulls
Tuncer Cebeci,
Ke Gree Chiang,
and Kalle Kaups
Douglas Aircraft Company,
Long Beach, California
ABSTRACT
A general method for representing the flow proper-
ties in the three-dimensional boundary layers around
ship hulls of arbitrary shape is described.
use of an efficient two-point finite-difference
scheme to solve the boundary-layer equations and in-
cludes an algebraic eddy-viscosity représentation
of the Reynolds-stress tensor. The numerical method
contains novel and desirable features and allows the
calculation of flows in which the circumferential
velocity component contains regions of flow reversal
across the boundary layer. The inviscid pressure
distribution is determined with the Douglas-Neumann
method which, if necessary, can conveniently allow
for the boundary-layer displacement surface. To
allow its application to ships, and particularly to
those with double-elliptic and flat-bottomed hulls,
a nonorthogonal coordinate system has been developed
and is shown to be economical, precise, and compara-—
tively easy to use. Present calculations relate to
zero Froude number but they can readily be extended
to include the effects of a water wave and the local
regions of flow separation which may stem from bul-
bous-bow geometries.
1. INTRODUCTION
A general method for determining the local flow
properties and the overall drag on ship hulls is
very desirable and particularly so with the present
need to conserve energy resources. Et is difficult
to achieve for a number of reasons including the
turbulent nature of the three-dimensional boundary
layer, the complexity and wide range of geometrical
configurations employed, the possibility of local
regions of separated flow, and the existence of the
free surface. In addition, and although these dif-
ficulties may be overcome in total or in part, the
resulting calculation method must have the essential
features of generality, efficiency and accuracy.
It makes
188
The purpose of this paper is to describe a general
method which is capable of representing the flow
properties in the boundary layer around ship hulls
of arbitrary shape. It is based on the general
method of Cebeci, Kaups, and Ramsey (1977), developed
for calculating three-dimensional, compressible lami-
nar and turbulent boundary layers on arbitrary wings
and previously proved to satisfy the requirements
of numerical economy and precision. To allow its
application to ships in general, and to double-
elliptic and flat-bottomed hulls in particular,
an appropriate coordinate system has been developed.
Previously described coordinate systems, for example
a streamline system such as that of Lin and Hall
(1966) or the orthogonal arrangement of Miloh and
Patel (1972) are limited in their applicability and
the present nonorthogonal arrangement is similar
to that of Cebeci, Kaups, and Ramsey (1977).
The numerical procedure for solving the three-
dimensional boundary-layer equations makes use of
Keller's two-point finite-difference method (1970)
and Cebeci and Stewartson's procedure (1977) in
computing flows in which the transverse velocity
component contains regions of reverse flow. This
is in contrast to previous investigations, for
example those of Lin and Hall (1966) and Gadd (1970),
which are limited either to zero crossflow or to a
unidirectional and small crossflow. It is also in
contrast to the previous methods of Chang and Patel
(1975) and Cebeci and Chang (1977) which did not
have a good and reliable procedure for computing
the flow in which the transverse velocity component
contained flow reversal.
In representing turbulent flow by time-averaged
equations, a turbulence model is required and an
algebraic eddy-viscosity formulation, similar to
that of Cebeci, Kaups, and Ramsey (1977), is used.
This is in contrast to the two-equation approach
which Rastagi and Rodi (1978) have applied to three-
dimensional boundary layers and which, in principle,
should be better able to represent flows which are
far from equilibrium. The previous comparisons pre-
189
sented in Cebeci (1974, 1975) demonstrated that the geodesic curvatures of the curves z = const and x
present eddy-viscosity model allows excellent agree- = const, respectively. They are given by
ment between measurements and calculations but did
not include comparison with the three-dimensional il t) dhy
NG) rey lee Unvacrersi8) 5
boundary-layer measurements of Vermeulen (1971). hyhgsinO | dx Oz
Since this data includes a strongly adverse-pressure
gradient case which allows a stringent test of the a 1 ioe Rey Ie dh 3)
present model, corresponding calculations and com- D hyhgsind | dz 1 ox
parisons are reported.
The calculation method is described in detail The parameters Kj)» and K» are defined by
in the following section which states the three- =
dimensional, boundary-layer equations in curvi- asy ell i BG) 1 306
linear, Se ea Meeaheaeae and describes B12 sind | (x: i 1 2) pe cose (x, ei 22) ie)
and discusses the required initial conditions, a
turbulence model, and transformations in separate al ( dy ee) i 8
: ; : S : K = {| Kp =) 6 (Kk), += —
subsections. Section 3 is devoted to the coordinate ay sind 2 hg dz Soe 1 hy ox Ko9)
system which is an essential feature of the present
method. The numerical method is discussed briefly For an orthogonal system 6 = 7/2 and the parameters
in Section 4 and calculated results are presented Ki, Ko, K,2, and Kj), reduce to
in Section 5 which includes comparisons with the ss
measurements of Vermeulen (1971) and demonstrations ear Ste 1 Os eee 1 ohe 7
of the ability of the method to represent the geom- 1 hyhy 92 20 hyh, 0x (7)
etry of different hull configurations and to result
in realistic velocity and drag characteristics. eK on ee (8)
Summary conclusions are presented in Section 6.
At the edge of the boundary layer, (2) and (3) reduce
to
2. BASIC EQUATIONS
uy au, W du, 5
Cee: BAS a 2
Boundary-Layer Equations h), 0x ho dz Ne COS! i RCS Ce ‘i uae
The governing boundary-layer equations for three- as esc26 3 Pp cot@cscé 3 P 9
dimensional incompressible laminar and turbulent er hy exp x ho dz\ po @)
flows in a curvilinear nonorthogonal coordinate
system are given by: u_ ow w_ oW
e e e e 2 2
a ee a ee I SIONIELS) Se NCTBLCISIONS) ae TL Ay
Continuity Equation hy 0x hg oz S 5 cas
5 ; 5 5 _ cotescé 2-(2) _ csce76 (2) (10)
x (uh 2sin®) + jg (Wh isin®) + peamnns ing) =O (1) hy ax \p ho az \pe
x-Momentum Equation The boundary conditions for Eqs. (1) to (3) are:
9 3 3 y = 0: u,v,w = 0 (11a)
i 4y% Y+yt- K,u~cot6 ap Kow“cscd + K)2uw
hy ox.) hip) dz oy
7 = OB u = Ug (x,2), W = We(x,z) (11b)
ae esc28 Of i cot8csc8 9 /p
hj dx \p ho dz \p
Initial Conditions
) du ;
Ww By (% 2 Wy ) (2) The solution of the system given by (1) to (3),
subject to (11), requires initial conditions on
two planes intersecting the body along coordinate
lines. In general, the construction of these
initial conditions for three-dimensional flows on
cscO + Kp)uUw arbitrary bodies such as ship hulls is difficult
due to the variety of bow shapes, which may be ex-
tensive and complicated. For this reason, assump-
cot8csco al) a esc*8 mG) tions are necessary in order to start the calculations.
hi ax \P he 3z \p In our study we choose the inviscid dividing
streamline on which dp/dz = 0 to be one of the
a ( ow vu") (3) initial data line (see Figure 1). In the case of
z-Momentum Equation
a ow )
e wl & WV 2s = Kone
u Aue 2
hy ox ho dz dy
cot@ + Kyu
+ — (v—
OY oy rectilinear motion of a ship, this streamline runs
along the plane of symmetry. Because of symmetry
Here, h, and h2 are the metric coefficients and conditions, w and dp/dz are zero on this line causing
they are, in general, functions of x and z; that is, (3) to become singular. However, differentiation
with respect to z yields a nonsingular equation.
hy = hy (x,z); hg = ho (x,2) (4) After performing the necessary differentiation for
the z-momentum equation and taking advantage of
Also, 9 represents the angle between the coordinates appropriate symmetry conditions, we can write the
“x and z. The parameters K; and Kp are known as the so-called longitudinal attachment-line equations as:
190
Zz
~<|
INITIAL CONDITIONS ON
A CROSS SECTION (x = xq)
DP
Lug
FIGURE 1. The nonorthogonal coordinate system and the
initial data lines for the ship hull.
Continuity Equation
3 : ; a F
9q (un2sin®) + h)sinéw, + oy (vhjhgsin@) =O (12)
x-Momentum Equation
u_ du ou 2
hy ox SV; By cotéK)u
Us du +
= a - K,uzcot6 + 2 (8 - wv") (13)
z-Momentum Equation
dWy Ww ow
u
— ~—— + — + vy —— +
h, ox ho M oy LOE
Ue BW i Wze ane peels ( owe =
ia OE ia 21UgWoe By Coa - (w'v Ne (14)
Where wz = dw/dz and (w'v')z = d(w'v')/dz. These
equations are subject to the following boundary con-
ditions:
y = 0: u=ve=w, =0 (15a)
y= 6: WS Tap We SMa (15b)
The other initial data should be selected near
the bow of the ship along the line perpendicular to
the z = const coordinate (see Figure 1). However,
because of the variety of possible bow shapes,
approximations are necessary. For a simple, smooth
bow section, where curvatures are small and no
separation is expected, the flow along the initial
line can be successfully assumed to be two-
dimensional without pressure gradient, and the
governing two-dimensional equations for a flat
plate are solved. However, for most general mer-
chant ships, the bow section is complicated and
flow separation and reattachment are expected be-
cause of large curvature variations and adverse
pressure gradients; as a consequence, the boundary-
layer calculations can only be performed downstream
of the attachment line (or point) where turbulent
5 INITIAL CONDITIONS (z = 0) ON
ve THE PLANE OF SYMMETRY
flow is presumed (since it is unlikely that the flow
remains laminar after separation and reattachment
with high Reynolds number). Generation of the
initial data for turbulent flows is much more in-
volved if there are no experimental data available.
It requires sound mathematical and physical judgment
and tedious trial-and-error efforts. We shall
discuss this aspect of the problem later in the
paper.
Turbulence Model
For turbulent flows, it is necessary to make closure
assumptions for the Reynolds stresses, -pu'v' and
-pv'w'. In our study, we satisfy the requirement
by using the eddy-viscosity concept and relate the
Reynolds stresses to the mean velocity profiles by
' ’ du ' ' ow
UNA NG Ri ty
Hh et (16)
We use the eddy-viscosity formulation of Cebeci
(1974), and define €, by two separate formulas. In
the inner region, €, is defined as
au A
(emi = | (3)
where
L = 0.4 y [1 - exp(-y/A) ] (18a)
v Cae We
NS 26 wh. S|) (18b)
T p
2 2 L
Trac ) + (2) + 2cos@ (2) (=) (18c)
Y/ w YY. <n NIMS
In the outer region €_ is defined by the following
m
formula
Eq = 0.0168 at (una u,) dy (19)
0
where
= (as bP aweh 2 Q)2 (20a)
Daa S (US We UgW.Cos
UE = (u2 + we + es) e (20b)
The inner and outer regions are established by the
continuity of the eddy-viscosity formula.
Transformation of the Basic Equations
The boundary-layer equations can be solved either
in physical coordinates or in transformed coordinates.
Each coordinate system has its own advantage. In
three-dimensional flows, the computer time and
storage required is an important factor. The trans-
formed coordinates are then favored because the
coordinates allow larger steps to be taken in the
longitudinal and transverse directions.
We define the transformed coordinates by
x
u
xX =X, Z= Z, dn= (=) dy, Ss) = Jonjax (21)
1 0)
and introduce a two-component vector potential
such that
uhysing + oun wh ; sing =e) (22a)
ay oy
vhyh sing = - (Be ae) (22b)
ox ox
where y and » are defined as
y= (vs jug) thf (x,z,n) sing (23a)
$ = (v8 )Ug)% (Uo ¢/U,)h) g(x,2,n) sing (23b)
and Uyef is some reference velocity.
Using these transformations and the relations
given by (9), (10), and (11), we can write the x-
momentum and z-momentum equations for the general
case as:
x-Momentum
(b£")' + m)f£" - mp (£')2 - msf'g'
+ mgf"g - mg(g')* + mj]
z-Momentum
(bg")' + mjfg" - m,f'g' - m3(g')* + mggg”
ay2
= fake) (GEN) sp ial
ag! af ag' ag
4 @ OB gf OE ; ee os
= m0 (« es =) wad (s BD We eBeay) ate
and their boundary conditions as
i) = Wkap 287 = dbo Gl = WEAtas (26)
Here primes denote differentiation with respect to
nN, and
€
The coefficients m, to mj, are given by
s 3u, s
m, = L @ Ee — =) fe pee NE See (hy sing)
2 hyu, 8x hjhosin@ 3x
S] dU Vref
a ae = -s,K t
M5 hug Ox s,K,cote, M3 S)K5 Fi cot¢
¥ _ 8a Gress dUe mR Uref
M, = S)Ko1, ne as ae aon SiRio ou
e
Sal Crete
Me = pects ee a Daa Oe (4 s; hy sing (28)
h hosing aera dz e Ue
eS 1
i 2
Si ref
m7 Se, Mg = 51K) ( ) escé
2 Ys Ue
u S$}
Mg = S )K) csc8, mig = rae
ref 1
191
We We
nn, Stiles oe Ms + m (
ef Uref
w We 2
m)2 = my m3 ( ) + M9
ref Uref
M19 We M7We IWe
Uref ox u2 if az
To transform the longitudinal attachment-line
flow equations and the boundary conditions, we use
the transformed coordinates given by (21) and de-
fine the two-component vector potential by
uhgsin@ = oe,
: oY)
hyh =-(— +
vh;h9sin6o (2 ®) (29)
with ¢ and still given by (23). With these vari-
ables, the longitudinal attachment-line equations
in the transformed coordinates can be written as
(bE")' + m, ff" = Ty (£1)2 ar mefi"g + Mm) }
Ss '
1 @ O20 a at ) Go
(bg")' fy mg"£ pa myf'g! - m3 (g!)2 + megg" tas Mg (£')2
= ag! of
+mjp = — (= a a) (31)
The boundary conditions and the coefficients m, to
m)2 are the same as in (26) and in (28) except now
Mien 7.8
N = Neo? gr
Yet
m3 = af Suet me =m
3 ho us , 6 3
mg = 0, M)}] = m2
2
Wze Wze 3 1 dWre
m)2 = m3 + my oF A 5 (32)
Uref Uref Ip Uret ox
In terms of the transformed variables, the alge-
braic eddy-viscosity formulas as given by (17) to
(20) become
2
i 2 y\ 2
(a= S01 |, = Seo . 2) (Ey)
vRy B
2 L
u a u 3
+ aa (g") 4 + 2 nee ogcost (33)
Ge Ue
he We 2 We 4s
(Sale = 0.0168 VR, if 1+ e + 2\ —]cosé
é Ue Ue
2
u
- ee (Ens
Ue
u A
+ 2 ee eeticose dn (34)
Ue
Here R, = u,S}/v and
Une ‘3
+2 mamgawewccce (35)
3. COORDINATE SYSTEM
Since, in general, a ship hull is a complicated non-
developable surface, a Cartesian coordinate system
is not suitable for boundary-layer calculations.
Most existing merchant and naval vessels possess
the following features: a flat bottom [y = £(x,zZ)
is not a single-valued function]; a bottom which is
not parallel to the water surface; and a bow which
has a submerged bulb extending toward the origin.
In addition, the problem is further complicated by
the existence of a free surface, corresponding to
the water level of a partly-submerged hull. The
chosen coordinate system must be sufficiently general
to allow these various features to be represented in
the boundary-layer calculations.
The streamline coordinate system is superficially
attractive but the determination of the streamlines,
the orthogonal lines, and the associated geometrical
parameters requires considerable effort. They are
dependent on the Froude number, and also on the
Reynolds number if the displacement effect is taken
into account. Consequently, and in addition to
being hard to compute, this coordinate system be-
comes uneconomical to use when the effect of the
Froude number and the Reynolds number are to be
systematically examined. -
A desirable requirement of a coordinate system
for the boundary-layer calculations is that it be
calculated only once. Miloh and Patel (1972) pro-
posed an orthogonal coordinate system which depends
only on the body geometry and is calculated once
and for all. This coordinate system has been applied
by Chang and Patel (1975) to boundary-layer calcula-
tions on two simple ship hulls: ellipsoid and double
elliptic ship. One of the coordinates is taken as
lines of x = X = constant and the other as z(X,Z) =
constant, which is orthogonal to x = constant lines
everywhere on the ship hull, and is obtained from
the solution of the differential equation
dz fzts
ax | 1 + £2 Ee
Zz
Here y = £(X,Z) defines the ship hull, and (x,y¥,Z)
denote the Cartesian coordinates. The major ad-
vantage of this coordinate system is its simplicity.
Because one of the coordinates is subject to the
condition (36), there is no guarantee that the
boundaries of the ship hull are coincident with the
coordinate lines. Furthermore, for a ship with flat
bottom for which y = £(xX,Z) is not a single-valued
function, one of the coordinates cannot be calcu-
lated from (36). The coordinate system is limited,
therefore, to some special geometries only.
In this study we adopt a nonorthogonal coordinate
system similar to that developed by Cebeci, Kaups,
and Ramsey (1977) for arbitrary wings. It is based
on body geometries only and, hence, it is calculated
once and for all. In addition, the system can deal
with the peculiar features of most merchant and
naval vessels discussed previously. The details
of this coordinate system are described briefly
in the following paragraph.
Now consider the ship hull as given in the usual
Cartesian coordinate system; that is, x along the
ship axis, y and z in the cross-plane (see Figure
1). We select x = x = constant as one of the co-
ordinates and the other coordinate, z, lies in the
yz-plane. Because the coordinate system is non-
orthogonal, we are free to select the values of z
in the plane to satisfy the condition that the
boundary lines of the ship hull are coincident with
Z = constant coordinate lines. There are several
ways of finding the z-values. Here z is determined
by mapping each yz crossplane into a half or hull
unit circle depending on whether the crossplane in-
tercepts the free surface or is completely submerged.
The polar angle, normalized by tm or 27 on the unit
circle, is taken as z-values. The z-values then
range from 0 to 1 on each crossplane. The advantage
of the mapping method is that equi-interval, z =
constant coordinate lines are automatically concen-
trated in the region of large curvature where the
boundary-layer characteristics are expected to vary
greatly. Hence the number of z = constant coordinate
lines can be reduced without loss of accuracy.
There are several methods available for the map-
ping of an arbitrary body onto a unit circle. Here
we use the numerical mapping method developed by
Halsey (1977). It makes full use of Fast Fourier
Transform techniques and has no restrictions on the
shape of the body to be mapped. To map a smooth
crossplane onto a unit circle, the procedure is
fairly easy. If there are inner corner points, or
trailing-edge and leading-edge corner points (see
Figure 2) caused by the reflection of the cross-
plane, they must be removed before mapping is per-
formed to improve numerical accuracy and to provide
rapid convergence. The inner corner points are
rounded off by using Fourier series expansion tech-
nique and the leading-edge and/or trailing-edge
corner points are removed by using the Karman-
Trefftz mapping. For details see Halsey (1977).
To use the mapping method to find the coordinate
system, it is only necessary to define the ship hull
as a family of points in the x = constant planes,
to locate the intersection of the ship hull and the
free surface, and to indicate whether corner points
exist. The data in each plane is then mapped into
a unit circle as ¥ vs z and z vs z and interpolated
for constant values of z. Another set of spline
fits, in the planes z = constant for y vs x and Zz
vs x, completes the definition of the coordinate
[aie le aN
LEADING-EDGE
CORNER POINT
W.L.
TRAILING EDGE f y
CORNER POINT:
SHIP HULL
«(NNER CORNER POINT
FIGURE 2. Notation of corner points used in the
Mapping procedure.
system. The lines formed by the intersection of
the planes x = constant and z = constant with the
hull constitute the nonorthogonal coordinate net
on the surface, and the third boundary-layer coor-
dinate is taken as the distance normal to the surface
in accordance with first-order boundary-layer approx-
imation.
Since the spline-fitting also yields derivatives,
the metric coefficient and the geodesic curvatures
of the coordinate lines can be calculated from the
formulas given below.
The metric coefficients:
D) 2.
4 i
mene (24) « (2 (37a)
ax / 2 SY
2 (SEN> (ES?
hp = (=) oF (2) (37b)
x x
The angle between the coordinate lines:
i =) (=) Gy &)
cose = (2 as ap — = (38)
hyho oz 5% ox i ox 2 Cr4 .
The geodesic curvature of the z = constant line:
fe, 1 oy 22 | _ (ax oz
1“ huhosind oz ox ox dz
1 x Zz Zz x
(2 a4y oy ay
ox ox2 ox ox2
oF 29
‘ @) e “A
x Zz
(39)
az (eS
az ax2
x 2
The geodesic curvature of the x = constant line:
(ase jaceaee oy) OB. on (oe 82
a hy hi sind Zz) \ox/ , ax/ , \dz
x
(40)
The other parameters K}2 and K2] are calculated from
(6). It may be noted that K; and K» can also be
obtained from (5). This provides a check on the ex-
pressions given by (39) and (40).
In the boundary-layer calculations, we need the
invisid velocity components along the surface
coordinates. Let V be the total velocity vector
on the hull, (Uu,v,w) the corresponding velocity com-
ponents in the Cartesian coordinates, and (Ue,We)
in the adopted surface coordinates. As can be seen
from Figure 3,
V- ie, = ¥ o ee cos®
pean sin26 a
v- ty = Vv 2 t) cos¢é
eS kee aa: oe
> =>
Here t; and t2 are the unit tangent vectors along
x and z coordinates and are given by
ne kles @)s a) |
t) = Taq 1+ Cz 5] (2) k (43)
Zz Z
= = Lee) se oz\ 2
tre (=) 4) (2) k (44)
x x
193
FIGURE 3. Resolution of the velocity components.
With the definition of V and with the use of (43)
and (44), Eqs. (41) and (42) can be written as
= {92
MN Ba (45)
x
1 atl |le x) B (2
= + —<
“OT Sines ho @ © \G2 ee
4. NUMERICAL METHOD
We use the Box method to solve the boundary-layer
equations given in Section 2. This is a two-point
finite-difference method developed by Keller and
Cebeci. This method has been applied to two-
dimensional flows as well as three-dimensional
flows and has been found to be efficient and accu-
rate. Descriptions of this method have been pre-
sented in a series of papers and reports and a
detailed presentation is contained in a recent
book by Cebeci and Bradshaw (1977).
In using this numerical method, or any other
method, care must be taken in obtaining solutions
of the equations when the transverse velocity com-
ponent, w, contains regions of flow reversal. Such
changes in w-profiles will lead to numerical in-
stabilities resulting from integration opposed to
the flow direction unless appropriate changes are
made in the integration procedure. Here we use the
procedure developed by Cebeci and Stewartson (1977).
In this new and very powerful procedure, which fol-
lows the characteristics of the locally plane flow,
the direction of w at each grid point across the
boundary layer is checked and difference equations
are written accordingly. At each point to be calcu-
lated, the backward characteristics which determine
the domain of dependence, are computed from the
local values of the velocity. Since the character-
istics must be determined as part of the solution
a Newton iteration process is used in the calcula-
tion procedure to correctly determine the exact
shape of the domain of dependence.
To illustrate the basic numerical method, we shall,
194
at first, consider the solution of the longitudinal
attachment-line Eqs. (30) and (31) and then the
solution of the full three-dimensional flow equations
as given by (24) and (25). We shall not discuss the
Cebeci-Stewartson procedure for computing three-
dimensional flows with the transverse velocity, w,
containing flow reversal since that procedure will
be fully described in a forthcoming paper.
Difference Equations for the Longitudinal
Attachment-Line Equations
According to the Box method, we first reduce the
Eqs. (30), (31), (32), and (26) into a system of
five first-crder equations by introducing new depen-
dent variables u(x,zZ,n), v(x,Z,n), w(x,Z,n),
t(x,zZ,n), and 6(x,zZ,n). Equations (30) and (31)
then can be written as
ey yy (47a)
w'=t (47b)
(bv)' + 6v - mou* + mj] = mg u =e (47c)
(bE) 4 + 8 = miuw = m3w- = mou + mj)2
ow
= mM) 0 u Bee (47d)
6' = mju + mew +m au (47e)
1 6 10 ox
The boundary conditions (26) and (32) become
n= 0: u=w= 0 = 0 (48a)
N = No: u=l1, we Woe/Uref (48b)
We next consider the net rectangle shown in Figure
4 and denote the net points by
x0
i]
(o)
6
=}
|
SS Peesy) an ky
ll
(o}
=)
no = -
j-1 + hy a ek Bp acon a
We approximate the quantities (u,v,g,t,9) at
points (x05) of the net by functions denoted by
(uh, v9 wh, th, 65) . We also employ the notation for
points and quantities midway between net points and
for any net function SH
(x) n-1/2
(xq) n-1 (x,)
n
FIGURE 4. Net rectangle for the longitudinal
attachment-line equations.
,
na ph
aia = 2% * na} yaya = 2 ("3 i "j-)
nS /2 5 a(n y-1 ) eS pean n
ei, =" (33 + s3 yEE i fo =D S5 + 5-1) (49)
The difference equations which are to approximate
(47) are formulated by considering one mesh rect-
angle as in Figure 4. We approximate (47a,b) using
centered difference quotients and average them
about the midpoint (%y N5-1/2) of the segment P )Po2.
Olid S sit enya!
hy (uy j-1) 5-1/2 (50a)
il soy Gal — Aig
Similarly, (47c,d,e) are approximated by centering
them about the midpoint x,-1/2+15_1/2 of the rect-—
angle P)P )P3P,. This gives
-1 n n n
he by) =" (bv ar (Chi)
: ( XA) ( Deon ( Deep
a n 2\n _ aril meer
(m5 a a) (u 5 -a/2 Reey/2 M)] (50c)
= ak n n n
ite || Coren = “Cee)) + (Git)
J 5 5-1 ‘5-1/2
= n n sd n 2 n = n 2 n
(m, + O,) (UW) a7 m3 (Ww eee Mg (u Vee
a, sat we uM yntl wi
Sn) Y4-1/205-1/2 5-1/2" 5-1/2
=cut n (50a)
jai72 ~ ie
1 n n n n TT
hj G = 1) -( nj + 2a) 8-1/2 ea
n-1
aT (50e)
Here
n-1 pa 2,n-1
jo a ey
Sil n= n-1 ig —ill
= hie bv).
{5 [( v) (bv) 1] = COR.
=a! ey a. n-l
2: 3-172 aH a (51a)
n-1 a ( n-1
y= SO Bp
=e n-1 n-1 n-1
=< he bie)is 3 (VoKe)) a + (6
o [ ( ); ( 5-1] ( ) 5 1y2
TAN 4 Dol n-l, 9\n-1
—m (w ts 2
3 ewe 9 (u Boy (51b)
n-1 n-1
ey. gooey
Ly ek 1 n-1 n-1 n-1
£5 (@ i ai) Ty Mile
TS =i jaa
6 j-1/2 (Slc)
(j,n,i-i) (j,n,i)
(j-1,n,()
(jn b—
a
Ha
“aA
(j-1, n—1, i)
(j-1, n=1, i-1)
f— Cierra
FIGURE 5. Net cube for the difference equations
z(i)
for three-dimensional flows, wj > 0.
n-1/2
w@
= (51d)
n xk
n n=
Difference Equations for the Full Three-Dimensional
Equations
The difference equations for the full three-
dimensional equations, as given by (24) and (25),
are again expressed in terms of a first-order sys-
tem. With the definitions given by (47a) and (47b),
they are written as
(bv)' + @v - mju? - mcouw - mgw? + m))
=Mj9 u oy m7w x (52a)
(bt)' + 6t - myuw - m3w* - mgu2 + m)9
=™)9 u oy m7w a (52b)
6' = mju + mew + mo a + m7 se (52c)
Their boundary conditions, (26) become:
n= 0: u=w=60=0 (53a)
in) = wWs3 u=1, W = We/Uref (53b)
The difference equations for (47a) and (47b) are
the same as those given by (50a) and (50b): they
are written for the midpoint [(x) ye (2) 55-770] of
the net cube shown in Figure 5; that is,
al Ca - ut ) = yori
j jon) > “3 ai7a°
SW ciogah L Gaelpsl\\ = Afalpal
my G “ie osy2 ee)
The difference equations which are to approximate
(52a,b,c) are rather lengthy. To illustrate the
difference equations for these three equations, we
consider the following model equation
(bv)' + 0v + m,)] = mo u oe 4 m7w ~— (55)
The difference equations for (55) are:
195
-l) — — —— n-1/2
A a ; 8
te |=, Wa + ( 5 fp + Cn) 5 yp
u-wu
a yn 25 n n-]
10° 4-1/2 3-1/2 k
Bly 1 ln
INI / 2 al i-l
+ See
O21 7295-12) xy oe
Here, for example,
rae al n,i i ynriml n-1,i-1 va)
J @\9 j j j
a ak Flat n,i-1l alae ,i-l
=a +
a a G j el oP 4; 1
Sn dh pal. n=), n,i n=1,i
na 7 a (o! io ene eo ) ee)
and
n-1/2
(m1) 5-1/2
1 n n n-1 n-1
S 7 inde w Cina ony Cade |e in) aaa
Zz) = 0 Fag Sag on eB a a Bo acbop 28 (58)
Solution of the Difference Equations
The difference equations (50) for the longitudinal
attachment-line flow and the difference equations
for (52) are nonlinear algebraic equations. We use
Newton's method to linearize them and then solve the
resulting linear system by the block-elimination
method discussed by Keller (1974). A brief
description of it will be given for the streamwise
attachment-line equations.
Using Newton's method, the linearized difference
equations for the system given by (50) are:
h,
5 al =
ORS Oy, Sa Way > Og) = Ga) (59a)
Ay
- ‘ -=— me taOiter. = F 59b
by > Ota oe Oe, Oh, 4) = Ge (59b)
aOWin oF év., + 200.5 4 NOOR
(S1) 5 Nea (G2) Ve (63), 5 (Su) 5 5
1
+ “OU ou, = ; 5S)
(os), Bas (Se), Bag 5S) 2 (59c)
R a (Sh A a sor : :
(By), 8, + (B2),6t,_, + (Ba), 60, + (8,), 68
> i
_ W. _W. gu, + (Bg). cu,
+ (Bs), 8", + (Be), + (B75 S48, + (Bq); ou
1 J aaa
; (59d)
(ry),
+ fi A + z s
(01) 68, (oa) a OO5_ 5 (93). du
+ (6). 6,
j Si 5 Oe
j-1
+ Wop) Om + Woie)) 4 Ors 5 = (x5), (59e)
196
Here we have dropped the superscripts n, i and have
defined (1K) 51 (%) 5. (8) 54 and (oy) 5 by
r= Ble - ws to BeVe a75 (60a)
ro = a - we + Meee iV/e (60b)
= ann ay
ea) s = See M)]
= bv)! + :
[ ean” SONS,
(60c)
- (my + 2) saa
p=
wa ase =
(Ty) 5 Faia 7 2
- lone) Y ne) =
[ Veena eo ey Ot On) (UW) 5 a7
ag 2 = 2
Hee FOC n7e
ms! n=
z ("5-1/2 wj-1/2 “4-1/2 “a7 oe
_ ari
ity) 5 T 5-1/2
. ev SONS Ce ORE pn Petscaja | (60
fo i
a eo
(5 AT ies 8, (61a)
j
P51
(aS Se te Ones (61b)
j
(3). = ev 61
(E5) 7 2 73 ( c)
(ty). = 2G
Bia ay Was (61d)
(e5), =-—- (mp + a )u (6le)
(56). = =) (GY) oC ae (61£)
Sie =
( 1), (21), (62a)
Bo). = :
( 2), (2), (62b)
Fa deep ae
(83), = ks (62c)
(Baya it
Tepe (62a)
A at il n-1
(Bs). = 5 (m, + 4 _)u. - m3w. - hon Y5_1/2 (62e)
(Be) =- = (my, + a_)u i maa
tg =
Deana eZ ozs)
= + a ple n-l
(87), = (m, + @ Ns mgu. + 5 a W512 (62g)
(Go, = oS Gu eg. om
eee) aan ipa ean
ee ee 62h
DS S/2 (ny)
1
(91), = (63a)
J
all
Kop) Baie (63b)
J
(93). = - Ga + 2a_) (63c)
3) > (mM a c
(Oy). = - Tr ae 2x01) 63d
4 5 = 2 1 n ( )
(C5) ai 2 63
Sa > M6 (63e)
(Ge). =o z m (63£)
The boundary conditions become
Sug = Swo = 58 = 0, bus = ow, = 0 (64)
The solution of the linear system given by (59)
and (64) is obtained by using the block elimination
method. According to this method, the system is
written as
fA gé=r 65
aes (65)
Here
Ao nS
Sr eae
NG
w= : Cc
J 3) J
8 Ey (xy), Oe
§, Ko (r)5 év
goa] B [a to | Os ||
: : (5 OE.
a5 Xs (ae 80,
The A., By, C. in /A denote 5x5 matrices. The
solution of (25) is obtained by the procedure
described in Cebeci and Bradshaw (1977).
5. RESULTS
Turbulent Flow Calculations for a Curved Duct and
Comparison with Experiment
The turbulence model described in Section 2 has been
used with considerable success to compute a wide
range of two-dimensional turbulent boundary layers
[see for example Cebeci and Smith (1974)]. The
model has also been used to compute three-dimensional
flows and again is found to yield accurate results
[see for example Cebeci (1974, 1975) and Cebeci,
Kaups, and Moser (1976)]. To further test the model
for three-dimensional flows, we have considered the
experimental data taken in a 60° curved duct of rect-
angular cross section. Figure 6 shows a sketch of
the flow geometry. The experimental data are due
to Vermeulen (1971). Here z denotes the distance
from the outer wall, measured along normals to the
wall; x denotes the arc length along the outer wall;
and y denotes distance normal to the plane x,z.
To test the computed results with the data, it
is necessary to specify the initial profiles given
by experiment. This can be done in a number of ways.
In the study reported by Cebeci, Kaups, and Moser
(1976) the profiles were generated by using Coles'
velocity profile formula. That formula, which repre-
sents the experimental data rather well for two-
dimensional flows, was not very satisfactory for
three-dimensional flows. Here we abandon the use
of Coles' formula in favor of Thompson's two-
parameter velocity profiles as described and im-
proved by Galbraith and Head (1975). According
to this formula, the dimensionless u/ug velocity
profile is given by
INITIAL
CONDITIONS
MEASURING LOCATIONS
eee
FIGURE 6. Coordinate system and notation for the
. curved duct.
197
u
ei Ye (=) 15 (hy S 57a) (66)
= €/ inner
Here yg is an intermittency factor defined by the
following empirical formulas:
‘LoS =
Oia Oe
y y Z
405 < °*S4 Oo, = 1 - 2.64214(— - 0.05
Sn” hes (5 nh )
y y d
poi Ss = PE
O25 <a 8 Oo ie 4.4053(2 0.5)
- 1.8502( 4 = 0.5) + 0.5
89
y 2
0.7<*%< 0.95 yg, = 2.64214 (= 0.05)
50 0
y A
nee 0.95 Ys = 0.0
The dimensionless velocity profile for the inner
layer, that is, (u/ug)inner, is given by
y <4 ut = y*
4< yt < 30 ut = c) + coln yt + c3(1n yt)?
+ cy (in y*)3
y > 30 u’ = 5.50 In y’ + 5.45
Hene en = 4 Si co =" Sey 45y Cape— OP elOF ici
=0. 767) yu = WY, B= (t,,/2) 2, ut = u/u,, and 6,
is a parameter which is a function of 6, Ce, and H.
To find the functional relationship between On
CE 68, and H, we use the definitions of displacement
thickness, 6*, and momentum thickness, 9. Substitut-
ing (66) into the definition of 6*, after some alge-
bra, we get
A
SE i ae red a
55 Rae
c Cc
£ 5* f )
= 0.5 + — — - = —
0.5 5 Ay, In oe A3 Ao In ce 5 (67)
where
A, = 50.679, Ap = 1.1942, A3 = 0.7943, Ay = 1.195.
An expression similar to that given by (67) can also
be obtained if we substitute (66) into the defini-
tion of 8. However, the resulting expression is
quite complicated. For this reason, the expression
for 8 is obtained numerically, and for a given value
of © and H, the corresponding values of c¢ and 6,
are computed from that equation and from (67).
Equation (66) is recommended for two-dimensional
flows. Here we assume that it also applies to the
streamwise velocity profile by replacing u/u, by
u,/ug, with c, now representing the streamwise skin-
friction coefficient.
In order to generate the crossflow velocity com-
ponent (a)/ug,) + we use Mager's expression and
define Up/Us, by
198
08
0.2
FIGURE 7. Comparison of gener-
ated initial total velocity ° sesh
profiles with Vermeulen's data. y 0
Yn Us ;
aan ee ( - x) tang, (68)
Se "se
with the limiting crossflow angle fy, obtained from
the experimental data.
Once the streamwise and crossflow velocity pro-
files are calculated by the above procedure, we
compute the velocity profiles u/u, and w/wWe in the
orthogonal directions x and z by the following rela-
tionships
u Uu. WwW -
u s n e
Tee Gow eMOS feo)
e Se Se e :
u u
Ww Ss Un e
a (69b)
We Us, Us, We
Figure 7 shows a comparison of generated and
experimental total velocity profiles along the line
A. As can be seen, the procedure discussed above
for generating the initial velocity profiles from
the experimental data is quite good. This is impor-
tant for an accurate evaluation of a turbulent
model, especially for three-dimensional flows.
Here
Oo DATA
— PRESENT METHOD
844 (mm)
©
iT
o
}
\e)
°
°
°
=O
SS SS SS
> oa ao
°
x (m)
FIGURE 8. Comparison of computed momentum thickness
with Vermeulen's data.
AN
ANZ
A10
oo AN AB
GENERATED PROFILES
GENERATED PROFILES
DATA
° DATA
— ) 0 —— 4 —————
30 40 0 10 20 40 50
y (mm) y (mm)
(0) u2 + w
Oa wee ee (70)
— e
The solution of the boundary-layer equations also
requires the specification of the metric coefficients
and the geodesic curvatures. They are calculated
from the following expression:
(a: straight section
hy =
1 - 2/R curved section
le fe)
hy = 1.0, Ko = 0 (71)
(a : E
(0) straight section
K] =
1/ (Ro-2) curved section
S
A comparison of calculated and experimental values
of streamwise momentum thickness, 8 ),, shape factor,
H}1, skin-friction coefficient, cf, and limiting
crossflow angle, 8y, is shown in Figures 8, 9, 10,
and 11, respectively, along the lines B, C, D, E.
Here the limiting crossflow angle is computed from
° DATA
— PRESENT METHOD
x (m)
FIGURE 9. Comparison of computed shape factor with
Vermeulen's data.
© DATA
——PRESENT METHOD
x (m)
FIGURE 10. Comparison of computed skin friction
coefficient with Vermeulen's data.
We Ate urea) oe 5 f
EE (72)
2 " "
a) (eed a) Se bg
tanBy =
Figures 12 and 13 show a comparison of calculated
and experimental total velocity profiles and cross-
flow angle profiles along the lines C and E. Here
the crossflow angle is computed from
w/oa L ee) 5" ie £"]
(73)
9/2, (2/4)?
sinpy, =
As in Figures 8 through 11, again the agreement
between calculated results and experiment is very
good. The computed results follow the trend in
the experimental data well and indicate that the
present turbulence model, as in two-dimensional
flows, is quite satisfactory for three-dimensional
flows.
Results for a Double Elliptic Ship Model
To test our method for ship hulls, we have con-
sidered two separate hulls. The first one, which
is discussed in this section, is a double elliptic
ship whose hull is given analytically. The second
DATA
PRESENT METHOD)
1.0 15 2.0 25 3.0
x (m)
FIGURE ll. Comparison of computed limiting crossflow
angle with Vermeulen's data.
nie})
one, which is discussed in the next section, is
ship model 5350 which has a rather complex shape.
Its hull is represented section-by-section in tabu-
lar form and contains all the features of most
merchant and naval vessels. It proves an excellent
test case to study the computational difficulties
associated with real ship hulls.
The double elliptic ship model can be analytically
represented by
pi ty UAL Ne le BY? (le
Y = 2 (s4) = TON ae 1 -{ = (74)
It has round edges except for the sharp corners at
x = +L and z = +H. The body of L:H:B = 1.0:0.125:0.1
together with the nonorthogonal coordinate nets on
the hull is shown in Figure 14.
The potential-flow solutions were obtained from
the Douglas-Neumann computer program for three-
dimensional flows. To get the solutions, 120 control
elements on the surface were used, 12 along the x-
direction and 10 along the z-direction.
Before we describe our boundary-layer calculations,
it is useful to discuss the pressure distribution for
this body shown in Figure 15. As can be seen from
the figure, the streamwise pressure gradient is
initially favorable in the bow region and then ad-
verse up to the midpoint of the body. This is fol-
lowed by a region of favorable pressure gradient and
then by a shape adverse pressure gradient very close
to the stern. The crosswise pressure gradient varies
in a more complex manner. Near the bow the pressure
decreases down from the water surface to a minimum
and then increases as the keel is reached. As the
flow moves downstream, the location of the minimum
pressure moves up and reaches the water surface at
about x/L = -0.80. The minimum pressure remains at
the water surface to about x/L = 0.80 and then moves
toward the keel. As a result, near the bow and the
stern, one may expect flow reversal of the crossflow
across the boundary layer does not reverse direction
from the keel to the water surface. This conclusion
is drawn from considering the pressure gradients only.
The real situation may be somewhat modified because,
in addition, there are the upstream effects and the
curvature effects on the flow characteristics.
The boundary-layer computation starts with turbu-
lent flow from X/L = -0.90. We have tried to start
the computation from X/L = -0.97 and X/L = -0.95.
However, flow separation was observed at X/L = -0.90
near the keel due to the sharp curvature and adverse
pressure gradient in the bow region and can be seen
from Figure 15. In the previous calculations of
Chang and Patel (1975) and Cebeci and Chang (1977),
the flow separation near the bow was not found due
to the orthogonal coordinate system they adopted in
which the second net point from the keel is so far
from the keel that the region of adverse pressure
gradient is omitted.
In our boundary-layer calculations, we have used
40 points along the x-direction and 16 points along
the z-direction. In the normal direction, we have
taken approximately 40 points. The nonuniform grid
structure described in Cebeci and Bradshaw (1977)
is employed in the normal direction so that the grid
points are concentrated near the wall where the
velocity gradients are large.
Some of the computed results for R; = 10’ are
shown in Figures 16 and 18. Figure 16 shows the
spanwise distributions of the pressure coefficients,
Cp, local skin-friction coefficient, cre, the shape
200
0.2
0.2
FIGURE 12. Comparison of com-
puted total velocity profiles ° 1
C6 0.8
Cc! 02
PRESENT METHOD
DATA
PRESENT METHOD
DATA
3% 20
y (mm) y (mm)
E4
£3
PRESENT METHOD
PRESENT METHOD DATA
DATA 0
with Vermeulen's data.
factor, H,;,], the Reynolds number based on the momen-
tum thickness, Rg, and the limiting crossflow angle
for x/L = -0.85, 0.0, and 0.75. As can be seen from
these figures, the boundary-layer parameters vary
greatly near the keel where the curvatures and the
pressure gradients are large and remain almost un-
changed near the surface where the curvatures and
the pressure gradients are small. Except at x/L =
-0.85, the limiting crossflow angle is positive.
This implies that the crossflow near the wall moves
from the keel to the free surface as predicted from
the pressure distribution. Figure 17 shows typical
longitudinal and transverse velocity profiles at
z= 0.6 for several values of (x/l), and Figure 18
shows typical transverse velocity profiles at (x/L)
= -0.2 for several values of z. As can be seen from
Figures 17(b) and 18, the transverse velocity compo-
nent undergoes drastic changes in the longitudinal
and transverse directions under the influence of
pressure gradient and body geometry. As was dis~
cussed before, when the transverse velocity changes
30 20
y (mm) y (mm)
sign across the boundary layer and contains regions
of reverse flow, numerical instabilities results from
integration opposed to flow direction unless appro-
priate changes are made in the integration procedure.
The new numerical procedure of Cebeci and Stewartson
(1977) handles this situation very well and does not
show any signs of breakdown resulting from flow re-
versal of transverse velocity component.
Results for Ship Model 5350
The ship model 5350, unlike the one discussed in the
previous section, is a realistic tanker model. The
geometry of the hull is so complicated that it is
represented in tabular form section by section.
The model possesses all the special features of
existing merchant and naval vessels, that is, a
bottom which is flat and not parallel to the still-
water surface and an extended bow completely sub-
merged under the water surface, and consequently
201
20) Cé = PRESENT METHOD
——— PRESENT METHOD
° ‘DATA
° DATA
30
y (mm) ‘S
nN
c
20 +
i \
\
’ \
NS
10
" Ny \
A Sie. .
9 As as we rt AS a
0 10 15 20 10 20 30 40
J (degrees) J (degrees)
60
£10
£16
E14
50 5 50 |
£12 i
PRESENT METHOD
fo} fe)
40 + ° DATA 40 + ?\ \ ©
c6 Es OF o\ O\
y (mm) o \ o\ a ~ PRESENT METHOD
&4 a\ ° aX }
30 | 0 \ ° } ON ° DATA
BS om ° ) °
y (mm) g o\ ° ° °
€1 } g °\ Ne °
20 20 + 2 \ 9 °
. Boe eens as
\ ve ‘e aX : s ;
Na} \e ) ° ° 0
10 7 QR 10+ \° ° ° °
\e So \o f) °
hoe o 6 Xe > GJ °
<2 SO'g 0
Bs On } 5 See oe plo oe as .
0 Ji fan os fu = 5) 0 o— ~Baed.c- 0 6, pa SS
0 0 0 0 5 10 15 20 0 0 ty) () 0 10 20 30 40 50
J (degrees) J (degrees)
FIGURE 13. Comparison of computed crossflow angle with Vermeulen's data.
serves as an excellent case on which to apply our
method.
Figure 19 shows a three-dimensional picture of
this ship model together with our nonorthogonal co-
ordinate system. We see from this figure that, as
a by-product of the mapping method discussed earlier,
the z = const. coordinate lines are concentrated
in the bow and corner regions where the curvature
is large. Figure 20 shows different cross-sections
(indicated by solid lines) and interpolated values
obtained by a cubic-spline method (indicated by
circles) from which the geometric parameters are
obtained.
The inviscid velocity distribution for the model
is obtained by using the Douglas-Neumann method
treating the model as a double ship model. Figure
21 shows the pressure distribution for the entire
ship and Figure 22 shows a detailed pressure distri-
bution for the bow region. We see from these figures
that the longitudinal pressure gradient near the keel
FIGURE 14. Three-dimensional picture of double ellip-—
-tic ship model with the nonorthogonal coordinate system.
FIGURE 15.
GIRTH, %
Pressure distribution for the double-
elliptic ship.
is)
in)
Cc, x 103
Hay
nN
(b)
40
2.0
Rg x 10-4
(c) = 425
(a) 1.0
FIGURE 16.
20 40 60 80 100
GIRTH, %
model for Ry; = 10° at (a) x/L = -0.85, (b)
|
0 20 40 60 80 100 |
GIRTH % 4
KEEL
zZ
———)
100 |
KEEL
Computed Cp, Ce, Hii, Ro, and By for
x/L =
x/L = 0.0
x/L = 0.75
_>y
x/L = —0.85
—_-—~Sy
the double-elliptic ship
0.0, (c) x/L = 0.75.
<
1.0 7
0.8
0.6
0.4
x/L = —0.85
4 1
0) 10 20 30 40 50 60 70 80
n(=(u,/¥s,) 2y)
FIGURE 17.
10’ at z = 0.6.
is favorable and then later becomes adverse. The
pressure gradient in the transverse direction de-
creases rapidly from the keel to a minimum value
and then increases continuously up to the free sur-
face. Due to this rapid pressure variation in the
bow region, preliminary boundary-layer calculations
showed flow separation and required an approximate
procedure to generate the solutions for x < 22.5 m.
After that (x > 22.5), the three-dimensional boundary-
FIGURE 18. Computed transverse velocity profiles.
203
*/L\= 0.25
x/L|= 0.50
x/L = 0.75
—0.05 0 0.05 0.10 0.15
—0.10
Computed longitudinal and transverse velocity profiles for the double-elliptic ship model for Ry =
layer calculations were performed for a given invis-
cid pressure distribution. The initial conditions
at x = 22.5 m were generated by solving the boundary-
layer equations in which the z-wise derivatives for
a constant z were neglected.
Figures 23 to 25 show some of the computed re-
sults for R; = 3 x 108. Figure 23 shows the varia-
TOM Cr Co, Cp Rg, Hy], and 8, at the cross-planes
of x = 30 m, 105 m, and 210 m. Typical streamwise
velocity profiles at x = 105 m and z = 0.2 are shown
in Figure 24 and typical crossflow velocity profiles
at x = 60 m are shown in Figure 25. As can be seen
from these figures, the crossflow velocity profiles
show great variations and indicate clearly the flow
reversal that takes place in the crossflow plane.
This implies that differential methods based on two-
dimensional and/or small crossflow approximations as
well as methods based on integral methods are not
adequate to boundary-layer calculations on ship
hulls. Other interesting results that emerge from
these calculations are the sudden jumps of the limit-
ing crossflow angle from positive to negative, and
the thickening of the boundary layer in the corner
region of the crossplanes. The jumps of the cross-
flow angle indicates the convergence of the flow from
FIGURE 19. Three-dimensional view of ship model 5350
with the nonorthogonal coordinate system.
204
INPUT SHIP FORM
° INTERPOLATION BY CUBIC SPLINE FUNCTION
BOW SECTIONS
FIGURE 20.
Body plan for ship model 5350.
both sides of the corner region and, hence, enhances
the thickening of the boundary layer. This thicken-
ing of the boundary layer in the corner region of
ship hulls has been verified experimentally by Hoff-
mann (1976).
6.. CONCLUDING REMARKS AND FUTURE WORK
According to the studies presented in this paper,
the three-dimensional boundary layers on ship hulls
can be computed very efficiently and effectively.
The turbulence model, as in two-dimensional flows,
again yields satisfactory results for three-
dimensional flows. This has been demonstrated
by Soejima and Yamazaki (1978) who also have applied
the present turbulence model to compute three-
dimensional boundary layers on ship hulls. However,
there are additional studies and problem areas that
need to be considered and investigated before the
present method can become a more effective tool to
design ships. They are briefly discussed below.
WATER SURFACE
STERN SECTIONS
Generation of Initial Conditions on Arbitrary
Bow Configurations
In Section 5, we presented calculations for the ship
model 5350 and mentioned that due to flow separation
in the bow region, we had to start the boundary-layer
calculations at some distance away from the bow.
Additional studies are required to generate the ini-
tial conditions on the bow. These studies can lead
to a better design of bow configurations and to
better handling of bilge vortices, which contribute
to the total drag of the ship. However, this is by
no means an easy task. Consider, for example, the
ship model 5350 discussed earlier. A sketch of the
bulbous nose with a plausible inviscid streamline
distribution is shown in Figure 26. We assume
that the ship is symmetrical about the keel plane
and there is a nodal attachment point on the bulbous
nose at B. If the ship is floating, then the water
line is determined by conditions of constant pressure
and zero normal velocity. Hence the intersection A
of the plane of symmetry with the water line and the
1.0 (WATER LEVEL)
ia 0.85
\
<[
0.50 ‘
—0.6 y
7
0.30 \
Px ry ‘
l] 7 0.15 \
0.2 \
7 1200 (KEEL) \ ‘ \
) \ \ \
FIGURE 21. Pressure
entire
distribution
for the ship’ model 5350.
’
Yi
205
evidence for this is based on a successful scheme
that we have already worked out for the prolate
spheroid, Cebeci, Khattab, and Stewartson (1978).
Other aspects that need further study include the
condition at the water-line section. It has been
usual to assume that the normal velocity is zero at
the undisturbed free surface. This is not quite
correct and the error may have implications for the
nature of the solution near A and especially the
question of separation along BA. Even if separation
does occur, it may be possible to handle the post-
separation solution, since it probably extends only
GIRTH, % over a limited region of the ship, by means of an
interaction theory, i.e., modifying the inviscid
0.2 flow by means of a displacement surface.
Viscous-Inviscid Flow Interaction
0.4
The present boundary-layer calculations are done
for a given pressure distribution obtained from an
inviscid flow theroy. In regions where the boundary-
layer thickness is small, the inviscid pressure dis-
tribution does not differ much from the actual one;
as a result, the boundary-layer calculations are
satisfactory and agree well with experiment, see,
for example, the papers by Cebeci, Kaups, and Moser
(1976) and by Soejima and Yamazaki (1978). When
the boundary-layer thickness is large, which is the
case near the stern region, the effect of viscous
flows on the inviscid pressure distribution must
be taken into account. One possible way this can
be done is to compute the displacement surface for
a given inviscid pressure distribution and iterate.
Such a procedure is absolutely necessary to account
bow is a saddle point with the streamlines of the for the thickening of the boundary layer as was
inviscid flow converging on A along the line BA and observed by Soejima and Yamazaki (1978).
diverging along an orthogonal direction. It is
known that the boundary-layer equations can always
0.6
0.8
1.0
FIGURE 22. Pressure distribution for the bow region
of ship model 5350.
be solved at B but that at A the situation is more Prediction of Wake Behind Ship Hulls
complicated and furthermore it is still not entirely
clear what their role is in relation to the general The present boundary-layer calculations can be done
solution. It is likely, however, that provided no up to some distance close to the stern; after that,
reversed flow occurs at A in the component of the flow separation occurs. Since one, and probably the
solution along the direction BA, then separation can biggest, reason why one is interested in boundary-
be avoided along this line by appropriate choice of layer calculations on ship hulls, is the calculation
design. Furthermore, if separation does occur, its of drag of the hull, additional studies should be
effect may be limited. The recently developed Cebeci- directed to perform the calculations in the separated
Stewartson procedure (1977), however, can be applied region and in the wake behind the ship. Recent calcu-
to the present problem but there are some hurdles to lation methods developed and reported by Cebeci,
be overcome. Keller, and Williams (1978) for separated flows by
Of particular difficulty is the choice of coordi- using inverse boundary-layer theory and recent calcu-
nate system on which to compute the solution and to lation methods developed and reported by Cebeci,
join it with the already well-established method Thiele and Stewartson (1978) for two-dimensional
downstream of CD. We have seen that in the case of wake flows are appropriate for these purposes.
the prolate spheroid (see Cebeci, Khattab, and
Stewartson (1978)) it is helpful to have a mesh
which is effectively Cartesian near the nose and the PRINCIPAL NOTATION
methods which were used to produce it in the earlier
study are applicable to any body which can be repre- A Van Driest damping parameter, see
sented by a paraboloid of revolution in the neighbor- (18b)
hood of the nose. Now here we have a paraboloid near A, ,A2,A3,Ay constants
B but not one of revolution, but we believe that the CE local skin-friction coefficient in
necessary generalization is possible. The mesh now streamwise direction
has to match with that which has proved convenient C1, ,C2,C3,Cy constants
downstream of CD. Again we believe that a smooth £ transformed vector potential for wp
transition can be achieved by building into the g transformed vector potential for
mesh sides, right from CBA, an appropriate spacing hy, ,ho metric coefficients
such that the points of a uniform mesh on CD are hy net spacing in n-direction
also points of this mesh although not, of course, at H,Hj] boundary-layer shape factor along
~ a constant value of one of the coordinates. Our streamwise direction, 6*/8)}
FIGURE
Rg, and 8, for ship model
for Ry,
(b)
x
2)
3
3
105m,
23. Computed c
pr
10° at (a)
and
(c)) x
Cc
1.0 a a
_50 GIRTH, %
x = 105m
Cy x 103
$
Hiy
(b)
a
eee 9
KEEL
—5° GIRTH, %
—0.05 4 —l. —| z
X = 210m
o>
SS SS SSS a
me eee ee EE EEO
3
°
Fad
m
m
-50 GIRTH, %
10) 50 100 150 200
nl=(u,/¥s,)"/y)
100 150 200
nl=u,/vs,)"/2y)
FIGURE 24. Computed streamwise velocity profiles for ship model 5350 for Ry, = 3 x 10” along (a) z = 0.2 and
(b) x = 105m coordinate lines.
150.0
nl=(u,/vs,) '/2y)
QOOOQOGO
—0.04 —0.02 0 0.02 0.04 0.06 0.08 0.10
FIGURE 25. Computed crosswise velocity profiles
for ship model 5350.
A.D FREE SURFACE D
_——Se ae Cc
207
- 250
FIGURE 26. Pattern of streamlines near the bow of ship
model 5350.
ky net spacing in x-direction
geodesic curvatures, see (5)
geometric parameters, see (6)
L mixing length, see (18a), or refer-
ence length
mM] ,Mo,---M)2 coefficients, see (28) or (32)
p static pressure
Q total velocity in the boundary layer
Ry Ry, Reynolds numbers, ugs)/v and u,L/v
Rgx Reynolds number, ug _6*/\v
Ro Reynolds number, Us,811/Y
s arc length along coordinate line
1 ,t2 unit tangent vectors along x and z
directions
u,V,W, velocity components in the x,y,zZ
directions
u,V,W velocity components in the Cartesian
coordinate
Ug Uy velocity components in boundary layer
parallel and normal, respectively,
to external streamline
u. friction velocity, see (18c)
Ugo freestream velocity
ref reference velocity
XT VIn nonorthogonal boundary-layer coor-
dinates
Se Cartesian coordinates
-pu'v',-pv'w' Reynolds stresses
B crossflow angle
By limiting crossflow angle
) boundary-layer thickness
6* displacement thickness,
oc
Sa- us/Us_,) dy
Exp eddy viscosity -
eu dimensionless eddy viscosity, E/Y
n similarity variable for y, see (21)
811 momentum thickness,
(oe)
al Us/us, (1 = us/Ug,,) dy
u dynamic viscosity
v kinematic viscosity
p density
rT shear stress
o,w two-component vector potentials, see
(23)
Subscripts
e boundary-layer edge
s streamwise direction
t total value
Ww wall
primes denote differentiation with respect to n
ACKNOWLEDGMENT
This work was supported by the David. Taylor Naval
Ship Research and Development Center under contract
NOO014-76-C-0950.
REFERENCES
Chang, K. C., and V. C. Patel (1975). Calculation
of three-dimensional boundary layers on ship
forms. Towa Institute of Hydraulic Research,
Rept. sNo 7.6).
Cebeci, T. (1974). Calculation of three-dimensional
boundary layers, pt. 1, swept infinite cylinders
and small crossflow. ATAA J., 12, 779.
Cebeci, T. (1975). Calculation of three-dimensional
boundary layers, pt. 2, three-dimensional flows
in Cartesian coordinates. ATAA J-., 13, 1056.
Cebeci, T., and P. Bradshaw (1977). Momentum
Transfer in Boundary Layers, McGraw-Hill/
Hemisphere Co., Washington, D.C.
Cebeci, T., and K. C. Chang (1977). A general
method for calculating three-dimensional laminar
and turbulent boundary layers on ship hulls. 1.
Coordinate system, numerical method and pre-
liminary results. Rept., Dept. of Mech. Engg.,
California State Univ. at Long Beach.
Cebeci, T., and A. M. O. Smith (1974). Analysis of
Turbulent Boundary Layers, Academic Press, New
York.
Cebeci. T., and K. Stewartson (1977). A new
numerical procedure for solving three-dimensional
boundary layers with negative crossflow, to be
published.
Cebeci, T., K. Kaups, and A. Moser (1976). Calcula-
tion of three-dimensional boundary layers, pt.
3, three-dimensional incompressible flows in
curvilinear orthogonal coordinates. ATAA J-,
147, hOIO
Cebeci, T., K. Kaups, and J. A. Ramsey (1977). A
general method for calculating three-dimensional
compressible laminar and turbulent boundary
layers on arbitrary wings. NASA CR-2777.
Cebeci, T., A. A. Khattab, and K. Stewartson (1978).
On nose separation, paper in preparation.
Cebeci, T., H. B. Keller, and P. G. Williams (1978).
Separating boundary-layer flow calculations, paper
submitted for publication.
Cebeci, T., F. Thiele, and K. Stewartson (1978).
On near wake laminar and turbulent shear layers,
to be published.
Gadd, G. E. (1970). The approximate calculation of
turbulent boundary-layer development on ship
hulls. RINA paper W5.
Galbraith, R. A. McD., and M. R. Head (1975). Eddy
viscosity and mixing length from measured bound-
ary-layer developments. Aeronautical Quarterly,
2OMAlbn. AB Shs
Halsey, N. D. (1977). Potential flow analysis of
multiple bodies using conformal mapping. M.S.
thesis, Dept. of Mech. Engg., California State
Univ. at Long Beach.
Hoffmann, H. P. (1976). Untersuchung der 3-
dimensionalen, turbulenten grenzschicht an einem
schiffsdoppelmodell in windkanal. Institut ftir
Schiffbau der Universitat, Hamburg, Bericht Nr. 343.
Lin, J. D., and R. S. Hall (1966). A study of the
flow past a ship-like body. Univ. of Conn.,
Civil Engineering Dept., Report No. CE66-7.
Miloh, T., V. C. Patel (1972). Orthogonal coor-
dinate systems for three-dimensional boundary
layers with particular reference to ship forms.
Iowa Institute of Hydraulic Research, Rept. No. 138.
Rastogi, A. K., and W. Rodi (1978). Calculav.on
of general three-dimensional turbulent boundary
layers. AIAA J., 16, 151.
Soejima, S., and R. Yamazaki (1978). Calculation
of three-dimensional boundary layers on ship
hull forms. Trans. West-Japan Soc. Naval
Architechs, 55, 43.
Vermeulen, A. J. (1971). Measurements of three-
dimensional turbulent boundary layers. Ph.D.
thesis, Univ. of Cambridge.
Study on the Structure of Ship
Vortices Generated by Full Sterns
Hiraku Tanaka and Takayasu Ueda
Ship Research Institute
Tokyo, Japan
ABSTRACT
Many attempts have been made to measure the vortic-—-
ity distribution of vessels tested at the Ship
Research Institute. This led to the successful
development of the rotor-type vortexmeter and a
method for its calibration. In order to investi-
gate the structure of the full ship stern vortices
and gain an understanding of interaction of the
vortices and propeller, the wake flow behind two
geosim models was studied experimentally.
Using this vortexmeter, detailed diagrams of the
vorticity distribution are presented for the dis-
cussion of the structure and scale effects on the
stern vortices. The authors found the existence
of a separating vortex sheet in the vorticity dis-
tribution and indicated that, by using the vorticity
concentrated on the vortex sheet (Max. line), it
was possible to simulate the original vorticity
distribution. With these experimental results the
relation between the vorticity distribution and
the propeller performance on the geosim models was
also analyzed.
1. INTRODUCTION
In recent years, the knowledge of the wake structure
including stern vortices has made it essential for
the ship builder to obtain a better understanding
of the stern vibration with full stern forms.
Nevertheless, the stern vortex characteristics such
as its geometry and structure as well as the scale
effect remained obscure. This situation may be
partially due to the fact that the stern vortices
do not cause serious problems in the resistance
augmentation or in the self-propulsion factors.
To overcome this lack of detailed knowledge,
systematic investigations have been made concerning
the problems of full ship models with unstable
propulsive performance. This research was begun in
.1975 under the Research Panel SR 159 of the Ship-
209
building Research Association of Japan (Chairman,
Prof. H. Sasajima) which was mainly concerned with
the following areas: sources of the unstable
phenomenon, the unsymmetrical flows accompanying
this phenomenon, and the procedure for testing model
ships exhibiting this kind of phenomenon.
Throughout the Panel discussion there was great
interest in the behavior of the stern vortices as
the basic approach to understanding this phenomenon
and this led to the request for quantitative data
regarding the stern vortices. The major part of
this paper was completed during the course of this
Panel's activities in which one of the authors was
placed in charge of developing a technique for
measuring the fluctuating stern vortices. Asa
result of the discussions, a rotor-type vortex-
meter for obtaining a detailed description of the
structure of the stern vortices was adopted.
Needless to say, by obtaining an illustrative
model of the stern vortices it will be possible to
develop a mathematical model which will be extremely
useful for understanding the flow around the full
ship stern. Various vortex models have been sug-
gested by Tagori (1966), Sasajima (1973), and
Hoekstra (1977). The structure of the stern
vortices can be roughly described by a stream line
which, flowing upward around the bottom of the
hull, separates at a separation line formed at the
bilge. This flow rolls up at the boundary layer
around the bilge forming a separated sheet with
vorticity.
Sasajima has suggested a simplified model of
conical separating sheets as shown in Figure 1-1.
He assumed that the separating sheet could be
described by a triangular plane with which he
attempted to explain the basic character of the
stern vortices. This vortex model shown in Figure
1-1 has a core enclosed with a separating line
(S-S'), an attachment line (A-S') and the surface
of the separating sheet. In this model it was
assumed that the direction, velocity, and vorticity
of the flow along this developed vortex sheet would
DEVELOPED
SEPARATING
| SHEET
-ATTACHMENT LINE
SEPARATION LINE
7S) ie
FIGURE 1-1. [Sasajima (19
FIGURE 1-2.
SEPARATION LINE
[Hoekstra (1977)].
Illustrative models of stern vortices
have the same values as they had at the point the
flow passed on the separation line.
Hoekstra's vortex model also had a conical
separating sheet with a cusp as shown in Figure 1-2.
Although the stream along the separating sheet
flows upward and rolls inside, it does not touch
the hull surface to form an attachment line.
In addition to the study on the scale effect of
the stern vortices by Huse (1977), the studies
based on the theory of the three-dimensional
boundary layer by Okuno and Himeno (1977) has made
it possible to discuss the detailed structure of
the stern vortices. However, these studies did not
pay much attention to the vorticity distribution.
The authors concluded through their study that the
prominent features of the stern vortices could be
revealed by studying diagrams of the vorticity
distribution.
2. ROTOR-TYPE VORTEXMETER
Although numerous efforts have been made to investi-
gate the stern vortices, the state of the art for
measuring the vorticity distribution in aft section
of a model ship remains less developed than the
techniques for measuring the wake distribution.
This is evident by the few papers in which the
complete data of the vorticity distribution has been
published. This is largely attributable to problems
in developing vortexmeters for towing tank measure-
ments.
In the authors' experience the problems in using
five-hole Pitot tubes for measuring the vorticity
have been in maintaining sufficient accuracy through-
out the measurements. The analysis of vorticity
distribution which includes finite difference
methods results in insufficient precision. Besides,
for one mesh point of a vorticity measurement, it is
necessary to use the flow velocity data from four
adjacent mesh points which makes it difficult to
perform measurements close to the hull surface as
well as to measure fluctuating vortex flows.
The study of stern vortices has been greatly
stimulated by flow visualization developments and
especially noteworthy contributions have been made
by researchers using tuft grid observations.
However, flow visualization for observing the vortex
flow has a weak point illustrated in the following
discussion.
Superimposing an arbitrary irrotational flow on
a vortex flow, the resulting total flow should have
the same vorticity as the original vortex. An
example is shown in Figure 2 which is a velocity
vector diagram of a circular vortex core super-
imposed on a parallel flow. Examining this figure,
it can safely be said that few people would be able
to estimate an exact geometry or locate the center
of the vortex from only this vector diagram of the
total flow (or from a photo or sketch of the tuft
grid observation).
One of the authors [Tanaka (1971)] suggested
adopting a rotor-type vortexmeter for towing tank
measurements. He applied this technique to analyze
the stern vortices generated by a submerged body
running near the free surface. The application of
the vortexmeter is reported in many aerodynamic
investigations dating back to the 50's, and it was
proposed for ship research by Gadd and Hogben [1962].
The vital problem in adopting the rotor-type
vortexmeter for towing tank research lies in the
accurate calibration of the rotor. This is mainly
due to the fact that no one has succeeded in gener-
ating a stable vortex useful for the calibration in
a steady flow field.
The rotor-type vortexmeter utilizes the principle
that four-unpitched vanes mounted on a rotating
shaft, shown in Figure 3, are not affected by any
parallel and shear flow and only respond to a
(A) (B) (A) +(B)
Parallel circular vortex flow pattern on tuft grid
flow flow
FIGURE 2. Tuft grid pattern due to a circular vortex
and a parallel flow.
rotational Sheor
Flow Flow
(rotor) (Velocity vectors on rotor )
FIGURE 3. Principle of rotor-type vortexmeter.
rotational flow. When the rotating shaft of a
vortexmeter is parallel with vorticity axis, the
rotor turns with angular velocity of W-S, where S
is the slip due to rotational friction of the rotor
shaft and W is the vorticity in the fluid. At
present, the slip S can be estimated using the
following technique.
Using the simple consideration of the elementary
wing, the torque Q due to a rotor element having
small length dr in a radial direction can be deter-
mined from Eq. (1), where C; , %, and U are lifting
derivative, cord length of vane, and advance speed
respectively.
Q(x) = Ap LURC, gy rar (1)
where
ali Ssh ass
LY Soya me oy souRUa)
R. and L.F. are the rotor radius and lifting force
respectively. In the flow with uniform vorticity
distribution, the magnitude of the torque acting
on the rotor becomes:
R
O(w) = 2 a Q(r)dr = (2/3) pLRPUC, ts (2)
oR
calibration - motor
cl" ?
(at calibration mode )
outer- tube
inner — shaft
rotor
miniature ball bearing
DIRECTIONS
angular velocity
modes
in fluid rotor
{
calibration
vortex—
measurement
FIGURE 4. Principle of rotor-type vortexmeter calibra-
tion.
211
Then, the slip of the rotor in a rotational flow
can be determined by following equation, where
q shows rotational friction of the shaft.
S = d
(273) p2R30C, (3)
af
For the calculation of S, the rotational friction
of the ball bearings q should be determined experi-
mentally. This problem will be briefly discussed
later.
As previously stated, since the generation of a
stable vortex for the calibration is presently not
feasible, a mechanical calibration was attempted in
which vorticities mechanically act on the rotor
through the shaft of the rotor. This principle of
the calibration is shown in Figure 4 where a newly
designed rotor shaft is composed of duplicate inner-
shaft and outer-tubes. The outer-tube is mounted
on the outer rings of the ball bearings and the
vanes are fitted on the outer-tube. The inner-
shaft is connected to a calibration-motor.
To obtain the slip S, the vortexmeter is cali-
brated in an irrotational flow in which it travels
along at a constant speed. The inner-shaft is
driven by the calibration-motor at an angular
velocity, w, and the rotor turns at an angular
velocity, S, in response to the condition of the
ball bearing's frictional torque and the hydro-
dynamic characteristics of the rotor.
From the measured vorticity, Wg, we can estimate
the vorticity in fluid as w = Wo + S. According to
the authors' experience, if the frictional torque,
q, is approximately 10-6 kg-m it is possible to
consider S = O except in the case of fairly slow
speed (cf. Figure 25). This means that the
calibration of the vortexmeter seems unnecessary
for ordinary test conditions.
Although ball bearings exhibiting frictional
torque values less than gq = 0.7 107-®kg-m in air were
chosen in manufacturing the vortexmeter, there was
no direct measurement of the frictional torque of
the miniature ball bearings in water. The frictional
torque, gq, also can be determined by measuring the
torque on the outer-tube generated by inner-shaft
turning in water. According to the results of these
measurements, it can be said that there is hardly
any difference between the frictional torque value
of the bearings when they are used in water or
air.
An example of a vortexmeter is shown in Figures
5 and 6. The diameter and length of the rotor are
30mm and 18mm respectively, section of the vane
is lenticular shaped with a thickness ratio t/2
= 1/q. A transducer for rotating the rotor is
used in connection with a photo-transistor which
makes 4 pulses-signals in one revolution. Assuming
Cry = 0.67, g = 107°kg-m Einel Gf = 155 Wei, ae aig
possible to make a rough estimate of the vortex-
meter's precision from the value of slip obtained
by Eq. (3). From these values, the slip value, S,
equals 1072r.p.s.- which corresponds to 1% error
relative to a normal vorticity of w = 1 r.p.s.
As will be mentioned later, the vortex cores of
the stern vortex near the hull surface have a very
steep gradient in vorticity distribution. There-
fore, it is useful to consider the vorticity values
measured by the rotor with a finite diameter at
such boundaries. It is clear from the Eq.(1) that
a mean value of a torque during a turn due to a
wind element dr (see Figure 3) corresponds to a
ise)
H
N
200
7380
H
Cs)
ies}
ea)
Rotor-type vortexmeter.
mean value of the vorticity in a path of a wing
element.
Q(x) = %p2UC, x w(x) dr (4)
27 277
Oe) = aah (9,r)d8 (x) = i w(6,r) ae
= aa Or Ohler DN Oe m
0 0
On the other hand, concerning the vorticity gradient
influence on the radial portion of the rotor, from
the following equation it can be understood that
the tip of the rotor has a higher sensitivity:
R R
2 ff Q(x) ar = pLUC, F f rw (x) dr (5)
-R -R
In practice, using only large models, the error due
to the finite diameter of the rotor can be eliminated.
Such a problem is also present when determining
the mesh interval in the vorticity measurement
by a five-hole Pitot tube.
3. EXPERIMENTS AND RESULTS
The ship models used in the experiments exhibited
an unstable propulsive performance in ballast
condition. In recent studies, it has been recog-
nized that the limiting stream line around the stern
and the pressure distribution change along with the
thrust fluctuation in the self-propulsion tests
of the model ship. The influence of these
phenomena on the ship design has been reported by
Watanabe and Tanibayashi (1977) and Watanabe et al.
(1972) .
A special feature of this phenomenon was that it
appeared only in the self-propulsion tests and was
not observed on the towing tests. Thus, while this
phenomenon easily appeared in the self-propulsion
tests at Froude number 0.18 and 65% full displace-
ment, at the same conditions there was no indication
of this phenomenon during the tests concerned in
this report. The body plan of the 4 and 7m geosim
models are shown in Figure 7 and the principal di-
mensions are summarized in Table 1.
The intent of the experiments was twofold: first
to determine the structure of the stern vortices
using the rotor-type vortexmeter, and secondly,
to investigate the performance of the propeller
working in the presence of these stern vortices.
FIGURE 6. Rotor-type vortexmeter and stern of model.
213
rey
\ \. \ ak
FIGURE 7. Body plan of model.
Also as a reference, the vorticity distribution
was measured by the five-hole Pitot tube for com-
parison with the rotor-type vortexmeter measure-
ments
The positions where the vorticity distribution
of the stern vortices was measured, are shown in
Table 2 in which sq.st. 1/8 correspond with the
section of the propeller disk. The effects of
velocities of the model ship are studied at several
mesh points.
In order to discuss the scale effect of the
vorticity distribution, the results of measure-
ments on both models are shown in Figures 8 and 9,
and the induced velocity vectors on the Y-Z plane
which are calculated with the vorticity distribu-
tion, are shown in Figures 10 and 11.
The interval of mesh drawn on both diagrams of
vorticity distributions and velocity vectors,
corresponds to a non-dimensional length of 0.5%
Lpp. The values of equi-vorticity contours in
diagrams of vorticity distribution are non-
dimensional vortices defined as follows:
Bue) $2
E = a ee (6)
where wy shows the vorticity in r.p.s. which
corresponds to twice the number of rotor revolu-
tions. Considering a diagram of vorticity dis-
tribution as a geographical contour map, the vortex
core can be compared to a typical plateau. The
TABLE 1 Principal Particulars of Models
Model Ship No. M-7 M-4
Length (m): Lpp 7.000 4.000
4 Breadth (m) 1.167 0.667
2 Breadth Draft Ratio 2.760
Block Coefficient 0.802
Longitudinal Prismatic Coeff. 0.810
4, Pitch Ratio (const.) 0.7143
“ Boss Ratio 0.180
B Expanded Area Ratio 0.665
Ai Number of Blades 5)
TABLE 2 Measurement Positions of Vorticity
Distribution
Sq. “Sit
Model 1/2 1/4 1/8 eb Bexs)
Port £37)
= Port
M-7 Port Port St aeboara or
eed Port x2.
Starboard
Notes
*]1 Corresponds to propeller position
*2 Measured by vortexmeter and 5-hole Pitot tube
*3 Corresponds to Sq.St. -1/8
fact can clearly be seen in the foreward detections,
especially sq.st. 1/2 in Figure 12. In this
connection, the vorticity distribution is sq.st. 1/4
and A are presented in Figures 13 and 14
respectively.
As a reference, the induced velocity vector
diagram on sq.st. 1/4 is shown in Figure 15.
Furthermore the vorticity distribution (for
M.No.M-7) obtained by the five-hole Pitot tube
(diameter 12mm, angle between center and side
FIGURE 8. Vorticity distribution of M.No.M-7 at
Scot lS
\e)
Y 2 Y
io 2 8 6 =o Bee ©) 4 8
8} 8
e Zz
> 6 3
4 4
Ot Oe: )
1! aon J} ~
\ RAO A) |)
-2} i An 1 Lf of f | 2 z
% Med ¥ f/f
- oF LSD) Of LY
-4 - = 4 Oh da f fy ' 4
\ \C Tape /
6 ~. = Si. 6
mh a XN oe
x SZ »
1% InSESS af ee Wie
et ae 8
Be co BN 4h 3 . \
y LE Lee | j re SS
2 =~ Mth tn — Sea
‘ ‘ Uae | é \ | |
f H a a " ‘ '
\ | ea, ,
o} ®, 4 44-4 Se Tele
i VW AS anes Vy iy PG
\ \ ae Le), ,
y Hs) = a = /-2
JN i me IT \ Lt Ww
Nt ol VENV aie
Rae ae dele Lie ,
Sa ff Se s | 4 a ea 2
a Ce
ae y | \ x ee aS
Qi Se Ay ye a ee.)
FIGURE 10. Velocity vectors on y-z plane due to stern
vortices (M-No.M-7, Sq.St. 1/8).
“ = zi z e 1
Pool A gS yh Op phe ete
PSS Sey Ray Yh ll edhe a RS Le
Oat oa mt CEN i Wa tea ae ae YB a no
Diet Diem Sy. | | a Lae reel eee
6 = a a I fe Te 6
a Se i Les SoS ~
TA -— a ae a ae S%
/ Naat Alearet (i Soar s ‘
2t Thole esi TH ise Lace A A152
I ate y 14 i Vy
n | "Ay Te ea a y e
HL TEEEBREE ah
\ J z
3) Ve fyeh | A| | | \ ‘
2 ~ = Se
Bas aC AL AR So d
; oa Kegel Hd bend MONS rcrsiece
=4s = | ye Wei, <4
Tee aan ae Te Rall le eee
ee Sa Hes wwsSN——- 6
8~ aa THAN = - «8
Velocity vectors on y-z plane due to stern
(M.No.M-4, Sq.St. 1/8).
FIGURE 12. Vorticity distribution of M.No.M-7 at
SqSt..- 17/2)
~
——
lye
FIGURE 13. Vorticity distribution of M.No.M-7 at
SquSteml/4p
CHIE
Ha
NY
SaS> 41 rees
‘\ m
FIGURE 14. Vorticity distribution of M.No.M-7 at
Seis. As
holes 25°) is shown in Figure 16 and the wake
distributions and the velocity vectors by the five-
hole Pitot tube are shown in Figures 17, LB} p | I),
and 20 respectively.
FIGURE 15. Velocity vectors on y-Z plane due to stern
vortices (M.No.M-7, Sq-St. 1/4).
215
A
APR P
lg
uy
Viet 4 ~
Olacmmat + i i
Y
8 Bes
Zz
6
\ hs
Os
-BL
FIGURE 16. Vorticity distribution measured by 5-hole
pitot tube (M.No.M-7, Sq.) Sit.) 178).
4. DISCUSSIONS AND APPLICATIONS OF THE RESULTS
Remarks on Vorticity Measurements
The rotor-type vortexmeter performed as expected.
As seen from a comparison between Figures 8 and 16,
the rotor-type vortexmeter is more sensitive and
can be used to obtain a finer vorticity distribu-
tion contour than the five-hole Pitot tube. While
both vorticity distribution diagrams appear to have
a similar shaped vortex core, they have fairly
different values. The distinguishing difference
is mainly in the pattern of the distribution. Al-
though the plateau-type distribution would be the
expected form of the typical vortex cores in the
vorticity distribution obtained by the vortexmeter,
the plateau-type is broken in Figure 16. It can be
said that the difference between these results
indicate the usefulness of the vortexmeter's resolv-
ing ability.
Contrary to the general opinion that a geometry
of the stern vortices is fluctuating, in the
authors' measurements, the vorticity and geometry
of the stern vortices were generally quite stable.
However, there is an unstable vorticity-zone at
the top of the main vortex core indicated in
Figures 8, 9, and others. Through these experi-
ences, it can be shown that the dynamic character
of the vortexmeter is one of its prominent features.
While the present diameter of the vortexmeter's
rotor was selected for maintaining its accuracy in
measurement, it is possible that the rotor diameter
is too large for the 4m geosim model (M.No.M-4).
Furthermore, it appears that there were some
4 6 8 10
nm
“10 -8 -6 Y -4 -2 0
06! 0.5) 0403! 10.30.4105 6 Oz.
FIGURE 17. Wake distribution (M.No.M-7, Sq.St. 1/8).
problems due to the presence of an oblique flow in
those experiments. It is recommended in further
work that the characteristics of the rotor ina
strong oblique flow should be studied.
Structure of the Stern Vortices
As stated in the previous section, the equi-
vorticity contours of the stern vortices can be
compared with plateaus in geographical contours.
Furthermore, in examining carefully the diagrams
of the vorticity distribution, there is a line of
concentrated vorticity on the "table of plateau,"
which is denoted by the "Max. line" in this paper
and indicated in the contours. The Max. line
can be clearly shown in a cross section of the
diagrams of the contour as seen in Figures 21 and
22.
The Max. line can be considered as a kind of
~10 -8 -6 Y -4 -2 0) 2 4 6 8 10
; '05 04103 02) 020304 05| | 06 |
FIGURE 18. Wake distribution (M.No.M-4, Sq.St. 1/8).
-10 -8 -6 -4 = ce} 2 4 6 8 10
Pee I, ties a ae eed ae
| Q__ 20%. of U
8 L { { X =
240 ae a aibe So TSS TS ~
6 : L- Tk er ae oe =o 6
i IN oe ay
s+ a en eee
te +
ir
—,
N
Sees | ee
FIGURE 19. Velocity vectors on y-z plane (M.No.M-7,
Sqisteawl/A8)) ie
a ridge on this plateau; it is steep in the forward
section and becomes gently sloping while shifting
afterward. It is noticeable that the Max. line
seems to show the existence of a separating vortex
sheet. As is well known, stream lines flow from
under the bottom of a hull up the boundary layer
at the bilge and turn into part of the vortex
sheet. Although the vortex sheet previously
mentioned has only been used as a hydrodynamic
description, the authors are able to show its
existence in the flow behind the full stern as well
as provide quantitative measurements.
The development of the vortex sheet depends
mainly on the potential flow and the induced flow
from the vorticity. Its development is strongly
affected by each ship form, with effects of model
ship velocity and the Reynolds number effect
mainly limited to the diffusion of the vorticity.
In a comparison between Figure 8 and Figure 9, the
forms of the Max. line which correspond to a form
FIGURE 20. Velocity vectors on y-z plane (M.No.M-4,
SS (Sa.5t.,
a ete \s, STARBOARD
(Sq.St.4, PORT) / ‘ :
0 = Vp 0) Sy NP NSS 7182)
qT OCR OF Zot oS oF
-20 ke v4 Z.Y=!= 35mm MS.NQM-7
-40 SNe = 20mm M.S.NOM-4
-60
FIGURE 21. Cross section of vorticity distribution
(M.No.M-7 and M-4, Sq.St. 1/8).
of the vortex sheet are fairly similar for both
geosim models. On the other hand, the difference
in breadth of each model's vortex core seems to be
due to the effect of difference in Reynolds number.
Furthermore, the suitability of adopting the
idea of the Max. line is shown by the following
facts. Assuming that all the vorticity of the
stern vortices are concentrated on the Max. line
for computing the induced velocities, the resul-
tant velocity vector diagrams are similar to the
complete flow field velocity. For instance,
Figure 23 is a diagram of velocity vectors, which
have the same circulation value as Figure 8 but
with the vorticity concentrated on the Max. line
divided by ten, of circular vortices with mean
strength on the original Max. line. It can be
seen that both diagrams of the velocity vector,
Figure 10 and Figure 23, are fairly similar. This
will allow not only simplified treatment of the
stern vortices but also should simplify future
numerical analysis of the stern vortices.
In order to predict the wake of full stern
ships, it is necessary to estimate the wake com-
ponent due to the stern vortices in addition to
the potential and frictional wake components used
in Sasajima's wake prediction method. The concept
of the Max. line in the vorticity distribution
also may lead to the wake component due to the
stern vortices.
In order to discuss the relation between the
stern vortices and the wake distribution, an
illustrative model of the stern vortices is
presented in Figure 24. A stream line flowing
under the bottom of a ship, separates around the
bilge and forms a part of the separating vortex
sheet. The vortex sheet crosses to the hull
surface near the propeller bossing where the
authors denote the secondary separation line.
And at the secondary separation line, the vortex
217
(Sq.st. +, PORT)
Y
-6 -4
Z-|
Z2-2
FIGURE 22. Cross section of vorticity distribution
(M.No.M-7, at Sq.St. 1/2 and 1/4).
sheet makes the cross flow with the limiting stream
line flowing aft passing through the tunnel of
the vortex sheet. The crossed flow generates a
reversed vortex at the secondary separation line
as seen in the diagrams of the vorticity distribu-
tion.
The flow passing through the tunnel of the
vortex sheet can be found at the section of the
propeller disk (sq.st. 1/8) which appears as an
eye in the wake distribution pattern in Figures 17
and 18. This fact may be proved by the Max. line
which just covers the eye.
FIGURE 23. Velocity vectors due to concentrated vor-
ticity on max. line (M.No.M-7, Sq-.St. 1/8).
\72
SEPARATING
FLOW
+LIMITING
STREAM LINE
SECONDARY
SEPARATION LINE > cCPARATION CINE
FIGURE 24. Illustrative model of stern vortices.
This tunnel vortex sheet is quite different from
the conical vortex sheet used in the model proposed
by Sasajima or Hoekstra. Considering the flow as
passing through this tunnel makes it possible to
discuss the relationships of the wake flow, limit-
ing stream line, attachment line, and the stern
vortices.
Regarding the wake patterns of vessel with a
full stern, the authors suppose that if the Max.
line can be considered independent of the Reynolds
number, then the "eye" in the ship's wake pattern
should be in approximately same location as shown
in Figures 17 and 18. The above mentioned facts
will lead to further studies for prediction of
ship's wake, using the potential and frictional
wake patterns estimated by Sasajima's method.
Actually, the authors cannot verify the
relationship between the stern vortices and
Reynolds number because the range of the scale
ratio used in geosim models tested is too small
for a discussion of the similarity of the stern
vortices. However, it can be said that the
alternation of the Max. line between both models
seems relatively smaller than that of the wake
pattern. Furthermore, the vortex center, which is
defined as the vanishing point of the induced
velocity vector due to the stern vortices, has
shifted a distance corresponding to only 4% of the
propeller diameter as seen in comparing Figures 10
and 11. While the model size has comparatively
for r/R= 0.50
Vx/U
Vx/U for r/R= 0.90
FIGURE 26. Circumferential distribution of
wake flow on propeller disk, V,/U.
06
0.4
0.2
0.6
0.4
0.2
Relation between vorticity and velocity
of model ship.
FIGURE 25.
small effect on the shape of the Max. line, the
model size causes differences in the diffusion of
the vorticity. Thus, from the calculation of the
circulation of the vortex cores presented in
Figures 8 and 9, it was found that the circulation
of M.No.M-4 was smaller than M.No.N-7. The magni-
tudes of these differences were 6% smaller on the
portside and 8% smaller on the starboard side of
M.No.M-7. However, even for the same model ship,
the difference in the port and starboard side
stern vortex circulation was on the order of 8%,
so it is not possible to reach a definite conclu-
sion about the significance of the differences in
the geosim tests.
Since the authors limited study to vessel speeds
corresponding to Froude number 0.18, the effects
on the velocity due to the stern vortices still
remains obscure. However, the authors can in-
dicate some examples in which the vorticity has
been measured at the several mesh points as seen
in Figure 25. If the free surface effect could be
neglected, the non-dimensional vorticity €y should
be constant. Although the cause of the different
results explicitly shown in Figure 25 remains un-
known, it may not be said that the rotor-shaft
friction of the vortexmeter can be safely considered
as negligible in a range of very slow speeds such
as ES 0.1.
r/R =0.50
Vx/U for (/R*0.70
Effect on Propeller Operation Due to Tangential
Stern Vortex Flow
In the previous section, the authors have mainly
discussed the structure of the stern vortices
obtained from the towing experiments. As was
reported by Hoekstra (1977) it can be considered
that the structure and geometry of the stern
vortices is strongly affected by the flow induced
propeller thrust. However, the authors have studied
the forces and moments on the working propeller as
a preliminary problem, assuming the structure of
the stern vortices is not changed by the influences
of the propeller suction.
The forces and moments on the propeller are
remarkably related to the pattern of the flow
distribution at the propeller disk location. The
flow distribution relevant to the present problem,
is composed of the wake component, V_/U, and the
tangential components, V_/U, which were obatained
by the five-hole Pitot tube. The authors assumed
that the tangential components could be further
decomposed into the component obtained by the
vortexmeter, V__/U, and other components. Although
each component has already been shown in previous
figures, for convenience the circumferential dis-
tributions of V_/U, V_/U, and V__/U at 90%, 70%,
and 50% of the Sisk radius are ENewn respectively
Vr/U :tangential velocity component
obtained by five-hole Pitot tube /R =0.50
219
in Figures 26, 27, and 28. Furthermore, the authors
have included the tangential velocity vector com-
ponent, Viny/ Ue in Figure 29.
In order to determine the propeller forces and
moments induced by the stern vortices, the authors
have performed the following calculations using
the unsteady lifting surface theory developed by
Koyama (1975). The authors thus calculated the
thrust and torque of the propeller, along with the
vertical and horizontal forces and moments imparted
by the propeller shaft of the working propeller
with and without stern vortex flow. The definitions
concerning the forces and moments are shown in
Figure 30.
The authors have assumed for the calculation that
the tangential flow obtained from the subtractive
procedure (V_/U - V__/U) simulates one eliminating
the effect oie the of Som vortices, and a common
wake flow can be used for both calculations with
and without the stern vortices.
Since the results of the calculation for M.No.M-4
are quite similar to the results of M.No.M-7, only
the results of M.No.M-7 are shown in Figures 31 and
32. Figure 31 indicates a comparison of the torque
and thrust on a blade of the propeller with and
without the stern vortex flow. Total torque,
thrust, and other forces and moments on the pro-
peller (indicating propeller turning angle 0° to
Top Starboard side
n
Top FIGURE 27. Circumferential dis-
{o) 90 180
03
Vrv/U tangential velocity component
obtained by vortexmeter
clockwise
anti -
tribution of tangential flow on
propeller disk, Vp/U.
clockwise
FIGURE 28. Circumferential dis-
tribution of tangential flow on
Top Starboard side Bottom Port side Top
L 1 i
{e) 90 180
@ (deg )
360 propeller disk (induced flow),
Vpy/U-
220
ne tl ba
ih ss eS Vv/U
YP Lits
180 5 as
FIGURE 29. Tangential velocity vector due to induced
flow on propeller disk (M-7).
72°) are shown in Figure 32. According to the
results, main effects of the stern vortices flow
appeared on the vertical force (F_) and the
horizontal bending moment (M_) of the propeller,
but the other components are almost negligible.
It can be concluded that the effect of the stern
vortices is fairly limited to a few components of
forces and moments generated by propeller. The
results may be attributed to the tangential flow
around the propeller caused by the stern vortices.
It is mainly concentrated at the underside near the
bossing, and does not severely appear on the
propeller tip as shown in Figure 29.
5. CONCLUSION
The authors developed the rotor-type vortexmeter,
giving careful attention to the calibration method
of the vortexmeter, and, by using it in these tests,
showed its high utility.
PIGURE 30. Definitions of forces and moments due to
propeller.
0.070
0.060
0.950
(T7e.n?- D4) 201
0.040
0.0070
ze
0.0060
0.0050
(a /p-r?. 08)
0.0040) , F
1
Top Starboard side Bottom Port side op
(e) 90 180 270 360
@ (deg.)
with stern vortices
areas without stern
vortices
FIGURE 31. Variations of thrust and torque
coefficients on one blade (M-7).
—— with stern vortices
---- without stern vortices
0.27- 0.029 -
Thrust
way
Pe
I=
\a
EN
. Horizontal bending Mt.
Vertical force i 9
iN
4 \
¢ XN ,
Vertical bending Mt.
Horizontal force
0005; , ,-~0:005
Top @(deg.)
(et ti)
0 30 60 72
clockwise clockwise
FIGURE 32. Variations of force and moment coeffi-
cients on propeller (M-7).
Measuring the vorticity distribution around the
full stern of the geosim models, the authors
determined the structure of the stern vortices
and found the presence of a concentrated vorticity
line in the vortex core which corresponds to the
separating vortex sheet of the stern vortex.
As an application of the results, the effect on
the propeller operation due to the induced flow of
the stern vortices has been studied. The effect is
fairly limited to a few components of forces and
moments generated by the propeller. Consequently,
it can be said that the effect of the stern
vortices on the performance of the propeller and
propeller excited vibratory shaft forces and
moments is relatively small. However, in the case
of this ship model, this effect appears to change
the direction of the vertical force and the
horizontal bending moment acting through the pro-
peller shaft.
ACKNOWLEDGMENT
This research program has been carried out mainly
under the foundation of the Research Panel SR 159
of Shipbuilding Research Association of Japan.
The authors would like to express their apprecia-
tion to Profs. H. Sasajima and I. Tanaka, who led
the Panel, for their valuable suggestions.
The materials described in the present paper have
been kindly prepared by the members of the Ship
Research Institute among whom the authors are
especially grateful to Mr. K. Takahashi, Mr. T.
Haraguchi, Mr. Z. Ishizaka, Mr. N. Sugai, and
Miss H. Handa, for their helpful support.
REFERENCES
Gadd, G. E., and N. Hogben (1962). An appraisal of
the ship resistance problem in the light of
measurements of the wake pattern. WN.P.L. Ship
Rep. 36.
Hoekstra, M. (1977). An investigation into the
effect of propeller hull interaction on the
structures of the wake field. Proceedings of
Symposium on Hydrodynamics of Ship and Offshore
Propulsion Systems.
Huse, E. (1977). Bilge Vortex Scale Effect. Pro-
ceedings of Symposium on Hydrodynamics of Ship
and Offshore Propulsion Systems.
Koyama, K. (1975). A Numerical Method for Propeller
Lifting Surface in Non-Uniform Flow and Its
Application. Journal of the Society of Naval
Architects of Japan, 137, 78 (Japanese).
Okuno, T., and Yoji Himeno (1977). Distribution of
Wall Shear Stress and Gross Flow in Three-
Dimensional Turbulent Boundary Layer on Ship
Hull. Journal of the Kansai Society of Naval
Architects of Japan, 165, 83 (Japanese).
Sasajima, H. (1973). On scale effect of ship
resistance, Text of Symposium on Viscous
Resistance. The Society of Naval Architects of
Japan, 213 (Japanese).
Tagori, T. (1966). Investigations on vortices
generated at the bilge. Proceedings of 1lth
Fen WAC BAG
Tanaka, H. (1975). A study of resistance of
shallow-running flat submerged bodies. Selected
Papers from Journal of the Society of Naval
Architects of Japan, 13, 15.
Watanabe, K., K. Yokoo, T. Fujita, and H. Kitagawa
(1972). Study on flow pattern around the stern
of full ship form by use of the geosims. Journal
of the Society of Naval Architects of Japan, 131,
Se
Watanabe, K., and H. Tanibayashi (1977). Unusual
phenomenon of the stern of full ship models.
Proceedings of Symposium on Hydrodynamics of
Ship and Offshore Propulsion Systems.
Session [IV
SHIP BOUNDARY LAYERS
AND
PROPELLER HULL INTERACTION
TAKAO INUI
Session Chairman
The University of Tokyo
Tokyo, Japan
eit
By
Wake Scale Effects on a
Twin-Screw Displacement Ship
Arthur M. Reed and William G. Day, Jr.
David W. Taylor Naval Ship Research and Development Center,
Bethesda, Maryland
ABSTRACT
The results of a wake survey and boundary layer
profile measurements on a full-scale twin-screw
displacement ship are presented. The corresponding
model-scale measurements are also presented. The
full-scale wake measurements consist of the three
velocity components which contribute to the nominal
wake in the propeller plane, at four radii. The
full-scale boundary layer profile was obtained at
three longitudinal locations with and without the
propeller operating. The model-scale nominal wake
was determined in a towing tank using five-hole
pitot tubes while the model-scale boundary layer
measurements were made on a double model in a wind
tunnel using hot wire anemometers.
In order to identify the scale effects between
the model and ship, the deviation of the velocity
in the propeller disk from a uniform axial flow has
been separated into the velocity field due to shaft
inclination in a uniform stream, the perturbation
due to the hull and its boundary layer, and the
viscous wake due to the appendages. The principal
contribution to this perturbation from the axial
flow is the effect of inclining the shaft in the
uniform stream. The perturbation of the flow due
to the potential flow about the hull is small, as
are the effects of the displacement thickness of
the boundary layer of the hull. The proposed
scheme for predicting the viscous wakes of the
shaft and struts meets with little success. Never-
theless, some conclusions are drawn as to how these
wakes will vary between the ship and model.
1. INTRODUCTION
If unsteady propeller force and hull loading pre-
dictions are to be precise, the inflow to the pro-
peller must be known accurately. At the present
time the nominal wake of a model is measured and
extrapolated to full scale assuming geometric
225
Similarity. The extrapolation fails to take into
account any of the scale effects which may possibly
exist between model and full scale. This paper
presents preliminary results from a series of full-
scale nominal wake and boundary layer velocity pro-
file measurements on a high-speed transom-stern
ship. In addition, the corresponding model-scale
measurements are reported, along with a series of
analytical predictions, which are intended to
identify the principal contributions to the wake.
This is not the first investigation of this
nature. However, it is the first project to suc-
cessfully measure the three velocity components in
the propeller disk of a high-speed twin-screw
transom-stern hull form. The British have per-
formed an extensive series of experiments on a
frigate, [Canham (1975)], and the Japanese and
Germans have performed flow measurements on
several full-form ships. The Japanese and German
experiments were conducted on single screw tanker
forms and are reported in an extensive series of
reports [see for instance: Namimatsu et al.(1973),
Namimatsu and Muraoka (1973), Schuster et al.(1968),
Takahashi et al.(1970), Taniguchi and Fujita (1970),
and Yokoo (1974)].
While the British measurements were obtained on
the ship type of interest, a high-speed transom-
stern ship, only the longitudinal velocity compo-
nent in the propeller plane was obtained. This
resulted in the loss of the important tangential
and radial velocity components. In the case of
twin-screw transom-stern ships, these velocity
components are generally very significant due to
the inclination of the shaft to the direction of
the free-stream.
The Japanese, on the other hand, were able to
measure all three velocity components in the wake,
but they had to make their measurements in a plane
ahead of the propeller disk. Due to the full
sterns of the tankers, the flow into the propeller
is highly influenced by viscous effects, and as a
consequence is highly affected by changes in
226
Reynolds number. Therefore, while treating a much
more difficult problem, the results of the tanker
experiments are not applicable to the scaling of
the wakes of high-speed hull forms.
The full-scale velocity component ratios which
are presented here were obtained at a speed of 15
knots; the corresponding Froude and Reynolds numbers
were 0.36 and 4.10 x 108 respectively. The model
wake survey was conducted in a towing tank at the
full-scale Froude number. This resulted ina
model speed of 5.22 knots, and a Reynolds number of
1.56 x 107. The full-scale boundary layer measure-
ments were conducted at four speeds between 6.2 and
16.5 knots. These speeds correspond to Reynolds
numbers between 1.7 x 108 and 4.5 x 108 respec—
tively. The model-scale boundary layer measure-
ments were obtained on a double model in a wind
tunnel at a Reynolds number of 1.68 x 107.
Significant differences are observed between the
model and full-scale velocity components, particu-
larly in the magnitudes of the radial and tangen-
tial velocity components. These differences are
in the regions away from the ship's hull and
appendages; therefore, these differences do not
seem to be due to Reynolds number effects. A more
likely explanation is a lack of ship-model simi-
larity, possibly due to unexplained differences in
hull form or initial trim.
In order to obtain an understanding of the com-
ponents which contribute most significantly to the
deviation of the wake from uniform axial flow, an
attempt has been made to predict the velocity com-
ponents as seen by the propeller. To make this
prediction, the velocity field (in shaft coordinates)
was decomposed into its major components as follows:
Velocity = Uniform Stream
+ Perturbation due to Hull
+ Perturbation due to Hull Boundary
Layer
Viscous Wake of Struts
Viscous Wake of Shafting
+ +
The results of this decomposition show that the
inclination of the propeller shaft to the free
stream is the most significant factor contributing
to the deviation of the velocity from a purely
axial uniform flow. In particular, approximately 70
percent of the measured radial and tangential flow
is contributed by the inclination of the shaft to
the uniform stream. The boundary layer of the hull
is found to contribute insignificantly to the per-
turbation of the free stream. Although the viscous
wake of the shafts and struts makes a significant
contribution to the nonuniformity of the flow, the
empirical technique proposed herein overpredicts
the wake of the struts and underpredicts the wake
of the shafting.
2. BACKGROUND
During the last ten to fifteen years there has been
a marked increase in the installed horsepower per
shaft on high-speed commercial and naval vessels.
This increase in power has led to increased steady
and unsteady forces on propellers, and increased
loads on the hull surface. If adequate structural
designs are to be developed for the propeller, its
shafting, and the shaft supports; then the un-
steady forces and moments on the propeller must be
known accurately. Similarly, if the hull is to be
habitable and to have minimal vibration, the
structural design must adequately account for the
propeller-induced surface forces. The propeller
forces and surface loads can in turn only be ac-
curate if they are determined using the full-scale
flow into the propeller.
Several theories exist for predicting the un-
steady forces and moments acting on a propeller in
a nonuniform flow, and the hull-surface forces
induced by a propeller. Tsakona et al. (1974) and
Frydenlund and Kerwin (1977) report on two of the
theories for the unsteady forces on a propeller;
Vorus (1974) reports on a theory for predicting
the hull-surface forces. In these theories, the
flow into the propeller is used in conjunction
with an unsteady lifting-surface theory to predict
the unsteady forces on the propeller and hull as
the propeller rotates through the nonuniform flow.
Typically, a propeller is wake adapted, that is,
designed to the radial distribution of the circum-
ferential mean velocity. The alternating forces
are determined by considering the propeller in a
nonuniform flow circumferentially. The variations
of the forces and moments in the nonuniform stream
from those in the uniform stream are then con-
sidered to be the unsteady forces and moments on the
propeller.
The longitudinal component of the velocity in
the propeller disk is the principal component of
the velocity on a transom stern ship with inclined
shafts. Typically the radial and tangential com-
ponents vary sinusoidally around the propeller
disk, and have peaks which are 20 to 25 percent of
the longitudinal velocity component. However, in
the process of determining the circumferential
average of the radial and tangential velocity com-
ponents, these components are reduced to 1 or 2
percent of the longitudinal velocity component.
Because of this, the tangential velocity component
contributes very little to the angle of attack on
a propeller blade as computed for the propeller
design. However, in unsteady force calculations,
the longitudinal velocity component varies from
its mean by 10 to 15 percent while the radial and
tangential components vary by 1000 percent from
their means. Thus the variation in the tangential
velocity component contributes significantly to
the changes in the angle of attack on a propeller
blade as it rotates through the wake. These
changes in angle of attack in turn result in the
unsteady forces and moments on the propeller.
Experiments by Boswell [Boswell et al. 1976)],
show that the maximum unsteady loads on the
propeller occur in the area where the tangential
flow velocities in the propeller disk are at their
maximum. As will be seen later, it is the tangen-
tial velocity components that are in least agree-
ment between model and full scale. It is this
fact that makes the issue of wake scaling important
to the accurate determination of the unsteady
forces on a full-scale propeller.
3. TRIAL VESSEL AND INSTRUMENTATION
A number of criteria went into the selection of
the ship on which the full-scale measurements would
be made. The hull form and appendage arrangement
of the ship had to correspond to that which is
typical of high-speed twin-screw commercial and
naval vessels. The ship had to be available for an
extended period of time and a means of propelling
the ship had to be available.
Of the ships which were in the U.S. Navy fleet,
four classes seemed to meet the geometric criteria,
and a means of propelling them could be identified.
These were the Gearing Class (DD 710), Forrest
Sherman Class (DD 931), Spruance Class (DD 963),
and the Asheville Class (PG 84). However, of these
classes, only the Asheville Class, which was being
decommissioned, met the criterion of long term
availability. As it tourned out, the David W.
Taylor Naval Ship Research and Development Center
(DTNSRDC) already had one of these ships under its
control, the Research Vessel (R/V) ATHENA.
The ATHENA had the added advantage that an ex-
tensive series of model- and full-scale correlation
experiments were already planned. Unsteady blade
loads, stresses, and pressure distributions were
going to be obtained full scale. The blade loading
measurements were also going to be repeated at model
scale. This blade loading data complement the full-
scale wake data, and would result in some of the
most complete correlation data of this type for any
ship and model.
The R/V ATHENA is a twin-screw aluminum hull
CODOG (COmbined Diesel Or Gas Turbine) propelled
high-speed displacement ship. Formerly designated
PG 94, the 46.9 meter LWL ship was decommissioned
in 1975 and placed in service as a high-speed
towing platform for DTNSRDC. The hull form and
propulsion arrangements are similar to today's
destroyers and frigates which are propelled by
Scale Ratio r
Block Coefficient
Prismatic Coefficient
Length/Beam Ratio
Beam/Draft Ratio
Displacement/Length Ratio
Coefficients
227
controllable-, reversible- pitch propellers using
gas turbines as prime movers. The principal di-
mensions and form coefficients for R/V ATHENA are
presented in Figure 1. Figure 1 also shows the
body plan, and bow and stern profiles of the ship.
Figure 2 shows a drawing of the propeller.
The ATHENA is equipped with two Cummins 750
V-12 diesels for low speed propulsion and a single
General Electric LM 1500 gas turbine for high-
speed propulsion. In the diesel mode, the ATHENA
is capable of speeds of around 14 knots. Under gas
turbine power, she can attain a speed of 40 knots.
The ATHENA is appended with twin shafts, struts,
and rudders typical of most high-speed transom
stern ships. In addition, she also has two anti-
roll fins located just aft of amidships.
Once the ATHENA was selected for the study of
wake scaling, the question of how to propel the
ship had to be resolved. The ATHENA is small
enough that she could be towed by either one or
two ships at speeds high enough to provide useful
data, or she could be propelled on one shaft and
measurements could be made on the other shaft. The
two-ship tow would have been the most ideal means of
propelling the ship during the experiments, because
it would have allowed the ATHENA to be towed with no
yaw angle, and outside the wake of another ship.
However, the logistics of this option made it much
less practical than propelling on one shaft.
A series of model experiments was instituted,
aimed at determining whether or not single shaft
propulsion could provide good course keeping
ability with minimal yaw angles.’ Flow visualiza-
Sf (825
on 0.48
Ce 0.63
Wo yee
B/T 3.89
A, 7.15
STATIONS
FIGURE 1.
228
rin riicn SECTION
LINtaR oer ne SHIP
cat
nOoeL
PROP. MO 01 oMIr
ar 70 pcy.|az 70 PCr.
INCHES | qq 5 8.727. |72.000 | 9.697
cul o710 | 8-250 221.67 |1828.80| 246.30 |2
INCHES | 47 5 727, [72-000 | 9.697
Bil 471 | 8-256 221.67 |1828-80| 246.30
a1
0 DIA.
nook shir
AY 70, PCr.
46.358
riven | wunaen F tant
nario. | or tor | eet awe, | Re | Be | gn, | ance | notation
Ar 70 pcr. BLADES ate 7 Tie
7 44
Beit tata | elcri2 | oO)
4 119.33
4.698
119533
46.358 3 |40.100 9
29908 .6 0.775 0.443 EE 0.670 | 0.048 | 0-000 LH. 4950
ponung 4.598
pomuagyt 0.670 | 0.048 | o-coo R:H. | 4950 :
+2
SHEPT OUTLINE
FIGURE 2. Controllable-pitch propeller geometry.
tion studies in the circulating water channel at
DTNSRDC indicated that yaw angles of less than
four degrees would provide satisfactory inflows
to the propeller disk, and still exclude the wake
of the roll fins. Subsequent self-propulsion
model experiments using only one shaft, indicated
that with the rudder set at one degree to port,
the ship would have less than one degree of yaw
and insignificant sway. Therefore, the decision
was made to propel the ship on one shaft rather
than to tow the ship.
The instrumentation which was installed on the
ATHENA consisted of three types. Five- and
thirteen-hole pitot tubes were used to determine
the velocity field in the propeller plane on the
starboard side, and ahead of the struts on both
the port and starboard sides.* A set of eight
boundary layer probes were used to measure the
boundary layer profile at four symmetric locations
on the port and starboard sides of the ship.
Finally a piezoelectric pitot tube, a five-hole
pitot tube with piezoelectric pressure transducers
mounted on its face, was used to measure the time-
varying flow ahead of the operating propeller.
The locations of the pitot tube rakes and bound-
ary layer probes are shown in Figures 3 and 4. The
location of the struts and the shape of the after
stations are shown in Figure 5. As can be seen in
these figures and in Figures 6 and-7, which show
photographs of the actual pitot tube rakes mounted
on the ship, two rakes of four pitot tubes each
were mounted on opposite sides of the propeller
hub. These rakes were attached to the crank disks
for two of the propeller blades. The details of
one rake with pitot tubes mounted are shown in
Figure 8.
*For the details of the instrumentation design and operation
see Troesch et al. (1978).
Q.25GR._ IN.
6-gk- An
2.727 IN.
69.3 7H
2
6
1.697 _ IN. 1.27
a3. Ah hag
FOR HUB ¢ PALM DETAILS
SEE P-4#7/09// Sh. 2
ALSO FOR HUB EXTENSION (wor smounm)
Figure 9 shows a close-up photograph of one of
the full-scale boundary layers probes. These
probes, which extended 0.46 meters from the hull,
contained 13 pitot tubes. Ten of the pitot tubes
were total head tubes, and three were Prandtl
tubes.
4. CORRELATION MODELS AND INSTRUMENTATION
The model correlation experiments were performed
using two fiberglass models designated DTNSRDC
Models 5365 and 5366. These models, which were
built to the lines of the ATHENA, had a scale
ratio of 1 to 8.25; the principal dimensions of
these models are listed with the ship dimensions
on Figure 1. A full set of appendages including
shafts, V-struts, rudders, roll stabilizer fins,
and a centerline skeg were fitted to each model.
Model 5365 was a ship model which was used for the
correlation wake surveys performed in the towing
tank to investigate the scale effects between the
model and ship wake surveys. Model 5366 was a
mirror image double model obtained by reflecting
the lines of the ATHENA about the mean water line
corresponding to a full-scale speed of fifteen
knots. This model was used for the boundary layer
correlation experiments which were made in a wind
tunnel.
The model-scale wake survey was made on the ship
model, Model 5365, using five-hole pitot tubes.
The pitot tubes were mounted on a rake, the shaft
of which was placed through the strut bossings and
stern tube on the model. Figures 10, ll, and 12
show the model which was used for the wake
surveys, and the details of the pitot tube rake
mounted on the stern of the model. Two papers, one
by Hadler and Cheng (1965) and the other by Hale
and Norrie (1967), give a thorough description of
229
-sayer zeAeT Aaepunog pue soeyeT aYEM JO UCTRIeETTeISUT HuTmoys TINY JO META STTFJOAq “€ FaNSIA
‘@310N SS3INN S3T0H S S38NL LOL!ld WV 310N
13) MOUS 06 38 01 JIVD |
ay
u |
3903 4314v
3903 OvvANOs
y30gny
w3oon
373) 310m Ss
1O1ld TOM Cl —
(Lud)
(ouvos
wy aE Ns
FROWd WAV] AMVONNO®
Wows wv) Awvonnos
a
su
as + +—t
a
j++ +++
he
(AINO) LuOd
TNGV) OWlLvEIeD
HIT wULNIO~
(ATWO) Lii0d HON
230
9 8 7 6 5 ‘ 3 2 1 Ww 0
. centering
Boundary Layer Positions scale. s16tto1-o"
FIGURE 4. Plan view of hull showing boundary layer rake locations.
Ruup @ 9.727 ft. (0.222 mD
L A
WNY
Zs
Va aK IAN
fp y
; He ns
as eae] r/R = 0.456
a ——! r/R = 0.633
r/R = 0.781
if r/R = 0.963
ae R>ROPELLER ~ 7°00 ft (0-91 m )
FIGURE 5.
Afterbody sections of ATHENA hull showing radii of wake measurements.
231
FIGURE 6. Starboard side view of R/V ATHENA
in drydock.
FIGURE 7. Port side wake rakes and propeller
on R/V ATHENA.
FIGURE 8. Close-up view of five-hole pitot
tube rake on starboard shaft on R/V ATHENA.
nN
WwW
to
ae 2
~
FIGURE 9. Close-up view of boundary layer rake Peres
on R/V ATHENA. eae
FIGURE 10. Fitting room photograph of DTNSRDC
model 5365 representing R/V ATHENA.
8g 10 124
ERS
4 6
CENTIMET)
FIGURE 11. After end view of DTNSRDC model
5365 fitted with a rake of five-hole pitot
tubes on the starboard shaft.
233
FIGURE 12. Afterbody profile view of DTNSRDC
model 5365 fitted with a rake of five-hole
pitot tubes on the starboard shaft.
FIGURE 13. Double model installed in DTNSRDC
wind tunnel.
the use and calibration of five-hole pitot tubes.
The boundary layer velocity profile measure-
ments on the double model, Model 5366, in the wind
tunnel were obtained using a hot wire anemometer.
The model was mounted on its side and the anemom-
eter was moved in the horizontal direction by a
rack and pinion drive. The rack and pinion, with
its stepping motor, allowed the position of the
anemometer to be set to within a fraction of a
millimeter.
Figure 13 shows the double model mounted in the
wind tunnel. The vertical strut at the stern of
the model is the support for the anemometry, and
the bottom horizontal bar is an arm to steady the
strut. The top horizontal bar is the traversing
arm on which the hot wire anemometer is mounted.
A close-up of the hot wire anemometer is shown in
Figure 14; a centimeter scale is shown in the
background of the photograph.
FIGURE 14. Hot-wire anemometer probe used for model
wind-tunnel boundary layer profile measurements.
°
wo
x
x
0.456 RAD.
FULL SCALE DATA
MODEL SCALE DATA
oe
FIGURE 15. Velocity component
ratios for R/V ATHENA and DTNSRDC
model 5365 at 0.456 radius.
-20 0 20
5. FULL-SCALE WAKE SURVEY AND BOUNDARY LAYER
MEASUREMENTS
The full scale trials were run in the Atlantic
Ocean off the Florida Coast near the mouth of the
St. Johns River. The conditions for the trials
were excellent as is shown in Table 1, which gives
the trial agenda and sea conditions. The full-scale
measurements were divided into four trials. Trial
1 consisted of a wake survey in the propeller disk,
and ahead of the struts on the port and starboard
sides.* The objective of the measurements ahead of
the struts was to determine the differences in the
wake both with and without the propeller operating.
Trial 2 consisted of a repeat of the wake survey in
the propeller disk. However, for this repeat
trial, the two rakes ahead of the struts on the
starboard shaft were removed to eliminate any
possibility of interference in the measurements.
Trial 3 consisted of boundary layer profile measure-
ments on the port and starboard sides of the hull.
Again, the purpose of these measurements on both
sides of the ship was to determine the effects of
propeller induction on the development of the
boundary layer. Trial 4 consisted of measurements
of the time varying pressures in a plane ahead of
the operating propeller. The results of Trial 4
are discussed in Appendix A.
*Note: The data from the wake surveys ahead of the struts
and in the propeller disk at a lower speed are not re-
ported in this paper, but will be reported in the future.
80 190 120 140 165 130
200 220 240
IN DEGREES
60 280 309 320 340 360 380
ANGLE
The pitot tubes on the rake in the propeller
plane were located at non-dimensional radii (local
radius divided by propeller radius) of 0.456,
0.633, 0.781, and 0.964. The angular position of
the rake was adjusted by turning the entire shaft
using the jacking gear. The shaft could be rotated
through approximately 230°, and because of this an
overlap of 50° could be obtained in the data
around 180°.
The data from the wake survey at 15 knots are
given, along with the corresponding model data, on
Figures 15 through 18. This ship speed corresponded
to a Froude number of 0.36 and a Reynolds number of
4.14 x 108. The data are presented as velocity
component ratios, where the velocities are given in
cylindrical coordinates centered about the pro-
peller shaft. The longitudinal velocity component
(VX) is positive for flow toward the stern. The
tangential velocity component (VT) is taken to be
positive in the counterclockwise direction when
looking forward on the starboard shaft. The radial
velocity component (VR) is taken as positive in-
ward. The angles are defined positive in the
counterclockwise direction, with zero directly
upward. The conventions for the angles and the
directions of the velocity components are shown on
Figure 5. These conventions are those of Hadler
and Cheng (1965), except that the data is presented
on the starboard shaft rather than on the port
shaft. Therefore, the angles increase in the
opposite direction from Hadler and Cheng, as do the
tangential velocity components.
0.633 Radius
N
i=)
QO
w
2
| 4
VX/V
i=)
@
235
oO
N
FULL SCALE DATA
MODEL SCALE DATA
Qo
an
ow
de
x
t=)
uo
i=)
i
oO
w
=)
N
oO
VT/V
ro)
D
Oo
1
i=)
ft
yes
FIGURE 16. Velocity component
40 150 186
GLE IN DESREES
-20 0
20
There are not sufficient data at any one radius
or circumferential position to adequately define
the limits of accuracy for the full-scale measure-
ments. A comparison between two different pitot
tubes at any one radius may be made in the region
between 150 and 200° where the data overlap. At
all radii the longitudinal velocity component
ratios show the greatest scatter in the full-scale
data. In particular at the innermost radius
(x/R = 0.456), the scatter in the longitudinal
velocity component ratios is greatest, approxi-
mately plus or minus ten percent. The scatter in
the longitudinal velocity component ratios at other
radii is significantly less than that, more nearly
plus or minus five percent. The increased scatter
in the longitudinal velocity component ratios is
due to the computation procedure which uses the
average of the longitudinal velocity components
from both the radial and tangential velocity
computations.
The full-scale wake survey provided a unique
opportunity to study the development of a turbulent
boundary layer on a ship, and also the effects of
propeller action on the boundary layer. The full-
scale boundary layer was measured at the eight
locations which are shown on Figures 3 and 4, at
four speeds. These speeds were 6.2, 9.1, 14.8,
and 16.5 knots; these speeds correspond to Reynolds
ed eh
200 220 240 260 280 360 320 340 360 330
ratios for R/V ATHENA and DTNSRDC
model 5365 at 0.633 radius.
numbers of 1.74 x 108, 2.56 x 108, 4.14 x 108, and
4.63 x 108 respectively.
The data obtained at location 1, for all four
speeds, are plotted on Figure 19. Except for the
data at 6.2 knots, which show a great deal of
scatter, the data are quite consistent with the
fullness of the boundary layer increasing as the
Reynolds number increases. The data obtained at
14.8 knots (RL 4.14 x 108) for location 1, 2, and
3 are plotted in Figures 20, 21, and 22 along with
the corresponding model data at the same Froude
number. The data from Locations 1, 2, and 3 are
plotted again in Figures 23, 24, and 25 along with
the data for the corresponding locations on the
port side with the propeller operating.
6. MODEL-SCALE WAKE SURVEY AND BOUNDARY LAYER
MEASUREMENTS
For the model-scale wake survey, Model 5365 was
ballasted while at rest to the drafts corresponding
to those of the ship during the full-scale wake
survey. The model was then towed at 5.22 knots
(2.685 m/s), the Froude-scaled speed which corre-
sponds to 15 knots full-scale. The velocity com-
ponent ratios were measured with a rake of five-hole
pitot tubes at radii corresponding exactly to the
N
w
or)
OnGSileRAD:
VX/V
VR/V
FIGURE 17. Velocity component
ratios for R/V ATHENA and DTNSRDC -20 0 20 640
model 5365 at 0.781 radius.
full-scale wake survey radii, slowing a direct
one-to-one comparison of the data. The data from
this wake survey are plotted on Figures 15 through
18.
It is customary to perform wake survey experi-
ments in the towing tank by towing the model at a
speed corresponding to the Froude-scaled speed of
the ship. In order to investigate the effects of
Reynolds number on the model-scale wake, a second
wake survey was run at an increased speed. This
second speed was the highest speed for which steady
data would be obtained, 13.5 knots (6.9 m/s). For
this second wake survey, the sinkage and trim of
the model were kept the same as at the 5.2 knot
condition. This was done in an attempt to separate
the effects of sinkage and trim, which is dependent
on Froude number, from other speed effects.
The data from the model-scale wake surveys at
5.2 knots (Fp = 0.36, R, = 1.56 x 107) and 13.5
knots (Fh = 0.93, Ry = 4.04 x 107) are presented
in Figure 26. The longitudinal and radial velocity
component ratios at these two speeds show no dif-
ference. However, the tangential velocity compo-
nent ratios obtained at 13.5 knots have peaks which
are 4 to 6 percent lower than those obtained at
5.2 knots. This is contrary to what might be
exepcted, in that the increased Reynolds number
should produce a thinner boundary layer and there-
fore, a flow which more closely approaches the
108 120 140 160 180 200 220 240 260 280 300 320 340 360 330
ANGLE IN DEGREES
potential flow around the hull. This anomalous
result is probably due to the increased Froude num—
ber and the corresponding change in the wave pattern
around the model.
The model-scale boundary layer profile measure-
ments were made in a wind tunnel using hot wire
anemometers. The double model was manufactured so
as to take into account the dynamic trim of the
ship. Although this cannot take into account the
effects of the free surface, it does account for
the angle of the shafting to the free stream, which
contributes significantly to the radial and tangen-
tial velocity components.
The model scale boundary layer profile was
obtained at a Reynolds number of 1.68 x 107, which
was intended to equal the Reynolds number of the
model in the towing tank at a Froude number of 0.36.
The Reynolds number in the wind tunnel in fact
turned out to be about 8 percent higher than the
Reynolds number in the towing tank. However, this
was not considered to be critical to the correlation
of the model and ship data.
The boundary layer profiles obtained in the wind
tunnel, without the propeller operating, at Loca-
tions 1, 2, and 3 are given in Figures 20, 21, and
22; where they are plotted against the full scale
data at the corresponding locations. The data
obtained at the same locations with and without
the propeller operating are plotted against the
"9.7 © FULL SCALE DATA }
ee ae MODEL SCALE DATA |
0.5 + { {+} 4 1 __} x) eed Ee
168 120 140 165 180 200 220
ANGLE IN DEGREES
40 635
Velocity Profile Data From R/V ATHENA
and Wind Tunnel Model 5366
Location |, x/Ly, = 0.90
E © RUN 209 m/s, Re= 1.74 x10°
= ; | G RUN 210 m/s, Re = 2.56 x10°
w © RUN 2il m/s, Re = 4.14 x 10°
i a] O RUN2I2 m/s, Re= 4.63 x10°
2 a
S
a a
Es 3)
x =
50
400
o
z
5 20
2)
300 4
=)
=)
25
= 30
{2}
a
200
WW
Z 20
fas
a
a
elo an
zx 10
=
a
fo}
2
oO Oo
fo} O02 04 o6 o8 1.0 12
U/Ue ° Ue = SHIP, MODEL SPEED
FIGURE 19. Velocity profile data from R/V ATHENA
measured at four speeds at location 1.
240
—
250 285 300 320 340 350 335
FIGURE 18. Velocity component
ratios for R/V ATHENA and DTNSRDC
model 5365 at 0.963 radius.
corresponding data from the ship in Figures 23,
and 25. This is the extent of the model scale
boundary layer data.
The accuracy of the model scale measurements with
five-hole pitot tubes is known reasonably well.
Model wake survey data have been repeated in past
experiments, with the circumferential mean longi-
tudinal velocity components repeating within 0.01
of the free stream velocity. The velocity component
ratios for the model data are repeatable to within
plus or minus one percent, except in areas where
steep velocity gradients occur. In the areas where
high velocity gradients exist, such as behind the
shaft struts, the five-hole pitot tube has much
lower accuracy. Experiments with hot wire anemom-
eters have shown that they are at least as accurate
as five-hole pitot tubes. In fact, in regions where
there are steep velocity gradients, hot wire anemom-
eters may be an order of magnitude more accurate
than pitot tubes.
24,
7. COMPARISON OF MODEL- AND FULL-SCALE DATA
A study of the velocity component ratios presented
in Figures 15 through 18 shows that the degree of
scatter of the full-scale data is higher than that
of the model data. This is due to the higher
variations in both pressure measurement and ship
speed. In particular, the full-scale data for the
longitudinal velocity component ratio at the inner-
most radius (r/R - 0.456) show the largest scatter,
Velocity Profile Data From R/V ATHENA
and Wind Tunnel Model 5366
Location | x/Ly, * 0.90
. EXPERIMENTAL MEASUREMENTS
7 | OO MODEL SCALE U, = 38.1 m/s, Re =1.68x107
Ww
| FULL SCALE Us=7.6m/s, Re=4.14x10°
a ’ 6
a oO EQUIVALENT BODY OF REVOLUTION CALCULATIONS
a n ———— MODEL SCALE
a a -——- FULL SCALE
a 6
me
* =
50
400
&
2
=) oy
a
300 4
=|
=)
Be
s 30
°
«
200 &
8
z 20
<q
EF
n
a
loos
=z lo
=
&
oO
a
° °
te) o2 o4 o6 os 1.0 12
U/Ue * Ue = SHIP, MODEL SPEED
FIGURE 20. Measured and calculated boundary layer
velocity profiles for R/V ATHENA and wind tunnel
model 5366 at location l.
Velocity Profile Data From R/V ATHENA
and Wind Tunnel Model 5366
Location 2 x/Ly, * O.77
iE EXPERIMENTAL MEASUREMENTS
E | MODEL SCALE Us = 38.1m/s Re = 168x107
E Ww
| = FULL SCALE -U, = 7.6 m/s, Re = 4.14x10°
oa a EQUIVALENT BODY OF REVOLUTION CALCULATIONS
a o ————MODEL SCALE
tr =I - —FULL SCALE
WwW
a re)
x
7 =
50
400
Ww
<
ire
5 40
yn
300 4
=)
=)
Be
= 30
3
©
200
Ww
Z 20
q
=
Cy
(=)
100,
=z 10
=
ac
fo}
z
() i)
ie} 02 04 o6 08 10 le
U/Ue | Ue = SHIP, MODEL SPEED
FIGURE 21. Measured and calculated boundary layer
velocity profiles for R/V ATHENA and wind tunnel model
5366 at location 2.
Velocity Profile Data From R/V ATHENA
and Wind Tunnel Model 5366
Location 3 x/Ly, = 0.98
5 EXPERIMENTAL MEASUREMENTS
e | MODEL SCALE U, = 38.1m/s, Re = |.68 x 107
ve}
| 2 FULL SCALE U, = 7.6m/s, Re=4.14x10%
ce Oo EQUIVALENT BODY OF REVOLUTION CALCULATIONS
a My ———-—MODEL SCALE
3 my -—— -FULL SCALE
Ww
a )
se
x =
50
400
Ww
<
w
= 40
a”
300 4
—
5)
=
s 30
fo)
a
zoo &
8
+2) | (40)
<q
(re
ca)
a
fete) © TF
a io
=
x
3
z
to) (
fo) o2 04 o6 0.8 1.0 12
U/Ue * Ue = SHIP, MODEL SPEED
FIGURE 22. Measured and calculated boundary layer
velocity profiles for R/V ATHENA and wind tunnel model
5366 at location 3.
Velocity Profile Data From R/V ATHENA
and Wind Tunnel Model 5366
Locations | and 8 x/Lw, * 0.90
Ie EXPERIMENTAL MEASUREMENTS
E | © MODEL SCALE U,= 38.1m/s, Re = 168x107
Ww OO w/O PROPELLER
| t| @ > FULL SCALE U,=7.6m/s, Re=4 14 x10°
Ww S|] @ W/O PROPELLER
= 7)
S
o a
& °
I
at =
50
400
3
rn
5 a0
Ww
300 4
a)
5
I
= 30
ro)
[vq
200) es
Ww
S 20
fas
7) @®
fas) @
100,
a 10
=
us e
z e@
6 6) See ye (CE ef ae
Cy 02 04 06 o8 10 12
U/U. | Uo = SHIP, MODEL SPEED
FIGURE 23. Measured boundary layer velocity profiles
for R/V ATHENA and wind tunnel model 5366 with and
without propeller at locations 1 and 8.
Velocity Profile Data From R/V ATHENA
and Wind Tunnel Model 5366
Locations 2and 6 x/Ly, * 0.77
E EXPERIMENTAL MEASUREMENTS
E | | GO MODEL SCALE U, = 38.1m/s, Re = 1.68 x10"
Ww e) w/O PROPELLER F
| a) FULL SCALE U,=7.6mM/s, Re® 49.14x10
w S| e W/O PROPELLER
=! ao
S
a wu
a
£ °
Fs =
50
400
6
E
5 40
n
300 4
=)
=)
a5
s 30
°
v4
200) se
o
eco)
ras
Ca)
a
100)
a lo
=
&
fo}
cA
° °
° 02 04 06 08 1.0 12
U/Ue > Ue = SHIP, MODEL SPEED
FIGURE 24. Measured boundary layer velocity profiles
for R/V ATHENA and wind tunnel model 5366 with and
without propeller at locations 2 and 6.
and the greatest deviation from the model-scale
wake.
In part, this scatter is also due to the fact
that the longitudinal velocity component ratios
presented are an average of the longitudinal velocity
component as measured in the tangential plane and in
the radial plane. Therefore, any scatter error in
either the tangential or radial plane measurements
will influence the calculation of the longitudinal
component. Another factor which probably contrib-
uted to increased scatter at the innermost radius
is the close proximity of the pitot tube to the
strut bossing.
The longitudinal velocity component ratio at the
innermost radius is about 10 percent lower for the
ship than for the model, while the peaks of the
tangential and radial velocity component ratios are
about 10 percent higher for the ship than for the
model. Although there are undoubtedly scale effects
on the shafting and strut bossing at this radius,
another significant factor is that the bossing on
the ship is proportionately much longer than on
the model. This is due to the collar to which the
pitot tube rakes ahead of the struts were attached.
At the outer radii the longitudinal velocity
component ratios for the ship are 2-4 percent lower
than those for the model. The peaks of the radial
and tangential velocity component ratios at the
outer radii are 8-10 percent higher for the ship
than for the model. At the two innermost radii,
the shift in the radial and tangential velocity
component ratios indicate that there is a stronger
upflow on the ship than the model, in the region
under and outboard of the propeller hub. This
effect is much weaker, and has shifted to the inside
239
on the two outer radii. One possible cause of the
shift at the outer radii is the fact that the full
scale trial was performed with a propeller operating
on the port shaft, while the model data were col-
lected without the propeller present. However, the
most likely source of the increased upward flow is
a difference in attitude between the ship and model.
The models were run at a number of Reynolds num-
bers in the towing tank and wind tunnel and the
longitudinal velocity component was measured at a
single location near the hull for these various
Reynolds numbers. The results of these measure-
ments are plotted in Figure 27. These results in-
dicate that for a Reynolds number greater than 107
there is very little effect of either Reynolds num-
ber or Froude number on the longitudinal velocity
component. Therefore, in cases where it is desir-
able to obtain accurate longitudinal velocity
component measurements, the model should be run at
the correct Froude trim, at a Reynolds number
greater than 107.
A comparison of the boundary layer profiles
presented in Figures 20, 21, and 22 shows that, as
might be expected, the model velocity profile is
not as fully developed as the full-scale velocity
profile at Locations 1 and 3. This is clearly a
consequence of the one decade difference in Reynolds
number between the model and ship. However, at
Location 2, the model- and full-scale boundary
layer velocity profiles almost coincide. This is
clearly an anomalous situation, particularly be-
cause even at 0.46 meters from the hull full scale,
the velocity has not reached the free-stream
velocity, let alone the potential flow velocity
which is even higher. The most likely explanation
for the low full-scale velocity profile is a mal-
Velocity Profile Data From R/V ATHENA
and Wind Tunnel Model 5366
Locations 3 and 7 x/Ly, =0.98
E EXPERIMENTAL MEASUREMENTS
E Fj MODEL SCALE U, = 38.1m/s, Re = 1.68x107
Ww W/O PROPELLER -
| zl FULL SCALE Us: 7.6m/s, Re=4.14 x10
Ww ss W/O PROPELLER
s 7)
S
a rr
a 3
=
Fs =
50
400
&
z
5 40
12)
300 4
a)
=)
=
s 30
°
[rg
200 &
8
2 20
es
&
(=)
TOO Mma,
=z 10
=
x
°
ra
0 0
) 02 04 06 08 1.0 12
U/Ue : Ue = SHIP, MODEL SPEED
FIGURE 25. Measured boundary layer velocity profiles
for R/V ATHENA and wind tunnel model 5366 with and
without propeller locations 3 and 7.
240
FIGURE 26. Velocity component
ratios for DTNSRDC model 5365 at
0.633 radius for model speeds of
5.22 knots and 13.5 knots.
function in the instrumentation but a check of the
data records indicated no obvious errors in the
data.
The results of boundary layer profile measure-
ment with the propeller operating, plotted in
Figures 23, 24, and 25 indicate that the data at
positions 1 and 8, just ahead of the propeller,
show a slight increase in velocity profile due to
the propeller suction. The increases are about
the same at both model- and ship-scale. The data
at positions 3 and 7, behind the propeller, show
rather significant increases in the velocity pro-
file for both scales. This is undoubtedly due to
the wake of the propeller. From the model-scale
data, at Locations 2 and 6, there is no noticeable
difference in the data obtained with or without
the propeller operating. This is consistent with
the separation between the boundary layer probe and
the propeller. There is no ship-scale data ahead
of the operating propeller at location 6 due to the
failure of that boundary layer probe.
In order to evaluate our ability to predict the
boundary layer of the hull, a series of boundary
layer calculations were instituted. For these
calculations, the ship was approximated as a body
of revolution, and the boundary layer was calculated
using the standard DTNSRDC method for bodies of
revolution [Wang and Huang (1976)]. Two methods
for generating the bodies of revolution were tested.
In one, the body was generated with radii equal to
the square root of twice the sectional area of the
$c 103 120 149 160
ANGLE IN DEGREES
135 205 220 240 250 280 309 320 340 360 33¢
ship; and in the other, the body was generated
using circumferences equal to twice the girth of
the ship. The boundary layer calculations using
the body of revolution based on sectional area
agreed best with the experimental data.
The results of the equivalent body of revolution
calculations are plotted with the experimental data
on Figures 20, 21, and 22. The calculations for
the ship at Locations 1 and 3 agree reasonably well
with the full-scale data. However, at the model-
scale, the calculations do not agree nearly as
well. This is probably due to the fact that at
lower Reynolds numbers, the boundary layer is much
more sensitive to errors in the flow velocity and
pressure gradient than at higher speeds. As stated
previously, the data at Location 2 is anomalous,
as is shown by a comparison with the calculated
boundary layer profile.
8. PREDICTION OF NOMINAL WAKE
Although the model- and full-scale wake of the R/V
ATHENA both agree qualitatively, there are some
substantial quantitative differences between the
model- and full-scale velocity components. To
develop an understanding of the origins of these
differences, it was necessary to predict the wake
of both the model- and full-scale ship analytically.
Since the hull of the ATHENA showed no separation,
it appeared that the presence of the hull could be
MODEL 5365 REPRESENTING R/V ATHENA
WAKE SURVEY SCALE EFFECT
4) AT O°
r/R = 0.63
lw =18 67 FT
Ls =1540 FT
O SHIP MODEL IN TOWING TANK
Q DOUBLE MODEL IN WIND TUNNEL
OD SHIP TRIAL
0.8
LONGITUDINAL VELOCITY COMPONENT RATIO (Vx
5x10° 10" 5x10" 10'
REYNOLDS NUMBER (Re,) BASED ON SHIP LENGTH
dealt with primarily by potential flow techniques,
combined with calculations of the boundary layer
displacement thickness. It was also assumed that
the viscous flow about the appendages could be
dealt with empirically.
The velocity in the propeller disk, expressed in
shaft coordinates, was decomposed as follows:
Velocity = Uniform Stream
Perturbation due to Hull
Perturbation due to Boundary Layer
Viscous Wake of Struts
Viscous Wake of Shafting.
oe eee
The principal factor contributing to the radial and
tangential components of the velocity in the pro-
peller plane is the inclination of the shaft to the
free stream. The shafting of the ATHENA makes an
angle of 8.9° with the baseline. In addition, at
15 knots Ga = 0.36), the ATHENA takes a bow-up
trim of 0.3° as indicated by model experiments.
Thus, the propeller shaft is inclined a total of
9.2° to the incident stream. The effect of resolv-
ing the incident stream into shaft coordinates is
shown on Figure 28.
The effects of perturbing the incident stream by
the presence of the hull were obtained by means of
potential flow calculations. For the purposes of
this study, the free surface was represented by the
zero Froude number condition, and the calculations
were made for a double model in an infinite fluid.
The hull was reflected about the mean waterline at
15 knots, and flow about the resulting body was
computed using the DTNSRDC potential flow program
[Dawson and Dean (1972)]. The results of this
computation are also shown on Figure 28. As can
be seen, the effects due to the perturbation of the
incident flow by the hull are small, on the order
of two percent of the ship speed.
The effects of the displacement thickness of the
boundary layer were considered next. The intention
was to increase the thickness of the hull by the
displacement thickness of the boundary layer, and to
repeat the potential flow calculations. However,
at its thickest point, the model scale boundary
layer determined from the equivalent body of
revolution calculations, would only have increased
241
FIGURE 27. Longitudinal velocity component
ratio at O-degree position of 0.633 radius
10? as a function of Reynolds number based on
hull length.
the thickness of the hull by 1 percent of the beam.
The full-scale boundary layer would have increased
the thickness even less. Since the complete hull
potential flow had only a two percent effect, the
revised potential flow was not computed for such a
small change in effective hull shape. The error
due to neglecting the displacement thickness of the
boundary layer is probably much less than the error
incurred by making the zero Froude number approxi-
mation for the potential flow calculations. There-
fore, the velocity component ratios based on only
the first potential flow calculations are presented
in Figures 29 through 32.
The velocity defect caused by the struts was
predicted using an empirical scheme based on data
from aerodynamics. The velocity defect was com-
puted using the following formula from page 584 of
Goldstein (1965).
(7R=0633
Sao 3-D POTENTIAL FLOW
UNIFORM FLOW
fe) MEASURED VALUES
VELOCITY COMPONENT RATIO
rt
° 40 80 120 160 200 240 280 320 360
ANGLE IN DEGREES
Velocity Component Ratios Predicted and Measured Full-Scale
Trial 2, Vs = 7.87m/s
FIGURE 28. Effect of shaft inclination and hull po-
tential flow on velocity component ratios for R/V
ATHENA at 0.633 radius.
242
T/R=0456
—-—-—-PREDICTED VALUES
O MEASURED VALUES
VELOCITY COMPONENT RATIO
° 40 80 120 160 200 240 280 320 360
ANGLE IN DEGREES
Velocity Component Ratios Predicted and Measured Full-Scale
Trial 2, Vs = 7.87 m/s
FIGURE 29. Predicted and measured values of velocity
component ratios for R/V ATHENA at 0.456 radius.
Ie
= 3/18a,x*
rend Oe Lo Gee
and
No 3/2a,, = 10D/n PU,
where Umax is the velocity defect, U, the free-
stream velocity, n_ is the nondimensional wake half
width, x is the nondimensional distance from the
strut, D is the strut drag, and p the fluid density.
These formulas predict the longitudinal velocity
defect in terms of the strut drag, wake thickness,
and distance behind the strut.
The shaft struts on R/V ATHENA are Navy EPH
sections with a chord-to-thickness ratio of 6.
r/R =0633
V/V
oo
®o0°
— — -0.2
5 oo
—-—-—-—PREDICTED VALUES
O MEASURED VALUES
VELOCITY COMPONENT RATIO
° 40 80 120 160 200 240 280 320 360
ANGLE IN DEGREES
Velocity Component Ratios Predicted and Measured Full-Scale
Trial 2, Vs = 7.87m/s
FIGURE 30. Predicted and measured values of velocity
component ratios for R/V ATHENA at 0.633 radius.
r/7R=0.78l
—-—-—-—PREDICTED VALUES
O MEASURED VALUES
VELOCITY COMPONENT RATIO
° 40 80 120 160 200 240 280 320 360
ANGLE IN DEGREES
Velocity Component Ratios Predicted and Measured Full-Scale
Trial 2, Vs = 7.87 m/s
FIGURE 31. Predicted and measured values of velocity
component ratios for R/V ATHENA at 0.781 radius.
Assuming that the drag on the EPH section would not
be too different from the drag on an elliptic
section of the same thickness-chord ratio, many data
for a number of elliptic sections were collected.
These data are plotted on Figure 33 as a function
of Reynolds number.
The nondimensional wake half-width was predicted
using Equation (4) from Silverstein et al. (1938):
os 45 5
n = sets) (G6 ce 415)
(0) D
In this equation n_ is again the nondimensional half
width of the wake, X is the nondimensional distance
from the strut, and C_ is the drag coefficient per
unit length of the strut.
Using the strut Reynolds numbers based on chord
length, of 1.46 x 10° for the ship, the correspond-
r/R =0.963
Vy/V
ie) °
=r ©" ~-o- ~ 688
’ SHIP
i
—-—-—-—PREDICTED VALUES
© MEASURED VALUES
VELOCITY COMPONENT RATIO
° 40 80 120 160 200 240 280 320 360
ANGLE IN DEGREES
Velocity Component Ratios Predicted and Measured Full-Scale
Trial 2, Vs = 7.87m/s
FIGURE 32. Predicted and measured values of velocity
component ratios for R/V ATHENA at 0.963 radius.
=6.0 WILLIAMS & BROWN (\937)
=40
£30 Fuinosey 11938)
2394
745
+ WARDEN (1934)
en ET AL (1929)
Cp = DRAG PER UNIT LENGTH /P/> cv2
104 5x10% 10° 5x10° 10°
REYNOLDS NUMBER (Re,) BASED ON STRUT CHORD LENGTH
ing drag coefficients are found to be 0.050 and
0.018 for the model and ship, respectively. Sub-
sitution of these drag coefficients into the above
formulas from Silverstein, et al. (1938) and
Goldstein (1965) yields the velocity defects which
are shown on Figures 29 through 32.
These computed velocity defects due to strut
wake are significantly greater than the velocity
defects which were observed at either model or full
scale. The cause of this over-prediction is
probably the fact that the formulas from Goldstein
are derived by assuming that the wake is being
calculated far enough downstream that the cross
flow terms in the momentum equation can be neglected.
This is an assumption which is undoubtedly violated
in the region near the struts, where the wake has
been predicted.
Although the empirical method for predicting the
wake of the shaft struts was not successful, it
does at least provide some insight into how the
wake should vary with Reynolds number. Both the
width of the wake of the struts and the velocity
defect in the wake of the struts are proportional
to the square root of the drag coefficient of the
section. Therefore, the velocity defect and the
width of the wake should both decrease (like the
square root of the ratio of the drag coefficients)
as the Reynolds number increases. However, the
full-scale wake survey data were not collected at
angular increments spaced closely enough to confirm
this scaling law.
The empirical method for predicting the wake
behind an inclined shaft is not as well defined as
the methods for predicting the wake behind the
struts. Following the methodology of Chiu and
Lienhard (1967), it was assumed that the separated
flow behind a yawed cylinder is a function of the
component of the velocity normal to the cylinder.
Following the method of Roshko (1955) and (1958),
an estimate of the velocity defect in the wake of
the shaft was developed based on the pressure
coefficient at the point of separation and the
Strouhal number. i
Data showing the base pressure behind a circular
*Note: The base pressure is not necessarily the pressure at
the separation point because there is usually some pressure
variation in the separated region.
5x10°
243
FIGURE 33. Drag coefficients of elliptical
7 section struts as a function of Reynolds
number based on chord length.
10
cylinder have been collected, and are presented as
a function of Reynolds number in Figure 34. Based
on this data and the Reynolds number based on cross
flow velocity, the pressure coefficients for the
model (R_ = 1.63 x 10*) and ship (R_ = 4.26 x 10°)
were found to be -1.1 and -0.2 respectively. These
pressure coefficients resulted in a predicted veloc-
ity defect, perpendicular to the shaft axis, of
0.25 for the model and 0.10 for the ship. However,
when resolved back into the direction of the flow,
the shaft wake is less than two percent of model
speed and one percent of ship speed. This is
significantly less than than the velocity defect
which is measured for either the model or the ship.
In fact, if the velocity defect in the direction
normal to the shaft were 100 percent of the forward
speed, the velocity defect in the wake would only
be seven percent, still less than the velocity
defect measured experimentally.
These results are not surprising when one con-
siders the discussion in Chiu and Lienhard (1969).
In this discussion, data are presented which point
out that the wake of an inclined shaft is in general
not parallel to the shaft. This is due to the
axial component of the flow along the cylinder which
develops a boundary layer which separates. The
Reynolds number for separation in the axial direc-
tion on the shaft is independent of the Reynolds
number of the flow normal to the shaft. In addition,
the data from Bursnall and Loftin (1952), show that
as a circular cylinder is inclined further and
further to the flow, the transverse Reynolds number
at which separation takes place becomes lower and
lower.
9. CONCLUSIONS
Significant differences have been found in the
tangential and radial velocity component ratios
between the ship and the model wake surveys. In
particular, the full-scale tangential velocity
component ratio has a peak amplitude approximately
eight to ten percentage points higher than that at
model scale. Similarly, the ship radial velocity
component peak is higher by six to eight percentage
points. These differences cannot be attributed
to scale effects. The most likely cause seems to
© BURSNALL & LOFTIN (195!)
Q_FAGE 8 FALKNER (193!)
© ROSHKO (1953)
0.8
2
°
b
Cp, = BASE PRESSURE /P, v2
° °
+ fo}
°
o
10% 5x10* 108 5x10°
REYNOLDS NUMBER (Rep) BASED ON SHAFT DIAMETER
be a difference in trim between model- and full-
scale. Because the model was ballasted to the
draft of the ship, further work will be required
to identify the source of these differences.
The longitudinal velocity component ratios for
the full-scale trial show a much greater scatter
than the tangential and radial components. For
this reason it is unclear that any difference is
shown by these data, when compared to model-scale
data. The innermost radius (r/R = 0.456) does show
that the high longitudinal velocity component
normally measured at these inner radii is not found
full scale. This may not be the result of scale
effects on the shafting and strut bossing, but the
fact that the full-scale bossing is longer than the
model-scale bossing. This is a result which will
have to be investigated by further model experi-
ments.
The results from model experiments in both the
wind tunnel and in the towing tank, and from the
full-scale trial indicate that for a circumferential
position near the hull, there was little difference
in longitudinal velocity component ratio for speeds
corresponding to Reynolds numbers greater than 10".
Therefore, when measuring only the longitudinal
velocity component ratios experimentally, the model
should be run at the trim corresponding to that of
the Froude-scaled speed and at a speed high enough
to yield a Reynolds number of greater than 107.
The attempt at predicting the wake for this high-
speed displacement ship showed that the most im-
portant contribution to the variation in tangential
and radial velocity component ratios was the shaft
angle to the flow. The calculation of the potential
flow around the hull and the resulting velocity
components showed that the effect of the perturba-
tion due to the hull was small. The effects of the
boundary layer of the hull on the wake were also
shown to be small.
In summary it may be stated that the full-scale
and model wakes differ by approximately ten percent
of the ship speed. These differences cannot be
adequately explained at this time. Further work
on wake of appendages is recommended as one step in
improving the understanding of these differences.
Chiu, W. S., and J. H. Lienhard (1967).
FIGURE 34. Base pressure coefficients of cylin-
10 drical shafts as a function of Reynolds number
based on shaft diameter.
ACKNOWLEDGEMENTS
This work was performed under the controllable-
pitch propeller research program sponsored by C. L.
Miller of the Naval Sea Systems Command (NAVSEA
0331G) administered by the David W. Taylor Naval
Ship Research and Development Center (DTNSRDC) .
The authors wish to express their appreciation
to personnel of the Ship Performance Department of
DTNSRDC, the University of Michigan, and the crew
of R/V ATHENA from MAR Inc. for their assistance
in conducting the full-scale trial and model experi-
ments which provided the data for this paper. CHI
Associates, Inc. and Rosenblatt Inc. are also
acknowledged for their assistance in the prepara-—
tion of this paper.
BIBLIOGRAPHY
Boswell, R. J., J. J. Nelka, and S. B. Denny (1976).
Experimental Determination of Mean and Unsteady
Loads on a Model C. P. Propeller Blade for
Various Simulated Modes of Ship Operation.
Eleventh ONR Symposium on Naval Hydrodynamics,
London, England, VIII 75-110.
Bursnall, W. J., and TR. K. Loftidne (1952))5" qxpemim
mental Investigation of the Pressure Distribu-
tion about a Yawed Circular Cylinder in the
Critical Reynolds Number Range. NACA Tech. Note
2463
Canham, J. J. S. (1975). Resistance, Propulsion
and Wake Tests with HMS Penelope. Trans. Royal
Inst. of Naval Arch., 117, 61-94.
Flow Over Yawed Circular Cylinders. Jour. Basic
Engineering, Trans. ASME, Series D, 89, 851-857.
___(1969) Discussion to: On Real Fluid Flow Over
Yawed Circular Cylinders. Jour. Basic Engineer-
ing, Trans. ASME, 91, 132-134.
Dawson, C., and J. Dean (1972). The XYZ Potential
Flow Program. NSRDC Report 3892.
Denny, S. B., H. A. Themak, and J. J. Nelka (1975).
Hydrodynamic Design Considerations for the
On Real Fluid
Controllable-Pitch Propeller for the Guided
Missile Frigate. Naval Eng. Jour., 87, 2; 72-81.
Fage, A., and V. M. Falkner (1931). Further Experi-
ments on the Flow Around a Circular Cylinder.
British A.R.C. Reports and Memoranda No. 1369,
186-208.
Frydenlund, O., and J. E. Kerwin (1977). The
Development of Numerical Methods for the Compu-
tation of Unsteady Propeller Forces. Norwegian
Maritime Research, 17-28.
Goldstein, S. (1965). Modern Developments in Fluid
Dynamics, Vol. II. Dover Publications, Inc.,
New York, pp. 331-702.
Hale, M. R., and D. H. Norrie (1967). The Analysis
and Calibration of the Five-Hole Spherical Pitot.
ASME Paper 67-WA/FE-24.
Hadler, J. B., and H. M. Cheng (1965). Analysis of
Experimental Wake Data in Way of Propeller Plane
of Single- and Twin-Screw Ship Models. Trans.
Soc. Naval Arch. and Mar. Eng., 73, 287-414.
Lindsey, W. F. (1938). Drag of Cylinders of Simple
Shape. NACA Report 619, 169-176.
Namimatsu, M., and K. Muraoka (1973). The Wake
Distribution of a Full Form Ship. JIHI Engineer-
ing Review, 7, 3.
Namimatsu, M., K. Muraoka, S. Yamashita, and H.
Kishimoto (1973). Wake Distribution of Ship and
Model on Full Ship Form. J. Soc. Naval Arch.
Japan, 134 (Japanese), 65-73.
INoKEIS7 So Mey Ws Wo @s iMG Wee bisdeyelovexen Jel 126
Young, and D. A. O'Neil (1977). Increased
Profits for Gas Turbine Container Ships by
Unique Applications of Combustion Technology and
Hydrodynamics. Trans. Soc. Naval Arch. and Mar.
Eng., 85.
Rains, D. A. (1975) DD 963 Power Plant. Marine
Technology, 12, 1; 1-24.
Roshko, A. (1953). On the Development of Turbulent
Wakes from Vortex Streets. NACA Report 1191,
801-824.
___(1954) On the Drag and Shedding Frequency of
Two-Deminsional Bluff Bodies. NACA Tech. Note
3169.
___(1955) On the Wake and Drag of Bluff Bodies. J.
MERA, Behe, 2A, NAAM.
Schuster, S., H. Grothues-Spork, H. Thieme, and
H. Schwanecke (1968). Meteor-Messfahrten 1967.
Jahrbuch der STG, 62, 159-204.
Silverstein, A., S. Katzoff, and W. K. Bullivant
APPENDIX A - TIME-VARYING PRESSURE MEASUREMENTS
AHEAD OF AN OPERATING PROPELLER
During Trial 4, the time-varying pressures at the
head of a piezoelectric pitot tube were obtained
as a function of shaft position. The pressures
were measured for each six degrees of shaft rota-
tion for the pitot tube at a fixed angle. Measure-
ments were obtained at four angular positions of
the pitot tube and at two ship speeds.
It should be noted that due to the fact that the
pitot tube is approximately one diameter of the
propeller forward of the propeller disk, the
amplitude of the pressure oscillation is only 1 per-
245
(1938). Downwash and Wake Behind Plain and
Flapped Airfoils. NACA Report 651, 179-206.
Takahashi, H., T. Ueda, M. Nakato, Y. Yamazaki,
M. Ogura, K. Yokoo, H. Tanaka, and S. Omata
(1971). Measurement of Velocity Distribution
Ahead of the Propeller Disk of the Ship. J. Soc.
of Naval Arch. West. Japan, 129 (Japanese),
153=J'68r
Taniguchi, K., and T. Fujita (1970). Comparison of
Velocity Distribution in the Boundary Layer
Between Ship and Model. J. Soc. Naval Arch.
Japan, 127.
Troesch, A., V. A. Phelps, and J. Hackett (1978).
Full-Scale Wake and Boundary Layer Survey
Instrumentation Feasibility Study. Dept. Naval
Arch. and Mar. Eng. Report, Univ. of Mich.
Tsakonas, S., W. R. Jacobs, and M. R. Ali (1973).
An Exact Linear Lifting-Surface Theory for
Marine Propeller in a Nonuniform Flow Field.
Jip Shipy Res, Li77, VI6—20)7 i.
Vorus, W. S. (1974). A Method for Analyzing the
Propeller-Induced Vibratory Forces Acting on
the Surface of a Ship Stern. Trans Soc. Naval
Arch. Mar. Eng. , 82), 186—21'0).
Wang, H. T., and T. T. Huang (1976). User's
Manual for FORTRAN IV Computer Program for
Calculating the Potential Flow/Boundary Layer
Interaction on Axisymmetric Bodies. DITNSRDC
Ship Performance Dept. Report 737-Ol.
Warden, R. (1934). Resistance of Certain Strut
Forms. British A.R.C. Reports and Memoranda
LG), BUY SQN
Wennberg, P. K. (1966). The Design of the Main
Propulsion Machinery Plant Installed in the
USCGC HAMILTON (WPG-715). Trans. Soc. Naval
Arch. and Mar. Eng., 74, 29-69.
Williams, D. H., and A. F. Brown (1937). Experi-
ments on an Elliptic Cylinder in the Compressed
Air Tunnel. British A.R.C. Reports and
Memoranda 1817, 103-112.
Yokoo, K. (1974). Measurement of Full-Scale Wake
Characteristics and Their Prediction From Model
Results--State of the Art. Symposium on High
Powered Propulsion of Large Ships, Wageningen,
pp. XI.1-28.
Zahm, Ao Be, Re He Smith, and FA. louden) (11929)
Forces on Elliptic Cylinder in Uniform Air
Stream. WACA Report 289, 217-232.
cent of the mean pressure signal. Due to the
failure of two of the pressure transducers in the
head of the pitot tube, it was impossible to obtain
any data on the variation of the flow velocity with
angular position.
During each of the runs with the piezoelectric
pitot tube, data were collected for a period of
time totalling between 5 and 10 minutes. All of
the data points for each angular position of the
shaft were then averaged to obtain a mean level for
each signal. Figures A-1, A-2, and A-3 show these
averaged pressure signals as a function of angular
position. Runs 209 and 205 were both obtained at
the same ship speed (15 knots) and shaft speed
246
i i th
0 40 80 120 160 20 240 | 280 20 360
T Wali aT I All T T
144-00
ie x os 9 As as Poa @
s ® i \ A @
13-95 i A o e
N [es St. ry Wad Odi @ 9%e J @ AR
13-90 L. r y b ® ° . Ve
5-30 | . -
me Pe e ry oes .
5-25 @ vee Seta ee { 28 al f wee
12 is \ ® e °, @ «6 ee, 9 ry ed J
} e
5-20 8 . 8 x
14-30 =
° pee a a ee ro” a* oot
a), ees p” * As Went 7 Ale SAR a Seay |
OU ° ves Vv, wee ® 8
14-20 A +
+ v of 2, +
97:0 v . +
ISS Oo She ° * a
A i +, + +
R2 96:5 |_ . 5 O86 4 + x
° ar * A *
96-0 |_ + * ‘7 + : ‘ pase
2-10 HH nee t orton ns TF HH eget Met eeee tht Meee teatangs ete reget ttt tenn et
FIGURE A-l. Circumferential dis- ii
tribution of piezoelectric pres- c &05 i | | =| | | !
sure transducer signals for 15- Q o) 0 129 @ ™m a » am ca
knot run 209. ott) BLADE POSITION @ IN DEGREES 360P= OME REVOLUTION (0.17SEC)
(345 rpm), but with the pitot tube in different signals. For Run 209, the amplitude of the eighth
angular positions, 180° for Run 205 and 300° for harmonic was 3 percent and those of the other har-
Run 209. Run 208 was obtained at a speed of 8.9 monics were generally less than 10 percent of the
knots and a propeller speed of 245 rpm, with the eighth harmonic. The only exception to this is the
pitot tube at 300°. sixteenth harmonic which is again of increased mag-
As can clearly be seen from the data obtained nitude. Although the magnitude of the harmonics
during Run 209, the pressure signal from the three from Run 205 were lower than those from Run 209 the
operating pressure transducers is periodic. There saem results apply. There is not obvious periodicity
is an obvious periodicity at twice blade frequency in the data from Run 208. However, a harmonic
(eighth harmonic in shaft frequency). A Fourier analysis of this data shows that the twelfth har-
series analysis of the data from the two 15-knot monic is dominant, although not nearly to the same
runs showed that the second harmonic in blade extent as in the cases of the high speed runs.
frequency was the dominant harmonic in all three
0 ® 80 1 10 m 200 0 30 360
i T mal al Ty TR Im iperaneee| 7 a1
im oe °@ / ° 4
ign oe ba eWieiues eo%e f Se gt 8 a Pe ae ore R. Hee:
Tl 96 ahs OY © 6 \ =e e s © oe” » eo e Z|
a0 - 4
3°40 Le =
e ° CV, 2 @ cy a ? a -?
7] 5B - simtiialiselve TS idee 0% Wa ee) FMI SPs =|
ee » 8 Oy ‘ oe” so, Poa 3 ae Lome id
5D el
BE re e 9 e 9 9 ee ae @ @ ° Pier aame
| . Q ° eo @ 0 ) g R
a) 13-D ee | a ea) eS) oN Oy Se Q a4
1 lat Yo ah ie aN os 6 %e o on
13-25 aU 0 é : 0 6 3 A
th 4
e
e
R2 = e on e ~J|
Le e e © e : e ae e? |
0 O eo
e e
26 me 6 eh s °
e ° ee
C 26-55 ite . SMe. Oras
v en Oi fe: FIGURE A-2. Circumferential dis-
aD ry \ i ¢° ane Le f iL 1 i tribution of piezoelectric pres-
0 i) 80 120 1 °° 20 0 320 360 sure transducer signals for 15-
BLADE POSITION 8 IN DEGREES 360° = ONE REVOLUTION (0.17 SEC ) knot run 205.
WW
12
RI
R2
0 40 80 120 160 z 40 280 320 360
14-49 = ommlps y aneareaia a ae
Q oo
og r coc, Gogo" 2 Q 9 \ J ° reg o
1448 = \y bord * ? doe Fie wig Nees 0"
° o o © og og
14-42 ia
a )
Q Po ° o%, 970900%>
7:28 a o® Poo 920
6 42 9%go oo @® Q eo a Q
ey er Bog 3° % 2s bg \ c:)
ry eo og
=
i 9
pa a ¢ o
pooo® PES) og Pa 9 2 a ow a
16:35 e009 24% eS ee, yf? Beoe™\ OH
“o. @ | eo / a
#9 o \9 Pa og <j »?
16°34 Ta
lass Cc)
5 ® o °° © ) ®
25:37 e ee @ ® @
5 OMe ® cc) aires of - OF ® oO.
e
25-36 ee AOS O16 0 .
: @ ® ©00°e 2%
ree | al ieee [eee | |
0 40 8 120 160 20 240 280 320 360
BLADE POSITION 8 IN DEGREES 360°= ONE REVOLUTION (0.17 SEC)
FIGURE A-3. Circumferential
distribution of piezoelectric
pressure transducer signals for
8.9-knot run 208.
247
Influence of Propeller Action on
Flow Field Around a Hull
Shunichi Ishida
Ishikawajima-Harima Heavy Industries Co, Ltd.
Yokohama, Japan
ABSTRACT
Flow field in the vicinity of a hull is analyzed
by using acceleration potential, and an approximate
calculation method is derived. The present method
can calculate the change of pressure on the hull
caused by a propeller action. Numerical results
by the present method are shown with experimental
results.
Wake far from a ship is analyzed by using Oseen's
approximation, and an optimum condition is given
for wake energy recovery by a propeller. This
condition is examined by the results of the self-
propulsion tests and the wake survey measurements
at distant positions behind a ship.
ibn INTRODUCTION
When a hull is towed in still water, a flow field
is induced around the hull. This flow field is
very complicated, and becomes more complicated by
propeller action. Many researchers have studied
experimentally and theoretically the phenomena
caused by the interaction of the hull and propeller,
[Yamazaki et al. (1972) ]. Unfortunately, however,
the number of practical uses of the study results
is less than those derived in other fields of naval
hydrodynamics. One of the reasons is because the
various suggested methods are themselves complicated
owing to the complexity of the phenomena.
It has been popularly known that both the equa-
tions and the boundary conditions which describe
flow field can be simplified, and analyzed easily
if disturbance by an object in the flow is a small
quantity of the first order. One of the typical
examples is the method of acceleration potential
in inviscid flow fields used for propeller theory
[Tsakonas et al. (1973) ]. Another example is
Oseen's method in a viscous flow field used for
the separation of hull resistance components [Baba
(1969) ].
In this paper, the above-mentioned concept is
applied to analysis of flow fields induced by the
interaction of the hull and propeller, and the
author derives practical methods relating to the
propeller-induced pressure change on the hull and
wake energy recovery by the propeller. Section 2
explains coordinate systems used in this paper.
In Section 3, the author applies the method of
acceleration potential for analysis of inviscid
flow fields in the vicinity of the hull, and derives
a method which can be used to calculate the change
of pressure induced by a propeller on a hull surface.
In Section 4, the author applies Oseen's method for
analysis of wake far from the hull, and derives a
method to predict recovery of wake energy by the
propeller. Then, this method is examined by the
experimental results obtained from self-propulsion
tests and the wake survey. Section 5 concludes
this paper.
Bo COORDINATE SYSTEMS
We assume that a ship with a single propeller is
moving with a constant speed on the free surface of
still water. At first, we define a coordinate
system O-XYZ fixed in space and a coordinate system
o-xyz fixed on the hull as indicated in Figure 1.
The coordinate system O-XYZ is an orthogonal coor-
dinate system, in which the XZ-plane coincides with
the still water surface and the positive direction
of Y-axis coincides with an upward vertical line.
The coordinate system o-xyz is a moving coordinate
system in which the origin o is moving on the X-axis
in the negative direction with a constant velocity
U, and this sytem satisfies the following relation-
ship with 0O-XYZ:
Mako WE WS we 4S Bp (1)
where t represents time.
Next, we define two more coordinate systems
FIGURE 1.
Coordinate systems.
0)-x1y1]Z and 0)-x)r8 related with the propeller
as indicated in Figure 1. In the coordinate system
01-X1]¥1Z1, the origin 0) coincides with the propeller
center and we assume that the x]-axis coincides
with the propeller axis and is parallel to the x-
axis. Further, the coordinate system 0]-x ]y]2Z] has
the following relationship with the coordinate
system O-xyz:
SF Op Oe SoS A oP Alp BS Bio (2)
where (x,, -f, 0) are the coordinates of the pro-
peller center on o-xyz. Moreover, the following
relationship is satisfied between 0]-x y ]Z] and
0)|-x)4r8:
X] = X1, y) = © cos®, 2] = r sind. (3)
3. PRESSURE ON A HULL SURFACE AND ACCELERATION
POTENTIAL
Pressure generated on the hull surface in the towed
condition differs from that in the self-propulsion
condition because of the influence of propeller
action. The time-independent part of this change
corresponds to the pressure component of the thrust
deduction and the time-dependent part corresponds
to the propeller-induced surface force. Now, with
conventional methods devised to calculate these
forces, numerical procedures tend to be extremely
troublesome. Consequently, a great deal of calcu-
lation time is required, especially in calculating
propeller induced velocity, and it is hard to apply
to a practical hull of a complicated form. Hence,
an easy method with which the calculations of pro-
peller influences can be reduced is needed.
In this chapter, the method which can calculate
change of pressure induced by a propeller on the
hull surface is explained. This method can be
obtained by using acceleration potential.
Fundamental Equation
In this section, we assume that the flow field
around the hull is inviscid. This assumption may
be considered reasonable in solving the problem of
pressure on the hull surface when the boundary layer
on the hull surface is thin.
At first, let us examine the flow field around
the hull in the towed condition. Denoting the
velocity potential of disturbance due to the hull
249
by $,(x,y,2), the velocity potential for the over-
all flow field can be expressed by U*x+},, and the
following equation must be satisfied for $<:
2 2 2
anes r OS a cats
ax2 ay2 az2
=o. (4)
Boundary conditions are given as follows. On the
hull surface, S,, the following equation must be
satisfied:
(o + 2) + sae) £ ais 0 , (5)
ny om Orn = :
x dy y 0z Zz on &
where Ny, Ny, and nz, represent xX-, y-, and z- com-
ponents of the outward normal unit vector on S..
On the free surface, we have two boundary conditions.
One of them can be obtained from the Bernoulli's
law and the condition of constant pressure there,
as follows:
1 (See p es; fs (2) t Oe ‘ | ee
2 | \ox dy dz ox SE eget 0
(6)
where (,(x,Z) represents the vertical displacement
of the free surface, namely, wave height. Another
boundary condition on the free surface is the
kinematical condition as indicated below:
a6 af ao ag ot
(o + 2)". s Ss Ss Ss
oe ho 8a... BE mies (7)
y=Ss
At infinity, the following boundary conditions
might be given:
Derivatives of bs > 0 when Vx? + y2 Foo => Cd
C5 * O when Vx* + 22 > . (8)
Next, let us examine the flow field around the
hull in the self-propulsion condition. We assume,
similarly to the towed condition, that the velocity
potential of disturbance exists. Then, we can
express the velocity potential of the overall flow
field by Uex + >, + o*. Here, the 9*(x,y,z;t) xrep-
resents the change of the velocity potential due
to the propeller action when the moving condition
is changed from the towed condition to the self-
propulsion condition, and $* must satisfy the
following equation:
2 2 2
ra) o* te A) o* ey A) o* A
ax? dy? 2°
We can also obtain the boundary conditions under
the self-propulsion condition in the same manner
as under the towed condition. In this case, however,
time derivatives appear in some conditons by the
influence of propeller rotation. On the hull sur-
face, the following boundary condition is given:
( Oy ag * (Ss ey
Wise ee ee OS De pe
OO. a
lee vie) te
Dio (9)
on So. =O. (10)
250
On the free surface, the condition of constant
pressure and the kinematical condition can be given
as follows:
/ Id*\ 9 3 ao* p) ap*
al ted Cte ea a
2 ax ax dy dy az dz
36. ee) ao*
ee — = ; 11
+ U (= tae) eae oul \ lees (0) (11)
Gis 2) es
u ox ax ox dy by
cle) ap* at ot
s sp sp
a Sie = 5 12
a3 (= + 22 ) Ozie w moe vaca. me oe
where Ssp represents the wave height in the self-
propulsion condition. At infinity, the following
boundary conditions might be given:
Derivatives of ¢, + >* > O when Vx2 + y? +2270,
G+ O when Vx? + 22 +m . (13)
sp
Finally, using the equations derived under the
towed condition and the self-propulsion condition
described above, let us derive the equation and
boundary conditions for $* which express the change
of the flow field around the hull due to the pro-
peller action. At first, $* must satisfy the Laplace
equation (9). Next, let us obtain the boundary
conditions for ~*. On the hull surface, the fol-
lowing relationship is given from (5) _and (10):
CUES ab* , OWE = 6
Fay es oy y i dz oz ee Ss, me oe
On the free surface, the following equation is
given from (6) and (11) in correspondence with the
condition of constant pressure:
oe (ee
2 ax ax ay dy az az
(ee es age
F Ui einer) ites ak OY, leer
8d_\2 (26.\2 /2¢.\ 2] a9
Ee ley a a
5 te ) *\ay Nae ee ON ace,
(15)
And, using (7) and (12), the following equation is
given in correspondence with the kinematical
condition:
*
Fg Alas Mas ag tance
\ ax ox ax oy dy:
ot
Pst eae Weed sD Sp
az oz / az at Yop
Uh ilies: BUN eect ea RAD We nce
5 \ ax ) ax oy Oz
dd *
(Zs Ch) Oo.
aan eee (16)
At infinity, the following boundary conditions
might be given:
Derivatives of »* > O when Vx? 4 22 aa Cie,
c* + 0 when Vx? + z2 +0 , (17)
where C* (x,z;t) represents change of wave height
due to the propeller action and the following
relationship must be satisfied:
Ge -SGSpe Gs. (18)
Acceleration Potential and Approximate Calculation
Method
Acceleration Potential
The purpose of this section is to indicate that the
equation and the boundary conditions for $* derived
in the previous section can be expressed in the
terms of acceleration potential on the assumption
of thin hull.
At first, using the assumption of thin hull, we
express the shape of the hull as follows:
Z = Of (o0,y,) in See p (19)
where € represents a small quantity of the first
order and S_* represents a projected plane of the
hull surface, Ss, in the xy-plane. And, it seems
reasonable to develop all our quantities in powers
of €, as follows:
o, = 1 + S40n' soe ; (20)
o* = ed] + [765 +... ; (21)
SoS en Seige ee ap (22)
a Boi PE Ba cbo 7 (23)
Thus, €* can also be developed as follows:
* 2 *
Ge 0 Gn ar EG Poco oo (24)
Next, we proceed to obtain the equation and
boundary conditions for $,* which correspond to the
first order of € by substituting the development
(20) ~(24) into the equation and boundary conditions
for ~* in the previous section. The following
equation in Q* can be obtained from (9) and (21):
24k 24k Jak
a2gt a2ge 92g
(25)
aa ay2 * 922
Let us consider the boundary conditions for )*.
First, using Eq. (19), we can estimate the magni-
tude of nye ae ne in the Eq. (14) as follows:
mim SO), mM = OS), mm = O@) - (26)
x y Zz
where O denotes the order symbol. In addition, we
obtain from (19) and (21)
ap*
wie | Se + O(e*) ,
2=€f£ (x,y) =)
(27)
ap* oi 2
al = 6 7 Ole) 5 (28)
Y z=ef (x,y) Y z=0
ag*
*
2 | ay = | + o(e2) (29)
ca z=ef (x,y) z=0
Hence, by substituting (26)~(29) into (14), we can
obtain
*
do
== = 0) sin SG (30)
dz s
z=0
Further, for the boundary conditions on the free
surface, the following equation can be obtained by
substituting (20)~(24) into (15):
* *
d¢1 91 A
bes so = al
y=0
And, in correspondence with the Eq. (16), the follow-
ing equation is also obtained:
*
os 9b. eal | = 0 (32)
ox Oy Oe rf ;
*
Hence, eliminating f, from (31) and (32), we can
obtain the boundary conditions on the free surface:
2k
3 $5
9x2
2 a> o* L oO gi
ag*
a 1
u2 9t2
+
U2 ay
= 0 (33)
i BeBe eo
Moreover, at infinity, boundary conditions are given
as follows by (17), (21) and (24):
A ee ee
Derivatives of $, > 0 when Vx2 + y2 + 2250, (34)
* —_——___——
en => © Winein (EOE ee a co 5 (35)
Now, let us denote the pressure of the flow field
in the towed condition and that in the self-propulsion
condition by pg(x,y,z) and Psp (*,¥,2;t) respectively.
By substituting (20) and (21) into Bernoulli's
expression, we can obtain
ag
Ps 1
SS = Ce 2
ae gy € ax OMES) i, (36)
) * *
Psp _ - -ey( vy paki) = a + O(e2
Or SY ox ox at (er) a (S)
where 9, represents fluid density. Hence, the
pressure change, i)(x,y,z;t), due to the interaction
of the hull and propeller is given by the following
equation:
1
a = a
ig if
This equation shows that the magnitude of Wy is of
( aor a)
sp ir P,) Geis Cae at /- (Ss)
251
the order of € Moreover, W/P. can be considered
as an acceleration potential as is obvious from the
relationship with $y:
Finally, we proceed to convert Eqs. (25),(30),
(33), (34), and (35) for b} to equations for \) by
using the relationship (38). Using (25) and (38),
y must satisfy the following equation:
a2y . a2p a2
Pal BY Bae 7S (22)
On the hull surface, So’ we can obtain from (26)
the following equation:
3
in | eae gona cil
on s y °y < on S
s s
25 a | + O(e7) in sé (40)
z=0
On the other hand, if (x,y) is a point on See the
following equation can be obtained from (30) and
(38):
a a 362
1
Chie
3 F) o*
-c +5): pew (cae alam ©
*
Thus, the hull surface condition for $, can be con-
verted to that for | as follows:
(41)
(42)
Similarly, the free surface condition (33) for )*
can be converted to that for W as follows:
2 2 2 3
a*w my Bo ha) Peal oa) a & v = 6 (43)
9x? U dxdt Ul dts Udy y=0
Moreover, for the boundary condition at infinity,
the following equation is given from (34) and (35):
W > O when Yx2 + y2 + 22 30. (44)
Integral Equation
We proceed to seek the solution of which is the
harmonic function in the region bounded above by
the plane y=0 and elsewhere by the hull surface and
satisfies boundary conditions (42), (43), and (44).
At first, we separate the solution into the two
parts and write it as follows:
W(x,y,Z;t) = V(x,y,z;t) + W(x,y,z;t), (45)
where both V and W are the harmonic functions in
the region as indicated above. Moreover, let W
represent the pressure induced by a rotating pro-
peller moving straight ahead with a constant speed
in still water and a free surface. Now, we have
252
many formulas for W*(x,y,z;t) which represents (52), (53), (55), and (56) with the method of Green's
the pressure induced by an N-bladed propeller moving function:
in infinite space. One of the formulas for W* is
given on the assumption of thin blades as follows ay (2) gy (2)
[Jakobs et al. (1972) ]: (e) ss ( ¥ Y )
anv *°" (Q,) = eS) (Sa 2 a) @ & (OpO5)
. 2 JvNt ns
Wty mW (a y771Z) ae
v=0 v y(t)
+(V
en ‘ ee (ans ‘ v ) 5am, i, a, (2; Q ) 5 (7)
= 25 1 fas Da L.(E",6,89)e> ese se at ;
v=0 47 a q=1 =0 rR where Q, denotes a point outside Ss, Q denotes a
e p Pp (46) point on S,, and the suffix, (i), means the inside
where 8g = Zu (GN) (47) of the hull surface. Then, we seek a solution of
N vy, (2) which satisfies the boundary conditon on S.,
as follows:
with j = imaginary unit, 9% = angular speed of the gy (2)
= 3W
propeller, Sp = lifting surface of propeller, L)' = v | eek | (58)
pressure jump across Sp, (&',0, 89) = point on Sp, dn Bie on Sats
is = normal unit vector at Spr and R = distance s s
between (&',9,89) and (x,y,z). Hence, using W* and ; . ‘
the method of the mirror image, we can obtain a W Then, using (53) and (58), we can obtain an internal
which satisfies the boundary conditions (43) and solution as follows:
(44). Then, we can write W as follows:
we? Sth, « (59)
© }V.
evince) See (48)
Therefore, by substituting (51), (58), and (59)
5 g e :
Next, let us consider V. Then, we assume that into (57), the external solution WS must satisfy
Vv and can be developed in correspondence with the
development (48) as indicated below: ( (e) ) 9 /
ATV (e) = pees Ts 4
ay ey) GSA aon ie ang a, (0:06 )
ya 8 ch Cegrpaen (49) S (60)
veo" *r¥s x eS
co JUL F 4 . 2
V = LOVy (ery, Ze & (50) Finally, we have the following equation by adding
v= B
anw, (Q.) to both sides of the Eq. (60):
Hence, using (42) and (44), we have
(e) az (2) (oy eelton
av, (e) | OW, a Amp (Q.) St MMO) ap ds (2) on5 ) (Q79,) :
22 on g Coons 8. (61)
Ss Ss
7 eS Olwhen Woe v2 pecs (52) In this equation, letting Q, be the limit of Q, on
oa we can get
where the suffix, (e), means the outside of the
hull surface. In the same manner, from Eq. (43), (e) 1 )
( ares da (e) Sees :
we have YS) to8) 20 S YW) 6 G,, (0790)
ONG, av, ov ee
+ = + Ka 2 i W, | = 05 (53)
gx? dy ox =0
= 2W (Q.) 62
where yee 2 ce
: 202
Cia i 2gwk -_ _ EV
Kos ee BS) = pO 5 : (54) because the singularity of first order exists in
U W U the ey Se chis Wle) (9 ) is exactly the change of
Now, we suppose that we know the functions G (&, pressure on the hull surface caused by the propeller
TVA ECAVIZ) ie U= Ole 2 een ) such that the Gare which we intend to calculate. If W. and G, can be
harmonic functions for n<O except at (x,y,z) where given a priori, Eq.(62) can be considered to be
G have a singularity of first order, and G., satisfy * an integral equation for the unknown )(e) (Q5)- Thus,
he boundary conditions: the problem of calculating the change of pressure i
on a hull surface caused by a propeller changes to
once dG 0G the problem of solving an integral equation.
Vv
+ Ko— + Ki— + KG =o, (55)
ae? an 0g ws
Vie Time-Independent Change of Pressure On the Hull
As 2 2 Zip
S,) Suen Vx Pe ihe ame are Meare KC (56) Now, we proceed to give Wo and Go for a steady case
(v=0). Go(&,n,0;x,y,2) can be written as follows
Then, we can obtain the following equation by using based on a wave making theory:
1
V(E-x)2 + (n-y)2 + (G-2)?
Go (EN, Gi X,Y 12) =
T
— foe)
2
ie k exp (k nty + ikp') (63)
T ae Aye Bees ss inn ee
k - Kgsec-6
= 0
2
where p' = (&-x) cos @ + (t-z) sin 0. (64)
We can get Wo by using Eq. (46) as follows. The
first step is to rewrite the integrated term in the
right side of Eq. (46) by the transformation
pe Ey EOL lay (65)
n
Then, using the rewritten expression, we can obtain
Wo as follows:
Wo(x,y,Z) = Wa (x,y 2) ae (66)
It should be understood from the above expla-
nation that L)' must be given to calculate Wo. In
order to obtain L)' precisely, we must consider the
boundary conditions on the propeller surface which
have been disregarded in the discussion up to this
step. To do so, however, requires complicated
calculations as seen in the conventional methods
for the problems of the hull-propeller interaction.
The complexity of the calculations have caused the
conventional methods to be impractical as described
in the Section 1. Hence, the author introduces the
following approximation. The steady change of
pressure on a hull surface which we are now examining
corresponds to a pressure component of thrust de-
duction. We can consider that obtaining the thrust
deduction is the same as obtaining pressure on the
hull surface as a percentage of the mean propeller
thrust, To. Hence, the relationship between un-
known Lg' and known Tg can be given as follows:
N
- 2 J esti eGo) I = @o > (67)
q=l es
Ss
Pp
where [ |]. denotes the component in x direction.
Now, the L,' cin be considered as the jump of the
pressure change across the propeller surface due
to the interaction of the hull and propeller, and
consequently, Eq. (67) may be considered as the
approximate boundary condition on the propeller
surface for vfey(g.). By giving an arbitrary
function, L,', which satisfies the auxiliary Eq. (67)
and calculating Wo by (46), (65), and (66), we can
solve the integral equation, (62). This is the
approximate calculation method proposed in this
paper.
253
Numerical Procedure
The purpose of this section is to describe the
numerical procedure for the method explained in
the previous section. Here, for convenience' sake,
let us denote §@) in (62) by wr.
Numerical Calculation
The integral equation, (62) is an integral equa-
tion of Fredholm type of the 2nd kind. Generally,
it is impossible to obtain analytic solutions of
the integral equation for S_ in an arbitrary form.
Thus, various approximation methods have been
suggested. In this paper, a definite integral is
approximated by a finite sum, the equation is con-
verted to a linear equation, and this equation is
solved numerically.
At first, the following approximations are used:
(i) A hull in an arbitrary form is replaced by a
polyhedron. The form of each surface named
"element" is a plane quadrilateral.
(ii) On each element, the unknown function Va (Or)
is assumed to be constant.
Using this approximation, the continuous function
VEMOR) is replaced by the discrete quantities, vF
(UST Di te rcketers , M), for the total number, M, of the
elements. A control point, Q , where Wo (Q_) must
be calculated, is selected fof each element. Thus,
we have the following transformation:
‘ pee ¥ 3G,
ds os Seeds Oar v, CB ang? iQue (68)
Ss aL element
where ase no, and Q' denote values on the elements.
The definite integral in the right side of this
equation is an influence function from point Q to
point Q 5 and we denote this function by Ag ,Q-
On calculating Ag,,Q, the existence of a singular
point, a so called doublet, becomes a problem. How-
ever, there are many numerical calculation methods
for this case. In this paper, the Hess-Smith method
is used [Hess and Smith (1967)]. Further, selection
of a control point is also a problem. However, for
this problem various methods have also been suggested
in the analysis of potential flow field. In this
paper, each element is selected to be similar toa
rectangle, and the point of intersection of its
diagonal lines is employed as the control point.
Finally, the hull surface after St.11/2 is taken
into consideration, and it is divided more nar-
rowly near stern in the longitudinal direction and
approximately equally in the depth direction.
Thus, each element, Aj, i'(i,i'= pera c.ccghy) 4
which corresponds to Ag,,Q can be calculated and
Wo (Qo), can be calculated for each control point.
Then, the integral equation of unknown function,
W*(Q_), is converted to a linear equation of un-
known, i*-
Now, in the calculation of Wo(Q_), the author
uses the approximation that the number of propeller
blades is infinite. Then, in correspondence with
(46), (47), (65), and ((6), we can get the following
relations:
254
ro PAL
. 1 3 ~
Wo(Q) =, race dr de@n?r Ce) anne ,
(69)
rE ie}
Go (rv, 93%,¥7Z) = Go(xp, xr cos@ -f, sinO; x,y,z), (70)
(71)
where H represents a mean pitch of the propeller
blade, and r, and rg represent respectively radius
of the boss and radius of the propeller. Moreover,
['(r,8) represents the thrust per unit length in the
radial direction of the propeller blade elements
and can be developed as follows:
P(e,8) = 42 Tie . (72)
We can also get the following equation in correspon-
dence with the Eq. -(67):
a2
°
To = dr NI'p (x) 6 (73)
J i
B
Further, for the calculation of Wo in Eq. (69), it
is approximated that I) is an elliptic distribution
against r, and [,, To ....are disregarded.
Examples
The numerical calculations are performed in the
case of two combinations of the hull and propeller
shown in Table 1. Figure 2 shows the body plan of
hulls. In order to examine the correctness of the
TABLE 1 Particulars and Operating Condition
U
SHIP Lop He B/T Cc D Z Ba Ou
L 6-00) (650) 92/586 SBW2 oS S 2505 oAo7 Oa'55
a 7.00 6.00 2.63 BYE) 52l@ aboA7 ailSs) Sos
Lpp = Length between perpendiculars (meter), B =
Breadth,
T = Draft at mid-section, Cp = Block coefficient,
D = Propeller diameter (meter), Z = Number of
propeller blades,
U = Ship speed (meter/second), Fn = Froude number,
nq = Propeller's number of revolution per second.
Ne
T ship L ship
FIGURE 2. Body plan.
approximation used in the calculation of Wo, the
procedure as follows is performed. First, perfor-
mance of the propeller in the nominal wake is calcu-
lated to obtain IT), and L,~. Next, by using the
five combinations of distribution forms of I), L)~
and the number of the propeller blades as follows:
(al) SN Eeimsktey asain Geli hy lly iereen beget Fy ltge eatin (SS)
(03) INP alweiotiee, meshing Waplng goss , IT7 in (69)
(c) N; finite, using Lo' only in (66)
(d) N; infinite, using Ig only in (69)
(e) N; infinite, using Ip: elliptic in (69) ,
The Wp are calculated. Then, by substituting these
Wo in (62), the pressure changes, W*, are calculated
and indicated in a non-dimensional form in Figure 3.
As shown in Figure 3, the * barely differ due to
the distribution form of I!,L', and the number of
propeller blades. Hence, the approximation of the
elliptic distribution is reasonable.
Experiment
The experiment was performed at the towing tank of
IHI by applying a standard hull surface pressure
measurement [Namimatsu, (1976)]. For the ships
indicated in Tdble 1, pressures on the hull surface
are measured under both the towed and the self-
propulsion condition. Differences of the measured
pressure between the towed and the self-propulsion
condition are used for the experimental values of
the pressure change caused by the propeller.
Figure 4 shows the comparison of the experimental
values to the calculated values, which are obtained
by approximating Ip as the elliptic distribution.
In addition, Table 2 shows the pressure component,
tor of the thrust deduction fraction, t, which is
the sum of the pressure change. The comparison in-
dicates better agreement for the L ship (a thinner
ship).
Discussion
The calculation method in this paper is derived by
expressing the equations and boundary conditions
(which determine the change of the flow field due
to the interaction of the hull and propeller) in
the form of an acceleration potential. For this
reason, this method nominally requires calculations
of pressures induced by the hull and propeller,
while the conventional methods, which express flow
fields in the form of a velocity potential, require
L ship
y=|.-61.3
mn
150mm 100 50 aCp =276.9
I
Mg
2
I
elliptic
p*
Pe
ACp=
100
Zz
150 mm 50
(y,z)=coordinate of a point on hull surface U(ship speed)=2.05m/sec
FIGURE 3.
calculations of pressures and velocities induced by
the hull and propeller. Generally, the calculations
of induced pressure require less time in comparison
with the calculations of induced velocity. Thus,
when the present method is used, the time required
for numerical calculations can be reduced to a
practical value. This method can also be applied
for the calculation of propeller-induced surface
forces [Ishida, (1975) ].
It is anticipated that the results derived by
this method may be worse as the calculation point
moves closer to the stern, because, in this method,
the assumption of a thin hull is used, propeller
boundary conditions are simplified, and the rudder
is disregarded. When the actual experimental values
are examined, it seems that the anticipation may be
correct. However, it iS more appropriate to con-
sider that the majority of the error is due to the
fact that the flow field around the hull is assumed
to be inviscid.
4. WAKE ENERGY RECOVERY BY A PROPELLER
A towed hull pulls still water forward, but when
the hull is self-propelled, the propeller acceler-
ates this forward flow toward the back, and thus,
the propeller recovers wake energy. Hence, it is
important for the improvement of propulsion effi-
ciency of a ship to know how the wake energy can be
recovered effectively. The present, self-propulsion
test method can give information for the wake energy
recovery as a propulsion factor. This method is,
however, insufficient to tell us how wake energy
255
T ship 4c,
y= [£139.8] mm =326. “4
Ye erate LEON, 5
200mm 150 100 50 :
(E186-5]
-0.1
pal
———_ + + 0
200mm 150 100 50
AC
P
(233-2) © 200mm 150
-0.2 39555 -0.3
2 + =
200mm 150 100 50
(y,z)= coordinate of a point on hull surface U(ship speed)=1.27m/sec
Numerical calculation of pressure change on a hull.
should be effectively recovered. This is due to
the fact that the balance of force is a basic prin-
ciple of analysis in the method, in which the balance
of energy is not given sufficient consideration,
and further, because almost no information on the
flow field can be given. To cover the fault of the
self-propulsion test method, a knowledge of the
overall flow field is necessary and the distribution
In the vicinity
of energy in the flow must be found.
of the propeller, however, the flow field is so
complicated that experimental measurement and
theoretical analysis are difficult.
consider, as a practical approximation, an attempt
to estimate wake energy recovery by a propeller
through an analysis of the wake at a position far
from the propeller.
In the next section, the phenomena of the inter-
action in a distant wake are analyzed by the use
of Oseen's approximation to determine under what
Hence, we might
TABLE 2 Thrust Deduction Fraction
SHIP t t ie
Pp Pp
L .- 166 .140 -109
ay oOul .-160 -200
t,, is obtained from pressure measurement.
tp* is calculated by present method.
256
L ship
* 150 mm
[E184.5]
CALCULATION
=o EXPERIMENT
3 150 mm 100 SO 0
(y,2)= coordinate of a point on hull surface U(ship speed)=2.05m/sec
FIGURE 4.
conditions the wake energy is effectively recovered
by the propeller.
Fundamental Equation
In this section, we assume that a ship is stationary
in a uniform flow of speed U. We proceed to examine
the balances of force and energy between the ship
and the flow field.
Now, for the surfaces where force and energy are
surveyed, we define six rectangular cross-sections
in addition to the hull and propeller surfaces.
These six rectangular cross-sections are indicated
in Figure 5. Two vertical planes are in right angle
to the direction of the uniform flow at the front
and rear of the hull. The free surface and the
bottom of the water are held between the two vertical
planes, and two more vertical planes are parallel
to the uniform flow at infinite distances to the
right and left of the hull. Further for simplicity,
we assume that the flow field is independent of
time even if a propeller exists and a coefficient
of diffusion, Ue, due to viscosity or turbulent
flow is constant. Moreover, notations used here
have the same meaning as those in Section 3.
At first, let us examine the input and output
of momentum at the individual surveyed surface in
the towed condition. Then, as a result, the total
resistance, R_, can be given by the integration on
the rectangular cross section, S., in the rear of
the hull as follows: *
T ship
y= [E1328] nm
*700 mm150 100
=186.5
“200 mm 150
=0.2 37305 023)
Zz 0 gal
200 mm 150
ACh
-0.4 Ss
-279.9 ———————
200 mm 150
-0.3
Zz
200 mm 150 100
(y,z)=coordinate of a point on hull surface U(ship speed)=1.27m/sec
Comparison between calculated and experimental value.
du
ds[po-p+2u, ae a
u_(Ut+u_) |
s s
(74)
b
aL 2
+ > Pd AZEGSi,
-b
where u, v, and w represent x-, y-, and z-components
of disturbance velocity and b represents the half
width of S. at the free surface. Further, po repre-
sents the pressure at x = -®. Moreover, when the
energy balance is examined, kinetic energy lost
when the uniform flow passes along the hull must
be equal to the sum of the energy dissipated to the
outside through the surveyed surfaces by heat and
work. Thus, we can obtain the equation as follows:
p 2
£ 2 —=—= 2 2
= ds[u* - (Utug + v_ + w.)](U + u_)
on
= av @(@) = ds {pop or)
Vv Ss
it [ du.
re a(u + 7) Ea
(Ge du 4
tig pe
2)
where V. denotes the flow field surrounded by the
Survey Surfaces, and
; dus) 2 av, \2 aw.) 2
® (e) = ue (—*) + =) + (5)
aw av.) 2 ow ane
(52 +3) (yet ae
(2s vs |
Vp EET
By using (74) and (75), the effective horsepower,
EHP, can be expressed as follows:
p
iis d(e) + cau dS (ug+vgtw2) (U+ug)
Sp
WE
(75)
N
(76)
EHP =
UP 5
Ps Gigs dS ug (Pg-Po)
=) Sa
jh du. (Se ou.
a Ws 4 dS |2u, cra a? Wes <a ay )
A
(ts **s)
qr Ws Deas a 6 (77)
The first term on the right side expresses heat
energy, the second term expresses the increase of
kinetic energy, the third term expresses the in-
crease of potential energy, and the fourth and fifth
terms express work toward the outside of Vs. This
equation, (77) gives the work, EHP, transmitted
to the fluid through the hull when the hull is
towed in still water.
Next, the self-propulsion condition can be
considered in the same manner as the towed condition.
The equation for the balance of forces is as follows:
du
= Bis
AR = ds E Days 5 P gu iy (OR )
or
b
i 2
7 a PS dz Crs! (78)
-b
where the subscript sp denotes the self-propulsion
condition and AR represents the skin friction
correction which is used for the ordinary propulsion
test at the towing tank. When the energy balance
is considered, we can get the following equation:
FIGURE 5. Survey surfaces.
DHP -{ dv oe (e)
Wee
Pe
-=— as |u2 =\ uraget + v2_ + w2 (Utu__)
2 s sp s sp
Sa
ie oe (Utu__)
Sa
Ons gives Usp
ar Be BUG) 5 + ul x oe )
3
" (2 P “sp ) (79)
sp ax az
where
du 2 av 2 ow 2
= LG) (Ge) Ge)
O55) = E ( re 12 ae
ee 262) @e: 53)?
Ne! OBE UAE ky TE
XG sp)?
a3 ( ox Oz (So)
Then, using
equation:
DHP + UAR =
Vv
p
~ ds(u2_ +
2 Ss
iS)
A
b
Up .g
pet fe ce +
sp
-b
dv © (e)
sp
ie
Cb OF VW 2 m_))
sp
ds Bap Bao 2)
Sa
PST
(78) and (79), we can get the following
ae Os ;
+a. ( ax -] 22) ji (o7))
This equation reveals that the work transmitted to
the fluid by the ship moving in still water with a
constant speed, U (sum of the delivered horsepower
and the work UAR caused by skin friction correction),
changes in the fluid and is dissipated as heat,
kinetic energy, potential energy, and work through
the surface si:
Oseen's Approximation and Problem of Variations
We assume that the hull is thin and S_ is placed
sufficiently far behind the hull. Then, the inte-
grations on S which appear in the right side of
the Eqs. (74), (77), (78), and (81) can be approx-
imated as indicated in the Appendix. Hence, the
following equations can be obtained:
WwW
2 2 2
Oe eee
ae om ds ay x ox
Sa bY
b
Og) 5
a if dztc, (82)
—b
AR = Pd J estes - 4H ey
Gd)
Pe i (Es) ( ‘sp )? (Es)
+ ds + =
2 S dy dz ox
A
b
Pg 2
rn £ dz csp , (83)
7
-b
where Ho, H_, and He represent the total head as
follows: P
Po u2
Hn = — + ay (84
0 Pd Y 2g )
B 2
HO ==—+y += (UFO + v2 + w*) A (85)
s [oye s
£
12) 2
= SS sy Ee (U+u vanes we) (86)
Further, W represents the sectional area in which
Ho-H is not equal to zero at S,-
And,
Po
EHP = av ® (e) + aS(Ho - H_)?
Ss Ss
V, mn)
pies o (eye Hee) (“s\?
2 dy az ax
Sa
b
P-gU
4 azine (87)
2 s
-b
Po 2
DHP + UAR = av a) + aS(Hjp - H_)
Ww
Ve
U a 3 3
lee e (ea (Gen) ( ‘ep)?|
2 ay az ax
ox
b
-JU 5
a) dz Usp (88)
-b
In the Eqs. (82) and (83) for the balance of force,
the forces Rt and AR, given to the fluid from the
outside are divided into the force related to the
viscosity expressed by the first term and the force
related to the wave making expressed by the second
and third terms. In Eqs. (87) and (88) for the
balance of energy, the energies EHP and DHP + UAR
given to the fluid from the outside are independently
divided into the first and second terms which repre-
sent the energy related to viscosity and into the
third term and the fourth term which represent the
energy related to wave making.
Now, using (87) and (88) which show that the
viscous energy and the potential energy are indepen-
dent of each other, it is obvious that the condition
for minimizing the viscous energy in (88) is a
necessary condition for minimizing the DHP. We
proceed, therefore, to obtain the minimum condition
of the viscous energy which corresponds to the
optimum condition for the energy recovery by the
propeller. For this discussion, we assume that in
the right side of Eq. (88), the first and second
terms change independently or that the increase
and decrease of the second term have, at least, a
positive correlation with the increase and decrease
of the first term. Based on this assumption, let
us consider the conditions required in minimizing
the following function:
a (89)
Using (83), the following equation is obtained:
da = = - ; 90
PI S(Ho Bao AR Rg, (90)
W
where R_ denotes a wave making resistance under a
self-propulsion condition. This R_ might be
approximately equal to a wave making resistance
under the towed condition. Furthermore, AR can
also be given by the total resistance under the
towed condition. Hence, it can be considered that,
under the self-propulsion condition, the following
equation is given:
g dS(Hy) - H_) = C, (91)
sp
where C is constant and can be decided by the towed
condition. Thus, the problem of minimization of E
is converted to the problem of variations for
minimization of E given by (89) under the constraint
condition (91). It is obvious that the following
solution exists for the problem of variations:
Ho - H = constant. (92)
sp
Furthermore, although it is omitted here, at least
the conditions that the ship speed and displacement
are constant are implicitly required in addition
to this constraint condition.
Let us consider the meaning of Eq. (92). Since
Ho - Hs and Ho - Hsp are proportional to the viscous
wake in a position far from the hull as indicated in
the Appendix, (Ho - Hg) 2 and (Hg - Hsp) * are propor-
tional to the kinetic energy of the viscous wake.
Hence, the minimization of E corresponds to the
minimization of the kinetic energy of the viscous
wake. And, it can be considered that the condition
259
(92) is the condition for minimizing the kinetic
energy left in the wake by recovering the kinetic
energy of the viscous wake with the propeller.
The optimum condition for this energy recovery
is obtained under the assumption that the constant
C of Eq. (91) is given as the constant decided by
the towed condition. In other words, it is con-
sidered that condition (92) gives only the condition
for the propeller to accelerate flow effectively
under the assumption. If, however, the wave making
resistance is zero under a purely self-propulsion
condition, then (AR=0) C can be expressed as C=0
regardless of the towed condition. Therefore, it
can be considered that this fact indicates condition
(92) applies not only to the optimization of the
flow acceleration by the propeller but also to the
optimization of the hull-propeller combination for
effective recovery of the wake energy.
The author proceeds to examine the correctness
of this condition in the following sections by
using results of the self-propulsion tests and
wake survey measurements.
Experiment
Total head at a wake far from the hull was measured
at the towing tank of IHI. The measurements were
performed for the ships and operating conditions
indicated in the Table 1 under both the towed and
the self-propulsion conditions. The measurement
cross-sections which correspond to plane Sag were
three vertical cross-sections of 0.3Llpp, 0.5Lppr
and 0.7Lpp behind A.P. Figure 6 shows the total
head loss distribution of the towed condition in
the non-dimensional forms and also shows H,* which
is the change of total head loss by the propeller
action. Here, Hp* is obtained as follows:
Bie = (ig Cel.) iy = 1) 6 (93)
Pp sp s
We observe that in the towed condition the wake of
the T ship spreads to the relatively lower region
of the flow field. Further, we can see that the
peak of the total head distribution in the towed
condition agrees well with the peak of the change
distribution for the T ship, but not for the L
ship. In addition, Table 3 shows results of the
TABLE 3 Self-propulsion and Towed Test Data and
Wake Survey
Sieg 1 te
A Ww Ths te Rg, AR EE a
L 5287 collGG aml DoW $557 .@O8 Agia, 5.42 150
ae 18S .20 OS 1.58 8.92 ails@ 1660 So os’)
Ww = Effective wake, Re = Total resistance from towing
test (kg.),
Ry = Wave resistance from wave analysis at towed condi-
tion (kg.),
AR = Skin friction correction (kg.),
fs = ptg J dS(Ho-Hg) at 0.7 LIpp behind ship in towed
condition (kg.),
f£sp = p£9 J ds(Ho-Hgp) at 0.7 Lpp behind ship in self-
propulsion condition (kg.).
260
L ship
323.1mm ——-+
(A) 5 Zuni
—— TOWED CONDITION
SS=oss CHANGE OF TOTAL
U=2.05m/sec
ey Oo Sim
1
g(Hp-Hs) /50"
dt
es bee
gH, [ru
U=1.27m/sec
FIGURE 6. Total head
distribution far from
a ship.
self-propulstion test and the towing test, and
viscous resistances obtained from the wake survey.
Discussion
By analyzing the wake at a distant position behind
a ship, an estimate of the recovery of the wake
energy by the propeller is made, and the optimum
condition (92) is given. Table 3 shows that hull
MEASUREMENT SECTION
0.3Lpp behind A.P.
efficiency is better for the T ship than for the L
ship. Results of the self-propulsion test, therefore,
indicate that the energy recovery by the propeller
is better for the T ship. On the other hand, results
of wake survey measurement far from a ship indicate
that for the T ship, the peak of the head change
distribution agrees well with the peak of the head
distribution in the towed condition. Hence, it can
be considered that the propeller of the T ship makes
the wake flatter in order to adapt the conditon (92).
261
TOWED CONDITION
—--—-—- CHANGE OF TOTAL HEAD
U=2.05m/sec
583. 3mm
U=1.27m/sec
Thus, condition (92) is not contradictory to the
results of the self-propulsion test.
Bo CONCLUSION
From the theoretical and experimental studies for
the interaction of the hull and propeller, the
following conclusions are derived:
(i) Flow field in the vicinity of a hull is
MEASUREMENT SECTION
0.5Lpp behind A.P.
FIGURE 6. (continued).
analyzed by using acceleration potential, and the
approximate calculation method is derived. This
method can be used to calculate the change of
pressure on the hull and has a higher practical
applicability than conventional methods.
(ii) For the analysis of the wake at a distant
position behind a ship Oseen's approximation is
used, and the optimum condition is given for the
wake energy recovery by the propeller. This
condition is examined by the results of the self-
propulsion tests and the wake survey measurements.
|L ship
|
eee
of
N
~
| U=2.05m/sec
ess
| % Ss SSE ae
| TOWED CONDITION ~s.>>--=72-~7
Sa CHANGE OF TOTAL HEAD
T ship
— 443. 3mm—
FIGURE 6. (continued).
ACKNOWLEDGMENT
Before closing this paper, the author would like
to express his deep thanks to Prof. R. Yamazaki of
Kyushu University who kindly examined the contents.
The author also thanks Prof. T. Jinnaka and all
members of IHI Yokohama Ship Model Basin who pro-
vided him with kind guidance and cooperation.
MEASUREMENT SECTION
0.7Lpp behind A.P.
REFERENCES
Baba, E. (1969). Study on Separation of Ship
Resistance Component. J. of the Society of Naval
Architect of Japan, 125, 9.
Hess, J. L., and A. M. O. Smith (1967). Calculation
of Potential Flow about Arbitrary Bodies.
Progress in Aeronautical Science, 8.
Ishida, S. (1975). On an Approximate Calculus of
the Propeller-induced Surface Force. J. of the
Society of Naval Architect of Japan, 138, lll.
Jacobs, W. R., J. Mercier, and S. Tsakonas (1972).
Theory and Measurements of the Propeller-Induced
Vibratory Pressure Field. J. of Ship Research,
N55 Bp Mac.
Namimatsu, M. (1976).
Pressure Resistance and Its Application.
A Measuring Method of Hull
Wigs Che
APPENDIX
Let us examine the definite integral in Eqs. (74)
and (78) for the balance of force and the definite
integral in Eqs. (77) and (81) for the balance of
energy. At first, we denote these integrals by F
E
and Ee as follows:
= du
FE as [bo 2 ? Qe qa p-u(U + u)
&
Sa b
1 2
AP > Pg dz c 1 (94)
-b
Hy
(0)
WW
S| mo}
Hh
n
Q sy
Q
n
(as ™
(S
nN
+
<q
i)
+
<=
nN
Sq
+
ft
+
Ss
7
ue
oO
(95)
‘ Up _g B
du dv du dw du 8 2
a Ue E + aa + =) + o( 2 + 22) + =f aziGe.
-b
If the terms to which uw is related are assumed to
be small, Ee and Ea can be rewritten as follows:
p
= i me 2 2B
Fe = Pg dw (Ho 130) ar 5 das (v°+ w-u*)
Wy S,
0.9
+ = a Gy (96)
=b
p,U
sae = “2,9 dw- u(Hg - H) + Sam das (v>+ we-u-)
Ww - oN
epi tae éa Gg (97)
2
-b
263
the Society of Naval Architects of Japan, 139, 13.
Tsakonas, S., W. R. Jacobs, and M. R. Ali (1973).
An "Exact" Linear Lifting-Surface Theory for a
Marine Propeller in a Nonuniform Flow Field.
J. of Ship Research, 17, 4; 196.
Yamazaki, R., K. Nakatake, and K. Ueda (1972). On
the Propulsion Theory of Ships on Still Water.
Memoirs of the Faculty of Engineering, Kyushu
University, 31, No. 4.
where H represents total head as follows:
= 2
Fo a Gar ke eS . (98)
Pd 2g
Now, using Oseen's approximation, the following
relationship can be written:
Meee bw we Os Wn Wi SS Se db ity A (99)
ox az
where ~ represents velocity potential, and u', v',
and w' represent velocity components of rotational
motion which are zero at other than W. Then, pres-
sure, p, and wave height, C, can be expressed as
follows:
cate a6 (100)
= SPA! = WAU eeein
SBE 6 oy (101)
12)
where Tf, is due to a potential motion and [' is due
to a rotational motion.
Substituting (99), (100), and (101) into (96) and
(97), we can get
dz Gar
-b -b!
+ (Vo)? + ow | (102)
a
b b!
0 _gU 0 -gU
f 2 iB 2
+ d '
+ 5 dz Ss 5 s Cc
-b —b\
b'
auiees ime 8 ty (103)
+ p,gU dz SAC ds 2
=b' W
where
0 _U
£ 12 12 12 cy 9d
p ie)
al 2 OO 4 ies a Foe? Ae ROD A 02 GOD}
13> 19 19d
+ (its v2 swt) |, (104)
and b' represents the half width of W at the free
surface. If only the largest terms in W are kept
in the definite integral in Eqs. (102) and (103),
the following approximate equations can be obtained:
SN
b
Pg 2
+ — Ch (Gs) = (910 dw u' , (105)
2 £
-b
Ww
OU 2 2 2
£ at.) (22 &
————— a ae ae =
Le 2 5 \@ 4 az ox
Sx
b
Crake 2 2
a azinG + 0U) 6lit) mm!
2 is) £ (106)
=b W)
Since the following relationship is approximately
satisfied in a wake far from the hull:
eo Sy = GG 8), (107)
Eqs. (82), (83), (87), and (88) can be obtained from
(74), (77), (78), and (81) by substituting this
relation into (105) and (106).
Prediction Of the Velocity Field in
Way of the Ship Propeller
Igor A. Titov, Alexander F. Poostoshniy, and
Oleg P< Orilov,
Krylov Ship Research Institute
Leningrad, U.S.S.R.
ABSTRACT
The paper covers the problems involved in determin-
ing the velocity field in way of the ship propeller.
The analysis is given for both the structure of the
stern viscous flow and its change due to the ship
propeller operation.
The method is offered for scaling the nominal
field of axial velocities based on the use of both
the semi-empirical theory of the boundary layer and
theory of free turbulence, and the engineering method
of estimating the action of the working propeller
upon the velocity field.
As an illustration, the data of studying the
influences of the scale effect and the working ship
propeller upon the velocity distribution and total
wake flow are presented in reference to a moderate
displacement tanker.
1. INTRODUCTION
The need for a reliable definition of nonstationary
loads acting on the propeller blades and shafting,
and also of the intensity of hull vibration and
cavitation phenomenon, has placed the wake flow
problem among the most important problems of ship
hydromechanics in the last few years. Though this
problem first originated mainly in connection with
the building of large full ships, it is of no less
importance in the design of modern high speed con-
tainer ships and some other classes of ships. In
this sphere of hydromechanics shipbuilders are facing
two main problems: a) prediction of the velocity
field in way of the propeller for a ship of given
lines as based on geosim model test results and
b) finding solutions which provide a more favorable
distribution of the wake flow. The rationalized
formation of the afterbody wake is also one of the
possible reserves of ship propulsion which do not
yet appear to be fully realized.
At present, the problem of the afterbody wake
265
and particularly its prediction attracts the atten-
tion of a growing number of specialists in research
centers of the advanced shipbuilding nations in-
cluding the USSR. In view of the extreme complexity
of the afterbody flow pattern in the presence of the
propeller-induced disturbances, the problem of the
wake flow is still far from being solved. The laws
regulating the development of wake flow and also
the dependence of the velocity distributions at the
propeller disk upon the shape of the afterbody lines
are not quite clear. The test methods of defining
the ship model wakes and model-to-ship correlation
methods are as yet imperfect. Therefore the ac-
curacy of the flow nonuniformity data obtained in
way of the propeller and used as a basis for calcula-
tion of the abovementioned hydrodynamic character-
istics does not satisfy the requirements of modern
practice. Hence, a detailed investigation of this
phenomenon is needed.
In our opinion the most important tasks are as
follows: First, comprehensive physical studies of
the afterbody velocity field. These would allow for
better understanding and proper evaluation of the
effects of different factors on the formation of
wake flow in that region and help create a flow
model exhibiting the main features of the phenomenon
and capable of being investigated by analytical
methods. At this stage the theoretical studies are
essential primarily for a better understanding and
more proper analysis of the test results, as well
as for improving the general knowledge of both the
flow laws and the scheme of breaking the wake into
components. Second, the results of the experiment
and the qualitative theoretical conclusions should
be the basis for the development:
- methods for simulation of the nominal wake or
methods for theoretical estimation of the scale
effect at early stages of designing;
- methods for experimental definition of the
effective wake and approximate methods for the evalu-
ation of propeller effect using the nominal velocity
field data. Since the velocity field in way of the
266
propeller is normally defined in the idealized con-
ditions of the towing tank, it is absolutely neces-
sary to evaluate and take account of the effect of
operating conditions, i.e., the effect that increas-
ing the roughness of the hull surface as well as the
ship motions and drift have on the extent of flow
nonuniformity at the afterbody. There are also
some additional tasks, such as improvement of the
method used for definition of the ducted propeller
velocity field, estimation of a possible change in
the wake flow over the propeller axial length, and
thinking over the practicability of the methods of
disturbing action upon the flow pattern with preset
requirements. The methods of experimental defini-
tion of the flow velocities in the vicinity of the
hull model are no less important. It is impossible
to cover the results of all the above studies ina
short report like this, so we shall restrict our-
selves to the following traditional problems: the
scale effect of the velocity field and the propeller
effect on the flow formation at the stern.
2. SCALE EFFECT OF THE NOMINAL VELOCITY FIELD
The decrease of the mean wake in a model--ship
correlation with sufficient accuracy can be at-
tributed to variation in total frictional losses.
The problem of simulating the local wake is far
more complicated. The flow in way of the propeller
is a combination of two three-dimensional flows:
the boundary layer in the upper part of the after-
body with intensive secondary flows characteristic
of this region and the initial part of the wake de-
model - ship correlation, the approximate methods
of the semiempirical theory of turbulent boundary
layer and of the free turbulence theory are of
great importance; also important are comprehensive
physical investigations of the afterbody flow which
are necessary for the refinement of the flow model
and formulation of the simplifying assumptions.
Such investigations should cover the whole of the
viscous wake region (Figure 1 and 2) and not be
limited to the disk propeller area as is usually
done in practice.
The phenomenon being too complicated, a general
approach to simulating the flow seems to be unat-
tainable at present. Therefore, it is expedient
to discuss some particular models of the flow. Some
of the flows may be considered as the most common
types which can easily be investigated. These are:
a) the velocity field of a single-screw ship of
moderate fullness with V-shaped or U-shaped frames
where the contribution of bilge vortices is not
Significant;
b) the velocity field of high-speed, twin-screw
container ships;
a more complex pattern and more complex scaling laws
are characteristic for
c) the velocity field of full ships (6 > 0.8)
with U-shaped frames where the intensive bilge
vortices are formed;
d) the velocity field of the very full ships with
the boundary layer separation at the afterbody.
Model "a"
The calculation data obtained for a three-dimensional
boundary layer lead to the conclusion that with moder-
ate transverse flows the variation in characteristics
veloping behind the hull which may contain discrete
vortices resulting from the boundary layer separa-
tion in way of the bilge where the flow lines from
under the bottom are extending to hull side sur-
face (Figure 1 and 2). As shown by experiments,
the contribution of each of these factors depends
on afterbody fullness, stern frame form, buttock
angles, and some other parameters.
The distributions of the relative axial veloci-
ties Uy/Ys(y/é;Rn) are different for the boundary
layer, the wake, and the vortex effect region, and
largely depend on the afterbody lines and the
history of the flow. The solution of the scale
effect problem by a purely experimental way is not
practicable, so when the general laws of variation
in the flow characteristics are established for
of the main flow accounting to Rn does not differ
markedly from those obtained for a two-dimensional
boundary layer. Hence, for practical estimation of
the axial velocity field in the upper part of the
afterbody (Figure 1) we can use, without introduc-
ing large errors, the boundary layer correlation
schemes developed to fit the two-dimensional flow
on the basis of the logarithmic law and the velocity
defect law. For simulating the wake flow use can be
made, with some assumptions, of the known Prandtl
asymptotic solution for a two-dimensional flow
which was obtained on the assumption that the flow
is barotropic and that the velocity defect, AU, is
FIGURE 1. Nominal velocity
field in the propeller plane
for a model of tanker with
moderate block coefficient,
cy = 0.73 (model 1).
art
/
VS SWS WV LS &/
S SSS SNS SOS SS i
SS BSS SRS SS!
BSNS SSS SSS SSS
& lye
Fale ONS SN WSN SONS SSS
a ESSA SAS SSNS v4
7 =S
NN EN / >See
St AANA
APE ww My WAVE
267
SS
INN
. ae ~~> SS
ESR ESRISS
ae =
we Q S ‘
I? ~ ‘ \
peor |
)
ys
FIGURE 2. Nominal velocity
field in the propeller plane
of a "Krym"-type tanker model,
cy = 0.83 (model 2).
insignificant as compared to the velocity at the Such a scheme of simulation makes it possible
boundary of the wake flow: to take into account the variation in both the
wake thickness and the form of the nondimensional
Wr = Wi 4
— 6 ae L profile U,/Us.
a Us een (C,/A%) S ASP2) ) Model - ship correlation data for a tanker of
Bela wm * (in (2)
them eae D = 1
Z, = 2,/R = 0.875 Ae
where AX is the relative distance between the body 0.85 Pee oi
trailing edge and the wake flow section under study.
Naturally, these relations do not provide a reliable
qualitative definition of the flow characteristics x
at the initial part of the three-dimensional wake IS) Osh
which develops with the longitudinal pressure gra-
dient. However, the above relations are considered
to be quite suitable for simulating the wake field 014) ae
velocity because the deviations due to the effect ) 1
of some factors ignored here can be mutually com-
pensating. The practical method of correlation is
based on the assumption of a negligible effect of
the potential component and of a free streamline 0.8
flow around the hull. The effect that the varia-
tion of the transverse velocity component has upon
the axial flow with the increase in Rn is also con- x
sidered insignificant. The initial experimental
data for the model are defined in the Cartesian
system as velocity or wake distributions against
the transverse coordinate, y = y/L, with the dif- 0.4 i ay
ferent constant values of %. The coefficients, Kj, 0 1 2 3
and Kj, in Eqs. (1) and (2) are assumed to be (Y/L) X 10?
constant in the geosim horizontal sections of the =
Z. = 0.438
wake.
cn Se |
= if =
W Wa Cr (BNg) /Cag (RT, A aioe 37/0) const (3)
© © O— According to Equations (1)—(4)
@ @ @— Jaking Account of the Boundary
Layer Scale Effect
= ID
b. = bo/l, = Cc Rn C
s s/ 5 by ao ad Fo {En,,) (4)
where “i me att |
1 a2 3
Co = frictional resistance coefficient (Y/L) X 10
in two-dimensional flow;
b = width of the wake; FIGURE 3. Velocity distribution in wake extrapolated
WwW = frictional wake ship, model. to full scale.
268
medium displacement are shown in Figure 3 as an
illustration. Isotaches (lines U = const) plotted
in Figure 1 show that the upper part of the propel-
ler disk is in the hull boundary layer region and
here the flow contraction will take place almost
normal to the constant velocity lines rather than
to the longitudinal center plane. In this connec-
tion an attempt was made to evaluate the variation
of the flow velocities in the upper part of the
propeller disk using the approximate method re-
ported at the 13th ITTC, which provides quite a
good agreement with the full-scale test data, and
those obtained by calculation of the three-
dimensional boundary layer [Boltenko et al. (1972) ].
The results of the refined model--ship correlation
for this model within the propeller disk practically
coincide. Velocity deviations of 3-4% V are ob-
served only in the vicinity of the viscous wake
boundary in its upper sections (outside the propel-
ler disk), Figure 3. However, in some cases (e.g.,
with pronounced V-shaped afterbody frames) the hull
boundary layer can play a more significant role in
the formation of the wake flow, and in that case
its effect should additionally be taken into con-
sideration. Similar practical methods based on more
general assumptions with respect to regularities
in the variations of the axial velocities were given
by the towing tanks of Europe and Japan [Sasajima
and Tanaka (1966), Hoekstra (1977), Dyne (1974) J.
For comparison Figure 4 shows the model--ship cor-
relation results obtained by the Japanese method*
for some specific profiles of the wake of the model
under consideration. As is seen, this method leads
to a greater contraction of the wake in model--ship
correlation and does not take into account the varia-
tions of the velocity defect in the centerline plane.
However, apart from some limited regions in the
vicinity of 6 = 0° and 180° the circumferential dis-
tribution of axial velocities U,, (78) calculated by
both methods differs slightly (Figure 5). For the
above reasons substantial discrepancies in the
vicinity of 6 = 0° and 6 = 180° can give rise to an
appreciable change in the harmonic spectrum of the
field especially in the amplitudes of the even har-
monics.
At present it is difficult to find an acceptable
practical method of simulating the transverse ve-
locities, though the semiempirical theory indicates
the possibility of a noticeable scale effect of the
secondary flow velocities in the three-dimensional
boundary layer of the ship.
Model "b"
The flow nonuniformity in way of the propeller of
the twin-screw ship is mainly due to the hull bound-
ary layer and the additional loss of velocity in the
wake behind appendages
WV eaMO PF Se SAU Stls ES Oe Os
*The method of Japanese researches was used as described by
Dyen (1974).
il
According to
Equations (1)—(4)
---- According to
Sasajima and
Tanaka, 1966
(Y/L) X 102 (Y/L) X 102
FIGURE 4. Full scale wake predicted by different
methods.
where
mana potential component of the wake;
Fo; viscous wake due to the effect of the hull
boundary layer;
Aw. = additional losses of velocity in the wake
behind the appendages;
U = horizontal local velocity
U = horizontal local velocity in the "bare"
hull boundary layer.
The investigation of the wake scale effect for a
twin-screw ship, with a probable interaction between
the wake components, involves a number of complex
hydrodynamic problems. They include that of the
hull three-dimensional boundary layer, also the wake
behind the propeller shaft fairing placed at an
angle of attack to the flow inside the boundary
layer, in which case not only is the mean velocity
Vy (y) changed but also the extent and the scale of
the "outside" flow turbulence. Then there is also
the wake--boundary layer interaction problem and,
finally, oblique flow around the circular cylinder
(shaft) placed in the turbulent boundary layer.
Many of the above problems are concerned with some
insufficiently known aspects of hydrodynamics of
viscous fluid and, therefore, cannot be completely
solved for the present. As with the previous case,
approximation schemes can be used for practical
estimations. By way of illustration let us con-
sider the model--ship correlation data for a twin-
screw ship equipped with propeller-shaft fairings.
1D
1 — Model R,, = 1.3 x 107
2 — ShipR, =1.5X 10°
(According to Equations (1)—(4))
3 — ShipR, =1.5x 109
(According to Sasajima and Tanaka, 1966)
1.0 PP
See SSX 2
a y
0.8 va \
\ .
/ \
x
i) 0.6
FIGURE 5. Full scale circumferential velocity
distribution predicted by different methods.
The experiments show that, in the vicinity of the
heavily-loaded blade sections which are at a dis-
tance from the hub, the interaction between the
boundary layer and the wake behind the fairing can
be considered insignificant; the effect of support-
ing vortices at the fairing junction is also negli-
gible or not found at all because provision is
usually made for a smooth transition of the fairing
to the shaft body. This enables simulation of each
component of the viscous wake Wro(Rn) and AW, (Rn)
to be investigated separately with the total scale
effect to be determined by the method of superposi-
tions. Here it is expedient to make measurements
in the Cartesian system of coordinates as well. For
the model--ship correlation of the wake behind the
hull the method described by Boltenko et al. (1972)
is used. When simulating a component of the wake
AWp caused by the flow around appendages, use can
be made of the relationships of the free turbulence
theory (1) and (2). According to data of the flow
visualization, it can be considered with an accuracy
sufficient for practical purposes that the stream-
lines on the fairing are arranged equidistant to
the hull surface, and that in evaluating the scale
effect the strip theory can be used. Then
U —_&
AW. = A v a
RS Wom _HS Cre) Com icone $a) a const
Unm
y= Zi const (6)
b. = oe Coe (7)
269
where
C.. = coefficient of the fairing resistance at
section at a given distance ¥% from hull
surface (Figure 6);
Uh = velocity in the hull boundary layer at a
given distance % from its surface;
b = width of wake behind the fairing at the
propeller.
From the model-ship correlation data shown in Fig-
ure 6 it is seen that the flow nonuniformity varies
almost equally due to the scale effect of the hull
boundary layer and the wake behind the shaft fairing.
The mean circumferential axial wake is reduced ap-
proximately by one half.
Model "c"
The discrete vortices, which develop due to separa-
tion from the bilge, with their axes oriented in
the direction of the main flow may have, in some
cases, especially where the flow is around the U-
shaped stern frames, a noticeable effect on the
afterbody flow pattern. Generally there are two
vortices arranged symmetrically in relation to the
center plane; however, sometimes more complex vor-
tical systems can be observed in the flow around
full ships. The development of the bilge vortices
leads not only to redistribution of the tangential
velocities at the propeller, but to the additional
nonuniformity of the axial wake as well due to
a) redistribution of the velocities of the main
flow in the hull boundary layer and in the wake
behind the hull under the action of the vortex-
induced transverse velocities and
b) variation of the axial velocities in the
vortex turbulent cores, the transverse dimensions
Diagram
Z/L Gh)
4X/0
~)
NS
LLL
Model (R,, = 1.0 X 10’) Ship (R,, = 6.0 X 10°)
9° 6°
260 0 100
FIGURE 6. Scale effect estimates for nominal velocity
distribution at propeller of a twin-screw ship with
shafting fairings.
270
of which can be rather large as shown in Figure 2.
Thus the whole flow field containing bilge vor-
tices can be divided into three parts:
1) the region of turbulent core,
2) the region of vortex effect on the hull bound-
ary layer and
3) the region of nondisturbed flow in the bound-
ary layer or in the wake (Figure 7).
The laws for changing the relative velocities in
each of these regions are different in model-ship
correlation.
Evaluating the scale effect of disturbances in
the boundary layer is rather a complicated task
partly due to the difficulty of distinguishing these
disturbances in the nonuniform three-dimensional
boundary layer of the hull. Therefore, at the ini-
tial stages of investigation the principal attention
was paid to the specific features of such kind of
flow in simplified conditions, i.e., under the as-
sumption that artificial vortex systems were pro-
duced by means of profiles of small aspect ratio
at the boundary layer of a flat surface [Poostoshniy
(1975) ]. For such simpler flows one can use the
approximate methods of evaluating the scale effect
of axial velocities in the region where influence
of the vortex is observed. These methods will be
based on a combination of experiment and theory or
approximate semiempirical schemes, which is most
important for having a general idea of the phe-
nomenon. :
Extra losses of axial velocities in the vortex
cores are rather high for some ship models (reach-
U = constant
Region of Vortex Influence
. . ; SS
on the Boundary Layer ~— Y Undisturbed Flow Region
Vortex Core -/
Uo
7 Velocity Distribution
in the Core
Circulation Distribution in the Bilge Vortex
“Core of Tanker Model (4 = 60000 t), T= 1.1 m2/s
2.0 a
Distribution of Circulation in a Vortex
Core of Free Flow, To= 0.07 m*/s
=
1.0 2.0
Ig(r/r,) +1
FIGURE 7. Velocity field in the boundary layer with
longitudinal discrete vortices.
ing 20-30% of the mean wake value); these losses
are also to be studied in detail.
As shown by the experiments (Figure 6) the
circulation distribution law for the cores of bilge
vortices is similar to that for the vortex cores in
the free flow. So, in order to evaluate the scale
effect of a relative defect of the axial velocity
in the core, i.e., the core allowance, use can be
made of the theoretical relationships derived for
linear turbulent vortices.
Calculated results which are based upon rather
a small amount of data on the variation in eddy
viscosity coefficients with Rn obtained during model
tank tests and fall-scale hydrodynamic experiments
lead to the conclusion that a model-ship correlation
involves relative decrease of the core size. How-
ever, far from decreasing, the wake allowance, unlike
that for the boundary layer, may even be markedly
growing. Some additional variation in the distribu-
tion of axial velocities in the core caused by an
increase in Rn may also be due to an increase in the
longitudinal pressure gradient at the stern owing
to the reverse effect of the hull boundary layer on
the external potential flow both on model and ship.
It is impossible at present to develop a flow
model of this complexity, define the component ve-
locities changing under different model-ship corre-
lation laws and, finally, determine these laws; in
other words it is impossible to develop a well-
founded method for simulation of a three-dimensional
wake flow with discrete vortices. The results of
the above-mentioned preliminary studies are of
qualitative character and need experimental verifi-
cation. A series of comparative model and full-scale
tests carried out mainly by Japanese researchers
[Namimatsu and Muroaka (1973), Taniguchi and Fujita
(1969) ] confirm the existence of bilge vortices in
full-scale conditions as well, though the data re-
ported in the above papers are inadequate to judge
the quantitative aspect of the phenomenon. We can
only observe that the disturbances induced by the
vortices in the flow around a ship are less notice-
able, i.e., the flow is cleaned up. Therefore the
attempt to use a more generalized model (model "a")
seems to be justified also in this case, i.e., in
the presence of developed bilge vortices, or at
least an attempt to establish limits for the appli-
cation of this’ flow model should be made. Compara-
tive data obtained from model and full-scale tests
are a decisive factor here.
Unfortunately no data of nominal wake distribu-
tion at the propeller are available. For an indirect
evaluation of the scale effect of nominal wake we
shall make use of the test data obtained in Japan
for a 36000 t (displacement) tanker and its 1/37-
and 1/20-scale models [Taniguchi and Fujita (1969) ].
The measurements were taken in the boundary layer
near the sternpost at a distance of 1.1D from the
propeller disk. In laboratory conditions the ve-
locity field was measured both during the towing
tests and self-propelled tests. The tests performed
with the model (\ = 1:20) allow the propeller effect
at the measurement plane to be considered as negli-
gible (~0.05 V) and practically constant within the
region equivalent to the propeller disk area. The
comparison between the velocity distribution in the
wake transverse section for % = 8, (where %, =
propeller axis level) and the circumferential dis-
tribution of the axial velocities (Figures 8 and 9)
for this tanker and those for a "Krym"-type tanker
shows that the simplified method of model-ship cor-
271
3. PROPELLER EFFECT UPON THE WAKE DISTRIBUTION
Consideration of the wake scale effect when using
the nominal velocity field as initial data will not
always improve the agreement between the calcula-
tions and full-scale measurements of nonstationary
loads acting on the shafting and, particularly, of
the constant bending moment component defined by
the analysis of the first harmonic. Systematic
model basin test results indicate that signigicant
variations of the velocity distribution at the stern
may be due to the propeller performance. Several
factors are to be taken into account when analysing
(Y/L) X 102 the causes of this phenomenon. The most important
LWL among these are the propeller-induced acceleration
= of flow and, hence, the decrease of the layer thick-
(+) ness upstream, and the effect of propeller-induced
radial velocity in the immediate vicinity of the
propeller.
Thus it becomes necessary to investigate the
ship-hull boundary layer and the wake taking into
account the transverse pressure gradient. Semi-
empirical theories do not permit this problem to
be solved and are adequate only for the most ap-
proximate estimations of the flow history. There-
fore, just as in studying some features of the
nominal wake flow mentioned above, preliminary
theoretical investigations of the velocity field
under simplified conditions are of great importance
here. Although these results are not directly
applicable to the ship, they may be useful for a
better understanding of the main relationships of
0.8 1.6 2.4. the phenomena under study and for the devleopment
of practical methods to obtain the effective wake.
In this connection one cannot but mention the
important contribution of American scientists to
the investigation of the axisymmetrical problem,
particularly, the latest works by Huang and Cox
(CL) Dc
To obtain approximate estimates of the effective
“KRYM"-Type Tanker
© 0 0 —Model, Experiment
eer Correlation
——
0.8 1.6 2.4
Taniguchi and Fujita
Experiment, 1969
oO 0 O — Model, Experiment
SS Ship, Experiment
(Y/L) X 10?
FIGURE 8. Comparison of velocity distributions for
model and ship wake.
relation reveals the characteristic features of
variation in the velocity field and its harmonic
spectrum. However, these conclusions cannot be “KRYM"-Type Tanker Tanker, Taniguchi and Fujita
considered reliable enough; they need further veri- Experiment, 1969
fication.
Model "d"
Several years ago, simulation of the velocity field
in the case of afterbody boundary layer separation
attracted the special attention of researchers in
connection with the development of very large tankers
with high block coefficients and a tendency to de-
crease the length-to-breadth ratio. Although this
problem has lost its vitality by now, studies in
this field are being continued. The attempts in
Japan and in the Soviet Union to theoretically and
experimentally evaluate the scale effect of separa-
tion of three-dimensional and even two-dimensional
boundary layers do not yet allow any definite con-
clusions to be made, even regarding the qualitative
aspect of the phenomenon, or the development of the
most approximate scheme of variation with Rn number,
not only in the velocity distribution, but also in
the mean value of the wake. Thus the problem of N N
simulating the characteristics of flow at the stern
with the boundary layer separation remains one of FIGURE 9. Circumferential velocity distribution and
the unsolved problems in ship hydrodynamics. harmonic spectrum for model and ship.
a_ X 102
272
wake, both in our practice and in the practice of
other model tanks, use has been made in recent
years of engineering procedures based on the results
of nominal velocity field measurements and propeller
theory relationships [Hoekstra (1977), Raestad
(1972), Nagamatsu and Sasajima (1975) ].
If we assume that the propeller effects are
mainly due to the factors mentioned above, the
propeller can be thought of as having a large diam-
eter when evaluating the mean wake field.
This assumption will result in a decrease of the
wake coefficient. The decrease of the frictional-
resisted wake due to the propeller effect can be
taken as inversely proportional to the square root
of the diameter. Then,
- 4
Ware = Wey/ (lL + wi/2V,) (8)
where
Con = (pv2/2)F
To define the potential component Wpe alte) Sk}
reasonable to apply the known propeller theory
relationship
= am (fea/2)) €s V
Woe Won ( o/ ) (w/ )
to =
= iio yal as G = Al
WON 2 [ Th ] (9)
where j
W No experimentally defined potential component
P of the nominal wake field,
tj = thrust deduction at zero velocity of
model.
Allowing for the smallness of the 2nd term in (9),
the thrust deduction fraction undergoing only minor
changes can be assumed for single-screw ships to be
tg = 0.07-0.10 (the last figure relating to fuller
hull shapes) .
The final expression for the mean effective wake
field (taking into account the scale effect) has
the form,
We = eon tF EGQ/QGAL Cr - 1)]
nh
mh Won Cao (RM)
fo) NS ES eee
V1/2(vV7l + C. +1 CEA (Ene)
VW/2(vE+ Cy + 1) Spon (aif
where
Cro = frictional resistance coefficient in two-
dimensional flow.
Relationship (10) displays good agreement with
the model test data (see Table 1) and W, values
close to those obtained from the full-scale test
analysis.
As can be seen from the Table, all known approxi-
mate methods yield practically the same results.
By making some additional assumptions, similar
methods can also be applied for an approximate
estimation of the circumferential distribution of
the effective wake, and in the main they correctly
reflect the variation trends of the flow at the
stern while the propeller is in operation. However,
they do not permit: taking into account and evalu-
ating some qualitative changes in the hull boundary
layer, which may take place due to propeller opera-
tion, such as variation in circulation of bilge
vortices and their positions in relation to the
ship hull; the possibility of preventing or reduc-—
ing the separation about the stern zone with the
propeller in operation; and, on the other hand,
the possibility of the boundary layer separation in
the vicinity of the stern above the propeller.
Therefore, when performing a quantitative analysis
of the effect the propeller has on the wake and the
harmonic spectrum of the velocity field, these
methods, in spite of their relative simplicity and
convenience, should be applied rather carefully, as
for most tentative estimates.
At the present stage of the wake problem in-
vestigation the development of experimental methods
is of decisive importance.
Both for the improvement of the general knowledge
of propeller effects on the flow pattern at the
stern and for the solution of problems associated
with ship form design, the accumulation of data
on the effective velocity fields for ships of
various types and the improvement of model test
methods is of great importance, especially those
taking account propeller induced velocities or
eliminating the same from measurement data.
A practical method for estimating the effective
velocity field, Uy, by way of flow velocity mea-
surements at some distance ahead of the propeller
in "open water" and behind the hull, was given in
Titov and Otlesnov (1975). For measured data
analysis the quasi-steady theory was accepted.
When the hydrodynamic flow angle, 8), of a
propeller blade section for the propeller operating
in "open water" is equal to that behind the hull,
— ' = te " a
tgBy WE oF Wi) Jot (Ura Wi) / (wr Use) (11)
where
W' and W" = axial induced velocities ahead
= of the propeller in "open water"
and behind the hull
U = circumferential component of the effective
Be velocity field
The axial component of the effective velocity field
ahead of the propeller is determined from the
relation
Comparison of the Mean Effective Wake
Calculated by Approximate Methods With
That Obtained from Self-Propelled Tests
(Model No. 1)
TABLE 1.
Titov - Poostoshniy method 0.345
Nagamatsu - Sasajima method (1975) 0. 340
Roestad method (1972) 0.355
Self-propulsion test data 0.350
Nominal wake 0.390
Ne ——
Wa + wi) = Wl We. ar wi) (12)
following from the equality of forces on the pro-
peller blade section.
However, another approach to the problem of
experimental determination of the effective wake
is also possible based on the data analysis of
measured flow velocities and total head pressure
immediately ahead of the propeller and behind it.
In this case, measurements are taken only with the
propeller in operation behind the hull.
As is known, the circumferential induced velocity
at propeller section, Wg, in "open water" is pro-
portional to the jump in the total head at the pro-
peller disk
PUTW, (78) = Ho(t0) —- Hy (Té) (13)
It can be shown that this relationship is also
valid for the propeller behind the hull, if the
variation of the circumferential induced velocity
of the hull wake, Ug, is negligible within the axial
length of the propeller or between the sections
where measurements are taken. In this case total
head pressures at sections 1 and 2 (see Figure 10),
ahead of the propeller and behind it are, respec—
tively, equal to
p 2 2) 2
= +> + US, + U
in (ie) al 2 Oe 61 11!
p -2 2
8) = Po +> + (W + U
Ha Mus) 2% a LOL) (Woy 82)
+ (W_, + U_,)7] (ale)
T2 T2
where
Wh. = UES + Wo = axial flow velocity at
i i v section 1
us = UA + We = axial flow velocity at
2 2 2 section 2
We G Wo and Whe = propeller induced velocity
z components at respective
sections
Theoretical investigation results of propeller in-
duced velocities and test data make it possible to
linearly approximate component variations of the
induced velocity, W3(x), within the limits of the
propeller axial length. It is believed that the
axial component variation of the wake in this re-
gion is small and also obeys the linear law.
With the above assumptions, in order to determine
the design effective velocity, U at section X9
where the condition
xOU
a
WaCg) =
(15)
is observed, we obtain the following set of equa-
tions:
U Xo) + wo 7/2
x
ee EE SS
Eoen WT — wy /2 + Us ee)
W
a. W,/2 (17)
2 tgBy
273
0 90 180
FIGURE 10. Circumferential distribution of velocity
components in way of propeller (r = 0.590).
w
a
wi (.) Su (% ) +See
Ok 0? sa ) 2
U =. Gj
Teen ete (18)
= SSS OK
x} AX 1
where
8; = hydrodynamic flow angle of a propeller
blade section
AX = distance between sections 1 and 2
AXg, = Xo =) Xa = distance between section 1
and the point of calculation
In propeller theory it is generally taken that the
above condition is met at the propeller disk plane
corresponding to the midspan section of the blade,
and, in the case of blade rake, corresponding to
the midsection of the blade at a relative radius,
Te O.76
However the calculation results of variations in
the anomalous induced velocity, W,(X), of the pro-
peller with the finite axial length indicate that
in fact the point must be found upstream of the pro-
peller disc plane.
This conclusion is confirmed by the experimental
investigation results of the propeller velocity field
in open water. Taking account of these data it is
more reasonable to assume the point of calculation,
corresponding to condition (15), to be on the lead-
ing edge of the blade.
274
Application of this procedure can be illustrated
on a medium size tanker model (Model 1).
Experimental studies of the velocity field for
the operating propeller were performed during free-
running model tests with the operational relative
speed, Fn = V/VgL) 1 = 0.22. Wake characteristics
ahead of and behind the propeller were measured at
equal distances from the propeller centre with a
6-point probe [devised at our model tank, Otlesnov
(1969) ], which enables simultaneous measurements of
total head pressure, (H), static pressure, (P), and
flow angles in the horizontal and vertical planes
in the immediate vicinity of the propeller. When
processing the measured data and analysing the nom-
inal wake, use was made of calibration relationships
which took into account the interference of flow
angles in the vertical and horizontal planes with
the readings of the probe. Figures 10 and 11 il-
lustrate the initial data and the calculated induced
velocities for the starboard-side of the propeller
disk (right-hand rotation) in the region where
sections experience maximum loading.
Comparison (Figure 12) of the nominal velocity
field with the effective velocity field calculated
from Eqs. (11)-(12) and (13)-(18) shows the pro-
nounced effect the propeller has on the wake at the
lower part of the propeller disk and the minor ef-
fect at the upper part of the same. This may be
accounted for by a better possibility for momentum
exchange between the external flow and the viscous
wake under the action of radial induced velocities
in a relatively thin wake at the lower part of the
propeller disk, and a worse possibility at the upper
part where the thickness of the viscous wake is much
greater (see isotachs in Figure 1).
0 90 6° 180
FIGURE 11. Circumferential distribution of velocity
components in way of propeller (r = 0.756).
=)
0 90 6 180
6
1 — Nominal Field
2 — Effective Field (Titov and Otlesnov, 1975)
3 — Effective Field (Proposed Method)
4 — Effective Field (Hoekstra, 1977)
x
ID
FIGURE 12. Influence of propeller operation on
velocity distributions.
The above two methods for defining effective
field axial velocities yield results which, as a
whole, show satisfactory agreement. However there
are some systematic discrepancies in the regions
of @ » O° and 6 * 80-160°, and additional analysis
is required to explain these.
Besides, the velocity distribution data obtained
on the basis of measurements ahead of and behind
the propeller in operation make it possible to find
the thrust distribution (load coefficient of pro-
peller, Cm,,) over the propeller disk area
We ar Wy 2
Cr G8) GG ) are (19)
xe
Figure 13 and the equivalent system of singularities
Q(68) 5 Wa ZU (20)
In its turn, the knowledge of this system of singu-
larities allows one to calculate the induced ve-
locities over the total wake region ahead of the
propeller, and perform a more detailed analysis of
the effect the nonuniformity of load distribution
over the disk has on thrust deduction.
The following conclusions can be drawn from the
comparison of Fourier transform coefficients for
the circumferential distribution of axial velocities
of the nominal field obtained for the model and ship
90° sB
180°
Down
FIGURE 13. Load distribution over the propeller disk
(based on effective velocity field measurements) .
(model-ship correlation), as well as of the velocity
field in model tests taking account of propeller
effects. The amplitudes of harmonics determining
the nonstationary hydrodynamic forces and moments
(Figures 14 and 15) may vary several times under the
influence of the above factors.
It should be mentioned that no definite regular-
ity could be observed here. With some relative
radii the amplitudes increase, with others they
decrease.
As the variation in harmonic spectrum of the
velocity field is of rather a complicated nature
let us illustrate the effect the variation of axial
velocities due to scale effect and propeller opera-
tion has on the constant component of the hydro-
dynamic bending moment in the vertical plane which
is mainly defined by the first decomposition har-
monic [Voitkunskiy (1973) ];
e
M. = = oS
Yo 7 “30 ~ Pyo ° 2 (2)
where
i =
ey = IC, /4S _ ft[a,+(1/FT) (J+2K,/C))a,, lat (22)
TO
1
Po = -JCoS— f/t[b + (1/FT) (J+2 iT
70 2S—F/T[b 1+ (1/FT) ( Ko/C,)b, lat (23)
oa V,/nD
tT) = relative radius of propeller hub
KprKo = thrust and torque coefficients at
design speed
a,,b, = Fourier transform coefficients for
the cosines and sines of the first
harmonic of axial velocity on a given
radius
agg = Fourier transform coefficients for
the cosine and sine of the first
275
harmonic of tangential velocity on a
given radius
2 = coefficients
f = coefficient depending on radius
e = distance between the design propeller
shaft section and the propeller disk
The distributions of transverse relative ve-
locities Uo = Ug/V were taken as equal.
Table 2 shows the design estimates of relative
values of the constant component, Myg/Kg, as based
on various initial data.
As can be seen, the calculated results based on
the nominal velocity field data may differ (even
qualitatively) from those obtained with considera-
tion for the scale effect or the effect of operat-—
ing propeller. Although the local variations of
the nominal field due to the scale effect or pro-
peller operation are quantities of the same order
(see Figures 5 and 12), the constant component
values of the bending moment in the vertical plane
determined from the effective field prove to be
4-5 times as large. Physically this may be due to
the fact that, in contrast to the scale effect,
the propeller effect on the viscous flow in the
upper parts of the propeller disk differs from that
in the lower part. In the upper part of the disk
(8 = 0 - 90°) the effective field distribution of
velocities in way of the heavier loaded blade sec—
tions differs only slightly from the nominal field
distribution, while in its lower part (@ = 90-180°)
the effective field velocities are much in excess
of the nominal field velocities (by a factor of
1.5-2). This increases the asymmetry of circum-
ferential distribution of the effective field axial
}
[ Ship, Nominal Field (Correlation Based |
on Equations (1)—(4)) |
0.1 [
y- Model, Nominal Field
Model, Effective Field
(Hoekstra, 1977)
Model, Effective Field
(Proposed Method)
Model, Effective Field (According
to Titov and Otlesnov, 1975)
FIGURE 14. Influence of scale effect and propeller
operation on harmonic spectrum.
276
FIGURE 15. Influence of scale effect and propeller
operation on harmonic spectrum.
velocities and results in an increase of the con-
stant component of the moment in the vertical plane.
The Myo/Ko values calculated from the effective
velocity field approximate those observed for full-
scale ships of this type under operational condi-
tions. This fact confirms the importance of taking
into account propeller operation when simulating
the velocity field at the propeller. The propeller
effect upon the velocity field is dependent on the
load, ship hull form and afterbody shape, initial
nominal field, and the relationship between pro-
10 a,
10 a,
10 a,
Model, Nominal Field
Ship, Nominal Field (Correlation Based
on Equations (1)—(4))
a —
Model, Effective Field (Titov and Otlesnov, 1975)
Model, Effective Field (Proposed Method)
0.6 F
peller screw size and wake thickness, i-.e., on the
propeller immersion into the viscous wake.
The full-scale conditions of effective field
formation are likely to differ from the model ones.
Hence, the next step in studying the prediction of
the flow velocity field in way of the propeller will
be the development of procedures which enable simul-
taneous consideration of both the scale-effect and
the effect of propeller operation on the wake at
the stern.
TABLE 2. Variation in the Constant Component of Bending Moment Depending on the
Velocity Distribution at the Propeller (Model 1)
Model. Esti- Model. Model.
mation of Experiment Experiment
: Propeller Consideration Consideration
Model. Model-ship Effect of Propeller of Propeller
Experiment. Correlation According to Effect by Effect by
Initial Nominal Using Equa- Hoekstra Using Equa- Using Equa-
Data Field tions (1)-(4) (1977) tions) ((22)i=(@i2)) stafon's (dis) =1('8))
M_ /K 0.04 -0.07 -0.08 -0.35 =(0) 57355)
REFERENCES
Boltenko, V. P., O. P. Orlov, and A. F. Poostoshniy
(1972). Determination of the boundary layer
characteristics of ships from model test data,
13th ITTC, Material of Interest, Berlin/Hamburg.
Dyne, G. (1974). A study of the scale effect on
wake, propeller cavitation, and vibratory pres-—
sure at hull of two tanker models, Transactions
the Society of Naval Architects and Marine En-
gineers 82, 162-185.
Hoekstra, M. (1977). An investigation into the
effect of propeller hull interaction on the
structures of the wake field, Symposium "Hydro-
dynamics of Ship and Off-shore Propulsion Systems,"
Oslo, March.
Huang, T. T., and B. D. Cox (1977). Interaction
of afterbody boundary layer and propeller. Pro-
ceedings of Symposium of Hydrodynamics of Ship
and Offshore Propulsion Systems, Oslo, March
20 =2)5y.
Nagamatsu, T., and T. Sasajima (1975). Effect of
propeller suction on wake. Symposium on Pro-
pulsive Performance of High Block Ships, Tokyo,
June.
Namimatsu, M., and K. Muraoka (1973). Wake distribu-
277
tion of full form ship.
7a Sp Massa
Otlesnov, Yu. P. (1969). New means for measuring
unsteady characteristics of ship wake ina
towing tank. Proc. 12th ITTC, Rome.
Poostoshniy, A. F. (1975). Features of flow around
the hull and full hull resistance structure, in
Problems of Ship Applied Hydromechanics, I. A.
Titov, ed., Izd. Sudostroeynie, 54-92 (in Rus-
sian).
Raestad, A. E. (1972). Estimation of a marine pro-
peller's induced effects on the hull wake field-
IHI Engineering Review,
scale effects on hull wake field. Report No.
72-3-M, Det Norske Veritas.
Taniguchi, K., and T. Fujita (1969). Comparison
of velocity distribution in the boundary layer
of ship and model. Proc. 12th ITTC, Rome.
Titov, I. A., and Yu. P. Otlesnov (1975). Some
aspects of propeller-hull interaction. Swedish-
Soviet Propeller Symposium, Moscow.
Sasajima, H., and I. Tanaka (1966). On the estima-
tion of wake of ships. Proc. llth ITTC, SNAJ,
Tokyo.
Voitkunsky, Ya. I., R. Ya. Pershits, and I. A.
Titov (1973). A manual on ship theory. Sudo-
stroyenie, Leningrad. (In Russian).
Recent Theoretical and Experimental
Developments in the Prediction
of Propeller Induced Vibration
Forces on Nearby Boundaries
Bruce D. Cox and Edwin P. Rood
David W. Taylor Naval Ship Research and Development Center
Bethesda, Maryland
William S.
Vorus
University of Michigan
Ann Arbor, Michigan
John P. Breslin
Stevens Institute of Technology
Hoboken, New Jersey
ABSTRACT
This paper concerns recent advances in the theory
and numerical solution of propeller induced pressure
forces acting on ship hull surfaces. The analysis
is formulated in terms of the diffracted potential
flow about general three-dimensional hull boundaries
in the presence of a free surface. The influence
of the propeller is derived from lifting-surface
theory, explicitly accounting for finite blade
number, blade thickness and skew, and radial and
chordwise loading (steady and unsteady, but sub-
cavitating). Two methods have been developed to
calculate the periodic forces. In the direct
approach, time-dependent source singularities are
distributed over the body surface with the strengths
determined for a prescribed propeller onset flow.
The force is then found by applying the extended
Lagally theorem. In the second approach, based on
a special application of Green's theorem, the force
is obtained by finding the velocity potential at
the propeller generated by the boundary executing
simple oscillatory motions.
A towing tank experiment is described in which
blade frequency forces were measured on a body of
revolution adjacent to a propeller operating in
virtually uniform flow. The simplifications of
body shape and propeller loading provided a physical
model which could be treated in a reasonably exact
fashion by the theory. The body consisted of two
parts. A heavy afterbody, attached to the towing
strut, acted as a seismic mass at all but very low
frequencies. The forces were measured on a light,
rigid forebody supported from the afterbody by a
specially designed strain-gaged flexure assembly.
Tests with two propellers differing only in blade
thickness revealed the separate contributions of
blade loading and thickness and the results obtained
agree favorably with the analytical predictions.
278
1. INTRODUCTION
Propeller induced ship hull virbration continues to
be a major source of uncertainty and, indeed,
frustration to the naval architect. Today we witness
a trend toward larger and faster ships with higher
power being delivered to the propeller. These
designs are inherently more susceptible to propeller
related vibration problems, as has been learned
from bitter and usually costly experience and this
situation has focused renewed attention on the need
for improved methods to predict propeller exciting
forces - methods which are both reliable and practi-
cal for application during the design process.
Two distinct, but related types of propeller
exciting forces (and moments) produce hull vibration.
Unsteady blade loads developed by the propeller
operating in the nonuniform ship wake and trans-—
mitted to the hull directly through the propeller
shafting are termed bearing forces. Periodic
pressure forces acting on the surface of the
hull, arising from the propeller unsteady veloc-—
ity and pressure fields, are called surface forces.
Various approaches have been developed to predict
these forces from model tests. For example, bearing
forces are measured on a model propeller in a water
tunnel using wake screens to simulate the flow at
the ship stern. Surface pressures can be obtained
from measurements of transducers distributed over
the surface of the model hull afterbody. Alterna-
tively, the entire hull afterbody can be cantilevered
on a flexure assembly instrumented to measure the
total surface force [separated stern technique,
Stuntz et al. (1960)].
The foregoing experimental techniques, and
others [most notably Lewis (1969)], have proven to
be costly and difficult to carry out in practice.
Moreover, a large number of experiments would be
required to examine all the pertinent physical
parameters, including hull form, propeller clearances,
blade geometry and loading characteristics. Con-
sequently, researchers are attempting to develop
theories and numerical procedures for calculating
propeller exciting forces. An analytical approach
offers a means to economically evaluate competing
propeller-hull design concepts as well as to diagnose
at-sea vibration problems and identify corrective
measures.
The present paper concerns recent advances in
the theory for propeller induced surface forces.
A general three-dimensional boundary intercepting
the propeller disturbance field poses a formidable
diffraction problem. As a first step, it is
necessary to determine both the time-average and
unsteady loading on the propeller. All of the
components of loading, together with blade thickness,
contribute to the propeller induced flow impinging
on the hull and the resultant unsteady pressure.
Fortunately, as a result of much past work in the
analytical prediction of bearing forces, there now
exist powerful theoretical methods for calculating
unsteady propeller loading in a prescribed nonuniform
flow. The analysis rests on a lifting-surface
representation of the propeller, explicitly account-
ing for number of blades, radial and chordwise
distribution of loading, thickness, and skew. While
further refinements and improvements, such as the
prediction of transient blade surface cavitation,
are needed, the calculation of blade loading can
now be done with sufficient accuracy to address the
surface force analysis. Also, as these improvements
in the propeller calculation become available, they
can be incorporated into the surface force calcula-
tion without fundamental changes.
Previous analyses of the surface forces are
formulated in terms of the diffracted potential
flow about the solid boundary in the presence of
a given propeller onset flow. To facilitate the
analysis, it was necessary to introduce simplified
representations of both the propeller and the
boundary as outlined by Breslin (1962) and more
recently, Vorus (1974). For example, analytical
expressions for the vibratory force produced on a
long flat strip and a circular cylinder adjacent
a propeller in uniform flow were derived some years
ago [Tsakonas et al. (1962) and Breslin (1962)].
These investigations provided useful insights regard-
ing the importance of propeller tip clearance and num-
ber of blades. However, such approximate treatments
neglect what are now known to be certain essential
physics of the propeller-hull interaction. The net
force on a long boundary may be deceptively small
because of cancellation of large out-of-phase force
components developed fore and aft of the propeller.
On a hull which terminated in the immediate vicinity
of the propeller, such cancellation will not occur.
Also, the components of unsteady blade loading at
or near blade frequency can produce much larger
surface forces than those arising from the steady
loading and thickness. Components of blade loading
at higher frequencies, while relatively smaller in
amplitude, generate field pressures which decay
much more slowly, encompassing a large portion of
the hull afterbody and resulting in a significant
integrated force. For this same reason, an experi-
mental determination of the total surface force by
measurement of pressures at selected positions on
the hull boundary can be disastrously misleading.
In view of these circumstances, it is now generally
accepted that a satisfactory theory must represent
the hull boundary in a reasonably exact fashion,
279
accommodate the presence of the free surface, and
account for all constituents of propeller loading.
This paper sets forth a comprehensive theory
for propeller-hull interaction and describes proce-
dures for calculating the periodic forces acting
on the hull surface. The paper is divided into
five sections. In the first section, the problem
for the diffracted potential flow about the hull
is formulated, in which the propeller unsteady
disturbance is assumed to be of small amplitude
and high frequency. In keeping with the desire for
first order results, the high frequency linearized
free surface conditon applies. However, the zero
normal velocity condition is satisfied exactly at
the hull boundary. Formulae for the surface pres-
sures and forces may then be expressed in terms of
the propeller velocity potential and the unknown
diffraction potential. The following section deals
with the representation of the propeller. Dipole
singularities with strenths related to the blade
pressure loading and thickness are distributed over
helicoidal surfaces approximating the geometry of
the actual blade surfaces. Based on this model,
expressions for the field point velocity potential
arising from loading and thickness are developed.
Examination of these formulae and their asymptotic
behavior at large distances reveals important prop-
agation characteristics associated with the unsteady
blade loading components at and near blade frequency.
In the subsequent sections, two methods of solu-
tion are developed for determining the surface
forces. The direct approach consists of distributing
time-dependent source singularities over the hull
surface with the source strenths determined for a
prescribed propeller onset flow using a modified
Douglas-Neumann calculation [Hess and Smith (1964)].
The force on the body is then found by applying
the extended Lagally theorem to the hull singulari-
ties. In an alternative approach, based on a
special application of Green's theorem, the force
is obtained by finding the velocity potential at
the propeller produced by the hull boundary executing
simple oscillatory motion.
In the final section, a towing tank experiment
is described in which blade frequency forces were
measured on a body of revolution adjacent to a
propeller operating in uniform flow. The simplifi-
cations of body shape and propeller loading provided
a physical model which could be treated in a reason-
ably exact fashion by the theory. Despite these
simplifications, certain classical problems were
encountered in the design of the experiment including
the measurement of a relatively small force, avoid-
ance of system resonances in the frequency range of
interest, and retrieval of the force signal from
background noise. A two-part body design was
developed, similar in concept to the separated
stern technique mentioned earlier. A heavy after-
body attached to the towing strut, behaved as a
seismic mass at all but very low frequencies.
Forces were measured on a light rigid forebody,
supported from the afterbody by a specially designed
and dynamically calibrated straingaged flexure
assembly.
Tests were performed with two propellers differing
only in blade thickness in order to reveal the
separate contributions of loading and thickness.
The measured forces (amplitude and phase) were
obtained for a range of speeds and advance coeffi-
cients and for two positions of the propeller
relative to the test body. The results agree
280
favorably with the theoretical predictions. It is
recommended that this experimental technique be
extended to study the effects of nonuniform flow and
intermittent blade surface cavitation.
2. FORMULATION OF THE PROBLEM
Consider a ship moving at constant speed U through
otherwise undisturbed water. We seek to determine
the periodic forces and moments exerted on the
ship hull surface arising from the unsteady propeller
velocity and pressure fields. The fluid is con-
sidered to be incompressible and inviscid and within
the domain bounded by the free surface, the hull
boundary, and the propeller blades (and trailing
vortex wakes), the flow is assumed to be irrotational.
Under these circumstances, a fluid velocity potential
exists which can be expressed in terms of steady
and unsteady components as
o(x,t) = Ux + d5(x) + 6) (Kt) + bp (x,t)
Here, x = (x,y,z) is a cartesian coordinate system
fixed to the ship with the x and y axes in the me
waterline plane, and z-axis directed upward. 9, (x)
is the steady disturbance flow about the bare hull
in the presence of the free surface, $,(x,t) is the
propeller potential, and bp Gx, t) is the potential
of the flow arising from the propeller-hull inter-
action, often termed the scattering or diffraction
potential. It should be noted that the presence
of the viscous, rotational wake of the ship is
ignored in the diffraction problem, i.e., it is
assumed that the unsteady pressure forces on the
hull can be derived from potential flow considera-
tions alone.
The propeller potential is periodic in time and,
by virtue of the symmetry of identical, equally
spaced blades, may be expressed as a Fourier series
with harmonics in blade passage frequency as
a = 1 a!
bp (x,t) = a bp, Gee Net (1)
with 6 being the complex amplitude of nth harmonic.
(In this and all subsequent expressions involving
einNwt the real part is understood to be taken.)
Similarly, the diffraction potential will be of the
form
co
bp Ge, t) =) ip, Ce (2)
n=0
We now consider the boundary value problem for
the potential $ = $, + $p, assuming the fluid
disturbance velocities to be small compared to the
ship speed, i.e., |V>| and |V¥$,| <(U. Within the
fluid domain, the potential must satisfy Laplace's
equation
V2o,(x) = 0 (3)
At large depth and distances upstream of the hull
and propeller the disturbance must vanish
x > -©
Vo, > 0
>0 Za OF (4)
and at large downstream distances, x > + 1d.
satisfies a suitable radiation condition.
The boundary condition on the hull surface,
denoted by S, requires that the fluid velocity must
be tangent to the surface, or
a a =
n° Voy (x) = 0 x on S (5)
a being the outward unit normal vector to the sur-
face (see Figure 1). Here we have assumed the
hull to be rigid and stationary with respect to the
translating coordinate system (i.e. hull motion
and deformation due to propeller excitation is
ignored) .
The linearized free surface boundary condition
May be written in the form
2
, 9b 3 oa 3b
-(nNw)“o_. + (2inNwU) —— + U 7 ej ——— =_
n ax 2
ax dz
on, z9= 0 (6)
In order to establish the relative magnitude of
terms the equation is recast in nondimensional form
using the ship speed U and propeller radius Rg for
reference length and time scales, obtaining
a 326 gR 3a
n 2 n oO D igh
ae + —_ — = =
ax € uD + G2 € NE O on z (0)
=) ar 2ie
where € = J/mnN, J being the propeller advance
coefficient. It may now be observed that typical
propeller applications, « <<_1 and the first term
will dominate. Thus, as a first approximation the
free surface boundary condition (6) reduces to
bn (x) = 0 on z =0 (7)
This completes the statement of the boundary value
problem for the diffraction potential as summarized
in Figure 1. It should be noted that by virtue of
(7), the function bn (x) can be analytically continued
into the upper half plane, z > 0, in a straight-
forward manner. As will be shown in subsequent
sections a solution can be constructed in terms of
VELOCITY POTENTIAL
Fz DXL) UX + QK)4+h (Kt)
= = >, inNwt
PRE Piwe
v2 py =0
RV, | ,70
Pa| 20)
B=0
lybal-= 0, |X|-e 00
E<O
FIGURE 1.
problem in propeller-hull interaction analysis.
Coordinate system and boundary value
appropriate “images" of the propeller and hull
singularity systems.
Upon solving for the velocity potential, all
other quantities of interest can be determined.
The linearized, unsteady component of pressure is
given by*
v6) , Wa Gc) = iu + Vo, Gx)
(8)
p(x,t) = -p (4449,
or from (1) and (2),
inNwt
p(x,t) eee
I
1
me}
[inNwd + Ue - Von]
(9)
i]
to
2}
zy
where the Py (x) are amplitudes of harmonics of the
unsteady pressure. The periodic force, F(t), and
moment, M(t) acting on the hull surface (see Figure
1) may be written as
F(t) = - pnds (10)
Ss
and
M(t) = - pxxnds (11)
Inserting the expression for p, one obtains the
amplitudes of the force and moment harmonics, as
a —S =>
r= 2 (inNwd, + Vs * Von)n ds (a)
s
and
—> A — oy —.
MO =o0 (inNw$, an Wa O Wun) BS 2 inh ols} (AES)
s
Until now, the propeller’potential has been regarded
as a known function. Before proceeding with the
surface force analysis, it is appropriate to discuss
the analytical representation of the propeller and
the velocities and pressures induced at arbitrary
field points.
3. REPRESENTATION OF THE PROPELLER
The primary source of propeller exciting forces is
the spatially nonuniform wake of the hull in which
*To be strictly consistent with the high frequency approxi-
mation, the convective pressure term should be discarded.
However, this term adds no serious burden to the ensuing
analyses and by retaining it, numerical calculations can
be used to demonstrate that the contribution from this term
is, in fact, negligibly small.
281
the propeller operates. As viewed in a coordinate
system rotating with the propeller, the flow
approaching the propeller consists of time-average
or circumferential mean component and an oscillatory
component. The oscillatory component gives rise to
unsteady loading on the blades in a manner analogous
to a hydrofoil encountering a sinusoidal gust. This
unsteady loading, summed over all the blades, yields
periodic shaft forces at blade frequency and integer
multiples. In contrast, the periodic pressure
forces acting on the hull surface arise from the
induced velocity and pressure fields from both the
mean and unsteady components of loading, as well
as the blade thickness, because of the varying
aspect of the rotating blades relative to the fixed
hull boundary.
Propeller theory for unsteady flow has developed
as a logical extension of linearized lifting-surface
theory for hydrofoils. It is assumed that the
oscillatory components of the wake velocities are
small compared to the mean, and can be resolved by
Fourier analysis into "wake harmonics," the funda-
mental harmonic being the shaft rotation frequency.
Each of these harmonics, within the linear approxi-
mation, will produce a component of unsteady blade
loading with the same frequency. By virtue of the
propeller's symmetry, upon summing over all the
blades, only certain harmonics of the loading will
contribute to the net force on the shaft. However,
all the harmonics of loading contribute to the
forces on an individual blade, and, as will be seen,
to the radiated pressure field of the propeller.
The propeller lifting-surface theory developed
by Tsakonas et al. (1973) is adopted in the present
work. This analysis and associated computer pro-
grams have been successfully applied in recent
propeller designs to minimize bearing forces, e.g.,
Valentine and Dashnaw (1975). In addition, the
analysis has been extended to compute field point
velocities and pressures, including the contributions
from the image of the propeller arising from the
presence of the free surface. As the details of
the development of these formulae have been largely
reported in the literature, we shall not burden
this paper by recounting them, being content to
outline the procedure.
Blade Loading Potential
The linearized equation of motion for unsteady flow,
referred to a non-rotating cylindrical coordinate
system (x,r,f) centered at the propeller axis
(Figure 2), may be written
36 36
Sent p 19)
2 i Dem Os (re)
=o (15)
where p is the pressure induced by the loadings on
the blades due to camber and incidence and p', for
later convenience, denotes the fluid density. Here
the angles of attack are produced by each axial
and tangential spatial harmonic of the nominal hull
282
FIGURE 2. Propeller coordinate system-projected
view looking upstream.
wake which is presumed to be known from wake survey
measurements.
The pressure induced at a field point by a single
blade is given by the following distribution of
pressure dipoles
M
LUpaat -ikut 9 1
epasy pe) = a a | Apr(&,p)e np Ros
Sp A=0
Pp (16)
where Ap, is the complex amplitude of the pressure
loading on the blade arising from the wake harmonic
order X and, as illustrated in Figure 3,
Sp is the surface of the blade, represented ap-
proximately by the helicoidal surface & = U/w
Ny is the distance directed normal to the surface
S
R = [(x-&)? + r2 + o2 - 2rp cos(§ +a -¥Y)] is
the distance from a point (&,0,89 + a) in the
surface Sp to the field point (x,r,/)
6 = - wt is the angular position of the blade
We note that the representation of the blade is
only approximate for a wake adapted propeller,
being correct for a constant pitch propeller in
uniform flow. Here we also assume that the pressure
jumps on the blades, Ap), have been previously
calculated by the unsteady lifting-surface theory
such as developed and programmed by Tsakonas et al.
(1973). ;
To place the harmonic content of 1/R in evidence,
the following identity can be used
Rea Jp pita lee 9 ars co 10
ana | (17)
where the amplitude A | is given by
,
Tq) 10K) yy (Le lx) p<xr<o
Tog) (HE[=)K) gy (110) 0<xr<op (18)
Im and K, being the modified Bessel functions of
the second kind of order m.
To secure the pressure field for an N-bladed
propeller, the blade position angle 9 is replaced
by 8 + 2m/N and the sum over n from n = 0 to N - 1
carried out. This sum yields a factor N and the
constraints on the frequencies A and m, given by
X =m = RN with & = 0,+1,4+2,+3...i.e., products of
terms for which A - m # 2N will sum to zero. ‘The
total induced pressure at any field is secured by
summing over 2 from -~ to +”.
Upon use of (15), (16) and (17) and looking after
the shifted time variable, using 8 = -wt which shifts
to -wt + w/U (x-x'), one obtains the velocity
potential in the form
co
N y i2Nwt
bp (x,r,P,t) = ~ ou S
M Q=-0
A , 12) rU,TG7Se ds
) a Py (E,0) Pq (Xr /PH E70) ae
A= Sp
in which the propagation function, Pr is given by
an2
aro “7 } ak (20)
where for each 2, m= A - &£N, and M is a practical
upper bound of the wake harmonic order number beyond
which the amplitudes of the wake harmonics are so
small as to render negligible values of Ap, for all
A > M. (A value of M = 8 is reasonable). Details
of further reductions of the integrals involved in
(19) and (20) may be found in Jacobs and Tsakonas
(1975).
BLADE
REFERENCE
LINE
p(t) —~+
7 HELIX: f= Ge
FIGURE 3. Propeller coordinate system-expanded view
of blade section at radius p.
To account for the presence of the free surface
which, at the frequencies of interest acts as a
zero potential surface (see Eq. (7), we merely add
to (19) the potential dp; =- bp (Yur pyr t) in
which
a = ly 24+ (2d-z.)2 =r when z =4d (21)
1 p p p
a |
fy = tan aE = 7” when Zp = d and
= Wf Sia S UPznP Leo) = © (22)
where d is the distance or depth of the propeller
axis below the free surface; y,, Z, are the transverse
and vertical coordinates of any field point (Figure
2). Thus, the total potential arising from the
loadings on an N-bladed propeller in the presence
of the free surface (neglecting the feed-back on
Ap, from the free surface) is
N =
=-—— 2Nwt
%D; e'U iy Srey
dp +
Q=-00
M
a A Ap) (€,0) [Pm (x,r,776,0) -
A=0 S
Pm(x,ri,fii&,0)] Aas (23)
and the spatial derivatives of this function yield
the velocities induced by the propeller and its
negative image in the free surface. Clearly bp +
oH = © seers Enlil Se Ehatel Yp for Zp = Glo
Blade Thickness Potential
The potential, $,, induced by blade thickness may
be constructed from a distribution of dipoles (with
axes tangent to the helical arc along the blade at
any radius) whose strenths are given by V_, V being
the local relative resultant velocity and T the
local thickness provided by the expanded blade
section drawing. Using the helical geometry as
before, one can obtain
Ro on
| ‘i U2 + (wo)? t(p,a)
Ry Oo
)
iL
aaa da dp (24)
or (x,r,f,t) = Te
where ae (Pp) and a;(p) are the angular coordinates
of the blade leading and trailing edges.
To allow for the free surface, 1/R is replaced
by 1/R - 1/R; with Ry being the distance from the
reflection of the dummy point in the free surface
to the field point on or below the water surface,
making use of relations (21) and (22). Again, to
place the harmonic content of 1/R and 1/R; in
evidence and to facilitate integrations over the
blade surface, the Fourier expansion (17) can be
applied.
283
Asymptotics of the Loading Potential
The fact that the disturbances induced by each of
the pressure jumps Ap, are propagated by widely
different functions of the space variables x,r, ~
must be emphasized as these behaviors have a most
significant impact on the pressure, velocities, and
the resultant forces generated on the hull. These
diverse characteristics can best be illustrated by
examining the asymptotics of the potential for
upstream locations which are large only with respect
to the x-wise extent of the blade surface. The
x-wise extent of the blades is given by the (chord)
5 Satin bpp being the local pitch angle which, in
the radial region of heaviest loading, is normally
of the order of 25° For merchant ships, the blade
chord in this region is of the order of one-half
the radius and, hence, the x-wise extent of the
significant position of a propeller is only about
0.2 radius. Thus, for axial distances of the order
of one diameter, the x-wise extent of the important
region of the blade can certainly be neglected in
an asymptotic analysis.
Using the expansion of Rae given by
Ree © Qm—a72 ') eae)
T pr
m=-0
where Q is the associated Legendre function, and
MAGE) eistelaeet tO
2pxr
and retaining only the leading term in the expansion
of Q for large Z, one can arrive at the following
behaviors for the ae ae of the loading poten-
tial, i.e., dp = Oe op being the part
associated ten oe OR and To) being that
arising from the torque-producting loading in the
forms:
Siete -i2Née
an2 p'w
Q=-2
een
Ro
r
m : anya |
Rh
(25)
eas czy a
San MIME eee
(x24r2402) |™ Eve [x?+r°+p*+4d (a-z,,) ] ae
inl -i2ne
pred | ct ae Ne oe
woe a
ae An2p'w* oe
Q=-0 =
(m=A-2N)
continued on page 284
284
R
oO
aN 2
HEE alta
Rh (26)
em? "3 elm fi a
L(x? +2407) Ree [x2424p?+4a (a-z,,)} mI#1/2
where
e flea Oct Cn ae a AZ) e_2) yeecp cheer
<Im| I (|}m| +1)
function (27)
and
ee
LA) =o Ap) (Pp, 0) asus da, the load density
(28)
ae
Here the effect of the free surface is included by
the last terms in each integrand. To exclude the
free surface, take the propeller depth of submergence
d=,
Limiting our attention to blade rate (£ = 1, - 1),
we see that, although the mean pressure jumps Apo
(A = 0) are much larger than those at all other
wake harmonics, the propagation functions for m =
X - &N = +N exhibit extremely rapid decay with
increasing x. In addition, we observe that the
radial loading for m = +N obtained from Apy is
weighted by the oscillatory function eiNa which
has the effect of producing an Iy(o) which is
inversely proportional to N. In contrast, the
contribution fon Ae— Ni, ase, m=O} as ofthe
form
(N)
Lo = ie da '
which has a "non-destructive" weighting function of
unity. Another feature which reduces the mean
loading contribution to the generation of forces
on the hull (wherein integration over the athwart-
ship variable yp is involved) is the presence of
the space angular function
: R -1
oiNy aya iN (tan Yp/Zp)
yielding pressures and velocities at different Yp
which are not in phase. In strong contrast in the
propagation mode for the blade frequency loading
Apy (for which m= 0), dp has no dependence on p
or p4, and all yp locations receive velocities and
pressures which are in phase with each other. On
the other hand, the coefficient C|,| is large for
} = 0 (being 6.5 for a 5-bladed propeller), whereas
Cc. = 1, the multiplier for the contribution from
the blade frequency Ap's.
These observations are succinctly summarized in
Tables 1 and 2 for the case of a 5-bladed propeller,
displaying the rate of decay with x, the variation
of the influence coefficients C]m| and mC|m|/1+2 m|,
and the dependence on the angular space coordinates
pand ~;, without and with the free surface effect
for the dominant terms at blade frequency arising
from the loading at wake harmonics i} = 0, N- 1,
Nand N+ 1.
One may observe in Tables 1 and 2 that the effect
of the free surface does not generally increase the
rate of attenuation of the potentials with x except
at or near all points in the vertical plane yp = 0
with the exception of the bp (N) and > (N) arising
from blade frequency loading on the blades, i.e.,
} =N and m =0, which show a change from x72 to x74
and x7! to x-3 everywhere, respectively.
A dramatic contrast in the force-generating
capabilities of the pressure field components arising
from the mean (the largest) and the blade-frequency
loadings on the blades can be found by integrating
the pressures
-p! Og) eng 59 94,7, (N)
ot the
over a rectangular region of half-breadth b arranged
symmetrically z, units above the propeller and
extending from -f radii forward to s radii downstream
of the propeller plane. Upon defining the coeffi-
cient of the vertical force on the rectangle as Ze (A)
= FZ) /o'n2p4, we can arrive at the following
TABLE 1. ASYMPTOTIC CHARACTERISTICS OF BLADE FREQUENCY COMPONENTS
OF THE THRUST-ASSOCIATED POTENTIAL ¢, FOR A 5-BLADED PROPELLER
FOR LARGE AXIAL DISTANCES
Wake Propagation Influence Relative
Order Order Coef. Loading*
r m C mil A)
2pQ,
0 =5 8.48 26.7
N-1=4 -1 4.71 4.8
N=5 0 3.14 1
N+1=6 1 4.71 2.1
Dependence on x, py and ¥,
Without With With
Free Surface Free Surface Free Surface
(yp =9)
x et ise x (cae 1) 26d (d-Zp)x
Ix|3 ix |!3 x5
x tiv x (el? -e “4 10d(d-z,)x
Ixi Ix [x7
x 6xd(d-zp) 6xd(d-zp)
ixP Ixf Ix
xe? x(eriP—e 10d (d-z),)x
7
ix/5 xf |x|
285
TABLE 2. ASYMPTOTIC CHARACTERISTICS OF BLADE FREQUENCY COMPONENTS
OF THE TORQUE-ASSOCIATED POTENTIAL ¢g FOR A 5-BLADED PROPELLER
AT LARGE AXIAL DISTANCES
Wake Propagation Influence Relative
Order Order Coef. Loading*
r m mC) | LO)
1+2|m| 2p),
0 -5 -3.86 26.7
N-1=4 -1 -1.57 4.8
N=5 0 0 1
N+1=6 1 IES; 2.1
Dependence on x, y and ¥;
Without With With
Free Surface Free Surface Free Surface
(yp =9)
eisy eid y_ li 22d(d-zp)
Ix }x |} |x ls
eiv civ _ eli 6d (d-zp)
IxP [xP Ix
ou 2d(d-zp) 2d(d=zp)
Ix! Ix? Ix
eiv paigmemGi 6d(d-zp)
IxP [xP Ix
*These are relative values as obtained from calculations of a 5-bladed propeller
using the wake of the SS Michigan.
** L O)
im
2 p04, 204,
a
~ a4,
expressions for the moduli of the blade-frequency
forces, viz.,
DO Nate
C. \(2e4b )
[Zp] = sin [ (Nt) 7]
an2p 'n2R2
(o)
1.0
N+1 :
fe) Ap, (p) sin Noy,
0,2 (29)
—(2N+ ~(2N+
{ (Vezeztep® NN) _ (Veep? "ONY yap
where
Gy, = dbeSaBoos (NEI) ig 2 tem! ica
N Zo
2 (N+1)!
for the contribution from the mean loading, (i =
0), and
10
N N
|z ( ai | 4p, | 2n
T 202 N
81p'n“R
)
0.2
(30)
(b+ [s*4+22+p2+b2)
SSE do
(b+, [£24+22+p24b7)
for the contribution from the blade-frequency loading
on the blades.
b 5
Ap, (p,a%) eM da = Apy (p) ar
b
sin may,
for Ap, independent of a
Evaluations of (29) and (30) were carried on a
hand calculator for various integration lengths f
forward of a propeller using assumed radial distri-
butions of Ap, and Apy and representative values
from computer calculations for a 5-bladed 22.5 ft
propeller in a single-screw ship (model) wake. The
calculations were made for a flat-bottomed hull of
half-breadth b = 2 Ry at 2, = 1.5 Ry (25 percent
tip clearance) and a stern overhang s = 1. Results
shown on Figure 4 show dramatically that the force
arising from the blade-frequency (b-f£) loading is
(asymptotically) 65 times larger than that from the
mean blade loading when the free-surface effects
are omitted (note that Ap, = 40 Ap;). Furthermore,
the total force due to b-f blade loading rises very
slowly to its asymptotic value as the integration
length is increased and even the force from mean
blade loading requires integration of the pressure
to three radii forward of the propeller.
To allow approximately for the effect of the
free surface, one can subtract terms of the same
form as (29) and (30) with z,* replaced by z,* +
4dh with d being the depth of submergence of the
propeller axis and h the hull draft in way of the
propeller. The reduction in force for d = 3.5 and
h = 2 is significant for Zp (N) but is found negli-
gible for the smaller force. As expected, the
asymptotic value Zp (N) (£09) is more quickly achieved
due to the presence of the free surface, but, never-
theless, requiring that one integrate to some 8
diameters to achieve the final value.
These results tell us that the current practice
in European model basins (in which b-f pressures
are measured on models in the vicinity of the
propeller and these are integrated in an attempt to
secure the b-f hull force) is highly suspect because
the slowly decaying pressures from b-f blade loadings
contribute large sectional force densities far from
the propeller. This effect is exacerbated by the
"growing" cross-sectional shape as one integrates
forward which is not accounted for in the constant
beam "ship" used in the foregoing analysis.
286
As an order of magnitude formula, one might use
(30) for £ = ~ with the correction for the free
surface included. This reduces to the complex
amplitude
1.0
Zn = pAp. Qn
(b+ s*+z *+p°+b?)
do
(b+ \s2+z 20 2 +4b2+4dh) (eat)
(which must not be used for hull drafts in way of
the propeller, h, which are small, as clearly Zep (N)
*+0O as h-+0O). In practice, Apy = ay(P) cos NO +
by(p) sin N8, ay, by being the chordwise average in-
phase and quadrature blade pressures given by the
unsteady lifting surface calculation.
With the foregoing considerations of the propeller
in mind, we now return to the surface force problem
for a general three-dimensional hull boundary and
prescribed propeller onset flow. In the following
section, a procedure is described for determining
the diffraction potential and the surface pressures
and forces in terms of singularities distributed
over the surface of the hull.
x10
FIGURE 4. Approximate moduli of B-F forces on
barge-like ship from pressures emanating from
mean and B-F loadings on a 5 bladed propeller
(in a single screw ship wake) as a function of Oo
integration length forward of propeller.
,
B-F FORCE COEFFICIENTZ
(EFFECT OF FREE SURFACE NEGLIGIBLE)
5 10 15 20 25 30
INTEGRATION LENGTH, f, FORWARD OF PROPELLER IN RADII
4. A DIRECT APPROACH FOR DETERMINING SURFACE FORCES
A "frontal attack" on the problem of predicting the
vibration forces generated on an arbitrary hull by
the induced flow of the propeller, (and its free sur-
face image) is to construct the potential of the hull
in the presence of these onset flows. This procedure
was first applied by Breslin and Eng (1965) toa
realistic hull form. At that time, however, only
the mean loading and the blade thickness were
accounted for in the flow impinging on the hull and
the computer time was observed to be excessive. In
contrast to these earlier efforts, the propeller
flow is now composed of all constituents of loading
and the (high frequency) images arising from the
presence of the free surface.
A solution for the potential, $,, which satisfies
equations (3), (4), and (7), is constructed by
distributing source singularities, on (x)einNut,
over the surface of the hull, such that
dn) = - A o,f") (—— - —4— Jas)
Ss |x-x" |x-x! |
ab
aC Dae O (32)
where the region of integration is over the submerged
portion of the hull and x; is the distance from an
ASYMPTOTIC VALUE
FOR f—=@ a
6.28
1 ARISING
FROM B-F LOADINGS Op, ON PROPELLER BLADES
WITHOUT FREE SURFACE
WITH FREE SURFACE CORRECTION
(d=3.5, h=2)
a NOTE EXPANDED
Z, —B-F FORCE ARISING FROM SCALE ———=
MEAN LOADING (4 P= 40 Ap, !)
WITHOUT FREE SURFACE
“image” hull point to the field joteulics abo, alse
x' = (x',y',2!), then xt =X AV Ze ee Source
strengths o,(x) can be determined by applying the
hull boundary condition (5) yielding an integral
equation
(x)
oO (x
n 1 — al iL
5 > it oy, (x")n o W ae SES as
+ n(x) “(Yop + Yop; ) = 0, Sons (33)
n
The integral teria gives the contribution from all
source elements other than at the point of interest
on the hull. The contribution from the source at
that point is given by the first term, 0, (x)/2.
Equation (33) with n- [Vop_ + Vop; _] as a known
i 7 ; n in 9
input is solved numerically by the generalized
Douglas-Neumann program [Hess and Smith (1964)].
In practice, the hull surface is divided up into
quadrilateral elements over which o, is considered
constant and the integral equation is replaced by
a set of simultaneous algebraic equations. Care
must be exercised to insure that the sizes of the
elements are small compared to the spatial "wave
length" of the propeller-induced velocity field.
This is particularly the case for field points just
downstream of the propeller since the velocity
components rapidly become proportional to sines
and cosines of N(w/U x-Y) so that the wave length
of these signatures is A =2mU/Nw, which, for J~1
and N = 5, becomes A = 0.4R,. In order to obtain
representations of an entire cycle, it is necessary
to take element lengths of one-quarter of this
length or about 0.10R,. Upstream, the induced flow
is monotonic in x and the element sizes can be made
much larger without loss of accuracy.
It is acknowledged that the above-described pro-
cess does not, in principle, completely solve the
problem since the feedback of the hull sources on
the instantaneous flow experienced by the propeller
is not included in the propeller loadings Ap). To
do this would require joining the integral equation
for the propeller loadings (with input from the
propeller generated hull sources) to Eq. (33) to
form a pair of integral equations for Ap) and oy,
which, when solved interatively to convergence,
would yield the complete solution. For the present,
we are content to ignore the hull feedback on the
propeller.
Once the source densities on the hull surface
are found, it is convenient to determine the force
induced on the hull in terms of simple integral op-
erations on these sources. Although the Lagally
theorem and its extension by Cummins (1957) is known
for submerged bodies, it is necessary to develop a
form which is suitable for use for floating bodies
beset by high frequency flows.
The force as given earlier by Eq._ (12) may be
considered as the sum of two terms F, ) and F, (2)
given by
ae = ipnNw [fe nds (34)
and Ss
F,(2) = 9 Jf Vz * Vb, nds (35)
287
Since $y = 0 on z = O, the region of integration
in (34) may be extended to include the hull water-
line plane S, (see Figure 1), thus forming a closed
surface about the volume ¥ inside the submerged
portion of the hull, and
F (1) = ipnnw bn nds (36)
S+S5
where the symbols ( )* and ( )~ are used to denote
a quantity evalutated on the outside and inside of
the surface of integration respectively. Noting
that for $,(x) given by (32), bn? = bn (i.e. the
potential is continuous across a surface distribution
of source singularities), and using the vector
identity n=n- Vx, one obtains
st
F,(1) = ipnNw dan * V¥ds(x) (37)
S+S,
By means of Green's reciprocal theorem applied to
the volume ¥, (37) becomes
Sy —
x
xn ° Vo_ dS(x) (38)
°
since V + V(x) = 0 and V*$," = 0 in ¥. A fundamental
property of a surface distribution of source singu-
larities relates the jump in the normal derivative
of the potential to the local source strength, viz.
Bl)
no) Vout = ni) Vduo cy, (39)
But since n° Vont = 0 on S by virtue of the boundary
condition (5), Eq. (38) may be written as
—— — —_ —_.
Fy, (1) = -ipnNw X On(x) dS(x) +
S
aS OO =>
ipnNw x a ds (x) (40)
Ss
oO
The first term in (40) has the same structure as
that derived by Cummins (1957) for submerged bodies
generated by internal singularities. The second
term arises from the capping of the volume by
extending the free surface through the ship (proposed
originally by Breslin in 1971). For the important
case of the vertical force, Fone we obtain
FZ (1) = ipnw i Op (x) aS (x) (41)
Ss
A similar analysis can be applied to the convec-
tive term F, (2) (see appendix A) to obtain
F,(2) = - p ile Wop. + op, ) eS) +
n
s
3
fe) I Vo5 = ds
So
(42)
in which again the first term exhibits the same
form as for a submerged body and the second term
accounts for the intersection with the free surface.
If it is assumed that 96 /dz = 0 on z = O (rigid
wall free surface condition for the steady flow
about the hull, i-e., low Froude number approximation),
then from (42)
FE, (2) = -9 o i108 (dp + op, ) ds (43)
=o S dz n in
S
and the total vertical force, F
becomes
; an
=> Res, Oi. oF Oma a + .
“Bg ff ee ca tr
Z
As noted earlier, the first term under the integral
will dominate because of the large multiplying
factor nNw. This will be confirmed in the calculated
example to be presented subsequently. First, how-
ever, we outline an alternative approach for
determining the vibratory hull force which avoids
the need to solve for the diffraction potential.
Zn + )
ds
(44)
5. AN ALTERNATIVE METHOD FOR DETERMINING THE
VIBRATORY HULL FORCES
Vorus (1971, 1974, 1976) has developed an alternative
procedure for determining the vibratory hull surface
forces which eliminates the need to solve for the
hull diffraction potential in the presence of the
propeller onset flow. The ith oscillatory force
or moment, Fin, exerted by the pressure on the
hull may be written from (12) and (13) as
: =x a Sas
Ech = (0) (inNwd, + Vs °* Von) my ctr ds (45)
Ss
> a
where the a, are defined as
= —> a = >
a, =i Cy, = yk =z 9
=> — > aS =
a2 = j Ch = 64 al be Us
a3 =k OF Gea) i ee (46)
Vorus has shown that the solution for Fjy, with no
additional approximation, is given by the formula
1 /Nw
-inNwt
F. =— dt e ds(&,p,9 + a)
-7 /Nw
SPyy
1 72 2 O A
p /U + (wp) ™ Wate
E (47)
All of the variables in (47) pertain to the propeller
except Hin- Hy, is the amplitude of the fluid
velocity potential due to the bare hull travelling
backwards with speed U across the water surface
and oscillating with unit amplitude in the ith
direction and at the frequency nNW. Since the
details of the derivation of this formula may be
found in the cited literature we will only outline
major steps as follows.
The second term in (45) can be rewritten using
the following vector identity
Wig 9 Woe) (ao i) SS Wo Ga) Wg 2 ta) +
> > > == = =>
Vx[¢n(@; x Vs)] * n - oy Vx (aq x Vs) * n (48)
Only the last term contributes to (45), because We
*n = 0 (steady flow hull boundary condition) and,
by Stokes' theorem
ae > a <=> > >
ff tes 1) (eno Ve)il om 6 =o tne x V5)d% = 0
Ss (49)
where the line integral is taken along the hull
waterline on which $, = 0. Consequently, Eq. (45)
becomes
FR — — a —>
Fin = 0 J fr [inNwo, - Vx(aj x Vs)] + n ds (50)
S
and, upon introducing the function Tela which satis-
fies
Uh = © in fluid domain (51)
H. =0 z= 0, outside S (52)
in
= > 4. es —>) —_
OY isla hae [inNw a; - Vx(a4;XVg)] on S(53)
Vu, +0 as [x| > =,z< 0 (54)
in
equation (50) is given by
— > 7
Fo? Oat AV TH eas (55)
s
This form can be identified as one of the terms in
Green's theorem applied to the functions $y, and
Hin in the fluid domain bounded by the hull surface
S, the free surface z = 0, and the surfaces of the
propeller blades Spyr and slipstream, Spy, which
yields
1 /Nw
N -inNwt
Fin a ra dte {f Hin
SPy
—7/Nw
ap + db -
ae eae) we, Ae 0 ;
an an lS}. ap (bp bp )n. Vv Hi ds
Bp Pp
SPyytSWyy (56)
where dp - bp is the jump in the propeller potential
across the blade and slipstream surfaces. The two
terms in (56) can be identified as the contributions
from blade loading and thickness, and with further
manipulation can be brought into the form of (47).
Equation (47) indicates that the velocity corres-
ponding to the potential Hj, is evaluated over the
propeller blades and slipstream. The propeller
representation by distributions of dipoles directed
normal and tangential to the blade pitch surface is
the same as previously discussed. In the formula,
the velocity induced by the bare hull, VH;,, is
resolved into components in the directions of the
dipoles, multiplied by the dipole strengths, and
the products integrated over the blade and slipstream
surfaces. The first integral in (47), in time,
extracts the nth Fourier harmonic. Both the blade
position and the dipole strengths are functions of
time.
In the case of vertical force analyses, an
approximation to the improper integral in (47) has
been found to yield acceptable results. Let I be
defined as
inNw
a WEE |) a
I= e im. © Wile lis 0 (57)
'o
p
3
E
If the oscillating exponential varies more rapidly
than VHjy, then the argument of the exponential can
be considered as "large" and I can be expanded in
an asymptotic series. VHj,, should vary relatively
slowly aft in the propeller slipstream for vertical
oscillation of the bare hull and an asymptotic
evaluation should therefore be valid. (Such a
treatment may not apply to an athwartship analysis,
for example, where a rudder is involved in the bare
hull oscillation.) To proceed with the asymptotic
representation, (57) is integrated by parts yielding
inNw
go ae ee) 2
=- ma OW a. |
inNw Ny win 'é
inNw
(57)
y e us a im oO W SI dé'
inNw ost I ~ alfa)
For the conditions stated, the integral term is
higher order. Hence, to one term,
intw “P ~ y He) (5)
and (47) reduces to
1 /Nw
Nw -inNwt
E. = — dt e dS [o'vtn,
-7 /Nw SP,
289
Ap as
inNw np! iY Ha (5?)
in which the induced flow is evaluated exclusively
on the surface of all N propeller blades Spy.
6. COMPARISON OF THEORY AND EXPERIMENT FOR A BODY
OF REVOLUTION
An experiment was conducted to measure the periodic
forces on a body of revolution adjacent to a propel-
ler loading provided a configuration which could
be treated in a reasonably exact fashion by potential
flow theory. As such, the experiment was intended
as a fundamental check on the theory and computer-
aided numerical procedures. However, it is believed
that the experimental technique can be extended in
the future to study more general hull geometries
and the effects of unsteady propeller loading and
transient cavitation.
In the following sections, the experimental
apparatus and procedures are described and the
force measurements are compared with the analytical
predictions.
Test Body and Propellers
The experiments were performed in the DTNSRDC Deep-
Water Basin [(22 feet (6.7 m) deep, 51 feet (15.5 m)
wide, and 2600 feet (792 m) long)]. Both the body
and propeller were supported and towed from Carriage
II which has a drive system capable of maintaining
speed to within 0.01 knot.
Forces were measured on the forward half of an
ellipsoid of revolution with a length/diameter
ratio of 5.65. This "half body" was mounted by a
specially designed strain-gaged flexure assembly
to the forward end of a massive streamlined after-
body, attached to the towing carriage by a single
strut. The propeller was driven by the DTNSRDC
35-horse-power dynamometer, separately supported
from the towing carriage and positioned so that
both the propeller shaft and body axes were aligned
parallel to the direction of flow as illustrated
in Figure 5.
ne half body consisted of a 0.25 inch (0.64 cm)
thick fiberglass shell measuring 36.0 inches (0.91
m) in length and 12.75 inches (0.324 m) in maximum
diameter. The shell was filled with polyester foam
in order to minimize the mass and obtain a high
natural frequency, sufficiently above the propeller
blade rate frequency range to reduce nonlinear
resonance effects. The aluminum, free-flooded
afterbody, together with its support strut had a
low natural frequency to prevent mechanical vibra-
tions from the propeller dynamometer gears and
shafts passing through to the body force dynamometer.
The towing strut was attached to a large frame,
mounted on the propeller dynamometer structure.
Slotted pads supporting the frame permitted trans-
verse and longitudinal adjustment of the body
location and orientation. Vibration isolating
mounts were placed in the framework to further in-
hibit “pass through" vibrations.
Vibratory forces were measured for two propellers.
DTNSRDC propeller 4118 is a 3-bladed, 12-inch (0.305
m) diameter aluminum propeller designed for uniform
flow. Propeller 4119 is identical to 4118, except
290
FIGURE 5. Experimental
arrangement.
AFT VIEW
that it has twice the blade thickness (and a slight
difference in pitch to correct for the added thick-
ness). The principal design characteristics of
the propellers are listed in Table 3.
were designed by lifting-surface methods and both
open water performance [Denny (1968) ] and field
point pressure measurements [Denny (1967)] have
been reported. It should be noted that the theoret—
ical predictions of field point pressures agree
very well with the experimental measurements (at
design advance coefficient) and the same propeller
theory is applied in the present surface force
calculations.
The Force Dynamometer
A dynamometer was developed to measure the horizon-
tal component of the unsteady forces produced on
the half body by the propeller. The half body is
cantilevered from the afterbody on five (5) flexures.
Forces are determined by measuring the strain in
one flexure, while the other four flexures absorb
the vertical force and moments as illustrated
schematically in Figure 6. The measurement flexure
transmits vertical forces and moments with miminal
The propellers
PROFILE VIEW
stress while resisting a large part of the horizontal
force (calculated to be over 90 percent).
Two competing requirements governed the flexure
design - the need to resolve small forces and the
desire to maintain the natural frequency of the
flexure-half body system far above the propeller
excitation frequency. Also the flexure was expected
to experience large (static) forces arising from
flow misalignment and hydrostatic loading.
From the relationships for stress and stiffness
of a simple cantilevered beam, it is known that
for a given force, the flexure should have a low
stiffness in order to produce maximum strain. This
in turn would require a small body mass to keep the
natural frequency high. However, if the body is
too small, the resulting propeller force signal
becomes difficult to retrieve in the presence of
background noise. Although sophisticated techniques
were employed to reduce electrical noise and boost
signal power, it was not possible to completely
eliminate mechanical noise generated by the rumbling
carriage. With these compromises in mind, the
flexure was designed for a frequency ratio of 0.5,
producing minimally acceptable stress levels of
1000 psi (6.9 uPa) for the one pound (0.454 kg)
force in this experiment.
TABLE 3. PROPELLER GEOMETRY
4118 419
DIAMETER, INCHES
NO OF BLADES
PITCH RATIO (0.7Ro)
EXPANDED AREA RATIO
BLADE THICKNESS FRACTION
NACA MEANLINE
—SUPPORT FLEXURE (4)
DIRECTION 0.750 IN. X 0.035 IN.
OF MEASURED (1.90 CMX 0.084 CM)
FORCE MEASUREMENT FLEXURE
O.500IN. X 0.005 IN.
(1.27 CM X 0.0127 CM.)
FIGURE 6. Schematic diagram of flexure arrangement.
For simplicity and economy, the flexure consisted
of conventional steel shim stock clamped between
the half body and the afterbody by sets of wedges.
The flexures were pinned and epoxied to the wedges
prior to insertion into the dynamometer plate.
Before assembly, eight strain gages were mounted
and waterproofed, with one gage placed at each
corner of the two large faces of the flexure. The
gages were electrically compensated for tension
(or compression) and torsion. In order to check
vertical alignment to the flow, two of the support
flexures were also strain-gaged.
Calculations indicated that the measured strain
in the flexure due to dynamic forces would be 135
percent of the strain due to a static force with
the same amplitude, assuming small damping. Also,
the phase angle of the strain relative to the applied
force would be affected by the large ratio of
excitation frequency to the natural frequency.
Consequently, the experiment incorporated an inter-
nally mounted electromagnetic voice coil to calibrate
the measurement flexure as a function of force
amplitude, frequency, and forward speed. Initally,
with a series of known static forces applied to the
body, a current was applied to the coil to return
the body to its unloaded position, as indicated by
the strain output from the measurement flexure.
These static calibrations revealed that the coil
current varied linearly with applied force and that
the flexure strain was virtually independent (less
than 2 percent variation) of the axial location of
the applied force.
Dynamic calibrations of the dynamometer were
performed using a frequency generator and amplifier
with the known sinusoidal current directly input to
the coil. (It is assumed that in the low frequency
range of interest, O to 60 Hz, the applied force
is independent of frequency). The response amplitude
(relative to the applied current or force) was
found to vary linearly with the applied force. By
averaging the data, the transfer function for each
frequency and forward speed was determined as shown
in Figure 7. These results revealed anomalous
behaviour for frequencies of 20 Hz and 50-60 Hz,
which were later identified as resonant frequencies
associated with the towing structure.
Instrumentation and Data Acquisition
During each data run the following physical quanti-
ties were measured (see Figure 8): the force on
the half body, the surface pressure at two locations
on the body, the distance between the body and the
propeller (tip clearance), propeller blade angular
position and rotation speed, the forward speed of
the towing carriage, and the horizontal accelerations
of the afterbody.
Pressures were measured by metal diaphragm solid-
state gages (KULITE XTMS-1-190) flush mounted to
Zo
the half body surface. The propeller tip clearance
which varied slightly with forward speed, was
determined by measuring the distance between the
35-horsepower dynamometer body and the test after-
body at two axial positions using linear variable
differential transformers (Schaevitz 1000 HCD).
These low friction devices recorded relative move-
ment without transmitting mechanical vibration.
The propeller blade angular position and rotation
speed were measured by a Baldwin Shaft Position
Encoder mounted on the 35-horsepower dynamometer
tachometer shaft, generating one interrupt per
degree of revolution and another interrupt once per
revolution. During the experiments each data channel
was sampled for each six degree increment of propel-
ler rotation, thus providing 20 samples per cycle
for blade frequency quantities. (The time lag
between successively sampled channels and the delay
between the encoder interrupt and capture of the
sample, together amounting to several degrees of
rotation, were later accounted for in the data
reduction). Analog data output from the measurement
tranducer was digitized and stored on magnetic tape.
Data for each angular position of the propeller
were summed and averaged over several hundred
revolutions in an attempt to reinforce the signal
of interest while self-cancelling random noise.
In order to determine the blade-frequency com-
ponents of the unsteady force (and pressure) on
the half body, a Fourier analysis was applied to
the averaged data to yield the coefficients of the
series
8)
35
F(0) = a + a, cos mO + be sin m0,-T7 < 0 <T
m=1
S)
a
=O! =
= 5 + Co cos (md vey) (60)
m=1
in which 6(t) is the blade position angle (Figure
8). For the three-bladed propellers, the nondimen-
O KNOTS (Om/s)
ro)
° 26.6 © O ©
4 KNOTS (4.1m/s)
ry 00 Oo
fc)
6 KNOTS (3.1m/s)
Ooo V2®o
TRANSFER FUNCTION
(~STRAIN GAGE VOLTAGE OUTPUT / COIL CURRENT )
8 KNOTS (4.1m/s)
°
q 9 ©
“10 20 30 40 50 60 70
FREQUENCY, Hz
FIGURE 7. Force dynamometer amplitude response as a
function of frequency for several forward speeds.
nN
oO
nN
FIGURE 8. Schematic diagram of
experiment.
sional amplitude and phase of the blade frequency
force F3, are given by
lF3| 5 5
is ahs vja3t + b4 (61)
Cc =
on2D
0 Se tea Gye) (62)
a = e ein 3/a3
where the phase angle, Ope is the position of the
reference blade when the force is a positive maximum
or, from Figure 8, 8, is the angle by which the
force leads the blade position.
Experimental Results
Force measurements with propeller 4118 located 16.0
in. (6.3 cm) aft of the nose of the body and with
a nominal tip clearance of 3.0 in. (1.18 cm) are
given in Figure 9. The force generally increases
in amplitude and lags further with higher propeller
loading. The data points at design J (0.83) for
speeds of 4 and 8 knots show good agreement. In
Figure 10, the blade frequency pressure induced on
tne body in the plane of the propeller [x = 16.0 in.
(6.3 cm)] shows a monotonic increase in amplitude
with increased propeller loading and repeats well
for different speeds.
Force measurements with propellers 4118 and 4119
positioned 10.0 in (3.94 cm) aft of the nose of
the body [4.5 inc. (1.77 cm) tip clearance] are
shown in Figure 11. Over the range of propeller
advance coefficient, the force amplitude tends to
increase with increased propeller loading and the
effect of thickness is demonstrated.
The data exhibit some scatter for reasons not
yet fully understood and further calibration experi-_
ments and data runs are needed. The variation in
the data for different speeds (and hence different
propeller excitation frequencies) is particularly
disturbing. It may be noted that a post-test
examination of the raw (unaveraged) data for the
flexure, displacement, and afterbody accelerometers
revealed three specific sources of difficulty.
First, low amplitude data, particularly for speeds
of 6 knots and a blade frequency of 35 Hz, was
AFT VIEW IN PROPELLER PLANE
PRESSURE
SDF TRANSDUCERS Z 12.75"
VD
be AFTERBODY
PLAN VIEW
difficult to process. An example of this type of
run and comparison with a good data run is shown
in Figure 12. Generally, the low amplitude data
resulted in force coefficients much below the
values obtained from the higher amplitude data.
Second, for certain runs the data were overscale on
the individual records, but not in the averaged
plot. These overscales, if abundant, produced
anomolies. Third, structural resonances of 18-20
Hz and 55-60 Hz grossly distort data for blade
frequencies with these values. To the extent
possible, data contaminated by these problems were
discarded and are not in the results presented.
DESIGN ZERO THRUST
(J = 1.16)
n
Ww
WwW
oa
oO
Ww
a
be
fo}
lu
nn
<=
Fe
a
°
4.0
oO
°
t= Sol)
>< Oo
wu
= 2.0 CALC. METHOD
Breslin (1964-1971)
Vorus (1974)
1.6
0.0 0.2 0.4 : Ws
J = U/nDd
Ay We ee!
FIGURE 9. Calculated and measured blade frequency
force for propeller located at 2 = 16.0 in. with
tip clearance C = 3.0 in.
SYM | SPEED
knots
PHASE ©, , DEGREES
N
ala
— i=
a
i}
a
oO
0.0 : 0.4 0.6 . 4
J = U/n
FIGURE 10. Blade frequency induced pressure on body
with propeller 4118 located at % = 16 in. anda tip
clearance C = 3.0 in.
Application of the Theory
Direct Approach - Extended Lagally Theorem
(Breslin and Eng, 1965)
The test body surface was divided into 154 elements
as shown in Figure 13 with finer subdivisions made
in way of the nearest approach of the propeller
blades. Panels 93 through 100 were used to close
the body. The geometry of these elements, together
with the normal velocity induced by the propeller
due to loading and blade thickness formed the input
to the generalized Hess-Smith program which inverts
Eq. (33) to yield the source densities on each of
the panels.
A typical velocity variation, as given in Figure
14, shows that, downstream of the propeller, the
loading contribution is oscillatory, requiring
great care as the body sections are becoming larger.
This test case presents a somewhat difficult appli-
cation of this technique for this reason. In the
ship case, there is only a small portion of the
hull downstream of the propeller, and the sections
are generally becoming smaller. As a result of
this non-ship arrangement, @ifficulty was encountered
in securing an accurate answer, requiring several
adjustments of the size and location of the source
panels.
A calculation for a single set of conditions,
specified by the geometry of DTNSRDC Propeller 4118
set at a tip clearance of 3.0 inches (1.18 cm) at
an axial distance of 16.0 inches (6.3 cm) downstream
of the nose of the body gives a blade-frequency
force coefficient Cp = 3.4 x10-3 anda phase angle
Op = -2.0°. These results are quite close to the
measured values shown in Figure 9. It should be
remarked that the evaluation included the Lagally
force corresponding to the integral of the convective
pressures, i.e., the action of the transverse pro-
peller velocity component on the sources which
generate the body in the uniform axial flow. This
contribution, as expected, is indeed small yielding
293
only 1.0 percent of the force arising from the
time rate of change of the potential. This surely
justifies the order of magnitude argument given
earlier.
Alternative Approach - Oscillatory Body Potential
(Vorus, 1974)
In order to apply Eq. (47) to the experimental
configuration, it is convenient to consider the
velocity potential Hj, of tne body travelling back-
wards and executing simple vertical oscillations,
so that a.=03 =k in Eq. (53). The free surface
condition Hj, = 0 on z = 0, Eq. (54), can be satis-
fied by reflecting the body surface into the upper
half space and satisfying the body boundary condition
additionally on the image surface, S;- In Appendix
B it is shown that the vertical force induced by
the propeller on a ship in the free surface is
equal to the force on the "double hull" deeply
submerged. If we make the further assumption that
the force due to the convective pressure can be
omitted, the problem for H;, now reduces to
7H sO amy (63)
in
= —
nav, He = inno (nek) oneSetaS: (64)
in i
Vu, +0, {x| >@ (65)
in
where ¥ is the whole space outside the “double-hull"
sumtacel,) Sica
This method is particularly convenient in the
present application because the velocity potential
of an oscillating spheroid is well known, e.g. Lamb
(1932). With slightly modified notation
SYM
DESIGN ZERO THRUST
(J = 0.83) (J = 1.16)
hoe: CBW RIE
rm
[aa
enes Rais LO aig & 4119
md a 4118
oe
20 U
2 fe)
a=
a
-40
®@
4.0 a2
@ 4119
8 ‘
= 3.0 o> ack
=e EXP. rs a
ct 2.04118 - Open Symbols ey avis
4119 - Solid Symbols a Le
CALC.
=—-= Vorus(1974)
QO Ws Oa MO POs EO sass
J = U/nD
FIGURE 11. Calculated and measured blade frequency
force for propellers 4118 and 4119 located at 2% = 10.0
in. with tip clearance C = 4.5 in.
294
Good Data Run
W
v
Cn
fe)
>
~
a
©
c
on)
“4
Ww
)
1
u
fe)
= High Frequency Contamination
oer aa |
ie
pee Vel des 2 a
too F = —s =
FIGURE 12. Examples of force measurement me 20 so 2c 12e «saad «21a
flexure signal output - data averaged over
several hundred propeller revolutions. Shaft Angle, Degrees
H = ae 2 al Vardl Y ; F f : 5
in (Orbe?) = -inNwe)c9S ,/l - u > on aan any cosf and (x,r,/) is a cylindrical coordinate system with
Y Bed the origin at the center of the spheroid and the
(66) major axis extending from x = L/2 to + L/2. The
5 constants, c] and c2 in (66) are readily determined
Here y = ,/1 + ¢% and (t,u,P?) are the spheroidal in terms of the spheroid's maximum diameter/length
coordinates defined by ratio, 6, as
oo 2 33 2
BS Say ie WS Sy = ee rhe We, ay?
. oh Ise.
with = = 5 , Cy = focal length (67)
OS Gs ep ot sme 1,30 2) <= an
a) sg Se
PROPELLER PLANE
FIGURE 13. Schematic of expanded surface of
DTNSRDC ellipsoidal test body divided into 154
source panels (dimensions in multiples of pro-
PANELS USED FOR CLOSURE OF BODY peller radius) .
SRNR TE fo sees
0.004
THICKNESS CONTRIBUTION
REAL PART
-2.0 -1.0
IMAGINARY PART (SINE COEF)
CONTRIBUTION FROM
MEAN BLADE LOADING Ww
U
il 2t1+ 1-621 _ 5, EBS") Vi-62
c2 2 § §2
(68)
and by a suitable coordinate transformation from
(u,t) to (x,xr), the velocity V Hj, can be calculated
at an arbitrary point on the propeller blades.
In general the propeller dipole strength repre-
senting blade loading is a function of blade position
O@(t), i.e., Ap = Ap(p,atO(t)). However, in the
present experiments the inflow to the propeller is
uniform so that the loading is steady and Ap = Ap
(9,a). The blades of propellers 4118 and 4119
employ NACA a = 0.8 meanline sections. For this
section, and assuming a radially elliptical distri-
bution of bound circulation, the pressure jump
across the blade is given by
8T hoo
Ap(p,a) = * F(a) (69)
0.9(ay- a)) ™(R,? - RN
in which
295
20
FIGURE 14. Variation of blade frequency verti-
cal velocities induced by 3-bladed DTNSRDC
propeller 4118 at r = TOR and > =
and T is the steady propeller thrust.
The calculated values of the forces produced on
the spheroid for conditions corresponding to those
in the experiment are summarized in Table 4 showing
the separate contributions arising from blade loading
and thickness as well as the total forces. The
latter are also displayed in Figures 9 and 11 and
agree quite well with the measurements.
Additional parametric calculations were performed
to study the effect of propeller location on the
force produced on an ellipsoid arising from propeller
mean loading and thickness. In Figure 15 the
attenuation in force (amplitude) with increasing
tip clearance is illustrated. (The phase was found
to be essentially independent of tip clearance) .
Calculations are presented in Figure 16 for a series
of axial positions of the propeller with the tip
clearance held fixed. As the propeller is moved
aft from the nose of the body, the force increases
TABLE 4. FORCE CALCULATIONS USING METHOD OF VORUS (1974)
4= 10.0 IN.
C= 4.5 IN.
4=10.0 IN
C= 4.5 IN.
PROPELLER LOCATION CONTRIBUTION | S, x103
MEAN LOADING
THICKNESS
TOTAL
MEAN LOADING 0.88
THICKNESS 1.58
TOTAL 1.52
MEAN LOADING
THICKNESS
TOTAL
1.50
Bou
2.90
0.88
3.16
2.97
PROPELLER 4118
= 16.0 IN
J = 0.83
C/R) = 0.25
IN EXPERIMENT
FIGURE 15. Modulus of blade-frequency force on ellip-
soid as a function of propeller tip clearance [calculated
using method of Vorus (1974)].
rapidly, largely due to the thickness contribution.
CONCLUDING REMARKS
The analytical methods given in this paper can be
applied to a wide range of problems in which it is
desired to determine the unsteady pressures and
forces generated by a propeller on a nearby boundary.
The formulation is quite general, being applicable
to arbitrary hull (and appendage) geometries, and
propeller locations, geometry, and loading charac-
teristics. The assumption of high frequency
propeller excitation, which greatly simplifies the
treatment of the free surface, is not at all
restrictive in most cases of practical engineering
interest. A severe limitation, to be sure, is the
restriction to subcavitating propellers. However,
researchers are actively pursuing this subject and
/
aN GaN!
- L f \i I Ip 4
Hew ase
SHO
' Ny iN pot
WA Se NS
C 3 Rees
MEAN penne 3
{e)
0.0 0.25 0.50 0.75 1.0
AXIAL POSITION OF PROPELLER, */(L/a)
FIGURE 16. Modulus of lateral blade-frequency force
produced on an ellipsoid of revolution (L/B = 6.0) as
a function of propeller axial position (constant tip
clearance, C/R_ = 0.25)-calculated for DTNSRDC pro-
peller 4118 using method of Vorus (1974).
as procedures for predicting transient blade cavity
geometry and the attendant pressure field become
available, this important feature can be incorporated
into the analytical representation of the propeller
and the analysis of induced forces.
As with any theoretical development of this kind,
the usefulness and limitations can only be fully
ascertained by comparison with a sufficient number
of experimental measurements. The comparisons
presented in this paper for the simple case of a
body of revolution adjacent to a propeller in uniform
flow represent an encouraging first check. This
experimental technique can be extended to examine,
in a systematic manner, the effects of nonuniform
flow (unsteady blade loading and cavitation) and
more general body shapes. For example, wire screens
selected to produce certain wake harmonics can be
towed upstream of the propeller. At the same time,
the need is evident to undertake calculations for
comparison with results of the many experiments
reported during the past several decades.
ACKNOWLEDGMENTS
This work was jointly supported by the American
Bureau of Shipping (ABS) and the Maritime Adminis-—
tration (MarAd). The continued interest and
encouragement by ifr. S. Stiansen and Dr. h. H. Chen
(ABS) and Mr. R. Falls (MarAd) is greatly acknow-
ledged. Prior support of the Office of Naval
Research, Fluid Dynamics Division, enabled the
development of the velocity field program. The
authors are also indebted to Mr. D. Valentine and
Dr. S. Tsakonas of the Division Laboratory for their
painstaking effort in developing the programs and
to Messrs. H. Saulant and M. Jeffers (DTNSRDC) for
invaluable assistance in the design and conduct of
the experiments.
REFERENCES
Breslin, J. P. (1962). Review and Extension of
Theory for Near-Field Propeller-Induced Vibratory
Effects, Proceedings Fourth Symposium on Naval
Hydrodynamics, ACR-92, Office of Naval Research,
Washington, D. C.
Breslin, J. P., and K. Eng (1965). A Method for
Computing Propeller-Induced Vibratory Forces on
Ships, Proceedings First Conference on Ship
Vibration, Stevens Institute of Technology,
Hoboken, Wew Jersey; available as DTMB Report
2002.
Cummins, W. E. (1957). The Force and Moment on a
Body in a Time-Varying Potential Flow, J. Ship
Research, 1, 1: 7.
Denny, S. B. (1967). Comparisons of Experimentally
Determined and Theoretically Predicted Pressures
in the Vicinity of a Marine Propeller, NSRDC
Report 2349.
Denny, S. B. (1968). Cavitation and Open-Water
Performance Tests of a Series of Propellers
Designed by Lifting-Surface Methods, NSRDC Report
2878.
Hess, J. L., and A. M. O. Smith (1964). Calculation
of Nonlifting Potential Flow about Arbitrary
Three-Dimensional Bodies, J. Ship Research, 8,
a9 BA.
Jacobs, W. R., and S. Tsakonas (1975). Propeller-
Induced Velocity Field due to Thickness and
Loading Effects, J. Ship Research, 19, 1; 44.
Lamb, H. (1932). Hydrodynamics, Dover Publications,
6th Edition.
Lewis, F. M. (1969). Propeller Vibration Forces in
Single Screw-Ships, Transactions Society of Naval
Architects and Marine Engineers, 77, 318.
Lin, W. C. (1974). The Force and Moment on a Twin-
Hull Ship in a Steady Potential Flow, Proceedings
Tenth Symposium on Naval Hydrodynamics, ACR-204,
Office of Naval Research, Washington, D. C.
Stuntz, G. R., P. C. Pien, W. B. Hinterthan, and
N. L. Ficken (1960). Series 60 - The Effect of
Variation in Afterbody Shape upon Resistance,
Power, Wake Distribution, and Propeller Excited
Vibratory Forces, Transactions Society of Naval
Architects and Marine Engineers, 68, 292.
Tsakonas, S., J. P. Breslin, and W. R. Jacobs (1962).
The Vibratory Force and Moment Produced by a
Marine Propeller on a Long Rigid Strip. J. Ship
Research, 5, 4; 21.
Tsakonas, S., W. R. Jacobs, and M. R. Ali (1973).
297
An "Exact" Linear Lifting-Surface Theory for a
Marine Propeller in a Nonuniform Flow Field, J.
Ship Research, 17, 4; 196.
Valentine, D. T., and F. J. Dashnaw (1975). Highly
Skewed Propeller for SAN CLEMENTE Ore/Bulk/Oil
Carrier Design Considerations, Model and Full-
Scale Evaluation, Proceedings First Ship Technol-
ogy and Research Symposium, Society of Naval
Architects and Marine Engineers, Washington, D. C.
Vorus, W. S. (1971). An Integrated Approach to the
Determination of Propeller-Generated Vibratory
Forces Acting on a Ship Hull, Department of
Naval Architecture and Marine Engineering,
University of Michigan, Report 072.
Vorus, W. S. (1974). A Method for Analyzing the
Propeller-Induced Vibratory Forces Acting on
the Surface of a Ship Stern. Transactions
Society of Naval Architects and Marine Engineers,
BAip AUK
Vorus, W. S. (1976). Calculation of Propeller-
Induced Forces, Force Distributions and Pressures;
Free-Surface Effects, J. Ship Research, 28, 2;
OME
APPENDIX A
THE LAGALLY FORCE ON A FLOATING BODY REPRESENTED
BY A SURFACE DISTRIBUTION OF SOURCE SINGULARITIES
The force F, (2) arising from the convective term
of the linearized unsteady pressure, Eq. (35), is
given by
= +
vot + Vo, n ds (A-1)
Ss
where, as before, the symbols ( ) einel ( ye denote
quantities inside and outside the hull surface, S.
We assume that the solutions for We and $y are
known in terms of distributions of source singular-
ities and images over the surface S as
bs (x) =- ma (ee face 4 G(x,x')|dsS
ese? |] fee? a |
vy
& ‘S)
Vz = iU + Voy (A-2)
i 1 = 1 1
n(x) = - aa Oy(x"') TSG Toate | Cay op,
x-x' | x-x' |
S)
+ op, (A-3)
1n
in which Og (x) and Op (x) are the source singularity
strengths, x! is the image point of x, and G(x,x')
is the "wave potential" of a source located at x!
and is regular in the half plane z < 0. The deriva-
tives of these functions on eacn side of the surface
S are related to the source strengths in the form
<> + a +E >
Wa = We 7 neg (A-4)
—_ > > ,
Vo = Von + no, x on S (A-5)
from which it follows that
— SS fs (A-6)
Ye OVO Sg = Wn = Os Ga
=
since vt Cin = Wear 0 ® = 0.
We now apply Green's theorem to the functions
Vs and Vd, in the closed volume ¥ surrounded by
the surface S and So, where Sp is the hull water-
line plane, obtaining
+
=> a >_
Viet O- Wisay saVelis) = V(Vg - Von )dv¥
+
S+S5 @
>_ 3S = > _
= [IVs > V(Vo, ) + Von * Wg ] d¥ (A-7)
¥
since V x We = Vx Vo, = 0 in ¥. Using Gauss'
theorem and the fact that V* V, =V-° Vo,- = 0
in ¥, (A-7) may be written as
Ah 9 - Vo, n dS =
S+S,
ane (Vo Vo" +n) + Von (VE + n)] as (A-8)
St+So
and hence
J
oe
5,
5
a
n
i
‘ab
a,
a
os
oa
5
s S
+ V- (Vg » n)] ds
+ Jf [ve (Vo + n)
So
— >_ ——
Ore Wctuen 1) h Sa Victi Geren aS
(A-9)
The last two terms in the integral over S, combine
to yield
= aS
- Vo. WS +n) =o, + Wy
—
n(Vg ° Voy )
~s -_=>
(Go, 23 WY) x bn Vo
(A-10)
The first term on the righthand side of (A-10)
vanishes since $n = 0 on So. The second term also
vanishes, since by Stokes' theorem
[fe 2s Gee = ¢ dx x $y Vg = 0(A-11)
iS) Co
where the contour cy is taken as the hull waterline.
Consequently, using (A-6), (A-9), and (A-10), the
expression for the force becomes
Fy” = p [Ve Woes a) + Vor We + n)
S
> — S
- dg on n] dS + p Ve (Voy * n)dS (A-12)
So
— —=
The contribution from the free stream, iU(in Vs),
vanishes since
iU(Vy, * n)dS = iU V7, a¥ = 0
StS, ¥
Also noting that Voy ° n=-o
(A-11) reduces to
= (2 ns aS
F(?) = - p [o, Vos + 65 Voy + og on n] AS
Ss
+p Vos Vor «nas (A-13)
So
or, upon defining
vot + Voz
V6 2 Ss S
a 2
Vo+ + Vor
A — n
Whey = 2 (A-14)
(A-13) becomes
Be sree Ga Vis = Te Vp,)4s
Ss
au 165 ane ds (A-15)
So
The first term has the same structure as the steady
flow Lagally force derived by Lin (1974) fora
linearized source sheet representation of a slender
strut piercing the free surface. The second term
arises from the intersection of the hull with the
free surface in unsteady flow.
In the low Froude number approximation, 0$</dz
= 0 on z = O (rigid wall representation of the
free surface), and G(x,x') > 0. In this case (A-2)
and (A-3) yield
—
2 1 = -x! ees
Woks = oP = Og (x") + ds
at > =>/3 ats ESS 3
DR | |x-x",
i
Ss
and
—s >
Ty = OS (x') ES Mees | a
ha 40 Sines = 5,3 s 3 Ss
|x-x'| |x-x',
al
‘S)
+V +V 2a-16
oP, op. ( )
n
for x on S, and where the integrals are to be
interpreted in the principal value sense. Inserting
these expressions into (A-15) and performing the
integrations, the equation for the force reduces to
Pasay =—- 6) Og (Vop + Von, a's:
n tn
S
ab
+ 0 Veg —,- a8 (A-17)
So
which is the result given as Eq. (42) in the text.
The reduction in the first term reflects the fact
that there is no net force arising from the mutual
interaction of the body sources.
APPENDIX B
REDUCTION OF THE ANALYSIS OF PROPELLER INDUCED
VERTICAL SURFACE FORCE TO AN INFINITE FLUID
PROBLEM
The linearized unsteady pressure at a point, x,
on the ship hull surface is given by (8) as
espe) = => (9 3 ie an Wis (G39 ved) (B-1)
and the vertical force acting in the hull, from
(10), is
F(t) = - p(x,t) n+ k as (B-2)
In the high frequency approximation, ¢ = O on the
free surface, z = 0, and this condition can be
satisfied by constructing an image of the hull
surface and a negative image of the propeller in
the upper half space and allowing the fluid domain
to extend to infinity in all directions. The
negative image propeller is identical to the propel-
ler proper, but rotates in the opposite direction
and the signs of the dipole singularities represen-
tating the effects of loading and thickness are
reversed from those of their images in the lower
half space.
The image hull surface, S;, is identical geomet—
rically to S, but the signs of the singularities
on Sj required to diffract the unsteady flow from
the "two propellers" will be reversed from these
on S due to the symmetry. The magnitudes of the
Singularities at image points will be equal.
The steady flow about the bare hull, Vc, in the
low Froude number approximation will satisfy the
rigid wall free surface condition Vs * k = 0. In
this case, the steadily moving hull can be reflected
into the upper half plane with a positive image
singularity system, i.e., the singularities on the
image surface, S; will be of the same sign as the
singularities on S to diffract the velocity iU.
Because of the assumed linearity, the unsteady
potential may therefore be considered as the sum of
contributions from the propeller and hull and their
respective images.
O = Oe © Oe Pe Ong 2 One (B-3)
where
bt) = — bys Gy, t)
— oN
bp (x,t) = - py (xy ,t)
—
So SOS pip) (B-4)
ab
for all (x,y,z) outside the surface § + Sz: If we
define bpy = bp + by then it follows that
(x,t) = opy (xt) - dp, Gg,t) all ¥ (B-5)
Therefore, the complete unsteady potential in the
fluid beneath the zero potential free surface can
be obtained entirely from consideration of the
propeller and the double-hull in an infinite fluid.
The unsteady pressure at a point on the hull
surface S is now given by
ab ie) Eas —_> —
at Gant) ee Vg (x) . Vopy (x,t)
p(x,t) =- p
299
f)
Bc TNE mel ig. | |
at (x5, ) g (x) gua |
9 => =
Now if Vg = (U + Ugr Vor Wo), the symmetry of Vo
is such that Uc and vg are even in z, while weg is
odd in z. It follows that
a — eS =>
Vg (x) Vobpy (X47 t) = Vo (x, ) Vopy (x47)
and hence
a =>
p(x,t) = Ppy(x,t) - Ppy(x;,t) (B-6)
in which
ap
PH =
Bay = 2 Fe ge ° We
Thus, the unsteady pressure at points on the hull
can be obtained from calculations, or measurements,
of pressures at image points on the double-hull,
with the double-hull and propeller deeply submerged.
Turning now to the formula (B-2) for the vertical
force, we obtain
F(t) = - J [renee - k ds
iS)
>
+ Pip (44 7t) 0 - k ds (B-7)
s
_> => =>
But since n(x) * k = —n (x; ) - k, (B-7) may be
written as
Pe (Ve) Ss Dip (Xr t) n (x) +k das
Ss
—2) — i =>
= Pryp (Xj ot) n(x, ) * k ds (B-8)
Ss
or, since the image hull S; is geometrically iden-
tical to the hull proper,
—>) >
aoe) S = Pyp et) n> k ds
S+S;
and consequently the unsteady vertical force on the
hull can be obtained from force calculations, or
force measurements, using the double model and
propeller deeply submerged.
A Determination of the Free Air Content
and Velocity in Front of the “Sydney-Express”
Propeller in Connection with Pressure
Fluctuation Measurements
A. P. Keller
Technical University Munich
and
E. A. Weitendorf
University of Hamburg,
Federal Republic of Germany
ABSTRACT
The Special Research Pool within the Institut ftir
Schiffbau and the Hamburg Shipmodel Basin (HSVA) in
collaboration with the Technical University Munich
and Det norske Veritas executed extensive full-scale
measurements on the Single-Screw Container Ship
"Sydney-Express." The main task of the project was
the determination of the free air content of the
seawater in front of the propeller during the voyage
from Australia to Europe.
Simultaneously the velocity was measured at the
control point within the Laser-beam, where the free
air content was measured by the scattered light
technique. Additional investigations were a deter-
mination of the water-quality, high speed films and
sterophotography of the cavitation at the blade,
and pressure fluctuation measurements above the
propeller.
ils INTRODUCTION
For several years the dynamic behaviour of small
gas bubbles or nuclei in hydrodynamic pressure
fields has been recognized as an important influence
on cavitation inception and its extent. Besides
other scale effects in the field of model propeller
testing, the importance of this influence of nuclei,
which also effects propeller excited pressure
fluctuation measurements, was often underestimated
and neglected. Thus, for instance, the results by
van Oossanen and van der Kooy (1973) have shown
that for equal non-dimensional flow conditions but
different absolute revolutions (i.e. n = 20 and
n = 30 Hz) the non-dimensional propeller excited
pressure amplitudes were different. After the
development by Keller (1973) of a practicable laser-
scattered-light (LSL) method for measuring the
undissolved air content, systematic cavitation and
pressure fluctuation measurements were carried out
300
in the medium cavitation tunnel of the Hamburg Ship
Model Basin (HSVA) with the model propeller of the
"Sydney Express" [Keller and Weitendorf (1975) ].
The results were similar to those by van Oossanen
and van der Kooy. Due to the additional application
of the (LSL) technique, the differences of the
nondimensional pressure amplitudes for different
revolutions could be clearly explained by the
influence of the free air content or nuclei on the
cavitation. A further finding was that the non-
dimensional pressure amplitudes and the cavitation
for a revolution of n = 15 Hz were increasing with
growing free air content, whereas the cavitation
and those amplitudes for n = 30 Hz remained more
or less constant. The different behaviour for
n = 15 Hz and n = 30 Hz were explained by Isay and
Lederer (1976, 1977). Using the theory of bubble
dynamics they found that the reactions of the
bubbles on the respective pressure gradient of the
propeller blades at n = 15 or n = 30 Hz were differ-
ent. Further, these investigations led to criteria
of cavitation similarity of such a kind that the
number of nuclei per unit volume of the model flow
had to be increased compared with the number of
nuclei of the full scale flow. By geosim tests
with hydrofoils or propellers it should be determined
to what extent these additional criteria for cavi-
tation similarity are applicable.
Keeping in mind these physical connections, the
full scale trials on the container ship "Sydney
Express" were planned. These investigations were
the first attempt to measure the nuclei distribution
in seawater around a ship by means of the LSL
technique. The nuclei distribution could serve as
a basic value for the geosim tests and perhaps as a
comparative standard value of the water quality for
model cavitation investigations. Furthermore, the
experiences, made during the almost adventurous
measurements on the "Sydney Express" with the LSL
technique in front of a full scale propeller, could
be of common interest because the introduction of
optical laser methods is a promising tool in the
research fields of boundary layers and propeller
flows.
The additional investigations on the "Sydney
Express" help in full-scale model correlation only
slightly; the main purpose of these measurements
was the securing and better interpretation of the
scattered light results. The following additional
measurements were performed:
1. Propeller-excited pressure fluctuation
measurements with six pressure pick-ups above
the propeller.
2. Cavitation observations for determination
of the thickness and extent of the cavity by
means of stereo photography.
3. Investigations of water-quality by means of
a simple scattered light method (Aminco-
colorimeter) for detecting suspended particles
and total air content by means of a Van-Slyke-
apparatus. For both measurements water
samples were taken.
4. Velocity measurements in the control volume
of the scattered light measurement in order
to estimate the bubble concentration.
The "Sydney Express", as one of the fastest
German single screw merchant ships, was chosen for
the investigations because its propeller has an
interesting cavitation extent.
2. SHIP DATA AND PREPARATION OF THE MEASUREMENTS
The single screw, turbine-driven ship "Sydney Express"
has been built by Messrs. Blohm and Voss AG, Hamburg
(No. 872) and belongs to the so-called second
generation of container ships.
The main data of the ship are given in Table 1:
TABLE 1 - "Sydney Express" - Data
Ship Data
Length b.p. L = 210.00 m
Breadth, moulded BPP = 30.50 m
Design Draft D = 121.00 m
Block coefficient c = 0.616
Displacement (Design) V 43,457 m°
Container about 1,600
Max. Power Pp 23,870 kw
Service Speed Ve = 22.0 kn
Propeller Data
Diameter Dp = 7.00 m
Pitch (mean) Ph 6.550 m
Blade-Area-Ratio Ae/Ag = 0.78
Number of Blades Z = 5
The necessary conversions of the ship construction
for the installation of the measuring devices in
the after peak of the ship were carried out at the
Hapag-Lloyd ship yard at Bremerhaven during the
latter part of September 1977. Figure 1 shows
allusively to what an extent the narrow steel
construction had to be cut free. The installation
of three windows for the reception of the scattered
laser light proved to be the most complicated of all
301
installations. For reasons of the ship's safety
and also to enable proper cleaning these windows
were pushed through 350 mm sluice valves together
with their tubular guide pipes. The windows, of
which only that opening was marked in Figure 1
which had been used for measurements, were arranged
between frames 12 and 13. Also, the fitting of the
three 350 mm sluice valves required skillful impro-
visation on the spot. The installations of the
sluice valves for the pressure pick-ups, dimensioned
in Figure 1, and of the cavitation observation
windows were carried out without any difficulties.
In addition, all electric lines were laid out
from the measuring pick-ups to the measuring con-
tainer during this period. The necessary amplifiers,
digital magnetic tape recorders, and computer (HP
2100 A) with its peripheral equipment were located
in this measuring container. The measuring container
was located in hole 6 directly on the tank deck of
the after peak, in the last bay. For the determi-
nation of the performance data of the ship, strain
gauges were attached to the shaft. In addition,
the shipborne electro-magnetic log (system Plath)
for determination of the ship's speed was connected
to the computer via an isolation amplifier. Thus,
the ship's speed and power could also be recorded
at each pressure fluctuation- and LSL-measurement.
A recalibration of the log was made on the outward
voyage in the North Sea by means of a speed measure-
ment carried out by the Hamburg Ship Model Basin
using their method with a resistance log.
Bo PROPELLER EXCITED PRESSURE FLUCTUATIONS AND
CAVITATION OBSERVATIONS
The measurements of the propeller excited pressure
fluctuations were started on the outward voyage
when leaving the English Channel and continued until
the arrival at Marseille (Tests No. 1-11). Further
details on these measurements as well as for the
pressure fluctuation measurements carried out in
13-16)
the Mediterranean (Tests No.
Tables 2a and 2 b.
are given in
Laser-Beam
Photo -
Propeller— multiplier Laser
Plane
Cavitation -
©) observation
N
© .U
o—-
Windows
me
815 + 815
1255 a4
Oo
a nu
Ov
Pressure
pick-ups
FIGURE 1. Locations of test setups.
., Laserbeam
1)
Frame 137 l
12 cS
iN
Control - )
Volume
1450 VX
Propeller -
circle
rame
6 8 10 13 ]
| Photom. Laser
Lo]
Longitudinal
cross section
Venu -Tip
FIGURE 2. Arrangement of test setups. -
The results of the pressure fluctuation measure-
ments for the Tests No. 1-4, 11 and 13-16 are given
in Figure 3 showing the dimensionless pressure
amplitudes of the blade frequency for the pressure
pick-ups Pl, P3, P4, and P6. They have been
harmonically analysed on the HP-computer in the
measuring container. As usual with right-hand
propellers the pressure pick-up on the starboard
Side (here: P3) clearly shows higher values than
that on the port side (P4). Figure 4 shows the
amplitudes measured by these two pressure pick-ups
up to the 15th harmonic. The harmonic analysis has
been carried out for a "representative" revolution,
resulting from the average of 60 propeller revolu-
tions.
Figure 5 shows the pressure fluctuations measure-
ments versus propeller rpm for the pick-ups P3 and
P4 for two drafts applied during the voyage in the
Indian Ocean. At this point in time the propeller
was already damaged. Further data of these measure-
ment runs can be found in the Tables 3a to 3h.
Examples of the results of harmonic. analyses up to
the 15th harmonic order for the pressure pick-ups
P3 and P4 are shown in Figure 6. In Figure 7 a
comparison is given of the pressure amplitudes of
these harmonic orders for the pick-up P4 (port) in
shallow and deep water. In shallow water the
pressure amplitudes are only slightly higher (5.8%)
than that in deep water. With the pick-up P3
(starboard) the difference was even smaller (1.0%
increase) .
TEST NO | 1-4|11 [13-16
SYMBOL | = | 0 |
is)
TEST NO
SYMBOL
0.04;
0.03
0.02
80 90
100 —»n [RPM]
Undamaged Propeller
Mediterranean
FIGURE 3. Pressure fluctuations.
The lower pressure amplitudes of the blade
frequency in the Indian Ocean (Figures 5 and 6)
compared with that in the Mediterranean (Figures
3 and 4) are to be attributed to a significantly
stronger, but mainly stationary cavitation of the
damaged blade (No. 3). A comparison between Figures
4 and © shows that due to the damage the pressure
amplitudes of the "not-blade-number" frequencies
have been strongly increased in opposition to the
blade frequency. It should be noted that the ship
superstructure vibrated strongly after the propeller
had been damaged. This damage resulted from a ground-
Pressure Pick-Up P3 (St-B)
Pressure Pick-Up P4 (Port)
1 5 (0 15
TEST NO 15; 17-10-1977
n = 100.4 RPM; V, =21.3Kn
Harmonic Order n
FIGURE 4. Harmonic components of pressure fluctuations.
303
Table 2a
Test No. iy. 2 3 4 1]
Date ———— 9.10.1977 ————— 125 No 77
Speed V, [kn] 17.2 18.6 19.8 21.4 20.7
Revolution n_ [RPM] 89.1 94.3 98.8 104.7 100.4
Power Pp [MW 11.4 13.0 14.6 16.8 14.9
Draft aft [m 8.94 8.94 8.94 8.94 8.33
Draft forward [m 6.35 6.35 6.35 6.35 6.96
Course 209° 231° 2 Bie 230° 37°
Sea region —— English Channel ——— Medit.
Wind [Beauf | 4 4 4 4 (6)
Wind direction 180° 180° 180° 180° )
Water Depth [m] 36 35 41 51 1040
Table 2b Measurements in the Mediterranean
Test No. 13 14 15 16
Date ——— 17.10.1977
Speed Vs [kn] 18.6 19.5 Dio) DD,,3)
Revolution n_ [RPM| BONS) 98n2) MOOG 105.1
Power Pp [MW Ook 12.2 15a Ws)
Draft aft [m O99 Bold Dats 9.73
Draft forward [nm] O63 9s08 9463) 9.63
Course 11s? MA . 1ae 114°
Sea region — 36°45'N; 18°49'E (Mediterr.)
Wind [Beauf] = | 806
Wind direction ——_———_ 90°
Water Depth [m] = 3500 =—
Table 3a Measurements in the Indian Ocean
Test No. 47 59 60 61 62 65
Date 30.11.77 1.12.77 —_—— Dol2ovT
Speed V, [kn] Die? 21.4 2S Dilo8) DNS Di33
Revolution n [RPM] 101.1 101.6 101.2 101.8 101.0 101.3
Power Pp [Mw Nod = = —= = —=
Draft aft [m 9.30 —————— _ 9.30 ——__=— 9.30
Draft forward [m] 7.620 — 7.62 — 7.62
Course 294° — 294° — 2gi”
Sea region or 1a 1S Onende Oe 9°09'S;
aasieien Hones | Ta Ne 20 88 Te San CATE
Wind [Beauf | 6 = 3 — 2 = 3)
Wind direction 100° ——__$_—_ 70°. —__—_=— 230°
Water Depth [m| 3300 ——__—— 4900 —————= _ 2000
Table 3b Measurements in the Indian Ocean
Test No. 70 7\ 72
Date ————— 4.12.77 ————
Speed V, [kn] 21.8 22.1 21.8
Revolution n_ [RPM] 101.7 101.9 101.3
Power Pp [MW = = =e
Draft aft [m — Qo S7 —
Draft forward |m —_ 8.08 —
Course — 314° —
Sea region or position = 2°58'N; 59°44'— ——=—
Wind [Beauf] = 142 —
Wind direction en BRO
Water Depth [] ——__§_—— 3250 ———_=—
Table 3c Measurements in the Indian Ocean
Test No. 73 74 75 76 77 78
Date — A V2o07 —_
speed V, [kn] Dito fl 20.9 — zee 16.9
Revolution n [RP] OT 2 96.8 95).4 92.6 B2I9) 82.5
Power P, [MW — 11.6
Draft aft [m —- 9.37 —
Draft forward [m) — 8.08 —
Course =— 314° =
Sea maglon Of POSa —————— 2°58'N; 59°44°— ———————e—
Wind [Beauf] = 32 —_
Wind direction = 235° —
3250) —
Water Depth {m]
Table 3d
Test No.
Date
Speed V. [kn]
Revolution n [RPM]
Power Pp [MW
Draft aft [m]
Draft forward [m]
Course
Sea region or pos.
Wind [Beauf]
Wind direction
Water Depth [m]
Table 3e
Test No.
Date
Speed V [kn]
Rexoilwe ton n [RPM]
Power Pp [MW
Draft aft [m
Draft forward [m]
Course
Sea region or pos.
Wind [Beauf]
Wind direction
Water Depth [m]
Table 3f
Test No.
Date
Speed V, [kn]
Revolution n [RPM]
Power Pp) [MW
Draft aft [m}
Draft forward {m]
Course
2°58'N;59°44'E;
Measurements in the Indian Ocean
79 80 81
—_—_____—_ 4.12.77
11.9 11.9 Nite’)
5 OFZ 60.1 61.3
7 Io8 =
7 9.37 9.75
08 8.08 7.82
—_____—_——._ 314°
——____—__ ] +2
2252
3250 3250 3250
82 83
Noe 12.0
0.9 61.3
ors) oS)
7.82 7.82
_————
—— 3°15'N359°27'E ——
————————— ee
a
3250 3250
Measurements in the Indian Ocean
84 85 86 87 88 89
as 1D 5.12.77 =>
17.5 17.5 17.5 19.7 20.4 20.4
85.3 85.1 85.2 95.4 96.5 96.3
13.9 — —
= 9.75 =
oe 7.82 _
a 8) eee
SBIR SOP OTN —
= te? =
— 230° —
— 3250 ——
Sea region or position
Wind [Beauf]
Wind direction
Water Depth {m]
Table 3g
Test No.
Date
Speed Vg [kn]
Revolution n_ [RPM]
Power Pp [MW
Draft aft [m
Draft forward [m]
Course
Sea region of pos.
Wind [Beauf]
Wind direction
Water Depth [m]
Table 3h
Test No.
Date
Speed V, [kn]
Revolution n [RPM]
Power Pp oa
Draft aft [m
Draft forward [m]
Course
Sea region or pos.
Wind [Beauf]
Wind direction
Water Depth [ml
Measurements in the Indian Ocean
90 91 92
SIZ
—_> 21.6 =
100.8 101.3 101.3
———_———- 9.75 eee
= 7.82 —
)
SS SG
8
15 9Ng SOS. ——
= Ori
= 23009 —_—_—>
— 3250
Measurements in the Indian Ocean
93 94 95 96 97
ae 95>.12.77—
Des 21.4 21.2 Die? Diath
101.0 101.0 101.0 101.2 101.2
== = = — DDD
— 8.63 Som
= 8.23 =—
<i ie) Fee
——————._ 8°28'N; 54°40'E —————
es 4
at SS ES
SS OOO
Measurements in
99 100 10]
G26
22.4 22.2 22.3
103.1 102.3 102.8
16.9 —
= 8
= 8.
— 273°
12°21'N; 47°03'E
ee ee
2.
si
the Gulf of Aden
103 104 105
——— Zoe
22.3 DP <3} 225)
101.8 101.9 101.
17.0 = —
2730, =
Bab-el-Mandab
37 38 40
304
{Kes ae Pressure Pick-Up P3 (St.-B)
0.03
TEST NO |70-80| 81-92
. +
0.02
0.01
0
Pressure Pick-Up P4 (Port)
0.03
TEST NO 70-80) 81-92
SYMBOL
0.02
0.0
0
60 70 80 90 100 Nn
[RPM ]
Damaged Propeller
Indian Ocean, 4-12-1977
FIGURE 5. Pressure fluctuations during laser-scattered-
light (LSL)-measurement.
ing due to a thunderstorm at the entrance of the
Suez Channel. The cavitation of the damaged blade
was so strong that it existed during the total pro-
peller revolution. This could be seen through the
cavitation observation windows. Unfortunately, no
photographies were made because the measuring crew
of Det Norske Veritas carrying out the cavitation
observations left the ship in Port Said.
In the Mediterranean, however, a large number
(about 800) of black-white photographs of the
undamaged propeller were made with the equipment
of Det Norske Veritas with stroboscopic lighting.
Since pictures were always taken with two Hasselblad
cameras it might be possible to carry out stereo-
Pressure Pick-Up P3 (St-B)
Pressure Pick-Up P4 (Port)
1 5 ae Re 15
TEST NO 70 rmonic Order n
Damaged Propeller V, = 21.8 Kn
Indian - Ocean n =101.7RPM
FIGURE 6. Harmonic components of pressure fluctuations
during LSL-measurement.
305
Deep Water with Low Nuclei Content
Test No 99: V, =22.4Kn; n= 103.1RPM
Shallow Water with High Nuclei Content
Test No 105: V.=22.5Kn;n = 101.7RPM
1 5 10 —» 15
Pressure Pick-Up P4 (Port)
Damaged Propeller,
Indian Ocean
Harmonic Order n
FIGURE 7. Harmonic components of pressure fluctuations.
metric measurements of the cavitation layers in
dependence of the blade positions. As an example
for the cavitation extension of n = 105 rpma
collection of photographs is shown in Figure 8.
These pictures were made with a camera with a fisheye-
objective. The photographed condition belongs to
Test No. 16.
4. INVESTIGATION OF THE WATER QUALITY
Measurements of Suspended Particles
In addition to nuclei measurements, which will be
described later, the content of suspended particles
was investigated as often as possible. This was
necessary for two reasons: the LSL-method does
not allow direct differentiation between solid and
gaseous particles. Thus it became necessary to
estimate the proportion of dirt or organic particles
(probably contained in the water) in the measured
nuclei sprectra. For these investigations a
scattered-light instrument (nephelometer) was used;
the J4-7439 fluoro-colorimeter of the American Instru-
ment Company (Aminco). The Aminco-scattered-light
instrument works on almost the same physical princi-
ple as the LSL instrument. Water samples of 1 cm,
investigated in the Aminco-colorimeter under a
scattered light angle of 90° were exposed to a green
light (514 nm) as in the laser control volume. The
geographical positions where the Aminco scattered
light measurements were carried out (as well as all
the other measurements described in this report)
are shown in Figure 9.
The results of the Aminco scattered light investi-
gations, given in Figure 10, were obtained in the
following way:
Water samples were taken from the condenser in-
flow of the ship's turbine during the voyage. One
part of this water was poured through a filter with
a pore size of 0.4 um. Another part was used for
unfiltered samples, which previously were roughly
degassed by stirring and shaking. Subsequently,
the unfiltered and filtered samples were investigated
306
KDrsesth 1 15°stb.
Ta =9.75m
Gy =0.22
FIGURE 8. Cavitation "Sydney-Expess."
in the Aminco-colorimeter. The deflection of the
meter for the filtered sample was adjusted on the
indicating scale to "0", which served as reference
value. Measured values of unfiltered samples are
shown in Figure 10; Relative Intensity is an
arbitrary unit.
The first measurements, at the end of October,
were made with a one-hole-aperture in the beam
path, the following ones with a four-hole-aperture
due to a thereby increased intensity.
In order to obtain a general idea of the sensi-
tivity of the Aminco scattered light method,
standard solutions were produced using the plastic
spheres also used for the calibration of the LSL-
instrument. It is apparent from this that five
parts per cm? with a diameter of D = 25.7 um could
still be measured.
Many results from investigations of sea water
did not show any difference between filtered and
unfiltered samples. The content of suspended
particles was thus very small in the Indian Ocean;
it was below the response level of the Aminco-device.
The samples taken on the 7th December 1977 contained,
however, suspended particles. They descended from
the shallow water region of the Bab-el-Mandab at
the entrance to the Red Sea.
The lack of knowledge about the back scattering
qualities of the particles appears to be a problem
when applying this scattered light method with the
£A.\ 20°Stb.
f
HAL 25°stb.
FAL 30°s tb.
Veg =22.3kn; n=105RPM
P,=17. IMW, Jkq=0.69
Aminco-colorimeter. A more expanded and intensive
investigations of suspended particles, for instance,
with coulter counter, could not be carried out within
the frame of this research work.
¥ Aminco - Colorimeter
T Total Air Content
+ Scattered Laser Light M.
® Velocity Measurements
FIGURE 9. Positions of measurements.
0 1,0 2,0 3,0 40 5,0 6,0 7,0 Relative Intensity
1 n 4
307
iL
Seawater, 27. 10.77 (One-hole aperture)
Seawater 28. 10.77
: 31. 10.77
2.11.77
2.11.77, 20 naut. miles before Fremantle
Swan-River 3.11.77; Fremantle
7.77
11.11.77, Swanson Dock West
7, White Ba’
212.77
612.77 (Gulf of Aden)
612.77
[_]Seawater 712.77 (Shallow water, Bab-el-Mandab)
71277 (Red Sea)
Seawater 10.12.77 Great Bitter Lake
11.12.77 (Mediterranean, 15 naut.miles behind Port Said)
Seawater 12.12.77 (lonian Sea)
Measurements of Total Air Content
Although the water should always just be saturated
at the surface, the gas concentration c_ of the
sea water was also continuously determined from
water samples with a Van-Slyke-apparatus. The
results are given in Figure 11, in dependence of
the temperature.
For the calculation of the gas content ratio,
Ee = c/e , the gas saturation capacity, c_, is
necesSary for the specific salt content and temper-
ature. Since the corresponding data were not known
some water samples were left overnight in a basin
with a large surface and the gas concentration c
was determined on the following day, which in this
case should indeed correspond to the saturation
concentration c_. The two values obtained for the
saturation concentration c_ are also plotted in
Figure 11 (with the symbol -O- ). They are within
the range of tolerance of the measured total gas
content, Co, for the voyage leading southward.
Subsequently the measured total gas content present
values which correspond to the gas content ratio,
€ = 1, i.e., to saturated water. Due to the
dissolved salt the total gas content values, Co,
for sea water should lie below the values for fresh
water. This is, in fact, the case with the exception
of some values of the voyage leading northward. It
must be left to other investigations to find out
whether the wind, seaway, and temperature "history"
of the sea surface has an influence on the total
gas content.
5. MEASUREMENTS OF THE NUCLEI SPECTRA AND LOCAL
VELOCITY
Device for Nuclei Measurement
The LSL method was applied to the measurement of
the nuclei spectra in front of the "Sydney Express"
Standard-Solution, D = 25,7 um; Cpart = 1,0 - 10 n/cm3 2810.77
St=Sol D=25,7 um; Cpa = 54 nicm3 31.10.77 cea for particle size
St-Sol D=25,7 um; Cport = 5 nicm? 1.1.77 D= 25,7 wm an
St=Sol D= 1011 wm; Coon = 176-108 niem3 2311.77 WA7Ret Int
aes 610 2311.77
Cpart = 176-108 n/cm? 23.11.77 = a a vale a ah
St.-Sol D = 1,011 pum; Cpart = 1,76 102 n/cm3 23.11.77
ae} 39
(ey Rel. Int.
a DPV 7 Ny ; same sample after 6 hours)
FIGURE 10. Measured suspended particles.
propeller. This method was also applied to the
model tests, described by Keller and Weitendorf
(1975). Detailed information about the measure-
ment principle has been supplied, for instance, by
Keller (1970, 1973). Thus, it is not necessary
to go into the details.
Compared with previous measurements carried out
in the laboratory the measuring distances were
essentially larger at these full scale investigations.
Thus, some new components for the measuring device
were required. The distance between the measuring
volume and the receiving lens amounted up to 2 m so
that the laser power and the diameter of the
receiving lens had to be markedly increased, in
order to obtain usuable measuring signals.
The arrangement of the measuring unit on board
is shown in Figures 1 and 2. The path of the laser
beam is bent three times and enters the water almost
horizontally; the path of the beam of the receiving
system is bent once and proceeds in the water
vertically. With this arrangement the flow direction
Co
[foo]
20 Se
(O59) aso= Saturation of air in pure water (70 mm Hg )
Measured total air content in seawater southbound voyage
Measured total air content in seawater northbound voyage
Saturation of air in seawater
2 Port Phillip Bay (Melbourne, 7. 11.1977 )
10 15 20 25 We
FIGURE 11. Measured total air-content.
308
and the direction of the laser beam as well as the
optical axis of the scattered light receiving system
are standing vertically, one upon another. This is
optimal for the measuring technique used.
The homogenization of the laser beam, i.e. the
conversion of the Gaussian intensity distribution
over the beam cross section into a rectangular
distribution, was made with a special filter. The
homogenous intensity distribution as well as the
shape of the laser beam (square or rectangular)
were maintained quite well by the very long focal
length of the laser system (about 6 m).
The control volume, optically defined, was
positioned in such a way that the stream line
through the control volume came into the range of
the propeller tip. The position of the control
volume in front of the propeller was determined by
the position of the reception window of the scattered
light between frames 12 and 13, i.e., 4.2 m in front
of the propeller plane. The additional geometrical
fixing of the control volume in the vertical direction
resulted from the laser window (located between
frames 13 and 14) with its horizontal beam outlet
into the water. Subsequently the positions for
the control volume was fixed as follows: 90 cm of
the ship's hull vertically downward and 145 cm from
midship on the port side between frames 12 and 13
(see Figure 2).
The Calibration Device
The relationship between the photomultiplier impulse
amplitude and the size of nuclei was determined by
a calibration with latex spheres. For this purpose
a special device was put through an opening in the
ship's hull when the ship stopped in calm water.
With this device it was possible to maneuver a
fine nozzle near to the control volume and to inject
the latex spheres into the control volume. The
apparatus was operated by means of small hydraulic
elements from the inside of the ship (Figure 12).
For the calibration latex spheres of 45 and 25
Um were used. The corresponding photomultiplier
impulse amplitudes fit excellently to the theoretical
curve of the scattered light intensity. The measuring
range was set to 8-117 Um for the nuclei diameter.
In addition to the scattered light intensity,
the dimensions of the control volume were important
data for the determination of nuclei spectra and
nuclei concentration. Since a direct measurement
or calculation of the cross section of the laser
beam in the control volume was not possible in this
case, a new method had to be applied to determine
the laser beam dimensions. By means of the above
mentioned hydraulic device a small rotating wheel
with thin platinum wires was adjusted in such a
way that the wires cut the laser beam vertically
at the location of the control volume. Thus the
light in the direction of the photomultiplier was
scattered. The dimensions could then be determined
from the width of the photomultiplier impulses, the
distance between the axis of the small wheel, and
the light point on the small platinum wires
(determined by crossed platinum wires) and the
revolution number of the small wheel. The diameter
(25 um) of the platinum wire had also to be con-
sidered. The exact knowledge of the control volume
dimensions was also important for the measurements
of the local velocity, as described below.
The dimension of the control volume in the longi-
Sluice Valve
Micro - Hydraulic Device
6mm Pipes flexible
6m long
Pipe 10cm (Length 2.5m )
Seawater
FIGURE 12. Calibration device and arrangement.
tudinal direction of the laser beam was adjusted as
usual through the measuring slit in front of the
photomultiplier, after the enlargement factor of
the reception optic was determined. This again was
done by means of the hydraulic device with which
an object, whose dimensions were known, was placed
in the control volume; its picture was measured in
the plane of the measuring slit.
For the nuclei measurement the dimensions of the
control volume were then fixed as follows: 0.86 mm
x 0.86 mm X 1.33 mm = 0.98 mm®. The cross section
of the control volume rectangular to the flow
direction amounted to 0.86 mm X 1.33 mm = 1.14mm’.
This detail was required for the determination of
the nuclei concentration.
Measurement of Local Velocity
When the cross section of the control volume and
the number of the nuclei measured per unit time,
were known, it was necessary in addition to know
the local flow velocity at the control volume in
order to determine the nuclei concentration. Since
the conversion of model test results from wake field
measurements to full scale appeared to be too in-
accurate for the determination of the local velocity
and because the measurement of the velocity with a
Prandtl tube, for instance, was not possible, a
new method was applied to measure the velocity and
flow direction. If the dimensions of the control
volume in the flow direction are known, the velocity
can be determined from the measured impulse width.
In order to estimate the local flow direction at
the control volume, an aperture is put into the
beam path of the laser (Figure 13), which gives
the laser ray a rectangular shape. This aperture
is turned until the photomultiplier impulses have
reached a maximum. Then it is possible to determine
from the position of the aperture the position of
the plane, formed by the flow direction and the
laser beam. If now the measuring slit is turned
until the half width of the distribution of the
impulse width has reached a minimum, it is possible
to read - from the position of the measuring slit -
the plane which is formed by the flow direction
and the optical axis of the reception system. The
flow direction in the volume results from the inter-
section of the two determined planes; the impulse
width gives the flow velocity, and the impulse
width spectra provides information on the degree
of the turbulence flow.
In this way flow characteristics can be determined,
undisturbedly and locally, with one measurement;
otherwise they could only be determined with a
three-component measurement. Furthermore, the
control volume simultaneously reaches the optimum
inclination for the measurement of the nuclei size.
Thus one signal provides data about the distribution
of nuclei size and about the flow field.
Large particles or bubbles require a longer
period to completely cross the control volume than
smaller particles at the same speed. This means
that besides the larger impulse amplitude there is
also a larger impulse width. These facts have to
be considered in the measurement of the velocity.
Therefore, a single-channel discriminator is inserted
into the impulse processing electronics. The
discriminator choses for the measurement only
impulses of the amplitude or a strongly limited
range of amplitudes. Thus it is possible to draw
a clear conclusion from the measured impulse width
on the speed of the particles in the control volume.
The new technique to measure the velocity is
illustrated in the Appendix. A rectangular beam
cross section whose breadth is the vertical to the
flow direction, has proved to be the optimum for
the measurement of velocity and the determination
of the flow direction.
4 Spectral Filter
309
General Remarks
Originally it was planned to shift the height of
the measuring point on the optical axis of the
reception system by different laser beam directions.
In addition, this axis should be shifted laterally
through two additional observation windows between
the frames 12 and 13. This would make it possible
to measure at several points in the plane between
the frames 12 and 13. Unfortunately, this could
not be realized due to lack of time, because the
installation of the measuring equipment at the
beginning of the voyage had taken too much time.
It is not intended to describe all the diffi-
culties which occurred at the installation of the
equipment. The problem of vibration, however, must
be mentioned.
To protect the laser, vibration damping should
be guaranteed as far as possible. It was, however,
observed during the outward voyage that the pneu-
matic vibration isolation, which had a resonant
frequency of £, = 1.8 up to 3.0 Hz, could not be
used, - even if the exciting blade frequency of the
propeller was within the range of 8 and 9 Hz.
Excitations occurred, of course, also at a propeller
speed of f = 1.8 Hz and due to seaway frequencies.
When it was obvious that different damper devices
also did not help, the support, on which the laser
and the photomultiplier were fixed, had to be
stiffly connected with the steel construction of
the after peak. This labor and the laser adjust-
ments required more than half the time of the voyage
to Australia during difficult climatic conditions.
The laser adjustment was carried out mainly when
the ship was stopped. The calibration of the
nuclei impulses and the determination of the control
volume, in which the nuclei were measured, were
also carried out during these periods. These were
kindly granted by the captain and his officers and
had to be regarded as a special concession since
the "Sydney Express" was on a fixed schedule. In
this connection it must also be mentioned that the
calibrations and later the measurements, made on
the return voyage, could only be carried out after
dark. For this reason, extra maneuvering watches
had to be set in the engine control room, usually
while the ship had a "16-hours-unattended-machinery-
space".
The above mentioned stiff support solved the
1 Laser 5 Grey Wedge Filter
2 Beam Expander 6 Rectangular Aperture
3 Aperture 7 Lens
Idealized Photomultiplier
signal from scattering
(ets objects.
A Idealized signals with grey
wedge filter(5) in rays.
(eee Slope indicates direction.
8 Microscope Lens
9 Flow Section
10 Control Volume
11 Receiving Lenses
12 Measuring Slit
13 Photomultiplier
FIGURE 13. Principle of LSL-measurements.
310
vibration problem almost completely. It provided,
however, the risk that the laser might fail.
Fortunately, this did not happen. The laser, a
Coherent-Radiation (4 Watt) product, achieved the
same performance (900 mW), to which it was adjusted
at the beginning, up to the end of the voyage with-
out any failure. The small vibration still observed
at the measuring point had no significant effect.
6. RESULTS OF MEASUREMENTS
Local Velocity
It was mentioned already that the number of measure-
ments originally planned could not be carried out
due to lack of time. Thus, for instance, the size
and the direction of the local velocity could only
be determined at one measuring point. This measure-
ment took much time since there was no special
electronic device available. It was the first
measurement of this kind and it was included in the
program at a late date, which made it impossible
to establish a special measurement before the
departure. The measurement was, therefore, partly
performed with the electronic device which was also
used for the scattered light measurements, and with
some special interfaces.
The velocity and one plane of the flow direction
at the place of the control volume could be deter-
mined for one velocity. At the ship's speed of
22 kn the velocity at the measuring point amounted
to 7.22 m/s and the direction was found at an angle
of 5° downward. The corresponding result from the
model test for the geometrically corresponding
position of the "Sydney Express" amounted to 7.47
m/s. This model test, however, was carried out for
the propeller plane of the towed model, without a
running propeller. - In full sale, on the other
hand, the plane formed by the flow direction and
by the optical axis of the reception device could
not be determined due to lack of time.
[eal
Relative Abudance —= =
0) 50 100 150 —» 200
T [asec |
FIGURE 14. Pulse width distribution and mean pulse
width dependent on the inclination of the flow.
NU Test 47
cm? Measuring Time:
100 try = 6-8 sec
= 3
1.0 €, = 37 N/cm
0.1 n =101.1RPM
0.01 Vs =21.2kn
0.001
0 20 40 60 80 100 ~m
— Diameter
30 -11 -1977; Wind Force : 6 Beauf.
FIGURE 15. Nuclei distribution.
The ratio, local velocity to ship's speed, 7.22
m/s to 11.32 m/s, and which corresponds to the
local wake in the control volume for the ship at
22 kn, was applied for all nuclei concentration
measurements. The nuclei concentration was then
calculated from the recorded ship's velocity, the
measuring period, and the measuring cross section.
Figure 14 shows examples of the velocity measure-
ment and also the change of the impulse width
distribution for the rotation of the rectangular
laser aperture. The value i = 0° corresponds to
the horizontal plane. At 5° downward (A = -5°) the
mean impulse width, evaluated on the HP-computer,
reaches its minimum at 59.6 Usec. The large half-
width of the distribution curve results from the
turbulent flow. With a laminar flow the distribution
curve would be smaller. (See Figures A 2.2 and
IX AoS)o
On the basis of these measurements a quantitative
statement about the turbulent degree of the flow
cannot yet be made. On the one hand we have no
experience with this measuring technique, on the
other hand the ratio, length to width of the laser
beam cross-section, was too small at this measure-
ment (2:1). At high turbulent flow the corners of
the beam cross-section were dispersed by a relatively
high amount of nuclei which resulted in shorter
photomultiplier impulses than with nuclei running
through the middle of the beam. A higher ratio,
length to width, would be more favourable.
The first practical experiences with this mea-
suring technique are so promising that its further
development is being promoted. The advantages which
this measuring procedure offers in connection with
the determination of the size of nuclei are quite
remarkable.
Nuclei Spectra
About one third of the spectra obtained between 30
November and 7 December 1977 are demonstrated in
Figures 15 through 24. The spectra contain the
respective sum of nuclei per cm? for the respective
range of diameters. In the diagrams one range of
diameter is marked by a horizontal line. The single
ranges of diameters do not have the same width.
The dissimilarity of these spectra, which obviously
results from different conditions, will later be
described in detail.
First, it has to be noticed that for all spectra
in the range of a bubble diameter from 20 to 40 um
(micron) there is either a relative maximum or an
absolute maximum of nuclei. The relative maximum
tea Test 59
10 Meas. Time :
; tm = 19.3 sec
0.1 8 = 13 N/cm3
0.01 n =101.6RPM
Vs =21.4kn
0 20 40 60 80 100 Hm
—~= Diameter
1-12-1977 Wind Force 3 Beauf.
Test 65
Measuring Time :
tm = 1.59 sec
B= 159 N/cm?3
n=101.3RPM
V=21.3 kn
0 20 40 60 80 100 Am
—= Diameter
2-12-1977; Wind Force: 2-3 Beauf., Heavy Swell
FIGURE 16. Nuclei distributions in seaways.
was detected in Test 47 - Figure 15, Tests 60 and
62 - Figure 17 and Tests 90 to 92 - Figure 20.
The absolute maximum was detected in Test 61 -
Figure 17 and Test 65 - Figure 16. The strong
fluctuation of the number of nuclei per em? (nuclei
concentration €9) can be read from the diagrams.
Figures 15 to 17 as well as 18 and 19 show spectra
which have been measured in different seaways.
During the performance of Tests 47 to 62 (Figures
15 to 17) there was a seaway and swell from astern.
The strong pitching motions of the "Sydney Express"
ae Test 60
Measuring Time:
tm = 17.03 sec
0.1 b= 15N/cm3
0.01 n=101.2RPM
Vez 21.3 kn
0.001
10.0 Test 61
Measuring Time:
10 t py = 1-16 sec
01 5 =219 N/cem?
n =101.8RPM
Co Ve= 21-3kn
0.001
Test 62
1.0 Measuring Time:
tr = 12.33 sec
041 i——| €)=21N/cm?
n=101.0RPM
a9) s= 21.3kn
0.001
0 20 40 60 80 100 ~§=©120 Bm
1-12-1977; —= Diameter
Wind Force: 3 Beaufort with Swell
FIGURE 17. Nuclei distributions in a seaway.
Shia
€5=15N/cm3
1 9
Cell 2000 17 21 25 No of Rev
Cell 0 — Time
Test 60 ; Measuring Time: t,,=17.03 sec
225
Cell 2000 2933 37 No of Rev
Cell 0
Test 61; Measuring Time : tm= 1-16 sec
€5=219N/cm3
FIGURE 18. Nuclei distributions in a seaway.
resulted in a strong fluctuation of the nuclei
concentration, i.e. from Co = 21 N/em? 10) (Gi) = AILS)
N/cm*? in the Tests 62 and 61 - Figure 17. Depending
on the quality of the water, either swarms of
bubbles or clear water, which hits the control
volume of the laser beam, 2000 nuclei were counted
more or less quickly. The process of counting the
2000 bubbles is demonstrated in Figure 25 for two
cases. There the analog-output voltage of the
memory is plotted against time and propeller revo-
lutions respectively. The output voltage of 1 Volt
is reached by the memory when its 2000 cells are
filled. In Test 61, 2000 bubbles were counted
within 1.16 s and in Test 60 within 17.03 s at an
almost linear processing of the output voltage.
Figures 18 and 19 show also a series of nuclei
spectra measured, one immediately after the other,
at a seaway of Beaufort 4. At this series the
direction of the seaway was, however, athwartships
up to "slightly from fore". The seaway motions of
the "Sydney Express" (length 210 m) were very
small in this case. Subsequently the number Co of
nuclei per cm? was higher than the smallest number
N
lon? Test 93
10.0 Meas. Time:
tm = 412 sec.
Lo -| €5=62N/em3
0.1 n= 101.0RPM
Ve=21.3k
0.01 s :
0.001
Test 94
1.0 Meas. Time :
tm = 5.63 sec
0.1 |__| § = 45N/cm?
n =101.0RPM
0.01 Vg = 21-4 kn
0.001
10.0 | Test 95
0 Meas. Time:
1. tm =4.20sec
01 b= 61N/cm3
n =1010RPM
0-01 Vs =21.2kn
0.001
0 20 440 60 80 100m
—e Diameter
5-12-1977; Wind Force: 4 Beaufort
FIGURE 19. Nuclei distributions in a seaway.
N
ae Test 96
10 Meas. Time:
tripsisgoisec
04 = 43N/cm?
0.01 n =101.2RPM
Ve =21.2 kn
0.001
Test 97
1.0 Meas. Time :
tm =6.5 sec
0-1 by =39N/cm3
n = 101.2RPM
Ve = 21-4kn
0 20 40 60 80 100 wm
—e= Diameter
5-12-1977 Wind Force: 4 Beaufort
FIGURE 20. Nuclei distribution for calm water.
with the pitching ship but still within the narrow
range of To = 39 up to To = 62 N/com?. Considering
the Tests 47 to 65 (Figures 15 to 17) and the Tests
93 to 97 (Figures 18 and 19) it can be said that in
a seaway the nuclei concentration G9 is higher than
in smooth water, and further, that the influence of
shipmotions on the concentration Cg superimpose on
the influence of the seaway.
The measurement series carried out with the
Tests 90 and 92 (Figure 20) under ideal weather
conditions, show on one hand the good repetitive
accuracy of the results for a constant speed in
good weather. In this case the number of nuclei
per em? amounts to So = 18, 19, and 18 N/cem? at
constant measuring periods of 13.6, 13.2, and 13.9
for 2000 nuclei. This, however, shows - on the
other hand - a clearly lower concentration, Co,
than with the Tests 61 and 65 in a seaway (Figures
17 and 16).
lens Test 90
Measuring Time:
try = 13.6 sec
b= 18 N/cm?
0.01 n =100.8RPM
Vg= 21.6 kn
0.001
Test 91
1.0 + Meas. Time:
try = 13.2 sec
0.1 Se 19N/cm3
0.01 n =101.3RPM
V,= 21-6 kn
0.001
Test 92
1.0 Meas. Time:
tm = 13.9 sec
0.1 “69 =18N/cm?
0.01 n =101.3RPM
Vo =21.6 kn
0 20 40 60 80 100 §=9120 Am
—» Diameter
4-12-1977
FIGURE 21. Nuclei distributions for different ship
speeds.
N
(5 Test 70
10 Measuring Time:
tm=13.55sec
ca = 18N/cm?3
n =101.7RPM
Ve =21.8 kn
0.001
Test 77
1.0 t———; Meas. Time :
tm =11.5 sec
0.1 = bo =27 N/cm3
0.01 n =82.9RPM
Ve =17.1kn
0.001
10.0 Test 79
1.0 Meas. Time:
tr =3.3sec
0.1 6, =137 N/cm?
n =59.2RPM
eo Vg =11-9kn
0.001
0 20 40 60 80 100 um
—=Diameter
4-12-1977, Draft aft Dg=9-37m; Draft forw. D¢ =8:08m
FIGURE 22. Nuclei distributions for different ship
speeds.
A further difference, previously mentioned, has
to be noticed when comparing measurements in a
seaway and in calm weather. Whilst with typical
spectra in a seaway (Tests 61 and 65) the absolute
maximum is between the nuclei sizes 30 and 40 um,
it can be detected for calm weather in the smallest
measured, nuclei range. This phenomenon will be
described later (in Section 7). The measuring
series of different speeds for two drafts are
as Test 90
1.0 Meas. Time:
tm =13.6 sec
01 € 5 =18N/cm3
n =100.8RPM
Ve=21-6 kn
0.001
Test 85
1.0 Meas. Time :
ta = 11.3 sec
bo = 28N/cm3
n =85.1RPM
Ve=17-5kn
0.001
10.0 Test 83
| Meas. Time:
1. tm= 3.2 sec
b= 140 N/cem3
n =61.3RPM
V2 12.0kn
(in 20) AOMGON MN GOMMNLIOO) rn
—= Diameter
4-12-1977; Draf aft Dq=9-75m; Draft forw.D¢= 782m
FIGURE 23. Nuclei distributions on deep water.
N
ae Test 99
1.0 Measuring Time:
tm = 20.04sec
01 9 =12N/cm3
0.01 n =103.1RPM
= 22.4kn
0.001 Vs
i Test 100
0 Meas. Time :
try = 16.90 sec
01 ===)
5 = 14 N/cm3
0.01 t——--
n = 102.3RPM
0.001 —| Vs = 22.2 kn
Test 101
velo Meas. Time :
04 tm = 22.06sec
C= tN /cm?
n =102.8RPM
Ve = 22.3kn
0 20 40 60 80 100 um
—e Diameter
6-12-1977; Gulf of Aden
FIGURE 24. Nuclei and particle distributions on
shallow water.
shown in Figures 21 and 22. In the second case the
"Sydney Express" was ballasted with 5,160 tons of
water. In both series it should be noted that with
decreasing speed the number T 9 of nuclei per cm?
increases. At the lowest speed of ca. Vg= 12 kn
the bubble range of a diameter between 20 and 40
Um contains the absolute maximum number of bubbles.
The differences between the two cases are, however,
{ N
‘cms
104 Test 103
Meas. Time:
1.0 tm = 1-56sec
= 3
O1 b= 155 N/cm
n = 101.8RPM
0.01 Vg = 22.3 kn
0.001
10.
oo Test 104
1.0 Meas. Time:
tm = 1.77 sec
0.1 bo = 137 N/cm3
n =101.9RPM
oo Vg = 22.3 kn
0.001
Test 105
10.0 Meas. Time :
10 tr = 0.98 sec
Go =270 N/cm3
o n =101.7RPM
0.01 Vs =22.5 kn
0.001
9 2 2 WW Go >P) |[/Aun]
7-12-1977 ; Bab-el-Mandab
FIGURE 25. Analog output of memory.
313
small. The differences between the drafts were
obviously not sufficient to provide stronger differ-
ences between the nuclei spectra.
The two measurement series shown in Figures 23
and 24 were made under ideal weather conditions,
the one 7 hours later than the other. The spectra
from Figure 23 were obtained in deep water in the
Gulf of Aden; the spectra shown in Figure 24 were
obtained from shallow water at the entrance of the
Red Sea at Bab-el-Mandab. With these two series
it was intended to clarify the point that the
propeller excited vibrations which occur on shallow
water result (apart from the shallow water effect)
to a higher extent from a stronger instationary
cavitation, which arises on occount of an increased
nuclei concentration in shallow water. It must be
said that this question could not be answered. On
the other hand a comparison of these two measurement
series shows that the number of nuclei per cm?
(nuclei concentration Co) increases from a fo of
11 to 14 N/cm? in deep water to a Z) of 155 to 270
N/cm*® in shallow water. This will be described in
the following Section. The absolute maximum of Tp
is here again in the range between the nuclei
diameter of 20 to 40 um. In this connection it
should be noted that hardly any nuclei with a
diameter of above 60 Um were detected.
7. DISCUSSION OF THE RESULTS OF NUCLEI SPECTRA
AND COMPARISON WITH OTHER INVESTIGATIONS
Simultaneously with the nuclei measurements in
shallow water - Figure 24 - water samples have been
taken. The results of the tests carried out with
these water samples with the Aminco scattered light
device appear in Figure 10. These samples from the
shallow water region at Bab-el-Mandab showed a
Relative Intensity of 0.4 for the difference between
unfiltered and filtered water. Even after six
hours the unfiltered sample still showed a Relative
Intensity of 0.28. From this it can be concluded
that the suspended particles, existing at this
coastal strip, settled in the samples within six
hours. From this Aminco scattered light measurement
it can further be concluded that the high nuclei
concentration shown by the LSL measurement - of the
shallow water measurement series, Figure 24 - results
mainly from suspended particles. There were probably
also solid particles concerned (it is likely to be
sand at the coast of Arabia) which show no inclusion
of gas. This is assumed because the cavitation did
not increase in the shallow water. The corresponding
propeller excited pressure fluctuations in deep and
in shallow water show practically no difference,
Figure 7.
In Figure 26 the results of the laser-scattered-
light technique and the Aminco scattered light
measurement for investigations in shallow water
(Test 105) are shown together. Figures 27 and 28
(in the diagrams marked with "Sydney Express") show
further results of the Aminco-scattered-light
measurements and the LSL measurements. In the
Aminco scattered light investigations the differences
between unfiltered and filtered water were equal
to zero [A(Rel.-Int.) = 0] in these cases. This
means that the concentrations of the suspended
particles were imperceptibly small; they were in
any case below the response level of the device.
In each top diagram of Figures 27 and 28, results
of the investigations of suspended particles from
“ Sydn.- Expr.“
n=101.7RPM — Test No 105
Latit. Longit. | Date
12° 28'N | 43° 55'E | 7.12.77
Aminco-Sc.-Light :
A(Rel.- Int) = 0.4
20 40 60 80 100 “m D+
Shallow Water
Bab-el-Mandab
FIGURE 26. LSL-technique compared with Aminco-
Sc.-Light measurement.
comparable locations are shown which were carried
out on the occasion of the Indian Ocean expedition
of the "Meteor". For the investigations, which
have been made by Krey et al. (1971), the so-called
inverted microscope and the Zeiss particle-counter
were used. These results lie always one magnitude
above the "Sydney Express" measurements for the
operating revolution (n = 101 rpm). In case of the
low revolution number of n = 60 rpm the nuclei
concentration measured at the "Sydney Express"
expedition (To = 15 N/cm?) reaches the values from
the "Meteor" expedition in the range 20 - 35 um
and exceeds in the range 35 - 92 um. Since, further-
more, the water sample tests carried out with the
Aminco scattered light device do not show any
difference between filtered and unfiltered water
(medium diagram - Figure 27) it is justified to
state that with the LSL measurement mainly bubbles
were recorded. The investigation of Keller et al.
(1974) of the optical qualities of the latex spheres,
applied for the calibration, supports this fact.
According to his investigation the latex spheres
“Meteor’’ Suspended
1964/65 Particles
Latit. | Longit. | Date
| 2°06'N[57°53'E [2.2.65 |
Station 179
“Sydn-Expr."’ Test No 79
Latit. | Longit. | Date
2°58'N | 59°44'E | 4.12.77
n=60RPM
Aminco-Sc- Light:
A(Rel-Int.) =0
Test No70
n=101.7 RPM
Latit.| Longit. | Date
Like Test No 79| 4.12.77
Aminco -Sc-Light :
A(Rel.-Int.) =0
" Sydn.- Expr."
0 20 40 60 80 100 “wm D—.
Deep Water
FIGURE 27. LSL-technique compared with other
investigations.
Meteor "
1964/65
Suspended
Particles
[ Latit. | Longit. | Date
12°43'N | 48°32'E |17. 12.64
13 Station 93
" Sydn.- Expr.”
n=103.1RPM _ Test No99
Longit.
12°21'N | 47°03'E | 6.12.77
Aminco-Sc-Light
A(Rel-Int.) =0
Deep Water
Gulf of Aden
FIGURE 28. LSL-technique compared with other
investigations.
show scattering characteristics similar to the
bubbles. Therefore, it can be said that the
sensitivity of the LSL measurements is - toa
certain extent - adjusted to the scattering
behavior of bubbles via the calibration. With the
LSL technique mainly bubbles are measured whose
number is always smaller than that of all solid
and gaseous nuclei. It is known, for instance,
that silica algae are almost transparent. It is,
therefore, understandable that there must exist
differences between the LSL method on the one hand
and the microscope method (with coloration perhaps)
and the conductivity measurement with Coulter
Counter on the other hand. The assumption that,
with the LSL method, mainly bubbles are measured is
supported by the good conformity of the LSL method
with the holographic method of an ITTC-comparison
measuring, Peterson et al. (1975). In this investi-
gation a holographic method, the laser scattering
light method, and a microscope method have been com-
pared with each other. The first two methods agreed
well with each other in the range of the bubble sizes
20-40 um, whilst the microscope method also showed
a nuclei concentration higher by one order of
magnitude. The higher concentration of nuclei
according to the microscope method apparently
results from mistakes arising from the focusing of
the nuclei. Similar difficulties might also occur
with the inverted microscope applied at the "Meteor"
expedition. This argument, however, does not say
that the highest nuclei concentration of the "Sydney
Express"-investigation, frequently occurring in
the smallest ranges of size, results from bubbles
only. (See, for instance, Test 70 - Figure 27 and
Test 99 - Figure 28 or all diagrams of Figure 20).
In the class of the smallest size nuclei solid
particles which always exist in the sea water have
certainly also been measured.
Oceanographic studies with the Coulter Counter,
for instance, carried out in the Gulf of California
by Zeitzschel (1970) show a strong increase in the
number of particles with a diameter of 14 to 4 um.
In addition, Zeitzschel cites the size distribution
of particulate carbon in the Indian Ocean by means
of fractional filtration investigated by Mullin
--e- Gordon (1970 )
Microsc., Organic matter
Surface Atlantic
—>— Carder et al.( 1971)
Coulter counter
Surface Pacific
“Sydney - Express ”
--- Test 79,n=60RPM
= — et 10; n=1017RPM
100 =um
Diameter
FIGURE 29. LSL-technique compared with other
investigations.
(1965). In Mullin's report the following average
percentage in the different size categories for
near surface samples (15 m) are given: 500 - 350
um: 3%; 350 - 125 um: 5%; 125 - 95 um: 4%; 95 - 60
um: 6%; 60 - 33 um: 6%; 33 - 10 um: 18%; and 10 - 1
um: 58%. The content of organic carbon can amount
to 4.5 - 34% of the particulate matter in the
different regions of the oceans [see Zeitzschel
(1970)]. Zeitzschel continues: "It can be concluded
from the results obtained at the Gulf of California
and the above mentioned references that small par-
ticles, mainly in the range from 1 to 10 um in diam-
eter, predominate in offshore surface waters of the
oceans." Investigations by Gordon (1970) and Carder
et al. (1971), which are compared with our results in
Figure 29, revealed the same results. It is obvious
that the "Sydney Express" results - ending at a
diameter of 10 to 20 um for reasons of intensity -
would probably show strongly increasing particle
numbers below this range. This can be seen from
the results of Gordon (1970) and Carder et al.
(1971) which have been published by Jerlov and
Nielsen (1974).
The fact that a large number of small particles
in sea water show every arbitrary geometrical shape
(according to Zeitzschel) also reminds one of the
shapes of particles from the water of a cavitation
tunnel, shown by Peterson et al. (1975) - Figure 6.
These sea water particles of different shapes
(diameter 1 to 10 Um), which according to Figure 29
are always available in a high concentration can
easily nucleate cavitation, as we know from many
investigations [(e.g., Peterson (1972) and Keller
(1973) ].
The problem of the difference between real shapes
of the nuclei, detected by the laser beam in the
sea water and the diameters evaluated for the
measuring results can only be mentioned here. In
this connection one should remember that the cali-
brations on the "Sydney Express" were performed
with latex spheres, whereas the real shape of the
nuclei in the seawater is unknown. This problem
also arises with the Aminco-method and with the
Coulter Counter measurements, the latter working,
however, according to the conductivity principle.
35
A further uncertainty is probably included in
the comparison of results obtained from oceanographic
studies carried out with water samples from the
open sea and those obtained from laser scattered
light measurements carried out in the flow and in
the boundary layer of the ship. The low-pressure
area of the boundary layer with its vortices of
different size most likely have a great influence
on the conversion of pore nuclei into bubbles when
they are moved from the calm free sea through the
boundary layer of the ship and thereby increase.
Due to the long running-time along the ship's hull
diffusion will also have an effect.
These physical processes accompanying the growing
of the bubbles in the low-pressure areas of the
boundary layers and the effect of diffusion could
be the explanation for the fact that the lower
speeds (12 kn, Tests 79 and 83) show a larger
bubble concentration Gg (due to the long running-
time along the ship's hull) than the higher speeds
(21.6 - 21.8 kn, Tests 70 and 90) with a shorter
running-time. (See measuring series with different
speeds - Figures 21 and 22). Thus - at a ship's
speed of about 60 rpm - a characteristic size of
bubbles has been formed. The measurements in a
seaway (Tests 61 and 65 - Figures 17 and 16) show
similar characteristic sizes of bubbles between 20
and 30 um. In a seaway the turbulence is larger
due to wave and ship motions. According to Sevik
and Park (1973) the turbulence can lead to character-
istic bubble sizes in connection with the pressure
history.
All considerations concerning bubble sizes must
finally lead to those bubbles participating in the
cavitation process. According to the calculations
by Isay and Lederer (1977), small bubbles, which
can also arise from pore nuclei, will grow faster
than big ones (Figure 30). The result of such
Distance from
suction side
Y= 0-005 Greatest
6-03 bubble Ro,
ug = 95 zm)
ad
2
k
om
| -
Smallest Bubble = Ro1= 54m
Py= 1 kp/cm?2
Smallest Bubble Ro j=5 am
=0.2
gees,
0.004 0.01 0.02 0.04 0.1 0.2
Chordlenght c=2A
FIGURE 30. Calculated growth of a single bubble in
a hydrofoil flow [Isay and Lederer (1977)].
316
calculations is valid for a hydrofoil of length
c = 10 cm, wherein the pressure distribution was
calculated by means of the profile theory for
incompressible flow with the completion of shock
pressures caused by the compressibility of the
water. With these calculations one question re-
mains unsolved: Up to which negative values can
the local pressures on the profile really decrease
in natural water? On the full scale propeller of
the "Sydney Express" the local pressure gradients
are probably steep and reach negative pressures,
causing bubbles with a diameter of 10 Um, or less,
to cavitate. Regarding the measurements, bubbles
with diameters of about 10 lim to 20 um were still
recorded in the results from Test 47 (Figure 15) up
to Test 65 (Figure 16). For unknown reasons, how-
ever, from Test No. 65 on nuclei with a diameter
of less than 20 um frequently could not be measured.
On the other hand one has to consider that, the
smaller the nuclei concentration U9 becomes, the
smaller the bubbles enlarged by cavitation.
It is apparent from these remarks that it would
have been desirable to record bubbles or nuclei
with a diameter below 5 um. But this was impossible
even with a 4 Watt laser which delivers 900 mW on
the green line. Therefore, it has to be admitted
that not all bubbles, which possibly are partici-
pating in the cavitation process, could be detected.
The question arises whether this will be possible
without any doubt in the future and if it is
necessary or not. Also the following aspects would
have to be considered: the required laser intensity
is limited; the exact local pressure distribution
on the propeller blades is difficult to determine
and on the other hand the tensile stress that can
actually be supported by the sea-water is quanti-
tatively unknown. :
Before closing this paragraph a personal impres-
sion in connection with the bubble sizes should be
mentioned which is supported by the collection of
photographs in Figure 8 and by numerous additional
pictures and propeller observations on the "Sydney
Express": The propeller will always find in the
flow a sufficient number of small nuclei leading
to cavitation. Therefore, the fullscale cavitation
will always be more stable than the model cavitation
with its smaller negative pressures and its different
nuclei distribution.
The white foam on the cavitation pictures of the
full-scale propeller clearly indicates a large
number of nuclei, which have led to cavitation and
grown together.
8. SUMMARY
The comprehensive laser scattered light measure-
ments on the "Sydney Express" showed the following:
1. The nuclei spectra measured in a seaway in
the Indian Ocean are quite different: In the range
of the nuclei diameter of 20 - 40 um either a
relative or an an absolute maximum of nuclei was
measured. (Figure 16). The motions of the ship,
especially the pitching motion, are in this con-
nection as decisive as the wave motiens on the sea
surface (Figures 18 and 19). The nuclei of this
range (diameter: 20 to 40 um), consist of bubbles,
since the scattered light method, carried out at
the same time with the Aminco-colorimeter did not
show any difference between unfiltered and filtered
water.
2. In good weather conditions the absolute
maximum of the bubbles with a diameter between 20
and 40 um (Figure 20) disappears. The nuclei of
smallest diameter show the largest nuclei concen-
tration. It probably consists of bubbles and
suspended particles, as the comparison with micro-
scope- and Coulter Counter measurements has shown.
3. Measurements made at different speeds
(Figures 21 and 22) have again resulted in an
absolute maximum at a diameter of 20 to 40 um for
the smallest ship speed at 12 kn. These nuclei
certainly consist of bubbles, since the Aminco mea-
surement in this case also did not show any dif-
ferences.
4. Measurements in shallow water show an
absolute maximum at a diameter between 20 and 40
um. The majority of these nuclei consists of
suspended particles, as the Aminco scattered light
measurement have shown. These suspended particles
probably do not contribute to cavitation, since the
comparison of propeller excited pressure fluctuation
measurements between deep and shallow water shows
practically no difference (Figure 7).
5. The ship's vibrations caused by the propeller
do pose a big problem for measurements of this type.
The insensibility of the laser against vibrational
stresses, however, after it was stiffly connected
with the ship, was suprisingly good. Even the
high loading caused by the temperature did not
create any bad effects in the laser.
6. Future laser measurements should possibly
anticipate diameter ranges below 5 um. A more
precise determination of suspended particles
requires a greater effort than the present method.
7. Further results of this trial will be
published later.
ACKNOWLEDGMENT
The comprehensive measurements on the "Sydney Express"
represented a project of the Sonderforschungsbereich
98 "Schiffstechnik und Schiffbau" (Special Research
Pool 98 "Marine Technology and Naval Architecture")
to which Det Norske Veritas (propeller observation)
contributed. The project was sponsored by the
Deutsche Forschungsgemeinschaft.
The authors wish to express their gratitude to
Hapag-Lloyd who made the "Sydney Express" available
for this investigation. Many thanks are expressed
to Captain W. Scharrnbeck, the Chief Engineer H.
Zwingmann, and the whole crew of the "Sydney Express",
who, by their excellent co-operation, made possible
the measurements and good results.
The authors are grateful to: Ing.(grad) L. Hoffman
(Hamburg Ship Model Basin - electronics, programming
and evaluation); U. Steidlinger and W. Folkers,
(Institute for Shipbuilding, Hamburg), and F. Meier
(Technical University Munich) - all three provided
for the mechanical construction and repair on board;
Mrs. U. Schmidt (Institute for Shipbuilding, Hamburg ~
- for drawing the diagrams); Miss A. van Blericq
(Hamburg Ship Model Basin - for the translation of
the German original into English); and Mrs. I. Jurschek
(Institute for Shipbuilding - for typing the manu-
script). Dr. R. Doerffer (Institute for Hydrobiology
of the Hamburg University, SFB 94) recommended the
Aminco scattered light device and made suggestions
concerning oceanography.
REFERENCES
Carder, K. L., G. F. Beardsley, and H. Pak (1971).
J. Geophys. Res. 76 5070-5077.
Gordon, D. C., Jr. (1970). Deep-Sea-Res. 17, 175-
185.
Isay, W.-H., and L. Lederer (1977). Kavitation an
Flugelprofilen. (Cavitation on Hydrofoils).
Schiffstechnik 24, 161.
Jerlov, N. G., and E. S. Nielsen (1974). Optical
Aspects of Oceanography. Academic Press, London
and New York.
Keller, A. P. (1970). Ein Streulicht-Zahlverfahren,
angewandt zur Bestimmung des Kavitationskeims-—
pektrums. (A Scattered-Light Counting Method
used for the Determination of the Cavitation
Nuclei Spectrum) Optics 32, 165.
Keller, A. P. (1973). Experimentelle and theore-
tische Untersuchungen zum Problem der modellma-
Bigen Behandlung von Stromungskavitation.
(Experimental and Theoretical Investigations on
the Problem of Cavitation in a Flow with Models).
Versuchsanstalt fur Wasserbau der Technischen
Universitat Munchen. Rep. 26/1973.
Keller, A. P., E. Yilmaz, and F. G. Hammit (1974).
Comparative Investigations of the Scattered-Light
Counting Method for the Registration of Cavitation
Nuclei and the Coulter Counter. University of
Michigan, Rep. UMICH 01357-36-T.
Keller, A. P., and E.-A. Weitendorf. (1975). Der
Einfluf des ungelosten Gasgehaltes auf die
Kavitationserscheinungen an einem Propeller und
auf die von ihm erregten Druckschwankungen.
(Influence of Undissolved Air Content on
Cavitation Phenomena at the Propeller Blades
and on Induced Hull Pressure Amplitudes) .
Institut fur Schiffbau, Universitat Hamburg.
Rep. 321A.
APPENDIX
DESCRIPTION OF THE NOVEL TYPE OF VELOCITY
MEASUREMENT
When particles or bubbles pass through a light beam,
they scatter a finite amount of light which is
dependent principally on the object shape, size,
index of refraction, and optical characteristics
of the beam. For this technique a small, homoge-
neously illuminated control volume (see No. 10 in
Figure 13) is optically defined by the cross-
sectional dimensions of the laser beam and the
optics of the system detecting the scattered light
(see No. 11 and 12 in Figure 13).
The amplitude of the electrical output pulses
from the photomultiplier (see No. 13 in Figure 13)
is proportional to the "nucleus" size, and thus is
the parameter used for "nucleus" spectrum determi-
nation.
The pulse width corresponds to the time in which
the scatterer remains in the scattering volume, and
therefore, by knowing the dimensions of the control
Siby/
Krey, J., R. Boje, M. Gillbricht, and J. Lenz (1971).
Planktologischchemische Daten der "Meteor"-Expe-
dition in den Indischen Ozean 1964/65. (Plank-
tological-Chemical Data of the "Meteor"-
Expedition to the Indian Ocean 1964/65). "Meteor"
Forschungsergebnisse, edited by Deutsche For-
schungsgemeinschaft, Reihe D-No. 9; Borntraeger-
Verlag, Berlin-Stuttgart.
Lederer, L. (1976). Profilstr6mungen unter
Beriicksichtigung der Dynamik von Kavitationsblasen.
(Hydrofoil Flow with Regard to Bubble Dynamics) .
Institut fiir Schiffbau, Universitat Hamburg.
Rep. 341.
Mullin, M. M. (1965). Size Fraction of Particulate
Organic Carbon in the Surface Waters of the
Western Indian Ocean. Limnol. Oceanogr. 10,
453.
Oossanen, P. van, and J. van der Kooy (1973).
Vibratory hull forces induced by cavitating
propellers. Transactions RINA 115, 111.
Peterson, F. B. (1972). Hydrodynamic Cavitation
and some Considerations of the Influence of
Free Gas Content. 9th Symposium on Naval Hydro-
dynamics, 2 1131, Paris.
Peterson, F. B., F. Danel, A. P. Keller, and Y.
Lecoffre (1975). Comparative Measurements of
Bubble and Particulate Spectra by three Optical
Methods. 14th ITTC, Ottawa. 2, 27.
Sevik, M., and S. H. Park (1973). The Splitting of
Drops and Bubbles by Turbulent Fluid Flow.
Transaction ASME, Journ. of Fluids Engineering,
95), Seriesh ay, Now 53.
Zeitzschel, B. (1970). The Quantity, Composition
and Distribution of Suspended Particulate Matter
in the Gulf of California. Marine Biology, 7,
4; 305.
volume, the velocity of the "nuclei", i.e., the
flow velocity, can be evaluated.
The sketch in Figure A 2.1 shows the shapes of
the optically bounded measuring volume for different
positions of the rectangular laser aperture and
the measuring slit in front of the photomultiplier.
The time the "nuclei" need to cross the control
volume is a function of the dimensions of the
volume in the flow direction, and of the flow
velocity. Therefore, the resulting photomultiplier
pulse width is a measure of the flow velocity if
the dimensions of the control volume are known.
To get an accurate relation between pulse width and
flow velocity, only nuclei of one known size,
defined by their pulse height, should be selected.
Example I in Figure A 2.2 displays an arbitrary
position of the control volume relative to the flow
direction. In that case, even for laminar flow one
gets a certain fluctuation for the pulse widths,
because the dimensions of the volume in the flow
direction are not equal.
318
Rectangular
aperture
Measuring
slit
FIGURE A2.1. Principle of velocity measurement.
FIGURE A2.2. Sketch of inclined control volume and
received photomultiplier signals.
FIGURE A2.3. Sketch of inclined control volume and
flow direction.
In example II Figure A 2.2 the main axis of the
rectangular aperture is positioned parallel to the
projection of the flow direction versus the plane
vertical to the optical axis of the laser, and
consequently the peak of the pulse width distribution
is at a maximum value of t.
In example III in Figure A 2.3 the direction of
the measuring slit is also parallel to the projection
of the flow direction versus the plane vertical to
the optical axis of the photomultiplier, so that
all dimensions of the measuring volume in the
direction of the flow are the same, and the pulse
width distribution therefore shows its most narrow
shape. The peak of the distribution indicates the
velocity in the main direction, whilst the shape of
the curve is a measure of the turbulence level.
The direction of flow can now be determined by
the position of the rectangular aperture and the
measuring slit. They each define a plane containing
the corresponding optical axis, whereby the line
of intersection represents the direction of the
main flow in this region.
Discussion
319
ORVAR BJORHEDEN and TORE DALVAG
We congratulate the authors of this very
interesting paper. For hull designers as well as
propeller manufactures the problem of predicting
the propeller induced vibration forces is a most
essential task indeed. In this context we wish to
inform you briefly about some recent developments
at the KMW* Marine Laboratory related to the model
testing technique applied in our cavitation tunnels.
The first item concerns the method of hull
wake simulation. For some time the well-known
dummy technique, involving ship afterbody models
and transverse net screens, has been used in our
tunnels for the purpose of simulating model wake
pattern. This is a rather time consuming process
since the net screens have to be adjusted step by
step until the correlation with the wake pattern
obtained in the towing tank appears satisfactory.
Moreover, the method has some technical drawbacks
as regards the stability of the wake as well as
the interaction between propeller and hull and the
influence of the propeller on the wake pattern.
In connection with hydro-acoustic tests, cavitation
occurring on the nets may worsen the background
noise level.
In order to eliminate the above drawbacks a
new technique involving longer afterbody hull
dummies has been introduced. The method aims at
simulating the full-scale ship wake pattern based
upon the concept of equivalent relative boundary
layer thickness, i.e., the frictional boundary
layer thickness in relation to some characteristic
length, e.g., the propeller diameter should be the
same in the model and in full-scale. For ordinary
cavitation testing purposes utilizing propeller
model diameters around 250 mm and tunnels speeds
of 4 to 8 m/sec this criterion results in hull
dummy lengths of 2.5 to 3.5 m for most types of
vessels. In principal, the model stern contour as
well as the aftermost water-lines are made to scale,
whilst the maximum breadth of the dummy is chosen
on the basis of 2-dimensional potential flow cal-
culations comparing the ship water-lines in unre-
stricted water to the dummy lines within the bound-
aries of the cavitation tunnel test section and
aiming at similarity in the potential wake
distribution.
Figure 1 shows a picture of a 3 m hull dummy
used for the testing of a 150 m, single screw, con-
tainer ship. In Figures 2 and 3 the model wake
distribution as obtained in the towing tank and
then corrected for scale effect according to the
so-called Sasajima method is given. In Figure 4,
finally, a comparison between the corrected model
wake and the wake distribution obtained in the cav-
itation tunnel is shown for a few radii close to
the propeller blade tip. As can be seen from the
diagrams, the agreement is quite good, particularly
as regards the wake peak in the 12 o'clock propel-
ler blade position.
*Karlstads Mekaniska Werkstad
Figure 1. Hull dummy for wake simulation in cavita-
tion tunnel.
Apart from the advantage of a quicker and more
direct simulation of the full-scale wake, the
method with long afterbody dummies results in a far
more stable wake distribution which in turn implies
more consistent recordings of fluctuating propeller
forces, propeller induced pressure pulses against
the hull, etc. Probably, the interaction between
propeller and hull is also more realistic with this
method of wake simulation as compared to the method
utilizing transverse nets.
The second item refers to the instrumentation
employed for recording of propeller forces and the
propeller induced pressure pulses on a ship's hull.
In both KMW tunnels a data collecting and evaluation
system consisting of an on-line connected desk com-
puter together with a printer and a plotter has
been used for several years. For the measurement
of propeller induced pressure pulses with the aid
of pressure pickups fitted into the hull, a pulse
sampling technique giving time averaged values from
a number of propeller revolutions at each blade
position has been the practice. With this method
the pressure signals are given in analogue form and
recordings can be obtained from only one pickup at
atime. Recently, a new data collecting unit was
put into service enabling simultaneous recording
on 6 channels and storing test results from every
r/R = 0.886
r/R = 0.709
r/R = 0.532
r/R = 0.355
r/R =0.177
r/R = 0.177
r/R = 0.355
r/R = 0.532
r/R = 0.709
r/R = 0.886
r/R = 1.063 ==
€
Figure 2. Model wake distribution as obtained in Figure 3. Wake distribution corrected for scale effect
towing tank. according to the Sasajima method.
W
Wake Distribution Corrected
1.0 Acc. to the Sasajima Method
6 m/s
r/R = 0.96 Simulated in Cavitation Tunnel
SOSSSe 3 m/s
r/R = 0.80
0.5
360 270 0 90 180
)
mparison between corrected model wake and wake simulated in cavitation tunnel.
321
Figure 5. Data collecting memory.
second degree of a propeller revolution in digital
form in a RAM semi-conductor memory controlled by
the desk calculator. With this instrument, instan-
taneous or time averaged test results can be stored
and are readily available for printing, plotting,
ERLING HUSE
The authors in their presentation draw atten-
tion to the problem of calculating cavity geometry
and thus the excitation force due to cavitation.
At the Norwegian Ship Model Tank in Trondheim we
are at present developing a procedure to overcome
this difficulty. In the cavitation tunnel we
Measure the propeller-induced pressure at only 4
positions on the hull model above the propeller.
The measurements are made for non-cavitating as well
as cavitating propellers. From the results of these
Measurements we calculate an equivalent singularity
O. RUTGERSSON
First I would like to congratulate the authors
on this interesting paper. The possibility of cal-
culating hull forces and moments and their distri-
butions directly on the body without the roundabout
way over freestream pressures and solid boundary
factors is especially elegant. Being somewhat in-
volved in calculations and measurements of pressure
fluctuations (with and without cavitation) at SSPA*
I would like to ask if the authors intend to use
this new method also to calculate solid boundary
*Statens Skeppsprovningsanstalt, Goteborg, Sweden
Figure 5. Desk calculator with printer and plotter.
transformation to full scale, and harmonic analysis
as well as integration of resulting hull surface
forces and similar calculations with the aid of
the desk calculator.
distribution to represent the propeller. This is
next combined with a theory similar to that of
Dr. Vorus to obtain the excitation force on the hull
referring to any given vibratory mode of the hull.
As a second comment on the paper I notice in
Figure 4 integration areas extending up to 30 pro-
peller diameters upstream. This is, in my opinion,
not very realistic because one is then passing one
or more nodal points of practically occurring modes
of vibration.
factors for different afterbody shapes and propeller
configurations?
Unfortunately the authors' investigation is
limited to non-cavitating propellers. This is a
severe limitation as the contribution from the
transient cavitation often is of a much higher mag-
nitude than the contributions from blade loading
and thickness. When discussing this subject the
authors declare that methods "for predicting trans-
ient blade cavity geometry and the attendant pres-—
sure field" are not available. I would like to ask
why the methods developed by Huse (1972), Johnsson
and Sgndvedt (1972), and van Oossanen (1974) have
322
not been considered? These methods have been used
in Europe for several years and the agreement with
experiments is usually good.
I agree that it is important that the integra-
tion of hull forces and moments is carried out over
a not too small part of the hull surface. This is
even more important when the forces from a cavita-
ting propeller are considered, as those pressures
have a slower decay than those induced by a non-
cavitating propeller [Lindgren and Johnsson (1977)].
Assuming that the hull forces should be used
for an estimation of the vibration level for a
certain ship project, I think that the problem is
far more complicated than just a matter of integra-
tion area. First, the described method is a near
field theory where the influence of the propagation
velocity of the pressure wave has been neglected.
When calculating forces far from the propeller this
could cause some difficulties. Secondly, the ship
hull is not a rigid body. The vibration response
will therefore be dependent not only on the hull
forces but also on their location relative to the
nodes of the vibration mode. Forces located close
to the nodes will contribute very little and those
located on different sides of a node will more or
less cancel each other. Calculations with the Fi-
nite Element Method have shown that hull forces aft
of the aftermost node are particularly efficient
in exciting high vibration levels. This could be
the explanation for rather good results often being
achieved in vibration calculations in spite of the
fact that the excitation forces have been obtained
by integration over a rather small area.
The correct treatment of the problem will, of
course, include vibration calculations, with a very
detailed Finite Element model with the complete ex-
citation forces and moments. Since this is very
complicated and expensive it is seldom done. In-
stead, different approximate procedures have been
developed by different institutions. Referring to
the integration problems the authors claim that
"the current practice in European model basins is
highly suspect." I very much doubt that this is
current practice. At SSPA for example, we use the
pressure fluctuations in a reference point above
the propeller as a basis for estimation of the risk
of vibration. On the basis of full-scale measure-
ments we have established an approximate relation
between excitation at this point and the vibrations
at another reference point [(Lindgren and Johnsson
(1977) 1].
REFERENCES
Huse, E.,(1972). Pressure Fluctuation on the Hull
Induced by Cavitating Propellers. Norwegian Ship
Model Experiment Tank Publ. No. III.
Johnsson, C. A.,.and T. Sgndvedt, (1972). Propel-
ler Excitation and Response of 230,000 TDW Tankers.
SSPA Publ. Wo. 70.
Oossanen, P. van, (1974). Calculation of Perform-
ance and Cavitation Characteristics of Propellers
Including Effects of Non-Uniform Flow and Viscosity.
NSMB Publ. No. 4657.
Lindgren, H., and C. A. Johnsson, (1977). On the
Influence of Cavitation on Propeller Excited Vibra-
tory Forces and Some Means of Reducing its Effect.
PRADS-Internattonal Symp. Tokyo.
323
Authors’ Reply
BRUCE D. COX, WILLIAM S. VORUS, JOHN P. BRESLIN,
and EDWIN P. ROOD
Our thanks to the discussers for their interest
and encouraging remarks. On Mr. Rutgersson's
question of calculating solid boundary factors, we
do believe it would be useful to perform computa-
tions for a series of hull afterbody forms and pro-
pellers. The results would illustrate sensitivity
to the various physical parameters and could pro-
vide guidance during the early stages of a ship
design. However, for realistic predictions of pro-
peller exciting forces, the complete calculation
should be carried out using the actual wake, hull
geometry, and propeller design under consideration.
As noted in the paper and by Mr. Rutgersson,
only the non-cavitating propeller case is consid-
ered which is a severe limitation in many pratical
applications. The principal purpose of the paper
was to present analytical methods and simple form-
ilae for predicting hull surface forces for a given
representation of the propeller and show compar-
isons with experiments. Future improvements in the
propeller theory, in particular, the allowance for
transient cavitation, can be incorporated quite
readily into the surface force analysis. It can
be shown [Breslin (1977)] that the time rate of
change of the cavity volume plays a crucial role
in generating the propeller pressure field. We
are familiar with a number of proposed methods for
predicting blade cavity geometry including those
cited by Mr. Rutgersson. These approaches for the
most part are empirical. An alternative procedure,
described in Mr. Huse's discussion, consists of
finding an "equivalent" singularity distribution
so as to produce agreement between calculated and
measured values of pressure at selected locations
near the propeller. The problem of analytically
predicting the proper singularity distribution to
represent the cavity volume dynamics is now the
subject of active research.
We agree with Mr. Rutgersson that compress-
ibility effects should be examined when considering
the far field pressures generated by a propeller.
A 5-bladed propeller operating at 100 rpm produces
a blade rate frequency disturbance with a acoustic
wavelength on the order of 600 feet. The relative
phase of the distrubances generated far ahead of
the propeller may be important in the integrated
pressure force amplitude and phase.
The theory presented in this paper assumes a
rigid hull boundary, intended to provide a first
estimate of propeller exciting forces acting on
the hull girder. Certainly for detailed stress and
vibration analyses, the interplay between fluid
loading and hull structural deformation would have
to be accounted for. In principle, the present
theory can be extended to satisfy the boundary
condition on a deformable body. The complete
analysis would then involve coupled equations des-
cribing the fluid loading and structural response,
and could be solved by finite methods.
REFERENCE
Breslin, J. P., (1977). A Theory for the Vibra-
tory Forces on a Flat Plate Arising from Inter-
mittant Propeller Blade Cavitation. Sympostwm on
Hydrodynamtes of Ship and Offshore Propulston
Systems, Oslo, Norway
' vr’
, € were £4
d ‘
5 i
s
1
4
Pe
’ “OL ay 4
Pat yi
<1 . - #1 «
— +
=. as Ah
. -
+ i
’ i 4.
r ~ =
i ‘
tt t
=
1 '
i % \
= t ~
i
> 2
om
E i
4 ’
r +
\
> y
- l
\
=I
.
Session V
CAVITATION
ERLING HUSE
Session Chairman
The Ship Research Institute of Norway
Trondheim, Norway
Ae)
=({
Mi
‘
ii
Pressure Fields and Cavitation
in Turbulent Shear Flows
Roger E. A. Arnd
University of Minnesota
t
Minneapolis, Minnesota
William K. George
State University of New York at Buffalo
Buffalo, New York
ABSTRACT
Cavitation in turbulent shear flows is the result
of a complex interaction between an unsteady
pressure field and a distribution of free stream
nuclei. Experimental evidence indicates that
cavitation is incited by negative peaks in pressure
that are as high as ten times the rms level. This
paper reviews the current state of knowledge of
turbulent pressure fields and presents new theory
on spectra in a Lagrangian frame of reference.
Cavitation data are analyzed in terms of the avail-
able theory on the unsteady pressure field. It is
postulated that one heretofore unconsidered factor
in cavitation scaling is the highly intermittent
pressure fluctuations which contribute to the high
frequency end of the pressure spectrum. Because of
limitations on the response time of cavitation
nuclei, these pressure fluctuations play no role
in the inception process in laboratory experiments.
However, in large scale prototype flows, cavitation
nuclei are relatively more responsive to a wider
range of the pressure spectrum and this can lead to
substantially higher values of the critical cavi-
tation index. Unfortunately, this issue is clouded
by the fact that higher cavitation indices can be
found in prototype flows because of gas content
effects. Some cavitation noise data are also
examined within the context of available theory.
The spectrum of cavitation noise in free shear
flows has some similarity to the noise data found
by Blake et al. (1977) with the exception that there
appears to be a greater uncertainty in the scaling
of the rate of cavitation events which leads to a
substantial spread in the available data.
1. INTRODUCTION
The physical processes involved in cavitation
inception have been studied for many years. Much
of this research has been directed toward an under-
327
standing of the dynamics of bubble growth and the
determination of the sources of cavitation nuclei
and their size and number in a given flow situation.
This research has led to a general understanding of
some of the environmental factors involved in
scaling experimental results from model to prototype.
More recently, considerable attention has been
paid to the details of the boundary layer flow over
streamlined bodies and the role of viscous effects
in the cavitation process. This research has shown
that viscous effects such as laminar separation
and transition to turbulence can have a major impact
on the inception process and that there can be
considerable variation between model and prototype
in the critical conditions for cavitation.
In the absence of viscous effects, the scaling
problem reduces to an understanding of the size
distribution of nuclei and the temporal response
of these nuclei to pressure variations as viewed
in a Lagrangian frame of reference. This was first
treated in detail by Plesset (1949). As already
mentioned, consideration of viscous effects shows
that the cavitation inception process can be
considerably altered by either laminar separation
or transition to turbulent flow. Obviously these
phenomena are interrelated and are strongly Reynolds
number dependent. The recognition of the importance
of these factors has had considerable impact on the
direction of cavitation research in recent years.
Several papers in this symposium deal directly with
this aspect of the cavitation scaling problem.
It is reasonably well understood that intense
pressure fluctuations, either at the trailing edge
of a laminar separation bubble or in the transition
region, can have a major effect on the inception
process on streamlined bodies. However, these
phenomena will be excluded from this review. The
focus of this paper will be on the relationship
between the temporal pressure field and cavitation
inception in free turbulent shear flows and fully
developed boundary layer flows. Scant attention
has been given to this problem, even though the
328
topic is of practical significance. Turbulent
shear flows are very common in practice and what
cavitation data are available for these flows
indicate that there can be significant scale effects.
For example, Lienhard and Goss (1971) present a
collection of cavitation data for submerged jets.
It is observed that the critical value of the
cavitation index increases with an increase in
jet diameter, with no upper bound on the cavitation
index being defined by the available data. The
cavitation index is observed to vary from 0.15 to
3.0 over a size range of 0.1 cm to 13 cm. Arndt
(1978) reviews the available data for cavitation
in the wake of a sharp edged disk. These data
increase monotonically with Reynolds number and
again no upper limit on the critical cavitation
index can be determined from the available data.
At present, it can be said that laboratory experi-
ments do not provide a reasonable estimate of the
conditions that can be encountered under prototype
conditions. From a practical point of view the
situation is much more critical than the scaling
problems associated with streamlined bodies since
at present there is no definable upper limit on
the cavitation index for these free shear flows.
There are a myriad of factors that enter into
the inception process in turbulent shear flows.
As a minimum, we need information on the turbulent
pressure field, such as spectra and probability
density. We require an understanding of the diffu-
sion of nuclei within the flow, and we need to
know how these nuclei respond to temporal fluctu-
ations in pressure. In taking into account the
bubble dynamics inherent in the problem, consider-
ation must also be given to gas in solution which
can have an influence on both bubble growth and
collapse. p
The theory of bubble dynamics is well founded
and reasonable estimates of critical pressure can
be determined under flow conditions that are well
defined. Needless to say, the flow conditions in
a turbulent shear flow cannot be defined in
sufficient detail. However, the problem of flow
noise has led to a more comprehensive understanding
of turbulence; in particular, recent aeroacoustic
research has provided a wealth of data on turbulent
pressure fluctuations. These data are a by-product
of the need for understanding turbulence as a source
of sound. At this point in time, it seems only
logical to review the inception problem in terms
of both classical bubble dynamics and the more
recent results of the field of aeroacoustics.
2. THEORETICAL CONSIDERATIONS FOR CAVITATION
Cavitation Index
The most fundamental parameter for cavitating flows
is the cavitation index
wherein p_ is a reference pressure, p_ the vapor
pressure, U_ a reference velocity, and p the
density of the liquid. The flow state of primary
interest in this paper is characterized by a
limited amount of cavitation in an otherwise Single
phase flow. There is a specific value of 0 associ-
ated with this flow condition, which for convenience
will be defined as the critical index:
If it is necessary to have completely cavitation
free conditions, one design objective for various
hydronautical vehicles is the minimization of 0 .
Cavity flows are assumed identical in model
and prototype for geometrically similar bodies
when O is constant, irrespective of variations
in physical size, velocity, temperature, type of
fluid etc. In practice O0_ is found to vary over
wide limits. Simply stated, these so-called scale
effects are due to deviations in two basic assump-
tions inherent in the cavitation scaling law; namely
that the pressure scales with velocity squared and
the critical pressure for inception is the vapor
pressure, p_. As will be shown, the two factors
can be interrelated, since in principle the critical
pressure is a function of the time scale of the
pressure field.
In order to provide a foundation for the ensuing
discussion, consider a steady uniform flow over a
streamlined body devoid of any viscous effects.
The following identity can be written:
wherein C_ is a pressure coefficent defined in the
usual manher. Generally speaking, C_ is defined
by the pressure on the surface of a Given boedy- east
is generally assumed that cavitation first occurs
when the minimum pressure, p_, is equal to the
: m™m 3
vapor pressure, p This results in the well-known
scaling law ys
Consider next the case where the pressure in the
cavitation zone is less than the minimum pressure
measured on the surface of the body, then
nt Pl E Py a Pl
Oo = ——— -C fb ee
Lou 2 1p me) W 2
2 Oo ts (e)
Here we have to distinguish between the pressure
at the surface of the body p, and the pressure
sensed by cavitating nuclei, p _. Assuming
cavitation occurs when Pal = Py we have
OL S=5= Cee (1)
Equation (1) is one version of the superposition
equation that is commonly referred to in the
literature.
Bubble Dynamics
It is generally accepted that the process of
cavitation inception is a consequence of the rapid
or explosive growth of small bubbles or nuclei
which become unstable due to a change in ambient
pressure. These nuclei can be either imbedded in
the flow or find their origins in small cracks
and crevices at the surfaces bounding a given flow.
The details of how these nuclei can exist have been
considered by many investigators. A summary of
this work is offered by Holl (1969, 1970).
Theoretically, liquids are capable of sustaining
large values of tension. However, the nuclei in
the flow act as sites for cavitation inception
and prevent the existence of significant tensions.
The mechanics of the inception process are adequately
described by the Rayleigh-Plesset equation, which
considers the dynamic equilibrium of a spherical
bubble containing vapor and non-condensable gas
and subject to an external pressure Boy ft)?
iN)
n
w\|we
co qo il
AER ———
RR 2 0 (2)
+ - -—-4
2. PG Ig SE) = Hl
wherein R is the bubble radius and dots denote
differentiation with respect to time. It should
be emphasized here that even for the case of steady
flow over a streamlined body, p_,(t) is a function
of time since we are concerned with the pressure
history sensed by a moving bubble. If the problem
is simplified to consider the static equilibrium
of a bubble, we find that there is a critical
value of p - p below which static equilibrium
is not possible. This is found to be
(ee = Dae = 45/3R* (3)
wherein R* is defined as the critical bubble radius.
Substitution of Eq. (3) into Eq. (2) with dynamical
terms identically zero will indicate that R* is a
function of the partial pressure of noncondensable
gas within the bubble. If p __(t) varies rapidly
in comparison to the response time of the nuclei,
then even greater values of tension are possible.
Thus in general we can write
PY y Pal Ss)
AS /SR*in my c 3
where
ll
o(o) =~, o(~) =1
The function ¢ depends on the flow field. The
argument of ¢ contains a characteristic time scale
of the pressure field (t_) anda GREACEOENSIELS
response time of the nuclei, (PR, 37s)* 7 En) the
case of a streamlined body in une absence of viscous
effects, t_ would be proportional to the quotient
of body diameter and velocity. In the case of
cavitation induced by turbulence, the characteristic
time scale could be any of the turbulence time
scales. For example,
u' OD
L4/ Vine 3
ae factor (PR, js)? is
derived from the asymptotic solution to Eq. (2)
for the case of negligible gas diffusion. Under
these conditions
is often appropriate.
329
and the growth rate stabilizes at a value given by
| 2) a (4)
3 p
Assuming a characteristic bubble response time
given by R*/R, with 1 = ea = 4S/3R*, we obtain
R*
= —— O.
qT, R 87 \ (5)
A typical variation of ¢ based on the theoretical
computations of Keller (1974) is given in Arndt
(1974).
The Influence of Dissolved and Free Gas
The discussion in the previous section is based on
the assumption of a healthy supply of free nuclei
which is generally the case in recirculating water
tunnels and in the field. Generally speaking, a
reduction in O_ due to bubble dynamic effects
usually only occurs on model scale. To some extent
the level of dissolved gas and the number and size
of free nuclei are interrelated. Some recent
experimental results are documented in Arndt and
Keller (1976). The level of dissolved gas can
play an important direct role when the time of
exposure to reduced pressure is relatively long.
Under these circumstances Holl (1960) has shown
that gaseous cavitation can occur at values of 6
much greater than those for vaporous cavitation.
Using an equilibrium theory, Holl (1960) deduced
an upper limit on on given by
wherein @ is the concentration of dissolved gas
and 8 is Henry's constant.
In summary, an overview of the effects of bubble
dynamics and free and dissolved gas indicates that
short exposure times such as are the case ina
model implies that cavitation will occur at pressures
lower than vapor pressure and OF is less than
expected. Long exposure time, Such as can occur
in vortical motion of all types, including large
scale turbulence, implies the possibility of gaseous
cavitation with © being greater than expected.
3. PRESSURE FLUCTUATIONS IN TURBULENT SHEAR FLOWS
Background
Considerable progress has been made over the last
five years in the understanding turbulent pressure
fluctuations in free shear flows in an Eulerian
frame of reference. Of particular importance is
the development of pressure sensing techniques
which under certain circumstances can lead to
reliable measurements of pressure fluctuations.
330
The first theoretical arguments on the pressure
fluctuations associated with turbulent flow appear
to be due to Obukov and Heisenberg [Batchelor
(1953) ]. Heisenberg argued that Kolmogorov scaling
should be possible for small scale pressure fluc-
tuations. Batchelor (1951) was able to calculate
the mean square intensity of the pressure
fluctuations as well as the mean square fluctuating
pressure gradient in a homogeneous, isotropic
turbulent flow. This work was extended by Kraichnan
(1956) to the physically impossible but conceptually
useful case of a shear flow having a constant mean
velocity gradient and homogeneous and isotropic
turbulence.
Apparently there were no attempts made to extend
this theoretical work until the 1970's when George
(1974a), Beuther, George, and Arndt (1977a, b, c)
and George and Beuther (1977) applied the concepts
developed by Batchelor and Kraichnan to the calcu-
lation of the turbulent pressure spectrum in
honogeneous, isotropic turbulent flows with and
without shear. When compared with experimental
evidence gathered in turbulent mixing layers, the
theory is found to be remarkably accurate. The
predicted spectrum (with no adjustable constants)
agrees with pressure measurements in turbulent jet
mixing layers from several sources, including
those of Fuchs (1972a), Jones and his co-workers
(1977), and the authors themselves. As shown in
Figure 1, the experimental data and the theory are
remarkably consistent, especially in light of the
fact that several different experimental techniques
and different flow facilities are involved.
The current state of knowledge of turbulent
pressure fluctuations can be summarized as follows:
1) Pressure fluctuations in a shear flow can
arise from three sources. The first two involve
interaction of the turbulence with the mean shear.
These are second order and third order interactions,
of which only the second order interactions are
important at small scales. The last involves only
interactions of the turbulence with itself.
2) Kolmogorov similarity arguments can be
applied to each of the spectra arising from these
-400 Uy X/D
198 15
19.8 30
30.5 1.5
5 30.5 3.0
po Michalke
% |X Fuchs
Ss (75)
=
ne -6.00
mo
2
RT) 200 300
log Kx
FIGURE 1. Experimental confirmation of the theoretical
pressure spectrum for a turbulent jet.
terms. These arguments are valid for the small
scale fluctuations.
3) If the turbulent Reynolds number is high
enough, there exists an inertial subrange in each
of the three spectra in which
us, D DPD gail
Ted (k) iP Kk
ses 2 =9/5
Uap ie) Se xs
= 2) WV} 7/3}
Tp (K) = 4,0 € k
wherein a . = 2, a. =0, a= 1.3, ¢ is the rate of
dissipation of ene Sailene enérgy per unit volume,
K is the mean shear, and k is the disturbance wave
number.
4) There is considerable evidence that coherent
structures play an important role in determination
of at least the large scale pressure fluctuations
[Fuchs and Michalke (1975), Fuchs (1972a, b), Chan
(1974a, b), and Chan (1976)].
Relation to Cavitation
Since the above spectral results are expressed in
Eulerian frames, they cannot be directly applied
to the problem of cavitation inception which is a
Lagrangian problem. Nonetheless, Kolomogorov scaling
has been successful in an Eulerian frame of reference
and therefore we can, with some confidence, infer
that similar scaling will be valid for Lagrangian
time spectra (i.e. the frequency spectra that would
be seen by a moving material point). The results
of such an exercise are as follows:
1) The Lagrangian turbulent spectrum can be
separated into interaction of the turbulence with
the mean shear and the interaction of the turbulence
with itself.
2) The high frequency (analogous to small scale)
will be well described by Kolmogorov scaling such
that
Apps (W)
i
A
i)
<
ow
SS
ie)
|
vr
Fh
n
—N
|E
Ss
L
es AppT (Ww) ye 6? fo Gal
2
p wr
where 1
2
Rete (oe.
cncy
3) In the inertial subrange these reduce to
-5
1
= 5 =i
Fees () se we 2)
fo) Ww
a
-3
3/2) es Ww
i AppT (W) = Vv / Se (=)
Ww
fe) d
In summary it appears plausible to assume that
the basic picture of pressure fluctuations arising
from mean-shear turbulence interactions will be
unchanged in a Lagrangian frame of reference,
although the actual spectra are different. The
postulated relations for Lagrangian spectra should
be directly applicable to any Lagrangian phenomenon;
in particular the relations should be applicable
to the inception of nuclei in a fluctuating pressure
field.
In relating the information on the pressure field
to the problem at hand, it is evident that two
criteria must be satisfied for turbulence induced
inception:
1) The pressure must dip to the vapor pressure
or lower.
2) The pressure minimum must persist for a time
that is long in comparison to the characteristic
time scale of the bubble, say Tp (taken to be the
time scale for growth at inception).
Both factors lead to scale effects. Consider
first the second factor. The preceding arguments
for the pressure field in a Lagrangian frame of
reference lead to the hypothetical spectrum shown
in Figure 2. For convenience we have normalized
the spectrum with respect to the mean square pressure
and the Lagrangian time scale JY. (c.f. Tennekes
and Lumley, Chapter 8). Requirement (2) for bubble
growth is plotted at the frequency W = 1/T,- te
is clear that as long as w << 1/Tp, any pressure
flucuation persists for a time longer than the
time scale of the bubble. Thus at frequencies less
than » = 1/Tg cavitation inception can occur with
minimal local tension. Moreover, by integrating
the spectrum from wW = O to W = 1/T, 1 we can deter-
mine that fraction of the mean square pressure
which can contribute to bubble growth without
appreciable tension (assuming a normal distribution
of nuclei).
Consider now the effect of maintaining Tg con-
stant while varying the Reynolds number. Taking
J~ &£/a' and noting that there are essentially no
pressure fluctuations of interest above the
Kolmogorov frequency, W = (e/v)2 we find that
after 1/Tp, exceeds (e/v) 4, the entire spectrum
can potentially contribute to bubble growth. This
will occur when the Reynolds number is roughly
3
Q
a
<
Ve Ve 0.2(%) "2
E B
FIGURE 2. Hypothetical pressure spectrum in a
Lagrangian frame of reference.
331
Mean Square Pressure Fluctuation (Lagrangian)
Sea eS Ses SS oo oS SS
Vy 0.2 () 2
wy —>
FIGURE 3.
spectrum.
Integration of Lagrangian pressure
ul/v ~ (2/uTR)?. By noting the spectral dependence
on frequency and performing a running integral, a
plot such as shown in Figure 3 can be generated.
This graph illustrates how rapidly the asymptotic
state is reached. This occurs when 7/T -ts J (€/v)
> (ut £/v) 2 or when £/u'T_ > (u'&/v) 72 as previously
stated. B
As an example,* cavitation is observed to occur
in submerged jets at an axial position, x, that is
roughly one diameter from the nozzle. Assuming
the dissipation rate to be approximately 0.05U;°/x,
where Uz is the jet velocity, results in a criterion
that the jet diameter must exceed the following
before scale effects are absent: d > 0.05U;°TR7/v.
Using typical values of Uz = 10 m/s and Tp = Om
sec., we conclude that the asymptote is reached for
d ~ 50 meters. Thus size effects could be important
in many model experiments.
1
i)
Effect of Intermittency at Small Scale
In 1947, Batchelor and Townsend concluded from
observations of the velocity derivatives in
turbulent flow that the fine structure of the
turbulence (small scales, high frequency) was
spatially localized and highly intermittant in
high Reynolds number flows. Subsequent work [c.f.
Kuo and Corrsin (1971)] has confirmed that there
is a decrease in the relative volume occupied by
the fine structure as the Reynolds number is
increased. Thus the spatial intermittancy increases
with Reynolds number. The effect of this phenomenon
on filtered hot wire signals is shown in Figure 4.
These data are derived from Kuo and Corrsin (1971).
It is obvious from these data that the signal is
increasingly intermittant as the filter frequency
is moved to higher and higher values.
Since the dissipation of turbulent energy takes
place at the smallest scales of motion, it is clear
from these observations that the rate of dissipation
of turbulent energy must vary widely with space
and time. It was this consideration that led
*Strictly speaking, these results are only applicable
when the Lagrangian turbulent field is stationary.
In most flows of interest this is seldom the case.
However, the smallest scales of motion can often
be considered to be in quasi-equilibrium.
FIGURE 4. Filtered hot-wire signals in grid-
generated turbulence [adapted from Kuo and Corrsin
(1971)]. (i) £ = 200 Hz, £/£ = 0.52, 20 ms/division
(horizontal scale); (ii) 1 kH2, OoSA, 4a (sists) Gp
0.52, 1; (iv) high-pass signal, f = kHz, 1 ms/
division. 2
Kolmogorov (1962) to reformulate his original
similarity hypothesis in terms of the average rate
of dissipation of turbulent energy <e> , and to
assume that the logarithm of € was governed by a
normal distribution. Later work by Gurvich and
Yaglom (1967) showed that any non-negative quantity
governed by fine scale components has a,log normal
distribution with a variance given by » =A+B
ln R, where A is a constant depending on the
structure of the flow, B is a universal constant and
Ro is the turbulence Reynolds number.
These results have implications for the cavita-
tion problem at hand. Beuther, George, and Arndt
(1977a, b) have shown that Kolmogorov similarity
scaling is applicable to the high wave number
turbulent pressure spectrum. As a consequence of
this and the observed intermittancy and spatial
localization of small scale velocity fluctuations,
it is reasonable to expect the same trend in the
small scale pressure fluctuations. This could
result in an important cavitation scale effect.
To make this point clear, a set of hypothetical
band passed pressure signals at high and low
Reynolds number are presented in Figure 5. For the
sake of argument, assume that the filter is set
around a range of frequencies which will result in
bubble growth (wTgp £1). Since the spectra of these
two signals will be identified in terms of Kolmo-
gorov variables and since the low Reynolds number
signal is less intermittant, there is a greater
probability that the high Reynolds number signal
FIGURE 5. Hypothetical band-passed pressure signals:
(i) low turbulent Reynolds numbers, (ii) high turbu-
lent Reynolds number.
will contain more intense deviations from the mean.
In particular, with all other factors held equal
it is more likely that the local pressure will fall
below the critical pressure when the Reynolds number
is high, even though the spectra are identical.
This is shown in Figure 5. If the log normal
arguments were applicable, then it can be expected
that this will depend on the Reynolds number.
The effect of intermittancy coupled with effects
cited earlier could be of considerable importance
to the problem of predicting cavitation inception
in the prototype from small scale experiments in
the laboratory. The Reynolds number in model and
prototype can vary by many orders of magnitude.
For example, experimental observations of boundary
layer cavitation by Arndt and Ippen (1968) were
carried out at Reynolds numbers, u'6/v, of the
order 5000. On large ships, Reynolds numbers of
10° and greater are not uncommon.
Coherency of the Pressure Field
An important factor related to cavitation in-
ception in jets is the existence of coherent
structure in the flow. Cavitation in highly turbu-
lent jets is observed to occur in ring like bursts,
smoke rings if you will. These bursts appear to
have a Strouhal frequency fd/U_ of approximately
0.5. This point is underscored by some recent work
of Fuchs (1974). Fuchs made 2 and 3 probe pressure
correlations as shown in Figure 6. His results are
summarized in Table 1. Signals filtered at a
Strouhal number of 0.45 were highly coherent. For
comparison, velocity correlations are shown in
parentheses indicating that the velocity field is
much less coherent than the pressure field.
The Turbulent Boundary Layer
Because of the relative ease of measurement, there
exists a considerable body of experimental data
Jet Nozzle
Probes (la2)
“|
Probe (0)—g\ A\
A \E 3d
General Arrangement
(a) (c)
(b)
a ae
PoP, Py (P,*P5) Py P, P, Po
FIGURE 6. Measurement of pressure coherency in a
turbulent jet [adapted from Fuchs (1974)].
for wall pressure due to turbulent boundary layer
flow. However, in many ways less is known about
the turbulent pressure field for boundary layers
than for free turbulent shear flows. Not only is
the theoretical problem made more difficult
(impossible to the present) by the presence of the
wall, the experimental problem is considerably
complicated by the dynamical significance of the
small scales near the wall.
Thus, in spite of over two decades of concentrated
attention we cannot say with confidence even what
the rms wall pressure level is, although recent
evidence points to a value of [Willmarth (1975)]:
mp = oo
c 2 ico) 3}
The basic problem is that the most interesting part
of a turbulent boundary layer appears to the region
near the wall where intense dynamical activity
apparently gives rise to the overall boundary layer
activity. While the details of the process are
debatable, most investigators concur on the importance
of the wall region on overall boundary layer
development. Unfortunately, under most experimental
conditions, the scales of primary activity are
smaller than standard wall pressure probes can
resolve [Willmarth (1975)]. Thus we have virtually
no information concerning the contribution of the
small scales to the pressure field, although we
suspect that the small scales are significant or
even dominant.
Pressure Spectra in Boundary Layers
Our knowledge of the pressure spectra may be
summarized as follows:
1) Pressure fluctuations arising from motions
333
in the main part of the boundary layer (y/é > 0.1)
scale with the outer parameters iB, and 6.
2) Pressure fluctuations arising from the inner
part of the boundary layer scale with the inner
parameters:
a) hydraulically smooth, uy, Vv
b) hydraulically rough, u h; where h is
roughness height
3) Pressure fluctuations arising from the
inertial sublayer (logarithmic layer) scale only
with u, and y, the distance from the wall.
4) The wall pressure spectrum is a composite
of all these factors and has a distinct region
corresponding to each factor.
A composite picture of the wall pressure spectrum
is shown in Figures 7a and 7b. The 1/k range is
evident in both the inner and outer scalings and
arises from the inertial sublayer contribution
[Bradshaw (1967)].
The pressure spectrum within the near wall region
should closely resemble the wall spectrum (although
this has never been confirmed). The spectrum in
the main part of the boundary layer, should, however,
resemble that obtained for a free shear flow at
high Reynolds numbers. Again there is no information
available to either prove or disprove this conjecture.
The Lagrangian model developed in the preceding
section depends in part on the assumption that a
material point is in a stationary random field.
As long as the Eulerian field is homogeneous, there
is no problem. This is approximately true in many
shear flows, but is never true in a turbulent
boundary layer. Thus our Lagrangian spectral picture
must be abandoned entirely (or used with great
restraint).
However, a number of features of the Lagrangian
model can be applied to this problem. In particular,
the "spectral peaks" in the outer flow can be
identified with the Lagrangian integral scale,
J ~ &/u'. The highest frequencies in the flow will
*!
Table 1.
Normalized correlation functions with pressure probes
arranged as shown in Figure 6 (corresponding velocity
correlations in brackets).
Signals
Unfiltered
Signals
Filtered at St = 0.45
Increasing
(cm!
m*
3 .
= Increasing
ee ud
a:
za
re
[oo
Do
2
log xv/u,
FIGURE 7. Wall pressure spectra: (a) outer scaling,
(b) inner scaling.
be e/a) or u*/h, depending on whether the wall is
hydraulically smooth or rough, and there will be
increasing intermittency with increasing Reynolds
number. The latter effect is most interesting and
is quite evident in the many observations of dye
streaks in the wall layer [cf Kim, Kline, and
Reynolds (1971)].
Effect of the Pressure Field on Cavitation
Whether or not the pressure fluctuations play a role
in the cavitation inception process, depends on
the previously cited criteria:
1) The minimum pressure must fall below a
critical level.
2) The minimum pressure must persist below the
critical level for a finite length of time.
The first criterion depends greatly on the yet
unresolved question of intermittency and its effect
on the probability density of the pressure fluctua-
tions. At this point in time we can say that the
critical cavitation index will increase with
Reynolds number because larger excursions from the
mean pressure are more likely. Without justification,
it is hypothesized that the effect on the pressure
variance will be approximated by a log-normal
dependence on the Reynolds number. ‘Detailed study
of the wall pressure such as that proposed by
George (1975) should aid considerably in resolving
this question.
The question of time scale is more easily con-
fronted. Since most of the energy in the pressure
spectrum scales with u, and 6 it is clear that the
criteria for bubble growth without appreciable
tension reduces to
u,T,/5 <1
In words, we again require a pressure fluctuation
to persist for a time which is long in comparison
to the response time of a typical nucleus.
Since v/u,? is the shortest time scale ina
smooth wall boundary layer, all of the pressure
spectrum is sampled by the nuclei when
2
u, T,/Y < il
This criterion is especially important in view of
the highly intermittant process near the wall.
For rough walls, the last criterion can be
expressed in terms of the roughness height h by
u,T,/n <a
Since in fully
that the small
rough flow u,h/v Sl ale GUS} Giles
scale criterion is more easily
satisfied with rough wall experiments.
In summary, the information we have on pressure
fields in turbulent boundary layers and its
relationship to cavitation inception can be
summarized as follows:
Significant scale effects can be expected when
u'T,/5 > 1. As the ratio of T. to the smallest
time scale in the flow decreases, the scale effect
would be expected to level off i.e. when uxTp/V or
u,T,/h <1. Further increase in the cavitation
n er with Reynolds number will be due to the
Reynolds number dependent effects on the probability
density of the pressure fluctuations as a result of
increased intermittancy of the small scale structure.
The latter effect should produce a more gradual
dependence of the cavitation index on Reynolds
number than the former effect.
The picture, as displayed above, is plausible
and perhaps even appealing, but it must be viewed
simply as conjecture until definitive experimental
information is made available. An important hint
of the relevance of these results can be found in
the work of Arndt and Ippen (1967) where it was
found that the region of maximum cavitation ina
rough boundary layer shifted inward with a decrease
in u,T /n. However, the change in this parameter
varied only by a factor of 15 in their experiments.
This will be discussed in more detail in subsequent
sections.
4. CAVITATION INCEPTION DATA
A rather limited amount of experimental data have
been collected under controlled conditions. The
types of flows considered to date include the wake
behind a sharp edged disk, submerged jets from
nozzles and orifices, and smooth and rough boundary
layers. There is a dearth of information relating
the observed cavitation inception with the turbulence ~°
parameters. Some of the earlier efforts in this
direction are summarized in a paper by Arndt and
Daily (1969) and by Arndt (1974b). A collation of
available data is presented in Figure 8. Here the
data are presented in the form of Eq. (1):
o fC =
Pp
* f£ (Cp)
Ke) T om Se T T Vapaly oe em Seer an au
© Smooth, Daily & Johnson ('56)
Boundary |@Sawteeth, Arndt & Ippen ('68)
Layer © Sand, Messenger ('68)
™ Sand, Huber ('69)
eo:
f
Jet 4 Rouse, ('53)
oo | Wake 4 Kermeen, Et Al ('55) |
+
co
OF a
Best Fit Curve 1
br ° >|
| 10 100
1000 C,
FIGURE 8. Collation of cavitation inception data.
wherein
2t )/pU* Boundary Layer Flow
Ce =
W145 Free Shear Flows
U
fe}
In this expression Cg is computed either from the
measured wall shear jn the case of boundary layer
flows or from turbulence measurements made in the
air at comparable Reynolds numbers for the case of
a free jet and a wake. The measured value of C, is
only significant for the case of the disk wake and
the pressure data was determined from the experi-
mental work of Carmodi (1964). The available data
seem tc be well approximated by the relation
which was originally proposed for boundary layer
flow by Arndt and Ippen (1968). These data would
seem to imply that a relatively simple scaling law
already exists and would further imply that the
previous discussion in this paper on turbulence
effects is superfluous. This is not the case.
Arndt and Ippen (1968) made observations of the
bubble growth in turbulent boundary layers. Some
of their results are depicted in Figures 9 and 10.
Figure 9 shows sample bubble growth data. The
growth rate is observed to stabilize at a constant
value during most of the growth phase. Using Eq.
(4), the levels of local tension are found to be
quite small, of the order 20 to 100 millibar. These
data correspond to observations in a rough boundary
layer. Of particular interest is the fact that,
in all cases, the life time for bubble growth is
a fraction of the Lagrangian time scale, J = d/u'.
In fact growth times were observed to be of the
order h/u,. Unfortunately there is not enough
*Tp was estimated from Eq. (5) using observed values
of R, reported in Arndt and Ippen (1967). For
convenience, the results are normalized to equivalent
Sand grain roughness, hg.
335
O16
ig) Fol
xo
[=
£
i)
OG 008
2
we}
Te}
J)
a "
004 © Run P35 k-0400
; © Run P47 k-0025
CORIEOMSIRaIN SMG II
Time (m sec)
FIGURE 9. Sample bubble growth data [after Arndt
and Ippen (1968)].
experimental evidence available to completely
illuminate this point. As shown in Figure 10,
cavitation occurs roughly in the center of the
boundary layer with a tendency for the zone of
maximum cavitation to shift inward as uxTp/h,
decreases from about 1.5 to approximately 0.1*.
In the cited boundary layer experiments, Cp is
negligible. Thus 0, = 16 Cf. Noting that p' is
approximately 2.5 pux* at the wall, we estimate
that cavitation is incited by negative peaks in
pressure of order 6 p'. This compares favorably
with Rouse's (1953) data for jet cavitation which
indicate that negative peaks of order 10 p' are
responsible for cavitation.
A strong dependence on Reynolds number can be
observed even in free shear flows. Figure 11
contains cavitation data for a sharp edged disk.
These data were obtained in both water tunnels
and a new depressurized tow tank facility located
at the Netherlands Ship Model Basin. The water
tunnel data are for cavitation desinence, whereas
the tow tank data are for cavitation inception
determined acoustically. The cross hatched data
were determined in a water tunnel at high velocities
by Keermeen and Parkin (1957). All the other data
were obtained at relatively low velocities (2 - 10
m/sec). There is considerable scatter in these
data and this is traceable to gas content effects
T T T T T T T T a reas |e
all bubbles all bubbles
30F ---- cavitating bubbles 30+ ---- Cavitating bubbles 4
Relative Concentration (%)
Relative Concentration (%)
2
FIGURE 10. Observation of cavitation in turbulent
boundary layers [after Arndt and Ippen (1968) ].
336
AL = 44 +0,0036R//¢
Od x 4
aS
Sur
Onna «5
Ow
High Reynolds Number Asymptote 4
ff Water Tunnel
B O Smooth
WY Kermeen & @ Rough
Parkin O Vacutank
(0) 2 4 6 8 10 l2 14 16 18 20
Reynolds Number x 109
Cavitation Index
FIGURE 11.
edged disk.
Cavitation inception data for a sharp-
which are dominate at low velocities as will be
discussed later. At low Reynolds number the data
appear to be satisfied by the empirical relationship
discussed by Arndt (1976):
o, = 0.44 + 0.0036 (Ud/v) 2 (7)
It was found that the tow tank data agree with this
relationship at relatively high Reynolds numbers.
Equation (7) was developed from a model which
assumes laminar boundary layer flow on the face of
the disk. It would be expected that this condition
would be satisfied at higher Reynolds numbers in
a tow tank than in a highly turbulent water tunnel.
At high Reynolds number (and also high velocity
where gas content effects are negligible), there
is a continuous upward trend in the data with
increasing Reynolds number. This underscores the
need for further work as suggested in the intro-
duction to this paper.
A systematic investigation of gas content effects
in free shear flow was recently reported by Baker
et al. (1976). Cavitation inception in confined
jets, generated either by an orifice plate ora
nozzle, was determined as a function of total gas
content in the liquid. The results are shown in
Figure 12. When the liquid was undersaturated at
test section pressure, the critical cavitation
index was independent of gas content and roughly
equal to that observed by Rouse (1953) for an
unconfined jet. When the flow is supersaturated,
the cavitation index is found to vary linearly with
gas content as predicted by the equilibrium theory,
Eq. (6). This effect occurs even though the
Lagrangian time scale is much shorter than typical
times for bubble growth by gaseous diffusion. For
example, in the cited cavitation data, a typical
residence time for a nucleus within a large eddy
is roughly 1/15 of a second. At a gas content of
7ppm and a jet velocity of approximately 10 m/s,
inception occurs at a mean pressure equivalent to
a relative saturation level of 1.25. Epstein and
Plesset (1950) show that for growth by gaseous 3
diffusion alone, 567 seconds is required for a 10
cm nucleus to increase its size by a factor of 10.
One additional point should be kept in mind here.
The local pressure within an eddy is much less than
the mean pressure and highly supersaturated con-
ditions can occur locally. Arndt and Keller (1976)
also reported extreme gas content effects in their
experiments with disks when the flow was super-
saturated. The magnitude of the effect also depends
on the number of nuclei in the flow. Gas content
effects were noted only in their water tunnel
experiments (where there is a healthy supply of
nuclei). No gas content effects on inception were
noted in the tow tank (where the flow is highly
supersaturated but there is a dearth of nuclei).
Thus the picture becomes more cloudy as the influence
of dissolved, noncondensable gas is taken into
consideration.
5. SOME REMARKS ON CAVITATION NOISE
A complete discussion on cavitation noise would be
beyond the scope of this paper. Recognizing the
unique features of cavitation inception in
turbulent shear flows, it appears appropriate to
review what is known about cavitation noise under
the same circumstances.
The general features of cavitation noise were
reviewed by Fitzpatrick and Strasberg (1956), Baiter
(1974), and Ross (1976). The spectrum of cavita-
tion noise can in its simplest form be defined as
the linear superposition of N cavitation events per
unit time. Thus we can write
S(£) = N G(£) (8)
The function G(f) is the spectrum of a single
cavitation event. If p, is the instantaneous
acoustic pressure due to the growth and collapse of
a single bubble, then by definition
J ccrar - ee dt
oO
—co
Fitzpatrick and Strasberg (1956) have shown that a
characteristic bubble spectrum can be written in
the form
a
b
g
= _047B(a-a,)
5 ; 2
5 1/2 pUG
S
S
my (a)
7]
Cc
Fy
a
(b)
Contoured Nozzle
FIGURE 12. Cavitation inception in confined jets.
wherein Tt, is a characteristic bubble collapse
time, Ry is the maximum bubble radius, and R is
the distance to the observer. In addition, it
appears reasonable to assume that N is related to
the number of nuclei per unit volume, n, the
velocity, the size of a given flow field, and the
relative level of cavitation. Therefore we write
N/nu a2 = £(c/fo_)
0) c
Thus a normalized version of Eq. (8) would be
——_——F = £(o/o |) G(fr 7) (9)
It is difficult to obtain appropriate scaling
factors for R_ and T_ in a turbulent shear flow.
The problem iS discussed briefly by Arndt and Keller
(1976). Lacking more detailed information, the
following assumptions can be used
mR os Gl
m
aiee
Be {UG
If we interpret S(f) as the mean square acoustic
pressure in a frequency band Af, Eq. (9) can be
written in the form
p 2/p2u “) 2
eee,
Afa/U 3
( / By ond
1
a2)
Se (a/o.) G(f£d/Uo ~) (10)
Blake et al. (1977) circumvented the requirement
of measuring n. They reasoned that
2 ——
= = 2
i G(£) dt i i Che SES Ish,
wherein Pp? is the time mean square of pp and y,,
is the total lifetime of the bubble (including
growth, initial collapse times and rebounding times) .
Further, they simply reasoned that
or that
S(e 32) = We eee _))
() (0) (o)
This results in the normalized spectrum
Di 2
pi” (£,4£) yt or
S(t, £) SSS (11)
A£N R 4op
Making the same assumptions as before, we would
expect that
a ee G! (£a/vo®) (12)
(hea/o,) 53/2” 2
Blake et al. were able to determine S(tgf) for the
case of noise due to cavitation on a hydrofoil
using measured values of Ry. They assumed N equal
to unity and found that Eq. (11) resulted in
excellent collapse of the data.
Arndt (1978) used Eq. (12) to normalize cavitation
data previously reported by Arndt and Keller (1976).
These data correspond to noise from cavitation in
the wake of a disk and were collected under a
variety of conditions in both a water tunnel and in
a depressurized towing tank. Both the level of
dissolved gas and the number of free nuclei were
monitored. As shown in Figure 13, the normalization
is not very successful. It would appear that Eq. (10)
would be more effective in taking all of the
variables into account. However, n could only be
measured in unison with acoustic observations in the
water tunnel. Because of the nature of the laser
scattering measurements used to determine n in the
depressurized towing tank, these measurements had
to be made separately from the acoustic measure-
ments. The assumed form for S(£T)) in Eqs. (10) and
(11) varies by a factor na3/o's. As an example, n in
the depressurized towing tank appeared to be rela-
tively constant and equal to about 15/cm?. Therefore
the factor nd3/o% was found to have a maximum varia-
tion of 23 dB. This does not account for the scatter
shown and one can only assume that there are other
complicating factors. It should be emphasized that
these data were collected under carefully controlled
conditions. This underscores the fact that the
current state of knowledge in this area is poor.
6. CONCLUSIONS
Cavitation inception in turbulent shear flows is
the result of a complex interaction between an
unsteady pressure field and a distribution of free
stream nuclei. There is a dearth of data relating
cavitation inception and the turbulent pressure
field. What little information that is available
indicates that negative peaks in pressure having a
magnitude as high as ten times the root mean square
4
(dB)
foe}
(eo)
joerc ere re are)
ae 2|5 100F
| Water Tunnel Dao U %/ed 1 4 Gas Content nd
Bcm 6M/sec 0.76 I5S/cem, 39ppm
I20F 2cm 4M/sec 0.80 4I/m> 45ppm
2cm 4MYsec 0.63 219/m> 7.5ppm 4
140 Bem 4/sec 064 75/em> 6 %ppm J
Vacu -Tank
ie 16cm 30™/sec 0.66 — 10 ppm 4
160 fem 3.eM/sec O66 — : 10 ppm ; ca
10 100 1000 10000 100000
fel 1
Uo ola
FIGURE 13. Normalized cavitation noise spectra.
338
pressure can excite cavitation inception. This fact
alone indicates that consideration should be given
to the details of the turbulent pressure field.
The available evidence indicates that two basic
factors related to the pressure field enter into
the scale effects. First, as the scale of the
flow increases, cavitation nuclei are relatively
more responsive to a wider range of pressure
fluctuations. Secondly, the available evidence
indicates that large deviations from the mean
pressure are more probable with increasing Reynolds
number. This would explain some of the observed
increases in cavitation index with physical scale.
In view of the almost total lack of information on
the statistics of turbulent pressure field (aside
from some correlation and spectral data) and the
potential importance of this knowledge to under-
standing cavitation, it is strongly recommended
that careful experiments be initiated to remedy the
situation. Such experiments have been proposed by
George (1974b, 1975).
Direct application
tion to cavitation is
of the pressure field informa-
unfortunately clouded by gas
content effects which also increase the cavitation
index with increasing exposure time. The fact that
a reasonably precise scaling law for cavitation
noise has not yet been found (perhaps a consequence
of the lack of knowledge about the pressure field)
further complicates interpretation of experiments
and theory. Therefore it is also strongly recom-
mended that the problem of the response of cavita-
tion nuclei to turbulence receive particular attention.
Such experiments have been proposed by Arndt (1978).
ACKNOWLEDGMENTS
R. E. A. Arndt gratefully acknowledges the support
of the Air Force Office of Scientific Research and
the Seed Research Fund of the St. Anthony Falls
Hydraulic Laboratory. W. K. George gratefully
acknowledges the support of the National Science
Foundation under grants from the Engineering (Fluid
Dynamics) and Atmospheric Sciences (Meterology)
Programs and the Air Force Office of Scientific
Research. Both authors are grateful to Mrs. Sandra
Peterson who typed the manuscript.
REFERENCES
Arndt, R. E. A. (1974a). Cavitation inception and
how it scales: a review of the problem with a
summary of recent research. Proc. Symposium on
High Powered Propulsion of Ships, Wageningen,
The Netherlands, December (available from the
Netherlands Ship Model Basin).
Arndt, R. E. A. (1974b). Pressure fields and
Cavitation. Trans. 7th IAHR Symposium, Vienna,
September.
Arndt, R. E. A. (1976). Semi-empirical analysis of
cavitation in the wake of a sharp edged disk.
J. of Fluids Engr., September.
Arndt, R. E. A. (1977). Cavitation and erosion:
an overview. Proc. Corrosion/77, Natl. Assoc.
of Corrosion Engineers, San Francisco, Calif.
Arndt, R. E. A. (1978). Investigation of the effects
of dissolved gas and free nuclei on cavitation
and noise in the wake of a sharp edged disk.
Proc. Joint IAHR/ASME/ASCE Symposium on Fluid
Machinery, Ft. Collings, Colorado.
Arndt, R. E. A., and J. W. Daily (1969). Cavitation
in turbulent boundary layers. Proc. of Symposium
on Cavitation State of Knowledge, ASME, 64-86.
Arndt, R. E. A., and A. T. Ippen (1967). Cavitation
near surfaces of distributed roughness. MIT
Hydrodynamics Laboratory Report No. 104.
Arndt, R. E. A., and A. T. Ippen (1968). Rough
surface effects on cavitation inception, J. of
Basic Engr., Trans. ASME, Series D, 249-261.
Arndt, R. E. A., and A. Keller (1976). Free gas
content effects on cavitation inception and
noise in a free shear flow. Proc. IAHR Symposium
on Two Phase Flow and Cavitation in Power
Generation Systems.
Baker, ©. B., J. W. Holl’, and R.\E. Aj Arndt (@o7o)e
The influence of gas content and polyethelene
oxide additive upon confined jet cavitation in
water. ASME Polyphase Forum, March.
Baiter, J. J. (1974). Aspects of cavitation noise.
Proc. Symposium on High Powered Propulsion of
Ships, Wageningen, The Netherlands, Dec.
Batchelor, G. K. (1951). Pressure fluctuations in
isotropic turbulence. Proc. Camb. Phil. Soc.,
4727) 359-314.
Batchelor, G. K. (1953). The Theory of Homogeneous
Turbulence, Cambridge University Press.
Batchelor, G. K., and A. A. Townsend (1949). The
nature of turbulence at large wavenumbers. Proc.
Royal Soc. (London), A199, 1057; 238-255.
Beuther, P., W. K. George, and R. E. A. Arndt (1977a).
Modelling of pressure spectra in a turbulent
shear flow. 93rd Meeting Acoustical Soc. of
Amer., State College, June.
Beuther, P., W. K. George, and R. E. A. Arndt (1977b).
Pressure spectra in a turbulent shear flow.
Bulletin Amer. Phys. Soc., 22, 12, December.
Beuther, P., W. K. George, and R. E. A. Arndt (1977c) .
Pressure spectra in homogeneous isotropic
turbulent flow. Bulletin Amer. Phys. Soc., 22,
12, December.
Blake, W. K., M. J. Wolpert, and F. E. Geib (1977).
Cavitation noise and inception as influenced by
boundary layer development on a hydrofoil. J.
Fluid Mech., 80, Part 4, 617-640.
Bradshaw, P., (1967). Inactive motion and pressure
fluctuations in turbulent boundary layers. J.
Fluid Mech., 30, 241.
Chan, Y. Y. (1974a). Pressure sources for a wave
model of jet noise. ATIAA J., 12, 2.
Chan, Y. Y. (1974b). Spatial waves in turbulent
Jets eRUST CSO Ger laTaS el 7a el.
Chan, Y. Y. (1976). Noise generated wavelike eddies
in a turbulent jet. JICAS paper, No. 76-42.
Epstein, P. S., and M. S. Plesset (1950). On the
stability of gas bubbles in liquid gas solutions.
J. Chem. Phys., 18, 11, November, 1505-1509.
Fitzpatrick, H. M., and M. Strasberg (1956). Hydro-
dynamic sources of sound. Proc. lst Symposium
on Naval Hydrodynamics, Natl. Acad. of Sciences,
Natl. Res. Council, Publication 515, September.
Fuchs, H. V. (1972a). Measurements of pressure
fluctuations within subsonic turbulent jets.
Ui, Wloletere! Etetol WAllaa np 2An So
Fuchs, H. V. (1972b). Space correlations of the
fluctuating pressure in subsonic turbulent jets,
Wn Clotdel Eveel WAllon 5» 2p Ip Tis
Fuchs, H. V. (1974). Analysis of circumferentially
coherent pressure fluctuations relevant to jet
noise. Colloquium on Coherent Structure in
Turbulence, Southampton.
Fuchs, H. V., and A. Michalke (1975). On turbulence
and noise of an axisymmetric shear flow. J.
Fluid Mech., 70, 1.
George, W. K. (1974a). The equilibrium range of
turbulent pressure spectra, Bull. of the Amer.
PAYS}. SOC PLO eS 8)
George, W. K. (1974b).
dynamics Research.
George, W. K. (1975). Proposal to ONR, Fluid
Dynamics Branch and presentation at David H.
Taylor Naval Ship Research and Development Center,
February.
Gurvich, A. A., and A. M. Yaglem (1967). Breakdown
of eddies and probability distribution for small
scale turbulence, Phys Fluids, 10, Supplement,
S59-S65.-
Heisinberg, W. (1948). Zur statistichen theorie
den turbulenz. F. Phisik, 124, 7-12; 359-374.
Holl, J. W. (1960). An effect of air content on
the occurrence of cavitation. J. Basic Engr.,
Trans. ASME, Series D, 82, 941-946.
Holl, J. W. (1969). Limited cavitation. Proc. of
the Symposium on Cavitation State of Knowledge,
ASME, June, 26-63.
Holl, J. W. (1970). Nuclei and cavitation. fran.
ASME, J. of Basic Engr., December, 681-688.
Jones, B. G., R. J. Adrian, C. K. Nithianandan,
and H. P. Planchen, Jr. (1977). Spectra of
turbulent static pressure fluctuations in jet
mixing layers, ATAA Paper No. 77-1370.
Keller, A. P. (1974). Investigations concerning
scale effects of the inception of cavitation.
Proc. Conf. on Cavitation, Inst. of M. E.,
Heriot-Watt University, September.
Kermeen, R. W., and B. R. Parkin (1957). Incipient
cavitation and wake flow behind sharp-edged
Proposal to General Hydro-
339
disks. Report 85-4, Hydrodynamics Laboratory,
California Institute of Tech., August.
Kim, H. T., S. J. Kline, and W. C. Reynolds (1971).
The production of turbulence near a smooth wall
in a turbulent boundary layer. J. of Fluid
Mech., 50, 133.
Kolmogorov, A. N. (1962). A refinement of previous
hypotheses concerning the local structure of
turbulence in a viscous incompressible fluid at
high Reynolds number. J. Fluid Mech., 13, 1;
82-85.
Kraichnan, R. H. (1956). Pressure fluctuations in
turbulent flow over a flat plate. J. Acoust.
Soc. Amer., 28, 3.
Lienhard, J. H., and C. D. Goss (1971). Influence
of size and configuration on cavitation in sub-
merged orifice flows. ASME Paper No. 71-FE-39.
Monin, A. S. and A. M. Yaglem (1975). Statistical
Fluid Mechanics: Mechanics of Turbulence, Vol.
II, MIT Press, Cambridge.
Obukov, A. M. (1949). Pressure fluctuaticns in a
turbulent flow. Dokl. Acad. Nauk., SSSR, 66,
1S el Os
Plesset, M. S. (1949). The dynamics of cavitation
bubbles. J. of Applied Mech., September, 277-
282.
Ross, D. (1976). Mechanics of Underwater Noise,
Pergamon Press.
Rouse, H. (1953). Cavitation in the mixing zone of
a submerged jet. La Houille Blanche, February.
Tennekes, H., and J. L. Lumley (1972). A First
Course in Turbulence. MIT Press.
Willmarth, W. W. (1975). Pressure fluctuations
beneath turbulent boundary layers. Annual
Review of Fluid Mech., 7, Palo Alto, Calif.
Secondary Flow Generated
Vortex Cavitation
Michael L. Billet
The Pennsylvania State University
State College, Pennsylvania
ABSTRACT
Secondary flow theories are employed to calculate
the secondary vorticity near the inner wall of a rotor
for several flow conditions. This calculated vortic-—
ity is used in a simple vortex model to calculate the
minimum pressure coefficient of the resulting vortex
behind the rotor. The influence of inflow velocity
distributions on the generation of secondary vortic-—
ity is discussed. Comparisons are given between the
calculated pressure coefficients and the measured
cavitation indices of the vortex.
1. INTRODUCTION
Secondary flows generate additional streamwise vor-
ticity when a boundary layer flow is turned by a
rotor. The apparent effect of this additional vor-
ticity is evidenced by the high cavitation numbers
of the vortex formed downstream of the rotor plane.
One example of the cavitation associated with a
vortex can be found in the draft tube of a Francis
turbine operating in the part load range. The
cavitation depends directly on the square of the
streamwise vorticity associated with the vortex. In
most cases, the critical cavitation numbers typical
of this vortex are often higher than those associ-
ated with any other type of rotor cavitation.
Previous experimental results have shown that a
cavitation inception prediction of this vortex is a
very difficult problem. All rotors operating with
a wall boundary layer have a vortex ‘along the inner
wall. The appearance of this cavitating vortex varies
from rotor to rotor. The critical cavitation number
can vary aS much as an order of magnitude. Small
variations in the wall boundary layer can cause a
significant change in the critical cavitation number.
Some confusion in cavitation inception data asso-
ciated with this vortex is due to a confusion of
types of cavitation, i.e., vaporous versus nonva-
porous cavitation. Vortex flows tend to be good
340
collectors of gas bubbles which can cause non-
vaporous cavitation. This often leads to confusing
nonvaporous for vaporous cavitation giving high
cavitation numbers. In general, results indicate
for vaporous- limited cavitation that
<
Tg = SC oee (1)
Thus, the minimum pressure coefficient is of partic-
ular importance in a study of vortex cavitation in-
ception.
It is appropriate then to find a simple descrip-
tion of the vortex in order to calculate its minimum
pressure coefficient. Unfortunately, the vortex is
composed of a finite number of vortex filaments
and a difficulty arises in specifying this number.
This is particularly difficult when the vortex exists
in the low pressure region near the inner wall of the
complicated flow behind a rotor. In this region,
there are vortex filaments in the primary flow in
addition to the secondary vortex filaments which can
influence this vortex. The combined effect of these
filaments is to induce a swirl velocity distribution,
Vg, which can be easily measured.
Some preliminary tests show that in many cases
small changes in the incoming velocity profile near
the inner wall cause large differences in the crit-
ical cavitation number of the vortex. Measurements
of the primary flow field show only a change in down-
stream velocity profile near the inner wall. This
is especially true if the rotor was designed to be
unloaded near the inner wall. For these cases,
changes in the critical cavitation number can be
directly related to changes in the secondary vortic-
ity near the rotor inner wall.
The secondary vorticity can roll-up into a vortex
like flow in the blade passage or it can simply com-
bine with other vortex filaments aft of the rotor to
form a larger vortex flow. In either case, there
will be a circulation and a characteristic dimension
of the passage vorticity which will determine the
critical cavitation number of the resulting vortex.
In this paper, a brief summary is given of the
method for calculating the secondary vorticity in
the blade passage with comparisons to flow field
measurements. Initially, the primary flow field
through the rotor had to be determined in order to
calculate the passage secondary vorticity. This was
accomplished by using a streamline curvature method.
Flow field results are given in detail for one basic
flow configuration so named Basic Flow No. 1. Com-
parisons between the calculated minimum pressure co-
efficients and measured critical cavitation indices
are given for several basic flow configurations or
inflow velocity distributions.
2. CALCULATION OF FLOW FIELD
Primary Flow Field
A schematic of the calculation procedure for the
flow through a rotor is given in Figure 1. This
outlines the iterative procedure for the calcula-
tions and indicates the point at which refinements
to the deviation angle are necessary and where
secondary flow calculations are employed.
It is important to realize that in this discus-
sion the flow field is being solved for a given
rotor configuration. For this case, the boundary
conditions are (1) the geometric or metal angles
of the blades, (2) the rpm of the rotor, (3) the
velocity profile far upstream of the rotor plane,
and (4) the bounding streamlines of the flow.
After solving for the bounding streamlines, the
iterative calculation procedure is started by
establishing the velocity profile far upstream of
the rotor. The initial conditions (Step 1) to the
solution for this boundary condition are (1) bounding
streamtube and (2) velocity profile in rotor plane
without rotor. With this information, the initial
streamlines without rotor can be calculated using
the streamline curvature equations (Step 2). The
result of this calculation is the boundary condition
of an initial velocity or energy profile at a station
far upstream of the rotor plane.
CALCULATION OF PRIMARY FLOW FIELD
STEP 1 INITIAL CONDITIONS
STEP 2 CALCULATION OF FLOW WITHOUT RoToR |
STEP 3 FIRST ESTIMATE OF ROTOR OUTLET ANGLE
Si? J—= CALCULATION OF FLOW FIELD WITH ROTOR
Sup 7 SECONDARY FLOW CALCULATION
STEP 8 THIRD ESTIMATE OF ROTOR OUTLET ANGLE
STEP 9—— | FINAL CALCULATION OF FLOW FIELD WITH ROTOR
FIGURE 1. Schematic of calculation procedure for
primary flow field.
341
AG
VELOCITY PROFILE
(B.C. #3)
y— BOUNDING STREAMLINES (B.C, #4)
METAL ANGLES OF —
BLADES (B.C. #1)
IER eT =
ROTOR RPM (B.C. #2)
FIGURE 2. Schematic of boundary conditions.
Knowing the blade metal angles, the first estimate
of the flow outlet angles (Step 3) can be calculated.
These flow outlet angles depend on the blade metal
angles and on a deviation angle. The deviation
angle correlation developed by Howell as discussed
in Horlock (1973) is initially applied. This
relationship considers only thin blade sections and
assumes that each blade secticn operates near design
incidence. As shown in Figure 2, all of the boundary
conditions are now known and the flow field can be
solved with the rotor included (Step 4) by using
the streamline curvature equations [McBride (1977)].
Once a converged solution is obtained for the
flow field using Howell's deviation angles (Step 4),
the axial velocity distribution is known whereby the
inlet angles can be estimated in addition to the
acceleration through the rotor. Now a second
estimate of the rotor outlet angles (Step 5) can be
made. For this deviation angle, the effects of
acceleration, Aéd', blade camber, 69, and blade
thickness, Aé*, are calculated separately. For the
calculation of the deviation term due to axial
acceleration through the rotor, an equation developed
by Lakshminarayana (1974) is applied. For the
calculation of deviation terms due to camber and
thickness effects, the data obtained by the National
‘Aeronautics and Space Administration [Lieblein
(1965)] are used. The result is an improved outlet
flow angle profile which can be used to again calcu-
late the flow field (Step 6).
The converged solution of the flow field (Step 6)
is then used to solve the secondary vorticity
equations (Step 7) and to determine a deviation term,
Aéds, which is due to nonsymmetric flow effects. The
details of the secondary flow calculations will be
discussed later in this paper. An improved outlet
flow angle profile (Step 8) is obtained by adding
this secondary flow term to the deviation terms
thus far calculated to obtain
Bo* = Bo — AS' + AS* + bo + AS. (2)
where 85* is the outlet flow angle and 8) is the
blade metal outlet angle. This outlet flow angle
distribution is then used as a boundary condition
in the calculation of the flow field (Step 9).
Finally, all of the deviation angle calculations
are checked based on the flow field calcvlated in
Step 9. If the angles did not change significantly
then the result obtained in Step 9 is used as the
final flow field.
In all, twenty-eight streamlines were calculated
342
© WITHOUT UPSTREAM STRUTS
WITHOUT SCREEN
WITHOUT ROTOR
© WITHOUT UPSTREAM STRUTS
WITHOUT SCREEN
DESIGN FLOW COEFFICIENT
(BASIC FLOW NO. 1)
DISTANCE FROM
SURFACE, Chie
R' (inches) CALCULATED PROFILE
inc
—_——
r or
02 OC HS WE a ke
AXIAL VELOCITY RATIO, Vien
FIGURE 3. Comparison between velocity profile with/
without rotor.
through the rotor with the first streamline being
at the inner wall and the last streamline going
through the rotor tip. The streamlines were spaced
more closely near the inner wall because the second-
ary flow calculations are most important near the
wall. Also, the streamline curvature equations are
inviscid so that there is a finite velocity at the
inner wall streamline.
A sample of the calculations for the flow field
is given in Figures 3,4, and 5 for the flow config-
uration called Basic Flow No. 1. For.Basic Flow
No. 1, the boundary layer entering the rotor is
axisymmetric with no upstream distribution such as
screens or struts forward of the rotor which is
operating at its design flow coefficient. In Figure
3, the calculated axial velocity profile in the
plane of the rotor without the rotor and the calcu-
lated axial velocity profile in front of the rotor
with the rotor operating on design is shown. In
addition, experimental data measured in the 48-inch
(a) WITHOUT UPSTREAM STRUTS
WITHOUT SCREEN
DESIGN FLOW COEFFICIENT
3 (BASIC FLOW NO. 1)
DISTANCE
FROM
SURFACE, 2 DATA
R' (inches)
ake
VE
l ined
CALCULATED
PROFILE
°
0 rt A (PAL al
0 0.2 #O4 0.6 0.8 1e Ome Zed
VELOCITY RATIOES, V5 NES AND Vy
FIGURE 4. Rotor outlet velocity profiles for basic
flow no. l.
WITHOUT UPSTREAM STRUTS
WITHOUT SCREEN
DESIGN FLOW COEFFICIENT
(BASIC FLOW NO. 1)
DISTANCE
FROM
CENTERLINE, 5 max
R’ (inches)
CALCULATED PROFILES
n°)
(0)
O° OF O41 G6 O86, RO be a
8
x
VELOCITY RATIOS, —— AND) —
Ves co
4
FIGURE 5. Tangential and axial velocity profiles
at cap.
water tunnel by a LDA system are given for a com-
parison. In Figure 4, the calculated outlet velocity
profiles are shown with comparison to measured data.
Finally, Figure 5 shows the calculated and measured
tangential velocity, component, Vg, downstream of the
rotor plane where cavitation occurs under certain
flow conditions. In general, the flow field calcu-
lations show very good agreement with the experi-
mental data.
Secondary Flow Field
The major equations used in the streamline curvature
method for calculation of the flow field were derived
from the principles of conservation of mass, momentum,
and energy. The fluid was assumed to be incompress-—
ible, inviscid, and steady. In addition, the flow
field was assumed to be axisymmetric.
The resultant equations allow for streamline
curvature and for vorticity in the flow. However,
it is important to realize that the solution to the
flow field does not contain all of the vorticity.
In particular, only the circumferential vorticity
is totally included. The other components of
vorticity contain derivatives with respect to the
circumferential direction which are assumed to Ee
zero. As discussed by Hawthorne and Novak (1969),
the neglected vorticity terms can be related to the
secondary flows that occur in the blade passage
along the inner wall.
Using the generalized vorticity equations, Lak-
shminarayana and Horlock (1973) derived a set of
incompressible vorticity equations valid for a
rotor operating with an incoming velocity gradient.
Their expressions for the absolute vorticities,
Ws', Wn', defined along relative streamlines, s',
n', were modified for the boundary conditions imposed
by this problem and were integrated. The resulting
equations are
W) ap}
Oe Sapte ead (3)
2 abo
and
2 2
AAW OY 22. Hue
Oo = Wo WR" ds' + Wo 3 ds!
1 1
2
1 '
W
-wW ecla ds' + w_! sts (4)
2 2 S$] Wy
where the primes refer to a rotating frame of
reference and the subscripts, 1, 2, refer to com-
puting stations along a streamline within the rotor.
As shown in Figure 6, s', n', b' represent the
natural coordinates for the relative flow, W is the
relative velocity, we' and W,' are absolute vorticity
resolved along the relative streamline, s', and the
principal normal direction, n', 2 is the rotor
rotation vector, and R' is the radius of curvature
of the relative streamline.
The means by which the streamwise component of
vorticity is produced in this relative flow are
similar to those discussed by many investigators
for a stationary system. However, it is important
to note that additional secondary vorticity is
generated when x W has a component in the relative
streamwise direction. Rotation has no effect when
the absolute vorticity vector lies in the s'-n'
plane and the rotation, 9%, has no component in the
binormal direction, B'.
These equations were employed to calculate the
secondary vorticity along a relative streamline
through the rotor. All of the quantities in the
equations were calculated by an iterative procedure
using the primary flow calculations. The initial
normal component of absolute vorticity, Wn , for a
streamline was calculated from the incoming axial
velocity profile to the rotor. In all, the vorticity
along twenty-eight streamlines was calculated.
As an example, Figure 7 shows the importance of
each term in Eq. 4 in the rotor exit plane for Basic
Flow No. 1. The sum of these terms is given in
Figure 8. The secondary passage vorticity is the
difference between the exit vorticity, Ws5, and the
inlet vorticity, Ws}, along a streamline.
CALCULATION OF FLOW FIELD THROUGH ROTOR IN RELATIVE COORDINATE SYSTEM
VELOCITY COMPONENTS
Ww
Vx,
U
BLADE ROW
VORTICITY COMPONENTS ROTATION COMPONENTS
STREAMLINE
STREAMLINE
Wr, Ty
| a2) a
1
/B2
s.
FIGURE 6. Description of relative coordinate system.
343
3 T T T Ta r t~—
WITHOUT UPSTREAM STRUTS
Q @ WITHOUT SCREEN
o 2u, ds 20,4, ds! DESIGN FLOW COEFFICIENT
3 = (BASIC FI i
FE - Wo liwRumenis C FLOW NO. 1)
€
z 2 22_,u, ds! a
=f |
2 qe W
z :
a
> O
=
Oo
&
wot =
r=
< c)
nn
a
CALCULATED DATA
0 n L |
6 5 “4 3 -2 -1 0 1 2 3
uy RR
RELATIVE ABSOLUTE STREAMWISE VORTICITY, G, =
FIGURE 7.
ge} dhe
Streamwise passage vorticity for basic flow
The effect of this additional vorticity, Ws -
Ws}, is to induce secondary velocities which are
assumed to occur at the exit plane of the rotor. It
is important to note that the normal component of
vorticity, Wndr is accounted for in the axisymmetric
flow analysis. Thus, only streamwise secondary
vorticity calculated as a function of radius influ-
ences the flow field.
The effect of the streamwise component of vorti-
city within the blade passage is similar to that
obtained in the flow through a curved duct [Hawthorne,
(1961), Eichenberger, (1953)]; however, there is
the difficulty of devising a reasonable approximate
method of satisfying the Kutta-Joukowski condition
at the exit of the rotor. The method used in this
investigation assumes that the flow is contained in
a duct defined by the blades and streamlines of the
primary flow leaving the exit of each blade. In
this exit plane, a flow solution devised by Hawthorne
and Novak (1969) was applied. The secondary stream-
Sire
WITHOUT UPSTREAM
STRUTS
WITHOUT SCREEN
re DESIGN FLOW COEFFICIENT
c (BASIC FLOW NO. 1)
fe C= =|
wwf
oO
fhe
[4
—)
wn
=
is)
&
seals |
= EXIT STREAMWISE
=< VORTICITY
= pe
QB a,
a )
1 i L
-4 3 =a = 0 1 2 3
Wo Rp
RELATIVE STREAMWISE VORTICITY, @, =
Sy We
FIGURE 8. Relative passage streamwise vorticity at
rotor exit plane for basic flow no. 1.
344
wise vorticity was divided into tangential and axial
components whereby the former, (We5-Ws 1) sinBo,
causes a radial gradient of axial velocity and the
latter leads to an equation for a stream function
describing the radial and tangential velocities in
the exit plane, r,6.
The form of the secondary stream function equation
is
2
ay ley 1 02y xd *
y2y = we OE i ue a) ee CE
Sed Soe Ste Da 1 pl ae (ete)
2 C)s) Yr
*
- Ge ")secBo = I5((Fe)) 5 (5)
where Vx is the secondary axial velocity and is
obtained from the solution of the tangential com-
ponent of streamwise vorticity. The solution to
Eq. (5) was found by applying standard differential
techniques. The solution and the necessary boundary
conditions will not be discussed in this brief paper.
The deviation angle due to the secondary flow
can be calculated using
N cos? B> UA
N\ = ou dr
s 271V Cha (S)
a 0
where N is the number of blades and ¥ is obtained
from the solution of Eq. (5). The axial velocity,
Vx, and outlet angle, 85, are determined in the
calculation of the primary flow field.
The results of the secondary flow calculations
for various basic flows indicate that the effects
are significant only near the inner wall where the
incoming vorticity is the largest. The deviation
angles calculated for Basic Flow No. 1 are shown
in Table 1.
3. CAVITATION EXPERIMENTS
The cavitation experiments were conducted in the
48-inch diameter water tunnel located in the Garfield
Thomas Water Tunnel Building of the Applied Research
Laboratory at The Pennsylvania State University. In
Correlation with cavitation data for basic
1 and 4.
FIGURE 9.
flow nos.
CAVITATION NUMBER
TABLE 1. Deviation Angles for Basic Flow No. 1
Normalized Distance
from Surface Deviation Angles
R/R AS
s
0.00 -5.4
0.04 =239)
0.14 -1.0
0.24
0.34
fo}
0.44 <0.2
0.54
0.64
all cases, desinent cavitation was employed as the
experimental measure of the critical cavitation
number. The cavitation in the vortex system occurred
on the rotor cap. Also, the occurrence of the
cavitation was very sporadic.
The air content of 3.1 ppm was chosen for all of
the cavitation experiments because gas effects are
reduced and the relative saturation level was always
much less than unity. Desinent cavitation number
data were obtained for different incoming velocity
profiles to the rotor. The incoming velocity profile
was varied by changes in the configuration of the
upstream surface in addition to varying the rotor
flow coefficient. Results were obtained with/without
upstream struts, with/without a screen on the upstrea
surface, and on/off design rotor flow coefficients.
In all, there were sixteen different flow configura-
tions or Basic Flow Nos. tested.
Figures 9-11 display the effects on the desinent
cavitation number over a range of velocities due to
variations in the inflow velocity distribution. In
general, the cavitation number increased for in-
creasing free stream velocity for all flow config-
urations shown. As shown in Figure 9, the addition
of upstream struts which consisted of four struts
placed at the 0°, 90°, 180°, 270° points on the
upstream surface caused the cavitation number to
8 T == T T
WITHOUT SCREEN
DESIGN FLOW COEFFICIENT
Le © WITHOUT UPSTREAA) STRUTS
GQ WITH UPSTREAM STRUTS
AIR CONTENT - 3.1ppm
(BASIC FLOW NO. 1)
(BASIC FLOW NO. 4)
st 5 Qg—
oad
A CALCULATED
fo}
T ec agent :
REFERENCE POINT
2
10 15 20 25 x0 35 4 45
VELOCITY ~ ft/sec
8 ior aT] Saree Teor
DESIGN FLOW COEFFICIENT
WITHOUT UPSTREAM STRUTS
© WITHOUT SCREEN (BASIC FLOW NO. 1)
7} 4WITH SCREEN (BASIC FLOW NO, 3)
WITH UPSTREAM STRUTS
G WITHOUT SCREEN (BASIC FLOW NO, 4)
LL. © WITH SCREEN (BASIC FLOW NO. 5)
wpsahe
l2pve
a
CALCULATED
CAVITATION NUMBER ~ 04
°
10 15 20 25 30 35 4
VELOCITY ~ ft/sec
increase. In contrast to this result, the addition
of upstream screens causes the cavitation number to
decrease as shown in Figure 10. Data in Figure 11
show that a decrease in the flow coefficient by 10%
causes a dramatic increase in the cavitation number,
whereas a 10% increase in the flow coefficient
causes the opposite trend which is not shown in the
figures. Additional cavitation results are given in
iBslililene, {(lS)745)) o
4. CORRELATION OF SECONDARY FLOWS WITH THE CRITICAL
CAVITATION NUMBER
Because of the complicated flow field where the
vortex exists, an absolute calculation of Cppiy of
the cavitating region would be very difficult. The
minimum pressure associated with the cavitation
occurs within the vortex which is located along the
inner wall. This minimum pressure is not only
© DESIGN FLOW COEFFICIENT (BASIC FLOW NO. 1)
aah © 10% LOW IN FLOW COEFFICIENT g
Malis (BASIC FLOW NO. 2)
a8|S
iT] >
© 3
2
a
S 4
5 8
Sool eae
S ° ae
= 3 fo} .e os) 8
S eS
=
Co
al WITHOUT SCREEN
WITHOUT UPSTREAM STRUTS
1
10 15 2 25 0 35 ry)
VELOCITY ~ ft/sec
FIGURE 11. Correlation with cavitation data for basic
flow nos. 1 and 2.
345
45 FIGURE 10. Correlation with cavitation data for
basic flow nos. 1, 3, 4, and 5.
determined by the vorticity associated with the
vortex but also by the location of the vortex in
the primary flow field.
Considering only the vortex, there are many fac-
tors which can influence the minimum pressure coef-
ficient. If one models a vortex by a simple
rotational core combined with an irrotational outer
flow, the Cpmin is found to be
2
IP
atm 7S a m=} V7)
where T is the circulation and r, is the radius of
the core. Thus, the factors which influence Cpmin
are those which influence the circulation or core
size.
Assuming that secondary flows control the vortex,
Eq. (7) can be used to predict changes in critical
cavitation number due to changes in the secondary
vorticity produced along the inner wall. Therefore,
Eq. (7) can be arranged into the form
T
Bef re Wf
A Ss CeOUA (8)
Ga. - 2
see) iP
we Von B
where [T is now the integrated component of stream-
wise passage vorticity and ro is approximated by the
characteristic dimension of the resulting passage
vorticity. The letters A and B refer to different
flow states.
The passage streamwise vorticity was calculated
along several mean streamlines in the blade passage
by the method outlined in this paper for four basic
flow configurations which are described in the left
hand column of Table 2. For all flow configurations
considered, the results show a large amount of
streamwise vorticity at the rotor exit plane near
the inner wall. An example of the exit streamwise
passage vorticity is shown in Figure 8 for Basic
Flow No. 1.
As can be seen in Figure 8, the vorticity near
346
TABLE 2 - Vortex Circulation and Core Size Calculated Flow Vorticity Data
Circulation Characteristic Nondimensional Planar Momentum
Basic Flows Th Dimension Ratio Thickness
(Gacsee) Ro (inch) Ue Wi) 6 (inch)
Basic Flow No. l
without upstream struts - 11.64 0.81 -0.080 0.85
without screen
design flow coefficient
Basic Flow No. 2
without upstream struts - 8.23 0.57 -0.091 O)g7/al
without screen
0.9 design flow coefficient
Basic Flow No. 3
without upstream struts =) 10599 0.20 -0.076 0.94
with screen
design flow coefficient
Basic Flow No. 4
with upstream struts - 8.29 0.45 -0.102 Gol
without screen
design flow coefficient
the inner wall has a characteristic dimension
associated with it. A measure of the circulation
associated with this vorticity can be found by
integrating the vorticity from the inner wall to
the radius where the vorticity changes sign. In
addition, the characteristic dimension of the
passage streamwise vorticity must be related to the
difference between the radius where the vorticity
changes sign and the inner wall radius. The results
for several basic flow configurations are shown in
Table 2. Also, the nondimensional ratio, T'/x Voor
which is a measure of the minimum pressure coef-
ficient of the vortex is given in addition to the
planar momentum thickness of the mean boundary
layer profile entering the rotor for each flow
configuration.
In order to make absolute comparisons between
calculated minimum pressure coefficients and
cavitation data, a reference point is necessary
and the effect of Reynolds number must be calculated.
A reference point for Basic Flow No. 1 of 6 = 2.8 at
a velocity of 15 ft/sec was chosen. The influence
of Reynolds number was determined by solving for the
relative streamwise vorticity at two different free
stream velocities. For these calculations, a bound-
ary layer profile at the reference Reynolds number
was used in one calculation and the boundary layer
profile at three times the reference number was
used in the other calculation.
Now using Eq. (8) with Basic Flow No. 1 as the
reference point, comparisons between cavitation
data and Cpmiyn calculated using the passage stream—
wise vorticity can be made. Some of the results
are shown in Figures 9, 10, and 11. As can be noted,
the changes in Cpmin or 06 for the vortex as calcu-
lated, using secondary flow theory, correlate well
with the cavitation results. Only the correlation
with the rotor operating off-design (Basic Flow No.
2) is poor at the higher velocities. It is felt
that this is due to primary flow problems.
5. SUMMARY
A secondary flow analysis has been developed which
can be employed to assess the effect of inflow
velocity distribution on the strength and core
size of a vortex. This analysis has been success—
fully applied to a rotor where the secondary flows
dominate the flow field near the inner wall.
NOMENCLATURE
ap' - streamline spacing in bi-normal direction
Rg - radius of rotor
WwW - relative velocity
Bo - relative outlet metal angle
g% - relative outlet air angle
Aé':- deviation angle due to axial velocity accel-
eration
AS, - deviation angle due to secondary flows
09 - deviation angle due to blade camber
og - cavitation number = (Pw - Py) /(1/2pVa~)
dg - limited cavitation number
dq - desinent cavitation number
Wg' - component of absolute vorticity vector in
relative streamwise direction
Wn' - component of absolute vorticity vector in
relative normal direction
Wp' - component of absolute vorticity vector in
relative bi-normal direction
Qn' - component of rotation vector in relative
normal direction
%!' - component of rotation vector in relative bi-
normal direction
ACKNOWLEDGMENT
This research was carried out under the Naval Sea
Systems Command General Hydromechanics Research
Program, Subproject SR 023 01 01, administered by
the David W. Taylor Naval Ship Research and
Development Center, Contract NO001773-C-1418.
REFERENCES
Billet, M. L. (1976). Cavitation results for a
secondary flow generated trailing vortex.
Applied Research Laboratory TM 76-234.
Eichenberger, H. (1953). J. Math. and Phys. 32; 34.
Hawthorne, W. R. (1961). Proc. Seminar Aero. Sci.,
Bangalore, India, 305.
Hawthorne, W. R., and R. A. Novak (1969). ‘The
aerodynamics of turbo-machinery. Ann. Rev.
Fluid Mechanics 1; 341.
347
Horlock, J. H. (1973). Axial Flow Compressors,
R. E. Krieger Company, New York, 55-60.
Lakshminarayana, B. (1974). Discussion of Wilson,
Mani, and Acosta - A note on the influence of
axial velocity ratios on cascade performance.
NASA SP-304, 127.
Lakshminarayana, B., and J. H. Horlock (1973).
Generalized expressions for secondary vorticity
using intrinsic coordinates. J. Fluid Mech. 59;
97.
Lieblein, S. (1965). Experimental flow in two-
dimensional cascades. NASA SP-36, 209.
McBride, M. W. (1977). A streamline curvature
method of analyzing axisymmetrical axial, mixed,
and radial flow turbomachinery. Applied Research
Laboratory TM 77-219.
On the Linearized Theory of
Hub Cavity with Swirl
G. H. Schmidt
Technical University of Delft
and
J. A. Sparenberg
University of Groningen
The Netherlands
ABSTRACT
In general, there is a cavity astern of the hub of
a ship screw. This cavity is rather stable and is
roughly in the shape of a long circular cylinder.
There is circulation about it, which occurs in the
case of a real screw propeller, when the circulation
around the blades at their roots is nonzero.
Because the divergence of the vorticity field is
zero, this circulation at the roots "flows" down-
stream in the form of circulation about the hub.
At the end of the hub the flow contracts and the
swirl velocity increases. The pressure becomes
lower and a cavity forms where the pressure decreases
to the vapor pressure.
We introduce the following simplifications:
First, we neglect the influence of the finite number
of blades and consider a half infinite axially
symmetric hub immersed in an inviscid and incom-
pressible fluid. The incoming flow consists of a
homogeneous part, parallel to the axis of the hub
in the direction of the endpoint, and of a swirl
which represents the circulation around the hub.
In the upstream direction the hub tends to a
circular cylinder while its radius tends to zero
towards the end point. Second, our theory will be
linear: The difference between the radius of the
hub and the radius of the cavity is assumed to be
small and quantities which are quadratic in this
difference will, in general, be neglected.
Using these simplifications we determine the
shape of the cavity for given values of, for
instance, the swirl, the incoming velocity, the
ambient pressure, and the vapor pressure. The
surface tension is also included in the general
formulation of the problem. The more detailed
considerations, as well as the numerical calculations,
will be confined to zero surface tension.
One of the unknowns of the problem is the
position of the point of separation. This position
can be determined by demanding that the pressure
exceeds the vapor pressure everywhere on the wetted
348
surface of the hub and by demanding that the flow
cannot penetrate the surface of the hub.
The shape of the cavity is roughly a circular
cylinder. There are waves on the surface of this
cylinder which are, within the limitations of our
theory, steady with respect to the hub, and their
crests and throughs are perpendicular to the axis
of the hub. We will give numerical results for
the wavelengths and amplitudes of the waves as
functions of, for instance, the incoming velocity
and of the shape of the hub.
1. INTRODUCTION
A long cavity generally begins somewhere at the
end of the hub of a ship screw. This cavity, which
has circulation around it, does not close or widen,
it has a rather stable mean value to its radius.
The circulation or swirl occurs in the case of a
real screw propeller when the circulation around
the blades at their roots is not zero. Because
the divergence of the vorticity field is zero, this
circulation at the roots "flows" downstream in the
form of circulation about the hub and then about
the cavity.
In order to gain some insight in this phenomenon
we introduce some simplifications. We neglect the
influence of the finite number of blades and con-
sider a half infinite axially symmetric hub immersed
in an inviscid and incompressible fluid. The
incoming flow consists of a homogeneous part paral-
lel to the axis of the hub in the direction of the
endpoint and of a swirl which represents the
circulation around the hub. In the upstream
direction the hub tends to a circular cylinder
while its radius tends to zero towards the end
point. Hence, near the endpoint the flow contracts
and the swirl velocity increases proportional to
the inverse of the radius. This means that the
pressure becomes lower and a cavity starts where
the pressure decreases to the vapor pressure of
the fluid. Another approximation is that our theory
will be linear. In order for this theory to be
valid it is necessary that there be no abrupt changes
in radius of the hub and cavity. In real fluids the
viscosity can have an important influence on the
point of separation [Wu (1972)], however, this
effect is too complicated to be treated by our
method. We will not take into account the dependence
of the local vapor pressure on the curvature of the
interface between vapor and liquid. Surface tension
is included in the general formulation of the prob-
lem. The more detailed considerations, as well as
the numerical calculations, will be confined to
zero surface tension.
One of the unknowns of the problem is the value
of the axial coordinate of the point of separation.
This value can be determined by demanding that there
is no place at the wetted area where the pressure
is lower than the prescribed pressure in the cavity
and by demanding that the flow cannot penetrate the
surface of the hub.
The problem is very similar to the shrink fit
problem, in the theory of elasticity, of an unbounded
elastic medium with a circular two-sided infinite
hole [Sparenberg (1958)]. This hole is occupied by
a half infinite axially symmetric rigid body and
the problem is to calculate the contact pressure
between the body and surrounding medium when for
instance shear stresses are supposed to be zero.
Also, in this case, the edge of the region of con-
tact has to be determined.
The way in which we solve our problem is
analogous to the way in which the aforementioned
elastic problem can be solved. First we determine
a Green function. This is, in our case, the
deformation of the two-sided infinite cavity with
swirl when a rotationally symmetric pressure of a
Dirac 6 function type is applied at the circular
cylindrical wall. By using this Green function as
a kernel we can write down a Wiener-Hopf integral
equation for the unknown contact pressure causing
the fluid flow along the hub. This integral
equation is solved numerically by the finite element
method.
2. EQUATIONS OF MOTION AND BOUNDARY CONDITIONS
First we consider a two-sided infinite circular
undisturbed cavity of radius Yor with swirl in an
inviscid and imcompressible fluid of density ~p.
FIGURE 1.
Undisturbed cavity flow.
The undisturbed velocity field and pressure field are
~ ~ ~ 1p ~
u=U, v=0, w==, p= Po(r), 12 Baap ((aL))
349
where u, Vv, and w are the velocity components in the
x, xr, and 9 direction, p is the pressure, and T is
27 times the circulation around the axis. From
Bernoulli's equation it follows that
po (x) = ppl? /2xr? (2)
Po is the ambient pressure in the fluid and p (r)
> p, for r+. On the wall of the caviity for
if oP ra we have
(x) = - pr? 2 = =
ID We Ie, OE fae a = eye. (3)
where p_ is the pressure inside the cavity and po
is the surface tension of the fluid. In the
following we assume
= > 0 4
5 Te (4)
hence, the ambient pressure at infinity is larger
than the pressure in the cavity. From (3) it
follows
- po + Vp2o2 + 2 pl? (p - P,)-
Sn et eo ee OY
c 2 - 5
(pela Pe) (5)
We had to choose the positive root under the
assumption (4). For (p, - Pp.) < 0 we would have
chosen the negative root, however, this would
yield an unstable situation. In the case of zero
surface tension (5) simplifies to
x, = 1¥p/2(p,-Pe) (6)
(o}
The equations of motion for a time dependent fluid
flow are
St, = Ot. . ot 1 ap
ae oP Wee
De Oe VY Be p dx ! 2)
= ©. = OF we 1 op
—+0—4+7V—- —-=- = =
at : ox M or Yr @ Or 9 (8)
ow . dw. ow. ww
—F+t1—+7—+—=
DE” “Oe” Y Oe 12 e f 2)
Also, we have to satisfy
diy (hy Yo te Be 2a Veo, (10)
x ag 1G)
For a disturbed motion which satisfies (7)... (10)
it remains true that (1)
Tj
Wheat (11)
otherwise a circular contour floating with the
fluid would change its circulation which is im-
possible when external force fields inside the
fluid are absent. This follows also from (9) which
is satisfied by (11). Hence substituting (11) into
CAD eseeverene (10) we are left with the following three
equations for the three unknown functions u,v, and
Pp,
350
Am = Bel ei 1 ap
oy Wa peseses , (12)
AVSuh a Ges ta fore eax
~ ~ ~ ~ 2
OU Rie On a SRO Cole ay (13)
at x or (0) he ~
dn Oy
— + — + — = 0 (14)
ax or ie
We now linearize these equations with respect to
the undisturbed swirl flow,
a=U+u, V=v,p=p,+ Dd, (15)
where the perturbation quantities widens nie) 6. WA(Ssp3e7
t), and p(x,r,t,) are supposed to be of OE) FSub-
stituting (15) into (12)...(14), neglecting terms
of O(c2) and using (2) we find
We Geog a Se, (16)
at ox p dx
ov av 1 ap
+ =a A aL7/
at uv Ox po or nT)
<¥+V%;42%=0 (18)
Because the (u, v) velocity field is without rota-
tion we can write
(a, v) = Ge, 2%) (19)
where ¢ = $(x, r, t) is a scaler potential function
satisfied by (18)
(20)
We now suppose the disturbed cavity wall to be at
fa Sac ab wep (21)
c c
where 6r (x, t) is O(€). On this axial symmetric
boundary we demand the difference between the
pressures inside the cavity and in the fluid to be
in equilibrium with the effect of the surface ten-
sion and with some still unspecified external
normal loading 0U*E£ (x, t) of the cavity wall,
i
R
= =\ 0}
12) 19 p (
where R,and Ry are the principle radii of curvature
of the boundary, reckoned positive when the centers
of curvature are at the side of the cavity.
Within the accuracy of our linearized theory we
can put
2) 4 Meee eee ef Oe, (a).
1 Ro Cc c
Substituting (23) into (22) and using (2) and (15)
we find
il
Pye (OR? /2r0) + P=) Re a. OPyS — sen On.) cua
[-)
16 SB ae Org. (24)
Expanding the functions of r in (24) with respect
to 6r,, neglecting second order quantities, and
using (3) the boundary condition (24) changes into
2 2 en
er op a 2
= (- ——+ = + =
p(x, roe t) ( im + 5 )ox. 0 a5 ox. + pUCE.
(co) Cc
From (16) we find, because p + 0 and $¢ > O for x
+ - om,
yp So? OW Seo Ss (26)
which is Bernoulli's law for the unstationary lin-
earized flow. Herewith the dynamical boundary
condition (25) becomes
32 >
+—= (—-- — -o——_ -
U5 5 (3 = ) br. oe 5 ox, Use (27)
The kinematical condition at the boundary of the
cavity is
a 3 ad
— + — 6 = —
at ore y ox ae Gre = We)
Hence, we must solve (20) under the conditions (27)
and (28) while » > 0 for r > © and for x > - ~.
3. THE GREEN FUNCTION
We suppose the dimensionless loading of the boundary
(22) to have the form
(GS 12) SY 229) Bere (29)
where € is a "Small" positive parameter which has
no connection with the linearization parameter ec.
Because our problem is linear we assume
ét =t
Di Crate) = (3705) Ny dx (x,t) = éx_(x)e~ Bn (3.0))
Then equation (20) and the boundary conditions (27)
and (28) change into
Gaz + goo t+ = gp) PORE) = 0, ay
ee Org (x) = U*E(x), (32)
To*
SURES ole elCxee) = Ole (33)
c or
a
edr (x) a Ux
We introduce the Fourier transform g (1) of a function
g(x) by
p00 WE
= ipx 1
= dx, =—
g(u) Von g(x)e x g (x) V2,
2
(= reals u? ¢(u,r) =0 . (35)
Hence, for real yu
¢ (ur) = Ay (u) K (ule) + Ag (wT (lulz), (36)
where Ky and Ip are modified Bessel functions.
Because ¢ > 0 for r > © we have
Aj (uv) = 0. (37)
Substitution of (36) with (37) into (32) and (33)
yields
2 —
(iw) x (lulz) aay - Go- 5+ woe ww
@ ta XG ©
= -U*F(u), (38)
Ju] Ky {ule ) ay) + (é-inv) 6x, (u) = 0. (39)
Solving (38) and (39) for 6x, (u) and applying the
inverse Fourier transformation we obtain
6x, (x) =
a = ans
i £(u) |ulK, (Julzye "du
a 2
Jon [(é-ino) 2x, (Lule) + - Sy +u?0) |ulx, (ulze)]
co c
(40)
We now choose
f(x) = 6(x) , (41)
where 6(x) is the delta function of Dirac, hence
£(u) = 1/V20. Next we split the range of integra-
tion into two parts namely - ~ < uw < O and O < 4u
< © and neglect terms of 0(&2) in the denominator,
then we find
def ee]
k (x) == 82 C2) | een ee)
ix
Ri(Be =" ale
1 fos)
Sis J
(E
2480 OK (E)-(a+BE~) Ky (E)
U {e)
351
~iEx
Ae{o]
ne Ky (E) Le dé
grate =
an (E+ “**¥o ) K (6) - (otBE2)K, (E) , (42)
U
where a and 8 are dimensionless quantities given by
T2
-< Bees. . (43)
ie Cc r U2
c
It can be easily proved that under the assumption
(4),
a/B = pf + [2r? w, - p,)/007| > Ale (44)
In order to find the Green function for the
stationary case we have to take the limit € > 0 in
(42).
We now make some remarks for the case o # O and
hence 8 # 0.
First, the integrals in (42) are absolutely con-
vergent for 0 < x < ». This means that when sur-
face tension is present Green's function k(x) is
finite even at the point of application of the
Singular loading (41). This could be expected
because the surface tension can be represented by
a membrane placed at the boundary of the cavity
and a membrane has the possibility to locally
sustend such a loading by a jump in its first
derivative while its deformation is still a contin-
uous function of x.
Second, we consider the denominators in (42) for
€ = 0 and look for positive real roots of
a K, (é)
(¢ ap {8 s) = Ki (a) . (45)
The left hand side of (45) is curved upwards for
— > 0, while the right hand side is curved down-
wards. The proof of the latter statement is rather
complicated and will not be given here. However,
taking this for granted, it means that there are
none or two real positive roots, which is analogous
to the case of ordinary gravity waves with surface
tension. One of the roots corresponds to a wave
primarily due to the swirl, the other one to
capillarity. [Whitham (1973), p.446]
4. THE CASE OF ZERO SURFACE TENSION
Green's function (42) in the stationary case for
zero surface tension, when we take a different
positive value for € which of course is irrelevant,
ILS}
, x
co 1g
A ig () © © ae
GS) dena) ee eee:
E>0 [(E-i E)K, (E)- a Ky (E)I
Ky (jive
[ (+i E)K,(&) = @ Kj, (&)]
352
First we investigate the number of poles of the
integrands for € = 0, hence, the number of positive
real roots of
K (&)
°
RG, 2 ae
(47)
where now (43) a We fire = 2(p, - p)/pu".
From the well known expansions of K,(&) and
K,(&) it follows that the right hand side of (47)
is zero for — = 0 and tends to infinity for § >>.
We prove that this function increases monotonically
with &, hence, we have to consider
2
re (U3)
= 65 == .
K, (&)
au poe alen
a— > Kj ()
K, (8)
K, (6)
(48)
Instead of proving that the right hand side of (48)
is positive we will show that
Ais (3) iG) =e Sole) | BR) 20. (49)
This is easily shown to be true for § >.
when the derivative of (49) is negative the
function itself has to be positive.
Hence,
2K, (6) (K, (E)+ & KE) I/E - Ky>(&) + Ky? (E), (50)
is negative, since K,(&) for 0 < §€ < ~. This means
that the right hand side of (47) increases mono-
tonically, hence, there is one and only one root
Ss Cre WAY) alin. SS C5
We will estimate the value of bee
show that
Therefore we
(€+1) K (8) SS LS) 2) Oe (51)
From well-known expansions for K
inequality holds for §€ + >.
left hand side of (51) being
and Kj, this
The derivative of the
(E+1) [Kp (&) - Ki (E)] , (52)
is clearly negative, and hence (51) holds in 0 <
— <e, From K,(&) > Kj(&) and (51) it follows
that the root BS of (47) satisfies
oe
24a)?
(Ns tae try OH DN CAM) (53)
Second we have to determine at which side of the
real axis this root is situated when € is small
but not zero. Consider the denominator of the
first integral of (46), hence a root of
(E-i €) Ee MS) 75 K, (€) = 0. (54) |
The zero in the neighborhood of the real axis of
(54) is assumed as
Ts) tn
ES ae (55)
where — satisfies (47) or (54) with € = 0. Sub-
stituting (55) into (54), expanding the modified
This derivative,
Bessel functions, and using the definition of Se
we find
ii Kee)
6 &= 5 5 (56)
(2K) (E,)K (60) -E Ki (E+E K (E,)}
Hence by (49) we find that
Im(é +6 5) 22 Oy, (57)
or the pole of the integrand of the first integral
in (46) is slightly above the real axis for € small
and € > 0. In the same way the pole of the inte-
grand of the second integral in (46) is slightly
below the real axis.
Now we want to give a different representation
of (46). We distinguish between two cases x > 0
and x < 0. In the case of x > 0 we rotate the
direction of integration of I, and Ip as follows.
1, =e M@ paris) pp By = (pmsl) p (58)
and in the case of x < 0
at =e (Oped) ip) ath == (Opa) 5 (59)
From the foregoing it follows, that for x > 0, a
pole has to be added to TI, as well as to In. The
question arises: are there still other poles in
the complex half plane Re — > O which are passed
by rotating the lines of integration? We now
shall give a proof that this does not happen. This
proof was kindly given to us by our colleague Prof.
Dr. B. L. J. Braaksma.
Consider the function
def
ENS) —— ee K, (6) = oh eS) = = [NY (iE)
+ (o+1)K, (€)], (60)
which is real for real values of €. Suppose sj
with Re sj > 0 and Im s; # 0 is a zero of F(&),
then also s» = 8; (complex conjugated value) is
such a zero. The functions K)(s.&), j = 1,2,
satisfy J
2. 42
‘da d :
a 2S Se Se eee Sy reG.s) = ©, J = 1,2
ae2 dé 5) J
(61)
Multiplying (61) by K,(s_&) with k = 2 for j =1
and k = 1 for j = 2, we find by subtracting the
results
2 2 a d
(s] - So) EK, (S16) Ky (so&) = ag £11 (S28) ae K, (s1&)
= K1 (818) $1 (S28) (62)
Hence =
2 2 a
(s] = So) ite K; ($7 &) Ky (sg&) dé = EL (Ky (S28) gpk (S18) 5
at
d (s &)]
= Lene) ae a |
(63)
It is easily seen that the right hand side
vanishes for — > © and because s) and Sp are zeros
1
of F(&) (57) this right hand side also vanishes
for § > 1. Because the integral is positive we
have found that the assumption Im s; = -Im so # 0
yields a contradiction. It follows that no other
residues have to be added to the resulting integrals
after the rotations as denoted in (58) and (59)
besides the two we mentioned for x > 0.
Using some formulae from Watson (1922) we find
f TL :
OG at 3) ar Ue J, 68) ak ween 0 & P Op (64)
iG & G) > len) 2 ea) 75 = Op GS)
Adding poles to the integrals in the case x > 0 we
can transform the Green function (46) into
Avsin, bx. yak > 0
k(x) = h(x) + , (66)
where
ro) _, Ie
2 e S dé
ind): SY ers | eee angen ee SES Qe
7 [ET (E) +05) (6) ) +[EY, (€)+a¥)(E)] (67)
oO
and
2
AS 28 (Murs ) 5. SB fs (68)
(o) (o) Omc
The function h(x) is symmetric, h(x) = h(-x).
For x > 0 it has a logarithmic singularity because
for x = 0 the integrand as a function of &, behaves
as 1/2€, hence
h (x) = cs Si Sop <a Or (69)
For x > © the behavior of h(x) depends on the behav-
ior of the integrand in (67). For & > 0, this
turns out to be as
(402/n2E2)+0(E32ne) . (70)
Then it follows from Doetsch (1943) p. 233 that
h(x) = Sin cies oc |x| +. (71)
Now suppose that for x < 0 the shape of the
cavity is prescribed.
FIGURE 2.
Flow with swirl along hub.
im = ie dp Oe (3) ,» Ow see © (729)
c c
and that the unknown pressure between hub and fluid
US} ie) oF pU2£ (x).
353
Then we have to solve the following integral
equation
)
[ k(x-x') £(x')dx'= Ox | (x) 7 $8 < O} (73)
which is of the Wiener-Hopf type.
5. THE EXPLICIT SOLUTION OF a K,(&) -€ K (&) =0
(0)
In order to find an explicit solution of Eq. (47)
we first have to make some preliminary considerations.
Assume the following loading of the otherwise undis-
turbed cavity boundary
f(x) = €,6(x), (74)
where €,; is a small parameter. By (66) we find for
the deformation of the cavity
€, A sin bx 7 2820, (75)
Sx (x) = €, h(x) +
(0) pm 3% S Oe
Next we consider
e(e3) 3 Sep , 1S O 9 GEG) S O7 3 2 Os (76)
The loading given in (74) is the derivative with
respect to x of the loading given in (76). Hence
the derivative of the deformation 6*r,(x) caused
by (76) has to be equal to (75), we take
BWeos lope 5 2. Op (77)
-e] 5
d*x (x) = =€) hi(Eyidercr
*
[>
, x <0,
o
where we have chosen the constant of integration
in such a way that for x > +” we have a harmonic
wave with mean value zero.
Finally consider
2G) SO »,*s<@ 3 5 82 Op (78)
£ (x) = €j
The loading given in (74) is also, in this case,
the derivative with respect to x of the loading
given in (78). Hence the same argument applies as
before. However, now the constant of integration
has to be chosen so that the disturbance tends to
zero for x > -~ , we find
(79)
A
oa p (i-cos lop) 5 38 E25
(0) pn 28S, Os
Subtraction of the disturbances (77) and (79) yields
354
+00
A
bar (x) - O*#r (x) = =e | Mae a=
Cole Xu LCON (80)
which is constant as could be expected because
belongs to a constant loading of magnitude -€1pU*
of the whole cavity.
This displacement however can be calculated in
another way by using (6) where we have to replace
Diy,
Py + pU*f = Die €)pU*. (81)
Expanding (6) with respect to €), we find
ip
2 ee
c Cc (p,-P te 1pU?) 2
T €,0U
= \[& ules ). (82)
Oe Sac 2p -p )
Po Po C6
Combining (80) and (82) yields
+0
u*r /2(p.-p_) = h(x) dx + A/b (83)
(o} 2 fe}
Substituting h(&), A, and b from (67) and (68) into
(83) and carrying out the integration with respect
to x we find after some reductions
-1 2 1
a = 2(2ata 5.) =L (84)
where
a dé
a 2 2
EL LET, (€) tad) (E)] FEY (E)+a0¥, (E)] }
(85)
Solution of (84) with respect to 5 yields
a 2L ys
bj(a) =a. {1 SF ene (86)
by which we have found the unique solution of (47)
for € real and & > 0. This derivation rests on
some mechanical considerations such as uniqueness
of the solutions in relation to radiation conditions.
The result however, which is interesting from the
point of view of zeros of transcedental equations
connected with Bessel functions, has been verified
by others in a more straight forward way and found
to be correct.
By (86) it follows that an axial symmetric wave
moving along the cavity with velocity U has a wave-
length A(U) given by
d(U) am/b =[mT/E (a)] [20/(p.-Pe) ]
oO
2 (P-Po) /pu* (s7)
’
Equation (87) describes the dispersion of these
waves when surface tension is neglected.
6. NUMERICAL SOLUTION OF THE INTEGRAL EQUATION
In the left hand side of (73) the function k(x) is
given by (66-68) and the dimensionless quantity
f£(x') is unknown. For x < O the right hand side
is determined by the geometry of the hub. Let this
geometry be described by
= = + 6 me) 9 88
r ry x) ae x ) (88)
where r(x) is a given function. Then the right
hand Sigs is known up to an unknown shift, s, of
the hub along the x-axis, since the position of
the point of separation is a priori unknown. Hence
for x < 0 we can write (73) as
)
{ kK (GcR Sexi") (Exe) xa or (Ge a> Sip x <= O
(89)
where the function f(x') and s are unknown. First
we will describe how f£(x') is computed numerically
from (89) for arbitrary values of s. Then s will
be determined by a condition to be satisfied by f
at x = 0.
We make some remarks concerning the behavior of
£(x') for x'tO and for x' > -©. As will be shown
in the Appendix, the behavior of f(x') near the
origin is, for arbitrary values of s,
B Ieee US H(0) (90)
eG) 2 Pi
where B is some constant which will be discussed
later.
The hub has a constant radius far upstream,
hence 6r (x) tends to a finite value for x > -~,
Since the kernel k(x) vanishes for x > -~, the
perturbation due to the end part of the hub van-
ishes far upstream. Hence, the pressure distribu-
tion there is the same as that of a two-sided
infinitely long hub with constant radius. This
case was also considered in the preceding section.
We' find £(-~©) from (82), with -e, = £(-~), and (6)
and (87);
£(-~) =a 6r (-~)/r . (91)
h c
In order to transform (89) into a discrete
function we choose n + 1 points on the negative
x-axis:
oO < a < x1 S boo S wy) < x) = 0, (92)
and construct n coordinate functions, f (x), ...,
f,(x), defined on -~ < x < 0 as follows: For
WS By .-, n-l the function f,(x) vanishes out-
side the interval (x41, Xm-1), and inside this
interval its value is
£ (G25 x
) Baal (xtc ) paex
™m ™m
ae, 9) (x = xy
) Ws S38. ayo
m m-
The function £ (x) vanishes for x SE5 a and:
1/2 1/2
fi, (x| = |x| 7 |x| 5 $y SR SO; (94a)
27) SB Reo sp) 7 Geo oc oS 8 Sear co (94b)
1 2
Finally £ (x) vanishes for X41 < x < 0 and:
= = Bi (95a)
a) ak a / ( Soe d * ee el
£ (x) (95b)
n
i]
a
*
1A
*
These functions are plotted in Figure 3. We approx-
imate the function f(x') in (89) by a linear com-
bination of the coordinate functions:
n
S@)S 6 2.62) 4 (96)
where the C_ are unknown coefficients. In order to
approximate f(x') well near the origin, we have
chosen f; in a special way and, besides, the points
x are more densily distributed near the origin.
Since f£(x') is almost constant for large negative
values of x', we have chosen f to be constant in
(© Qp 'F3_)o ie
Next we have to determine the coefficients C),
oetain Ch We substitute (96) into (89) and then
the C_ must be chosen so that the difference
between the right hand side and the left hand side
of (89) is as small as possible, in the sense of
some norm. The computed values of the C_ appeared
to depend strongly on which norm was chosen for this
difference; many of these norms give unreliable
results. We obtained reliable values of the C as
follows: m
Equation (89) with x = Kor 2=0,1,..., n-2 yields
n
M = + = Reni a
= on Cc ox, (x) s), R= O, n-2 (97)
m=1
where:
(0)
= = ' ' O
My ih ss, = a0) BG) Gh? (98)
—0oo
At the points x,-}; and x, we minimize the difference
between the right hand side and the left hand side
of (89). The expression
n n
z (eM EG ote be 2 S\iZ (99)
fe gen eek
is a quadrative, non-negative function of the C .
Now the Cj, -, Cy are determined so that they
minimize (99) with the constraints (97).
FIGURE 3. The coordinate functions
eG o >. £..
355
We have checked these numerically computed values
of the C, as follows. First, the computed approxi-
mation has the square root character (90) even in
the interval (x3,0). Second, the value of C, equals
the right hand side of (91) within an error of 0.5%.
Third, if we replace the kernel k(x) in (89) by a
kernel k(x), which has the same behavior (69) at
the origin, and which for x > ~ is also given by a
term A sin bx as in (66), then (89) can be solved
effectively by the Wiener-Hopf method. If we, apply
our numerical method to (89) with the kernel k,
then the numerically computed function, f, equals
the analytically computed solution within an error
of 1s.
We have tried to compute the Cj, ..., C_ in
different ways; for instance:
i) By collocating the points x), ..., x, with
the exception of one point xj, so that the number
of equations equals the number of unknowns. This
method had to be rejected because the computed
approximation for f(x) appeared to have oscillations
near Xj.
ii) By minimizing the sum of squares of the
differences between the right hand and the left
hand side of (88) at the points x, ..-., X,- We
have also rejected this method, because oscilla-
tion occurred in f(x) near the origin.
We make some remarks concerning the computation
of the matrix elements. M Inya(98) Seek orems— a5,
---, n - 1 the integrand iS non-zero only ina
bounded region. The kernel k(x) is written as the
sum of a logarithm and a function which is bounded
at x - 0. The integral over the logarithm is
evaluated analytically; the remaining term is
integrated numerically. For m =n the integrand is
non-zero in an unbounded region. For x < x, we
have f,(x) = 1 and we must evaluate
x
n
J k(x] = 220) s@bst s (100)
—o
Note, that the integrand does not tend to zero for
x' > -~, as follows from (66). However, the express-
ion (100) represents the deformation of the cavity
due to a loading which equals a step-function. This
deformation has been computed in (76,77).
We now come to the determination of the shift, s.
The pressure in x < 0 at r = r, must exceed the
vapor pressure. Hence, by (22) with o = 0, we
must have f 2 0, and by (90), B 2 0. As will be
shown in the Appendix, the shape of the cavity for
small values of x is given C
' L
6x (x) = 6x (0) + br (0)x - 4B(x3/m) 7/3, x + 0
c h h
(101)
This implies that the radius of curvature tends to
ZOO) EO exXiy) 0.
Since the fluid may not penetrate
356
the hub, we must have B < O for a hub with a smooth
surface. We found above that B > O and hence B
must vanish. For our numerical approximation this
implies that the coefficient, C,, must vanish. Now
the value of the shift, s, is determined by iteration
so that C, vanishes.
When f(x) has been computed we can compute the
shape of the cavity in x > 0 with a numerical
integration of (73) for x > 0. Using (40) with
o = 0, we can derive an expression for dr,(x) for
x > ©. The derivation is similar to the derivation
of (66) and, therefore we give the result only:
2E (102)
Oo j
OS (0) ee TASH (ERX) Wet A COS| (SX/.-m) he
c Beene ) ) 0) G
to)
where:
(0)
ex
A, = lim e cos (E x/r) oid (3x3) Gea (103a)
e+O -2
to)
IN = nin f e Sen (Ee) sm (5%) obren (103b)
e+O -2
7. NUMERICAL RESULTS
In this section we give computermade plots of the
shape of the cavity dr,(x) for a number of shapes
of the hub dr;},(x) and for a number of values of the
dimensionless parameter a. We consider the case of
zero surface tension, hence a is given by (87) or
by (43) with o = 0:
2(p, - P_) Tr
os = 5 (104)
It follows from (88) that 6r},(x) depends on rq for
a fixed hub. The value of r, is given by (6):
1
mae
= neue
Bese Gy) NG). raise) (105)
However we can vary 4 without changing drp,(x) by
varying U and keeping p, IT and p, - Poe constant.
In the Figure 4 the function dédrp(x) is plotted;
it consists of a straight horizontal line and part
of a parabola. The x-axis is chosen so that x = 0
at the point of separation. No scale-unit is given
in the vertical direction, since ér,(x) and 6r¢(x)
are the linearized perturbations of the undisturbed
FIGURE 4. The functions 6rp [(x+s)/r,]
(hatched curve), Sr¢(x/r,) and the asymptotic
expression (102) (a.e.). The values of a are
a)4, b)2, c)l, d)0.5, e)0.25. The point of
separation is at x=0. bry is given by
Sry (x/r,) = 1 for x < 0 and = 1- (x/xQ)? for
x > 0. The values of s/r, are a)1.092,
b)1.017, c)0.558, d)0.070, e)0.014.
357
cavity (1); see Figure 2. The dimensionless quantity the fluid particles leave the hub. This effect is
x/Yc is on the horizontal axis. In x > 0 the important in the case of a low speed.
numerically computed function érg(x) is plotted In Figure 5 we have plotted the same functions
and also the asymptotic expression (102) is given. for a different shape of the hub. Here bry, (x)
It appears that the asymptotic expression is a good consists of a straight line, a part of a parabola,
approximation for 6r,(x) also for rather small and another straight line. It appears that quali-
values of x/rc. tatively the same effects occur.
The Figures 4(a) through (e) correspond to In Figure 6 we have only one value of the param-
decreasing values of a. This is equivalent to eter, a, (a4 = 1), but we have plotted a family of
increasing values of the speed U with constant p, functions Oxy, (x). The plot of 6r,(x) is omitted, and
T, and p. - Pe- The length of the waves on the we have indicated the point of separation with a
cavity is an increasing function of a, as was stated dot. The amplitudes of the waves in dr,(x) at x = —
in Section 4. Further we observe from these figures are denoted in a table underneath Figure 6. These
the following: numbers are the amplitudes divided by dr} (-~).
i) An increase of the speed U induces an increase From this figure we observe the following:
of the amplitude of the waves on the cavity. iii) If dr, (x) decreases abruptly as a function of
ii) When U is relatively large, the point of x, then the amplitude of the waves on the cavity is
separation is near the point where dr,(x) attains relatively large.
the value 6r;(-”). When U is small, the point of iv) The sign of ér,(x) at the point of separation
separation is near the point where érp(x) = 0. can be positive or negative, depending on the
The latter phenomenon is easily understood, since function dry,(x). If é6r,(x) decreases more and more
we can imagine two reasons for which the fluid may slowly as a function of x, then the value of 6rp, (x)
separate from the hub: First, the radius of curva- at the point of separation approaches zero from
ture of the hub may be so small that the fluid below.
particles are unable to keep contact with the hub. We will compare the effects on the cavity by
This effect dominates in the case of a relatively changing a, or U with constant p and p_ - p_,, and
high speed U. Second, the value of dr},(x) may of the function 6ér, (x). In order to give a rough
become negative. Then the centrifugal force makes description of the dependence on Sr (x), we use the
T r El of a af Do Ot
-4.0C -2.0c o> 2.00 y 700 8.ac 10.a0
D> ae.
7?
=r T
-4. 0c -2.00 ON CO
CUUUUULU EU UE
“4.00 73.00
d
FIGURE 5. The functions 6r}[(x+s)/r,]
(hatched curve), 6x (x/r¢) and the asymptotic
expression (102) (a.e.). The values of a are
a)4, b)2, c)1, d)0O.5, e)0.25. The point of
separation is at x=0. dr, is given by
Srp (x/rQ)=1 for x < 0, = 1-(x/r,)*/2 for
0 < x < r,, and =1.5-x/r for x > r_. The values
of s/Xo are a)1.683, b)1.759, c)1.805, d)0.361,
e)0.052.
FIGURE 6.
at ile
2 al
3 1m
4 Alo
5 ING
6 0.
i QO.
8 Q.
fc) 0.115 0.607
10 0.0807 0.516
alal 0.0565 0.437
12 0.0395 0.370
13 0.0277 0.311
14
pany
oO
A family of functions Sr, (x/rc) with a = 1. They are given by Srp (x/r,)=1 for x < 0 and
=1-)(x/r¢) 2 for x > 0. The value of \ is given in the table. The point of separation is denoted by a dot.
The amplitude of the waves at x =“ divided by r,(-‘), denoted by A, is also given in the table.
DCU UL TU UE CU
7
7.
LD LLL EA MMM UA
Sot ee SCORE ema BI 5 a7 ayes
Sine tub -5 00 -4, 0
UU UU UU UU IE UY
xr
2 00 14. @¢
aa
15
7 c/f
: ES ae
2 (yt 77108 B25 (iy V08 5S. uv
Ly, x /r,
pa eee
ua OF
x/P,
1
6.00
VU in TTT.
T T T insert De
-3.00 =5. 00 -4 a0 -2. 00 a. 00
d
FIGURE 7. The functions 6r,(x/r,) (hatched vl
curve) and 6x4 (x/rQ) for vanishing waves at (HUA TVARINTE
x = ©, The values of a are a)4, b)2, c)l,
d)0.5, e)0.25. The point of separation is at aaa “6 00 4 00
x= 0.
e
-2.00 70’ 00
x/T
|
2.00 yaa 8 ud
x/r
TI
2.00 4.00 8.00
curvature kK of the hub, which must be interpreted
as some mean value of the curvature of 6ér, (x) in
the wetted region not too far upstream.
First we consider the amplitude of the waves on
the cavity. This amplitude is an increasing function
of both U and k, as follows from i) and iii). Next
we consider the point of separation. An increase
of U or k tends to shift this point from the point
where Sr}(x) = 0 to the point where éry,(x) = drp(-2)
as follows from ii) and iv).
an increase of U has roughly the same effect as an
increase of kK. However, the length of the waves on
the cavity is, as follows from the previous theory
(87), independent of k but is a decreasing function
of U.
Finally we consider a shape of the hub which
induces, for some value of a, no waves on the cavity
at x = ©. The existence of nontrivial shapes can
be shown as follows. Let dér,)(x) and 8x49 (x) be
two shapes of the hub and let 6X oe, (x) and 6X G5 (x)
be the corresponding shapes of the cavity for some
value of a. In each case the point of separation
is at x = 0. We choose A > 0 so that the amplitude
of ASXe, (x) at x = © equals the amplitude of OY Gy (x)-
(Notice that their wavelength is already the same.)
Next we choose a shift, s > 0, so that
lim AGL] (x+s) + 8X Qo (x) = 0% (106)
xo
Now we construct a shape, 6r, (x), which induces no
waves at infinity, as follows:
xcs (107)
Sx, (x) Adxy, , (xts) + Sry 2 (x),
Sx, (x) = ASX, (xts) + OY 5 (x) - of S sg <7)
The dimensionless load f was introduced in (22).
By virtue of the linearity of our equations we
obtain the load corresponding to the shape (107) by
shifting the load due to 8rpy over a distance s,
multiplying it by » and summing the load due to
6X49: This load is nonnegative in x < O and
vanishes in x > 0. The condition for the point of
separation described in the preceding section is
satisfied at x = 0. The shape of the cavity is
obtained by a similar construction as for the load
f. The value of ér (x) tends to zero for x + © by
virtue of (106).
Using this method we have constructed five
functions dr},(x) which induce no waves at x = © for
five different values of a respectively. The
functions 6r,) and 6r,2 are the functions plotted
in Figures 4 and 5 respectively. The results are
plotted in Figure 7. The point of separation is at
x = 0. The value of Sx (x) in x > O is unessential.
APPENDIX
SOME RESULTS DERIVED BY USING WIENER-HOPF-TECHNIQUE
In equation (73) we let the path of integration
be from -~ to ~ and we assume the load f(x) to
vanish for x > 0. Then applying Fourier transform
(34) to both sides of this equation, we obtain:
Hence in these respects,
859)
bx (E) = k(E) £(E), (A,1)
where:
‘i =
K(E) = Ky, ((E]) (a Ky, ({E]) - [é] Kole} (A,2)
Here we have chosen the unit of length to be equal
to ¥_, so that r. = 1. We have to solve (A,1) with
f(x) = 0 for x > O and with dr,(x) being prescribed
agoye og S Wr
In order to apply Wiener-Hopf-Technique, we need
a multiplicative decomposition of K(E). We define:
My ee
H(E) = -k(E) (E2 - cae) (E2 + : (A,3)
where € is the root of (47). This function is
continuous and positive in -~ < — < ~. By virtue
of well-known asymptotic expressions for the
Bessel functions, K, and K,, we have:
BS) S Ika © Gye)o Esra © (A,4)
Hence, we can decompose H(&) in the usual way, see
for instance Noble (1958); we find:
me) on) /m) 4 (A,5)
where:
2 i S4
H (&) = exp == | 7 in {H(Z) ] ae (A,6)
Cc
This represents two equations; the upper or the
lower of the + signs must be read. The contour of
Ci (resp. C-) is the real axis, indented into the
upper (resp. lower) half of the complex C-plane at
t = &. The function H*(E) [resp. H~(&)] is analytic
in the upper (resp. lower) halfplane. Using (A,2-3)
and (A,5) we can write (A,1l) as
1/2
Gag (5) (2 = 2) (BA) > ae) —
{o)
peli) =
AC) d: sai” (3). 28S) (A,7)
The function (E+i)% has a cut from -i to -i» and
(E-i) 1/2 has a cut from ito i~. They are both
chosen so that they are positive for §>°%.
The function Sr (x) is prescribed for x < 0.
First we assume:
Ax
bx (x) =e xI<0) (A,8)
for some positive \; later we discuss the general
case. Fourier transformation gives:
= = - +
Sale) =a On (Ey * + 8x (6), (A,9)
where
6x (E) = (Cay i! a Sx (x) dx (A,10)
O
360
is unknown. We substitute (A,9) into (A,7) and
separate functions which are analytic in the upper
half plane and respectively in the lower one. There-
fore we write:
@a)ne = (Gaas (sD eG) =
ne (&)) + he(é) , (A,11)
where:
h (&) = Seaer tits (CANFEIE2})
(27) (iA+i) g-id
is analytic in the lower half of the complex €-plane,
and:
+ “ H (in)
h (&) = —— [—*-
Hie) a
Qn) 4 (iA4+1)
5 (A,13)
(esi) 7? E-id
is analytic in the upper half plane.
and (A, 11) we can write (A,7) as:
Using (A,9)
SY sy +
[6x _*(E) (E+i) / H (é)]+h (&)] (E2 = Ea) =
- . 1 2 = =-
=Is\ (0S) (E - e) = (( > al) / Ist (15) 2e{(S)) (A,14)
+
The function 6r_ (&€) is analytic in the upper
half plane by virttie of (A,10). Hence the left
hand side of (A,14) is analytic in the upper half
plane. Since £(x) vanishes for x > 0, £(&) is
analytic in the lower half plane and hence the
right hand side of (A,14) is also analytic there.
Hence, both sides of (A,14) represent an entire
function. The H7™(&) tend to 1 and the nt (E) are
O(1/E) for — > ©. We assume ér (&) gl/2 and £(&)
gl/2 to be bounded for — + ~. Then the entire
function must be a first order polynomial C_ + Cj
—, where the values of the constants C and°C, will
be given later. We can now solve for the unknowns
6x0 and f:
pele (e Hens
ore) = Xe, tay
H (é) Be
= (Es - 2
£(E)) =) IG, en BP) EV Sey cus
H (E)
The value of Cl] is chosen so that £(£) is o(e 1/4
for —& +> ™. Hence, by (A,12):
)
er SHUG (net 5 EN) (A,17)
We choose Ce so that £(E) is an order smaller, i.e.,
0(£73/2), for — + ~; hence: :
c= - Opt NMEA aS BON) (A, 18)
The meaning of this choice for f(x) will be discussed
later.
We obtain f(x) with the inverse Fourier transform:
eo Léx (ei) 1/2
H (é)
£(x) =
Esa
(2m) 1/7
IG, Cie He (G) (l= Bees - (A,19)
Since the integrand is analytic in the lower half
of the E-plane, the right hand side of (A,19)
vanishes for x > 0, as follows from the calculus
of residues. In order to obtain an expression for
f(x) for x + 0 we investigate the integrand in
(A,19) for — ++. The function f(x) is continuous
at x = 0, since the integrand is 0(E73/2). Hence
£(07) = 0. For real values of € we have, by (A,6):
: a Pe at
+ Sy (A
H(E) = (H(E))-/? exp Neat ahi = a
where the integral is a Cauchy principal-value.
Hence for real — we have
ie (2) ~ Sn 2 Eee), (A,21)
where H(&) is a continuous function, which is 0(1/&)
for — > + ™ by (A,4). Therefore, if the factor
H7~(€) in the denominator in the integrand in (A,19)
is omitted, the value of the integral changes by a
term which is O(x) for x + O. The other factors
in the integrand in (A,19) are an exponential and
a rational function of €. Using the calculus of
residues and Tauberian theorems we can obtain an
asymptotic expression for the value of this integral
for x t 0. We do not go into details and give the
result only:
=P |x| 1/2
ie (5) SS Qur Ape ee 0) (A,22)
where:
1/2
(AE ey QUST Ae GR) - (a,23)
In a similar way we investigate dr (x) for x ¥ 0.
Substitution of (A,17) and A,18) into (A,15) and
then into (A,9) gives:
or, (8) =
i Hit (GA) (En) a NES
eee (A,24)
Ci) (ayy Fe) (Ee >) (=a)
This expression is OES) for §€ + © and hence
6r (x) and its first derivative are continuous at
x = 0. An expression for x ¥ 0 is obtained in the
same way as for f(x):
Ges (GF) Ss Gre (()) GE -WreY (@)) se = a B sel? x10}
c c (o} a2
(A,25)
At this point we return to our choice (A,18) for
Go. If (A,18) does not hold, then it can be shown
that:
£i(x)) 2) BS |x| 3 se P Oy (A, 26)
1/2
(Gre (9) wes _(@)) — rests 6% xe Ol (A,27)
c c
where the constants B* and B** have the same sign.
The condition f> 0 implies B* > O and the condition
that the fluid does not penetrate the hub implies,
in the case of a smooth hub, that B** < 0. Hence
they must vanish both, which is achieved only by
giving C_ the value (A,18).
Finally we consider a hub of arbitrary shape.
By virtue of the Laplace transform we can write:
oe Ax
Sr (x) Some g(A)e GA 5 SOG (A,28)
C-ic
where c is a positive number, and:
)
g(X ) = if mes bx (x) dx (A,29)
oo
First we assume that 6dr (x) is such that the integral
aay (VAp2S))) ats absolutely convergent iors JN SI Cp Joyihe,
our results will appear to hold for a more general
case. By virtue of the linearity of our questions,
the expressions (A,22) and (A,25) hold with B given
by:
Ctic
1/2
H (id) an.
(A,30)
g () 2 + es) (Remy)
GRales
Substitution of (A,29) into (A,30), interchanging
the order of integration, and applying partial
integration with respect to x twice, gives:
(o)
B= | L(x) {E7 Sx, (x) ae Oig UU) Ir Cbs p (A,31)
ic
where
Cctjio
1 Se EY aR
L(x) = Daa e Paks, Gy pp 32S Os. Hypa?)
(A+1)
c-ics
361
In this expression for L(x) we substitute ay SS lly,
take the limit c + 0 and use some symmetry-properties
of Ht (1). Then we find:
ete
il i H (u
L(x) = al Re e ae HSL, Chil, se = O, (H\,55)))
(1-in)
—ily/)
Since the integrand in this expression is 0(H / )
we can derive for L(x):
~ ly SLY
aeeeae* (sl 47 xt 0 (a, 34)
As stated in Section 6, the position of the
point of separation is determined by the condition
B=0. By (A,31) this condition becomes:
(0)
ff L(x) [e2 Sy (x) + Sr 0 | dx = 0.
= ° c c
We can give an interpretation to the two terms in
this integrand. There are two reasons for which
the fluid may separate from the hub. First the
value of é6r, may become negative, so that the
centrifugal force makes the fluid particles leave
the hub. This corresponds to the first term in the
integrand. Second, the radius of curvature of the
hub may be so small that the fluid particles are
unable to keep contact with the hub. This corres-
ponds to the second term in the integrand.
(A, 35)
REFERENCES
Doetsch, G. (1943). Laplace Transformation. Dover
Publications.
Noble, B. (1958). Methods Based on the Wiener-Hopf
Technique. Pergamon Press
Sparenberg, J. A. (1958). On a shrinkfit problem.
Applied Scientific Research, Section A, Vol. 7.
Watson, G. N. (1922). Theory of Bessel functions.
Cambridge University Press.
Whitham, G. B. (1973). Linear and Non Linear waves.
J. Wiley and Sons.
Wu, Th. Y. (1972). Cavity and Wake flows.
Review of Fluid Mechanics, 4.
Annual
Unsteady Cavitation on an
Oscillating Hydrofoil
Young T.
Shen and Frank B.
Peterson
David W. Taylor Naval Ship Research and Development
Center, Bethesda, Maryland
ABSTRACT
Bent trailing edges and erosion are often observed
on marine propellers and are attributed mainly to
unsteady cavitation caused by the nonuniformity
of the flow field behind a ship's hull. In order
to improve the physical understanding of the
cavitation inception and the formation. of cloud
cavitation on marine propellers, a large two
dimensional hydrofoil was tested in the DTNSRDC
36-inch water tunnel under pitching motion. Fully
wetted, time dependent, experimental pressure
distributions were compared with Giesing's unsteady
wing theory. The influence of reduced frequency
and pressure distribution on inception was determined.
A simplified mathematical model to predict unsteady
cavitation inception, was formulated. Good corre-
lation between theoretical prediction and experi-
mental measurements on cavitation inception was
observed. The reduced frequency, maximum cavity
length, foil surface pressure variation, and time
sequential photographs were correlated with the
formation of cloud cavitation. A physical model
based on the instability of a free shear layer
defining a near-wake region provides a reasonable
explanation of the observed results.
1. INTRODUCTION
Hydrofoil craft are typically designed to operate
both in calm water and waves; and marine propellers
normally operate in the nonuniform flow field
behind a ship. Unfortunately, due to the complexity
of the experiments, only a few experiments have
been specifically concerned with unsteady leading
edge sheet cavitation on hydrofoils and propellers,
Morgan and Peterson (1977). It is the intent of
this paper to report the results of experiments
concerned with leading edge sheet cavitation on an
oscillating two dimensional hydrofoil. Following
a brief review of the most pertinent experimental
362
data available in the literature, an analytical
method for the prediction of inception will be
developed and compared with the experimental data.
Once the cavity is present on the foil, cavity
instabilities develop due to the foil oscillation
and also due to the inherent instability of the
cavitation process. This general process of
instability in the leading edge sheet cavity is the
subject of this paper.
It has been observed by innumerable investigators
that a leading edge sheet cavity can, under certain
circumstances, be quasi steady with relatively few
collapsing vapor bubbles to produce erosion. How-
ever, if a propeller blade enters a wake field, the
inception angle of attack at the leading edge may
not agree with the uniform flow inception angle.
In addition, the developed cavity may exhibit
instabilities not produced in uniform flow fields.
One form of cavity instability is manifest by the
shedding of a significant portion of the sheet
cavity. This shed portion appears to be composed
of microscopic bubbles and is commonly referred to
as "cloud" cavitation, van Manen (1962). Cloud
cavitation is now considered to be one of the main
causes of erosion and bent trailing edges, Tanibayashi
(1973).
Model experiments have been performed by many
organizations in an attempt to simulate full-scale
wake fields in which propellers operate. One of
the first detailed experiments concerned with
unsteady cavitation was reported by Ito (1962).
These experiments were with pitching three dimen-
sional hydrofoils and propellers in a wake field.
A principle result directly applicable to the work P
to be reported here was that the reduced frequency
had an important influence on the cavitation. He
also concluded that the "critical" reduced frequency
at which a leading edge sheet cavity broke up into
cloud cavitation was 0.3 to 0.4. His latter con-
clusion will be considered in more detail in the
context of the results to be reported here.
A recent discussion of this subject was given
by Tanibayashi (1962). He concluded that the
occurrence of cloud cavitation in nonuniform flow
cannot be predicted on the basis of uniform flow
experiments. In earlier work by Tanibayashi and
Chiba (1968), it was concluded from experiments
with an oscillating two dimensional foil that an
unsteady flow was required for the formation of
cloud cavitation. However, unlike the earlier
results of Ito, no distinct critical reduced
frequency was found. Since these latter results
were for nominally hemispherical travelling bubbles,
instead of a leading edge sheet, it remains to be
established whether the type of cavitation in the
growth phase is of importance to cloud cavitation
formation.
Chiba and Hoshino (1976) carried out extensive
measurements of induced pressures on a flat plate
above a propeller. On the basis of comparing
results with and without a wake field and with and
without cavitation, they determined that strong
pressure impulses were detected on the flat plate
and these correlated with the presence of cloud
cavitation.
Strong pressure fluctuations of very short
duration have also been detected by Meijer (1959)
on the surface of a cavitating two dimensional foil.
He attributed these pressure fluctuations to a
stagnation point at the rear of the sheet cavity
passing over a pressure gage. Chiba (1975) has
attempted to correlate cavity collapse on a two
dimensional oscillating foil with the response from
a pressure gage mounted in the foil. He concluded
that, as expected, when the shed vapor collapses
large pressure impulses occur. The essential
points for both of these experiments are that foil
mounted pressure gages can be used in the presence
of cavitation and when correlated in time with
photographs can assist in the interpretation of the
physical processes involved. This technique was
also used in interpreting the results to be reported
here.
Two other oscillating foil experiments have also
been reported, Miyata (1972), Miyata et al. (1972),
and Radhi (1975), that demonstrate the importance
of the reduced frequency on the whole cavity
inception, growth, and collapse process. Both have
shown that for the particular conditions of their
experiments, inception could be delayed. The
greatest suppression occurred for reduced frequencies
in the range of 0.4 to 0.5. Both of these experi-
ments will be discussed later in more detail within
the context of the results to be reported in this
paper.
All of the experiments reviewed above describe
various aspects of cavitation instabilities that
are associated with the cavitation performance of
oscillating foils and propellers in a wake. This
cavitation performance appears to be uniquely
related to the unsteady flow field that exists
Pe (*/e = 0.033)
P, (*/e = 0.10)
P, (¥/e = 0.25)
PITCH AXIS LOCATION
alles ba C = 241 m
KULITE PRESSURE GAGE
363
over the cavitating surface. In the sections that
follow analytical and experimental results will be
presented in an effort to provide a better under-
standing of how these various results are related
and of the associated physical processes involved.
2. EXPERIMENTAL APPARATUS AND TEST PROCEDURE
Foil and Instrumentation
The foil was machined from 17-4 PH stainless steel
to a rectangular wing of Joukowski section with the
trailing edge modified to eliminate the cusp. To
simulate the viscous effects at the leading edge
as close to a prototype as possible, the model was
designed with a chord length of 24.1 cm and a span
of 77.5 cm. The maximum thickness to chord ratio
is 10.5 percent. The foil surface was hand finished
within 0.38 wm RMS surface smoothness.
Pressure transducers were installed at a distance
of 7.96, 24.1, and 60.3 mm from the leading edge.
These locations correspond to 3.3, 10, and 25
percent of chord length from the leading edge.
Kulite semiconductor pressure gages of the diaphram
type were mounted within a Helmholtz chamber con-
nected to the foil surface by a pinhole. With this
arrangement one could measure the unsteady surface
pressures due to foil oscillation and high frequency
pressure fluctuations inside the boundary layer
over a pressure range of +207 KPa (+30 PSI) anda
calibrated frequency range of 0 to 2 kHz. In order
to increase the spatial resolution in measuring
the local pressure fluctuations inside the boundary
layer, the diameter of the pinholes installed on
the foil surface were kept at 0.31 mm (0.012 inches),
(see Figure 1). This arrangement also reduces the
danger of cavitation damage to the pressure
transducers. Extreme care was taken to fill the
Helmholtz-type chamber through the pinhole under
vacuum with deaerated water to minimize the possible
occurrence of an air bubble trapped inside the
chamber. If a gas bubble was present within the
gage chamber, the resonant frequency of the chamber
would be reduced below its 3880 Hz value. For
example, with the above procedures for filling the
gage chamber at a pressure of 3.4 KPa, a bubble of
0.6 mm diameter at atmospheric pressure is produced.
This bubble will lower the chamber's resonant
frequency to 1100 Hz. The danger of becoming a
Helmholtz resonator was not observed in our dynamic
calibration tests up to 2000 Hz. The calibration
procedure used here was developed by the National
Bureau of Standards, Hilten (1972), modified to
the extent that water rather than silicone oil was
the fluid medium. Since it was very important to
determine the relative phase difference between the
foil angle and the pressure gage signals, all
amplication and recording equipment was selected to
minimize the introduction of unwanted phase shifts.
PINHOLE
HELMHOLTZ TYPE
CAVITY
FIGURE 1. A sketch of the foil
and three pressure gage locations.
364
Photographic Instrumentation
All photographs used to document the inception and
cavity instability processes were taken with two
35 mm cameras. Illumination was provided by strobe
lights having a light duration of 10 microseconds.
With the camera shutter open, the first frame of a
sequence was taken when a foil position indicator
triggered the strobe lights. Each succeeding
exposure was taken 10 and 1/25 foil oscillations
after the preceeding exposure. An electrical pulse
from a light detector was recorded on a channel of
the same magnetic tape that was used to record the
foil position, pressure gage responses, and a time
code. Oscillograph records then allowed a direct
correlation between these events. Both top and
spanwise photographs were taken simultaneously by
exposing the film with one set of flash lamps. In
order to focus the camera lens in the same region
as the location of the pressure gages when viewing
in the spanwise direction, the camera was elevated
at an angle of 4° and directed slightly downstream
by an angle of 10°.
High-speed 16 mm movies were taken at a rate of
9,300 frames per second to assist in the interpre-
tation of the 35 mm pulse camera sequential
photographs. Adequate exposure for these photographs
was achieved by using high intensity tungsten
filament flash bulbs of 25 millisecond duration.
Test Section
The closed jet, test section of the 36-inch water
tunnel was modified by the insertion of sidewall
liners to provide two flat sides as shown in Figure
2. On each end of the foil a disc was, attached.
This disc rotated in a sidewall recess. Thus the
foil could be rotated without gap cavitation
occurring between the end of the foil and the
sidewall of the tunnel. One sidewall assembly was
fitted with clear plastic windows to permit side
view photography.
The foil was oscillated by a mechanism whose
conceptual design is shown in Figure 3. With this
type of design the foil mean angle (a _) can be
adjusted statically and the amplitude of foil
oscillation (a]) can be continuously adjusted
between 0° < a; < 4° while in operation. The
oscillation frequency is continuously variable
between 4 Hz < £ < 25 Hz. Air bags, shown in
Figure 3, were installed to reduce the fluctuating
torque requirements on the motor drive system.
VIEWING PORTS
FOIL DISC
FIGURE 2.
Schematic of closed jet test section.
PNEUMATIC
AIR BAGS
FOIL
OSCILLATOR
ARM
ADJUSTABLE
PIVOT POINT
— FOIL SHAFT
SLIDE
CONNECTING
ROD —— ECCENTRIC CRANK
~ DRIVEN BY VARIABLE
SPEED D.C. MOTOR
FIGURE 3.
mechanism.
Conceptual design of foil oscillation
Water Tunnel Resonant Frequencies
The study of cavity dynamics in a water tunnel gives
rise to a fundamental question, namely, the effect
of tunnel compliance on transient cavity flows. If
the tunnel was perfectly rigid and if there were
no free surfaces other than that of the cavity
itself, then an infinite pressure difference in an
incompressible medium would be required to create
a changing cavity volume. To make sure that this
kind of tunnel effect would not be present in our
model tests, a hydraulically operated piston having
a frequency range of 0 to 45 Hz was initially
oscillated in a test section opening to simulate
the maximum expected change of cavity volume. A
sharp peak of fundamental tunnel resonance was
observed at 4.7 Hz. Consequently, all of the foil
oscillation experiments reported here were carried
out at frequencies either above or below this
resonant frequency.
Data Reduction
Due to the installation of two sidewall liners in
the test section, the tunnel velocity was corrected
according to the area-ratio rule. The tape recorded
time histories of foil angle and pressures were
digitized using a Raytheon 704 minicomputer and
reduced using algorithms implemented on the DTNSRDC
CDC-6000 series digital computers. The time histo-
ries were recorded on one inch magnetic tape at
15 inches per second (38 cm/s) using IRIG standard
intermediate band, frequency modulation techniques.
During digitization, these data were filtered using
eight-pole Butterworth low pass filters that have
a -3 db signal attenuation frequency at 40 Hz.
They were then sampled at 125 hertz. The run
lengths used in the data reduction were nominally
40 seconds. For the oscillating foil data the
computer output consists of values of mean and
standard deviations, sine wave amplitudes and
frequencies, and transfer function magnitudes and
phases. Mean and standard deviation values were
obtained from the stationary foil data. For the
transfer functions, the system input was foil angle,
where the pressures were responses to this input.
For the dynamic runs, foil angle was sinusoidal;
nominally one percent of this channel's signal
energy consisted of harmonics or noise.
The methods used in data reduction are now
described. The mean value, wt, and the standard
deviation, o_, were calculated in the usual manner.
The sine wave amplitudes and frequencies, and the
transfer functions were obtained using operations
on measured autospectra and cross spectra. These
spectra were obtained using overlapped fast Fourier
transform (FFT) processing of windowed data segments,
Nuttall (1971), where the following reduction
parameters were used: FFT size of 1024, 50 percent
overlap ratio, and full cosine data window. The
true autospectrum of a sine wave is an impulse,
0.5A°6(£ - £_); the measured autospectrum is this
true spectrum convolved with the spectral window.
The spectral window associated with the cosine has
the form:
sin mf
£1 (1-£7)
The wave frequency is, in general, not sampled at
a rate which is an integral multiple of the sampled
frequency. Thus, the measured spectrum consists
of this spectral window sampled at evenly spaced
frequencies where the location of the samples
relative to the sine wave frequency or spectral
window maximum is unknown. The sine wave frequency
and amplitude are found by fitting the spectral
window shape to the three largest samples that are
closest to where the sine wave is expected. The
transfer functions are given by the cross spectra
between the input and output data channels divided
by the autospectra of the input channel. The
transfer functions were evaluated at the frequency
of oscillation of the foil. Quadratic interpolation
between spectral samples was used to obtain the
cross and autospectrum values. Once evaluated, the
complex transfer functions were converted to magni-
tudes and phases. The transfer function magnitude
is then the output sine wave amplitude, and the
transfer function phase is the phase angle of this
output sine wave. Except for data runs when cavi-
tation was present, the cross spectra coherency
was always greater than 0.98; this high coherency
implies low noise and high linearity at the foil
oscillation frequency.
3. UNSTEADY HYDRODYNAMICS IN FULLY WETTED FLOW
Basic knowledge in the general field of unsteady
aerodynamics has been compiled, condensed, and
presented by several authors [for example see
Abramson (1967)]. Available experimental hydro-
dynamics information for oscillating wings and
foils is very limited, especially at high values
of Reynolds number. Most of the available experi-
mental data concern lift, drag, and moment
coefficients from flutter and craft control
investigations. For cavitation inception studies,
accurate determination of the pressure distribution,
especially around the leading edge, is of major
importance. In the present investigation, three
pressure gage transducers were installed on the
foil to measure the unsteady surface pressures.
Experimental data were then correlated with an
available unsteady flow theory with the intent
365
of providing adequate information to analyze unsteady
cavitation inception.
The foil was pitched about an axis at %% chord
length from the leading edge. The instantaneous
foil angle a is given by
= 1
6 + a) sin wt (1)
where & _, 4), and W™ are the mean foil angle, pitch
amplitude, and circular frequency of pitch oscil-
lation. Let Cp(t), Cys, and Cpy(t) denote the
total pressure coefficient, the magnitude of the
steady pressure coefficient at the foil mean
angle, and the magnitude of the dynamic pressure
coefficient, respectively. At a given location on
the foil, it is assumed that:
De ap 2 (Ge), oe
P oO 15 oo
c(t) = BE = Fe = 8 4
4p v2 A (e) W
(2)
= © a © (ie)
ps pu
where
B = EAS
ces Ey V2 (3)
and
P (t)
4
é re) = wl (4)
a 5p v2
where P(t), Ps, Py(t), and p are the local total
pressure on the foil, static pressure on the foil,
dynamic pressure on the foil, and the fluid density,
respectively; P, and V, denote the freestream
pressure and freestream velocity. We have:
sin(wt + 6) (5)
where |Acpu| and ¢ are the amplitude of dynamic
pressure response and phase angle, respectively.
A positive value of > means that the pressure
response leads the foil angle.
Let the Reynolds number, Rn, and the reduced
frequency, K, be defined by
Ww, €
nm = (6)
n v
and
wC
= 7
K 2 Voo 7)
where C, Vv, and w are the chord length, kinematic
viscosity of the fluid, and the circular frequency
of the oscillating foil, respectively. Fully wetted
experiments covered the range of Reynolds number
Rn = 1.2 to 3.7 x 10© and reduced frequency K = 0.23
to 2.30. The test results are given in Tables la
to lc. The phase angles and the amplitude of
dynamic pressure response per radian of pitch
oscillation are given in Figures 4 and 5 at values
Oe Ch = Os55 WsO, Etre BoOPs
An unsteady potential flow theory for small-
amplitude motion recently developed, Giesing (1968),
is used here to correlate the experimental results.
The unsteady part of the pressure coefficient is
(DEG)
PHASE ANGLES, ¢
-50
PHASE ANGLES, ¢ (DEG)
100
PHASE ANGLES, # (DEG)
-50
@ = 3.25 + 0.5 Sin wt
X/C EXP THEORY
@= 3.25 + 1.0 Sin wt
X/C EXP. THEORY
0.033 O ---
0.10 A ---—
0.25 [e)
a@ = 3.25 + 2.0 Sin wt
X/C EXP. THEORY
REDUCED FREQUENCY, K
REDUCED FREQUENCY, K
REDUCED FREQUENCY, K
FIGURE 4a. Phase angles of dynamic pressure
response at pitch amplitude a; = 0.5 deg.
FIGURE 4b. Phase angles of dynamic pressure
response at pitch amplitude a, = 1.0 deg.
FIGURE 4c. Phase angles of dynamic pressure
response at 4] = 2.0 deg.
obtained as the difference between the total pressure
coefficient minus the steady part. The steady
solution is based on an exact nonlinear theory.
The theoretical values obtained from Giesing's
program are plotted on Figures 4 and 5 along with
the experimental data.
The phase angles obtained from experiments and
calculations will be discussed first. As seen in
Figures 4a to 4c, the agreement between experimental
measurements and theoretical calculations of pressure
and phase angles is quite good for all three pressure
locations. The agreement is good between experi-
mental measurements and theoretical predictions of
Magnitudes of dynamic pressure for the cases of
X/C = 0.25 and 0.10, as seen in Figures 5a to 5c.
At low values of K the measured pressure coefficients
are seen to be slightly lower than the values
calculated for the case of X/C = 0.033. The exact
cause of this small discrepancy between measurements
and theoretical calculations has not been determined.
The cause of small discrepancies between the
theory and experiments requires further investigation.
Nevertheless, the overall good agreement observed
between our experimental measurements and Giesing's
method is extremely encouraging. It is noted that
Giesing's method is based on unsteady potential
flow theory. The combined theoretical and experi-
mental results by McCroskey (1975, 1977) indicate
that unsteady viscous effects on oscillating airfoils
are much less important than the unsteady potential
flow effects, if the boundary layer does not interact
significantly with the main flow. The present study
appears to agree with his conclusion for the case
of a fully wetted foil. On the basis of this
relatively good agreement between Giesing's method
and the experimental data, this method will be used
in the next section to predict cavitation inception
as a function of the reduced frequency, K.
4. UNSTEADY EFFECTS ON CAVITATION INCEPTION
The major objective of this section is to examine
what effect unsteadiness has on cavitation inception.
The question of the occurrence of cavitation is of
particular importance when comparing model test
results for marine propellers or hydrofoils with
the full-scale prototype data. We would like to
know whether a noncavitating model is also free
from cavitation in the prototype. When calculating
the flow about propeller blades or hydrofoils, it
is important to know whether the cavitation bubbles
form on the blades, and if so, under what circum-
stances. The cavitation number o, defined by
has proved useful as a coefficient for describing
the cavitation process. Here, p and P_ denote the
density and vapor pressure of the fluid and P_ and
V,, denote the freestream static pressure and the
freestream velocity, respectively.
In addition to the incoming flow properties such
as freestream turbulence and nuclei content, the
surface finish and boundary layer characteristics
on the body surface are also of paramount importance
to the cavitation inception process Acosta and
Parkin (1975). To limit the scope of the test
367
program, air content of the water was not varied.
The air content was measured with 70% saturation
in reference to atmospheric pressure at a water
temperature of 22.2° C and tunnel pressure of 103.6
kPa.
The foil was pitched sinusoidally around an axis
at the quarter chord location aft of the foil leading
edge. The cavitation tests were carried out by
lowering the ambient pressure from the previous
fully wetted tests. The determination of cavitation
inception was based on visual observations. For
every test condition, 30 pictures were taken to
record the cavitation process on the foil. A
picture was taken every ten oscillations plus 1/25
of the time period of the foil oscillation. Thus,
a series of high quality short duration photos
were taken that together simulate one and 1/5 cycles
of the foil oscillation. A pulse signal was
simultaneously recorded on magnetic tape when a
picture was taken. In this way, each cavity pattern
observed on the foil could be related directly to
the instantaneous angle of attack of the foil.
Analytical Prediction
A simplified mathematical model will be formulated
first to explore the possible effect of unsteadiness
on cavitation inception. A significant delay in
dynamic stall was observed experimentally and
discussed in a recent review paper by McCroskey
(1975), who showed that the pressure gradient AC, /dx
around the leading edge was of paramount importance
in dynamic stall. The studies by Carta (1971)
indicate that the mechanism involved in the delay
of dynamic stall is the large reduction of unfavor-
able pressure gradient dC /dx during any unsteady
motion. P
The mechanism involved in cavitation inception
is different from the mechanism of aerodynamic
stall. It is generally assumed that cavitation
occurs on a body when the local pressure, including
the unsteady pressure fluctuations within the
boundary layer, falls to or below the vapor pressure
of the surrounding fluid, Huang and Peterson (1976).
Aside from the effect of nuclei content of the
water, it is the value of the local pressure
coefficient that governs the occurrence of cavita-
tion. Prior to the occurrence of cavitation on
an oscillating foil, the foil is in a fully wetted
condition. Thus, the knowledge of pressure distribu-
tion on the foil in the fully wetted condition
can be expected to provide useful information for
unsteady cavitation inception prediction.
As previously mentioned, the combined theoretical
and experimental results reviewed and summarized
by McCroskey (1977) indicate that unsteady viscous
effects on oscillating airfoils are much less
important than unsteady potential flow effects, if
the boundary layer does not interact significantly
with the main flow. In the present study, as
discussed in the previous section, the three
pressure coefficients measured at three points
around the leading edge are predicted reasonably
well by Giesing's method both in amplitude and
phase within the range of reduced frequencies
examined. This unsteady potential flow theory will
now be used to investigate cavitation inception.
In the tests, the foil was oscillated about a
mean angle of 3.25°. The mean values of dynamic
foil loadings determined from measurements are
368
MAGNITUDE OF DYNAMIC PRESSURE RESPONSE, | py! fey (PER RADIAN)
FIGURE 5a.
oy
MAGNITUDE OF DYNAMIC PRESSURE RESPONSE, locpyl jen (PER RADIAN)
FIGURE 5b.
Oy
o
ow
@ = 3.25 + 0.5 Sin wt
THEORY
o) 1.0 1.5 2.0 2.5
REDUCED FREQUENCY, K
Magnitude of dynamic pressure response at
0.5 deg.
@= 3.25 + 1.0 Sin wt
x/C EXP. THEORY
0.033 a SoS
0.10 A - -
0.25 [e)} —
+ oa -,
1.0 125: 2.0 2.5
REDUCED FREQUENCY, K
Magnitude of dynamic pressure response at
1.0 deg.
@ = 3.25 + 2.0 Sin wt
x/C EXP. THEORY
15:
0.033 «3 ---
0.10 A a
= 0.25 Oo =
3
=
«
fre
=
Ss
a QO
3
_ 10
a
fS
=
a
is 4 2
& \
2 | \ BAS
ale Z
Co
ray F & ge
z Ne ee
> K A - 4
a en Te a
5 iene xe)
3° A é p25
2 -
cook eee)
= Ss oO”
S= 9-9
o |
0
5 1.0 1.5 2.0 2.5
REDUCED FREQUENCY, kK
FIGURE 5c. Magnitude of dynamic pressure response at
a) = 2.0 deg.
plotted in Figure 6. Aside from some scatter in
the data, they are seen to be independent of
frequency (or reduced frequency). The steady
pressure distribution calculated theoretically at
3.25° is given in Figure 7. A suction peak appears
at around 1.8 percent of the chord length aft of
the leading edge. Reasonably good agreement between
the theoretical prediction and the three experimental
measurements should be noted. Experimental data
confirm the basic assumption made in Eq. (2) that
the total pressure coefficient Cp(t) is the sum of
the dynamic pressure coefficient Cpu (t) plus the
static pressure coefficient Cys at the mean foil
angle, i.e.
CAGE) =n€ stn Can Gt)
1) ps pu
We will now proceed to examine the possible
relationship between the dynamic pressure coeffi-
cient Cpu(t) and the static pressure coefficient
Cps- Let the instantaneous foil angle be expressed
as in Eq. (1). The dynamic pressure response is
then given by
u | :
sin (wt + 6) (5)
Here |ACpul| and $ are the amplitude and phase angles.
They are functions of reduced frequency K and
location X/C. They can be obtained either from
experimental measurements or theoretical calculations.
In the following study, Giesing's program will be
used to compute these variables. In our oscillating
tests, the mean foil angle was always maintained at
a = 3.25°. The type of cavitation observed in our
369
TABLE la - MEASURED DYNAMIC PRESSURE RESPONSE AT PITCH AMPLITUDE %= 0.5 DEG.
x/c = 0.25 x/c = 0.10 x/c = 0.033
ue v f K = é [SC nul/ oy é JOC lay, é [SCou| /oy
m/s HZ x10 Deg Per Radian Deg Per Radian Deg Per Radian
7002 4.88 5.5 846 1.2 43.4 3.48 7.1 5.53 =3).1 9.92
7006 4.88 10.0 1.539 1.2 77.7 5.40 31.2 5.60 7.0 9.62
7010 4.88 15.0 2.305 1.2 95.7 8.51 48.3 6.77 15.5 10.10
7014 6.71 5.5 635 1.7 35.8 3.07 1.2 5.69 -6.4 10.15
7019 6.71 10.0 1.154 1.7 58.4 4.13 13.6 5.72 ol 9.72
7024 6.71 15.0 1.730 1.7 81.6 6.59 32.5 6.21 7.6 10.32
7029 9.75 5.5 +423 2.4 18.3 2.69 -2.0 4.74 =11.5 10.43
7034 9.75 10.0 .769 2.4 44.9 3.34 12.6 4.79 -7.1 9.93
7039 9.75 15.0 1.153 2.4 62.2 4.45 24.8 5.32 =2.3 10.21
8002 13.11 5.5 -317 3.3 5.7 2.61 =11'53) 5.89 -15.1 10.90
8006 13.11 10.0 -576 3.3 29.5 2.76 =3.2 5.33 -12.4 10.10
8010 13.11 15.0 -865 3.3 48.5 3.50 6.4 5.36 -8.2 10.19
8029 6.71 4.0 +462 1.7 24.9 2.65 -4.3 5.60 =9.2 10.09
8041 9.75 4.0 308 2.4 7.1 2.58 -7.6 4.78 =13 73) 10.64
8045 13.11 4.0 -231 3.3 =2.5 2.77 -13.3 5.86 -15.8 11.68
8057 14.94 5.5 «282 3.7 7 2.88 =1'5).1 6.62 -16.6 11.80
1121 11.58 4.0 +264 2.8 = 39) 3.05 -9.9 4.54 -15.0 11.85
1122 11.58 5.5 +363 2.8 10.5 2.86 -1.0 4.44 =13.3 11.12
1123 11.58 7.5 495 2.8 20.3 2.91 4.3 4.44 -11.6 10.77
1124 11.58 10.0 +660 2.8 32.1 3.03 10.0 4.53 =9.5 10.72
1125 11.58 15.0 -990 2.8 52.3 3.93 24.0 5.00 -5.4 10.69
TABLE 1b - MEASURED DYNAMIC PRESSURE RESPONSE AT PITCH AMPLITUDE = 1.0 DEG.
x/c = 0.25 x/c = 0.10 x/c = 0.033
Run v f K Rn é | ac. Jo, | bel /er, é [oC pul /oa
Rinses m/s HZ x10~° Deg Per Radian Deg Per Radian Deg Per Radian
7003 4.88 5.5 -846 1.2 42.4 3.39 10.5 5.46 -1.5 9.66
7007 4.88 10.0 1.539 V2 71.1 5.13 28.5 6.09 7.4 9.77
7011 4.88 15.0 2.305 1.2 93.6 8.51 47.0 7.40 16.0 10.73
7015 6.71 5.5 +635 1.7 37.4 3.04 5.8 5.28 -5.3 10.03
7020 6.71 10.0 1.154 1.7 62.8 4.57 23.6 5.67 2.0 10.05
7025 6.71 15.0 1.730 1.7 79.5 6.71 38.8 6.75 8.6 10.70
7030 9.75 5.5 +423 2.4 20.7 2.67 -2.6 5.26 -10.4 10.36
7035 9.75 10.0 +769 2.4 44.9 3.28 9.5 5.19 -5.3 9.91
7040 9.75 15.0 1.153 2.4 62.6 4.64 20.9 5.76 =.6 10.41
8003 13.11 5.5 -317 3.3 7.6 2.64 -9.5 5.91 -13.6 11.03
8007 13.11 10.0 576 3.3 30.9 2.81 clo) 5.45 -10.2 10.22
8011 13.11 15.0 -865 3.3 49.0 3.63 6.8 5.65 =659) 10.40
8030 6.71 4.0 - 462 57) 23.5 2.69 =1.1 5.13 -8.6 10.07
8042 9.75 4.0 +308 2.4 8.2 2.59 =7..9 5.34 12.4 10.72
8046 13.11 4.0 231 3.3 =.8 2.78 -11.8 6.13 -14.2 11.76
8058 14.94 5.5 +282 3.7 3.5 2.90 -11.8 6.46 -14.8 11.96
1017 11.58 4.0 -264 2.8 208} 3.08 -11.3 5.76 -13.9 12.09
1018 11.58 5.5 +362 2.8 9.4 3.03 -6.3 5.58 =12'57, 11.83
1019 11.58 7.5 +493 2.8 21.7 2.97 -6 5.22 =10.5 10.99
1020 11.58 10.0 +660 2.8 34.2 3.33 6.1 5.40 -9.0 10.83
1021 11.58 15.0 +988 2.8 52.2 4.15 15.5 5.69 -4.9 11.10
TABLE 1c - MEASURED DYNAMIC PRESSURE RESPONSE AT PITCH AMPLITUDE
@% = 1.5 DEG
x/ce = 0.25 x/c = 0.10 x/c = 0.033
aay f K a 3 [Ac I/, [c,ui/a, 6 14Cp ul /on
m/s HZ x10 Deg Per Radian Deg Per Radian Deg Per Radian
7031 9.75 5.5 +423 2.4 21.0 2.65 =3.2 5.50 -10.0 10.26
7036 9.75 10.0 +769 2.4 46.5 3.22 8.3 5.30 =5.1 9.76
7041 ous 5% Opel 53) 2.4 62.3 4.62 19.0 5.98 92 10.40
8004 13.11 5.5 -317 3.3 9.0 2.57 -8.6 5.72 -13.0 10.79
8008 13.11 10.0 -576 3.3 31.5 2.78 -1.0 5.39 -9.7 10.19
8012 13.11 15.0 -865 3.3 50.2 3.63 8.4 5.62 Bez 10.35
= 2.0 DEG
7004 4.88 5.5 - 846 1.2 46.7 3.36 16.9 4.93 Sai/ 9.34
7008 4.88 10.0 1.539 1.2 71.3 5.14 33.7 6.01 7.2 9.76
7012 4.88 15.0 2.305 1.2 90.7 8.47 49.0 7.96 16.0 10.98
7016 6.71 9o5 -635 1.7 33.0 3.01 7.4 5.06 -4.9 9.75
7021 6.71 10.0 1.154 1.7 58.5 4.16 23.2 5.59 2.0 9.83
7026 6.71 15.0 1.730 7, 77.8 6.30 36.4 6.91 8.7 10.87
7032 9.75 5/a15) =423 2.4 19.4 2.63 -2.8 5.38 =9.7 9.99
7037 9.75 10.0 -769 2.4 44.7 3.18 8.3 5.30 -5.0 9.68
7042 9.75 15.0 1.153 2.4 60.9 4.42 19.2 5.92 0.0 10.26
8031 6.71 4.0 -462 7, 18.8 2.70 =-1 4.86 -8.3 9.67
8047 13.11 4.0 +231 353) 2.0 2.50 -13.0 8.35 -16.5 10.16
O, = 2.5 DEG
7017 6.71 515 +635 1.7 32.8 2.89 6.9 4.91 -4.8 9.34
7022 6.71 10.0 1.154 aloy 57.7 4.01 21.6 5.54 1.6 972
7027 6.71 1550 1-730 slog 77.9 6.00 36.4 6.59 8.8 10.33
370
NEGATIVE PRESSURE COEFFICIENT,
~Cp
NEGATIVE PRESSURE COEFFICIENTS,
wo
FIGURE 6.
oscillating tests.
Seuie| ——+y— at
a z eo ol x
EXP x/C a@ = 3.25 + 1.0 Sin wt
Vx = 13.1 m/s
5 10 15 20 25
FREQUENCY, HZ
Mean pressure coefficients deduced from
- THEORY AT a@, = 3.5 DEG
THEORY AT a, = 3.25 DEG
EXP. AT aw, = 3.25 DEG
-05 -10 =) -20
x/C
test program always initiated near the foil leading
edge.
Let (dC,/da); denote the static pressure gradient
with respect to foil angle at a given location on
the foil. Similarly, let (dcp /da) 4 denote the
dynamic pressure gradient with respect to foil angle
at the same location on the foil with the reduced
frequency, K, as the parameter. To simplify the
writing, they will be referred to as the "Static"
and "dynamic" angular pressure gradients respectively.
Let &(k) be the ratio of dynamic angular pressure
gradient versus static angular pressure gradient
at a given location on the foil, namely
&(K) = (ac_/da) / (dc_/da) (9)
P u P s
This ratio §(K) and the phase angle $ for several
locations and reduced frequencies have been calcu-
lated and are given in Table 2. The static angular
pressure gradient (dC _/da)_ at a given location is
approximated for mean? foil angles of 3.3 to 4.3°
since leading edge cavitation inception typically
occurred within this range. As seen in Table 2,
for a given reduced frequency, the amount of
reduction in dynamic pressure ratio (&€) remains
almost a constant value in the range of 0.004 <
X/C < 0.06 which covers the foil region over which
leading-edge cavitation occurs. Consequently, if
the foil is oscillated around the mean foil angle
G54, the shape of the pressure distribution in the
neighborhood of the suction peak and the peak
location are essentially the same for both zero
25 FIGURE 7. Static pressure distributions at
foil angles of 3.5 and 3.25 deg.
TABLE 2 - THEORETICALLY CALCULATED DYNAMIC PRESSURE
RESPONSE AT VARIOUS (x/c) LOCATIONS
REDUCED FREQUENCY, K
0.05 0.1 0.3 0.5 0.75 1.0 1.5 2.0
de
At x/c = 0.0046, Gana = 33.52
é -7-47, 10.53 711.25 -9.53 -7.91 6.97 -6.26 -5.96
10 30.18 28.10 22.41 21.87 20.97 20.57 20.27 20.19
Ls «90 84 67 65 +63 +61 -61 -60
de
At x/e = 0.0073, (F#) = 30.25
8
4 -7.46 -10.51 -ll.11 9.25 -7.45 6.32 -5.26 -4.61
28.03 26.09 21.72 20.28 19.44 19.06 18.77 18.68
g +93 -86 +72 67 +65 -63 +62 -62
dc
At x/c = 0.0117, G®). = 26.59
da’s
é -7.44 -10.46 -10.88 -8.78 -6.66 -5.22 -3.56 -2.31
| apa! joy, 24.28 22.59 18.78 17.51 16.77 16.44 16.19 16.10
& -91 -85 -70 -66 -63 +62 -61 -61
dc
At x/e = 0.018, (;4) = 23.0
8
é -7.41 -10.38 -10.53 -8.08 -5.48 -3.58 -1.02 41.13
Joc} 20.72 19.27 15.99 14.89 14.25 13.96 13.75 13.67
pu /%
e 91 83 -70 +65 62 61 -60 -60
de
At x/e = 0.026,(7$), = 19.65
é -7.37 -10.28 -10.08 -7.19 -4.01 -1.52 +2.15 +5.40
Jac. | 17.72 16.47 13.63 12.67 12.12 1188s ella 71 ed. 68
pu’ /
E -90 84 69 65 62 61 60 -60
dc
At x/e = 0.035, (z£), = 16.79
é -7.31 -10.16 -9.53 6a LON = 2422 +.96 45.96 +10.51
Jac. | TGAY WAgiig)shlaehA 10.87 * 10.40 10.20 10.09 10.11
pu /Q%
z 91 -85 .70 65 62 -61 -60 -60
d
ic
ia os
At x/c = 0.058) Gab), = 12-78
6 -7.18 -9.84 =g510)-=3633' +2533. 47-23 415.41) | +22.84
, : 8.89 8.2 ; 7.78 7.83 8.0
Vou joy 11.68 10.84 3 7.88 5
g 92 85 .69 65 62 61 .61 63
and nonzero reduced frequencies. This is an
important conclusion which will be utilized later
in the analytical prediction of cavitation inception.
We will now proceed to develop a criterion to
define the unsteady leading edge cavitation inception.
Let 0;, denote the cavitation inception angle
measured in a stationary test for given values of
oO and Rn. As an example, at a cavitation number of
Oo = 1.15 and Rn = 3 X 10°, cavitation inception
occurred experimentally at ajo = 3.5°. The corre-
sponding pressure distribution calculated using
potential flow theory is given in Figure 7 with a
suction peak appearing at around 1.6 percent chord
aft of the leading edge. Let Cpsmin (jg) denote
the minimum value of the static pressure coefficient
Cops, at the foil angle 4 = aj,. It has been
generally assumed that cavitation inception occurs
when -Cpsmin (%is) = 9. Obviously, this simple
Sh7fal
relationship is not realized in the present test
results (See Figure 7). This kind of discrepancy
in applying the above scaling law for cavitation
inception is a classic problem and has been exten-
sively discussed in the literature [for example see
Morgan and Peterson (1977) and Acosta and Parkin
(1975) ].
One of the possible reasons for this discrepancy
is that a finite amount of time is required for
nuclei to grow. Thus, cavitation inception will
depend not only on the magnitude of the suction
pressure peak, but it will also depend on the shape
of the pressure distribution in the neighborhood
of the suction peak and the peak location. Since,
as shown previously, these two features of the
pressure distribution are essentially the same for
zero and nonzero reduced frequencies of interest
here, it will be assumed that the amount of time
required for nuclei to grow is approximately the
same for both a stationary and oscillating foil.
Consequently, it is assumed that cavitation incep-
tion occurs on the foil at nonzero reduced
frequencies when the magnitude of ~Cpsmin (O;5) is
encountered during the foil oscillation, for given
values of oO and Rn.
An analytical method will now be developed to
predict leading edge cavitation inception on a
oscillating foil based on inception measurements
made on a stationary foil. Let AC, be given by
Ke = |e. @ Jee, @). | (10)
p Psmin 1s psmin fo)
where Cpsmin (%) denotes the minimum value of the
static pressure coefficient at 4 = 49 and Cysmin (dis)
is the minimum static pressure coefficient at the
cavitation inception angle 4;,. According to our
assumption, unsteady cavitation occurs when the
difference in the static loading AC, between ig
and 4, is produced by the dynamic loading at some
instant of time tj. Thus, unsteady cavitation
occurs if
= ACp (11)
where Gan ea) | is the magnitude of the dynamic
pressure response at time t = tj. If the value of
Aig - % is small it follows from Eqs. (5) and (11)
that
a, (aC_/de). sin (Ob, + o)/=(@, = &))) (der /de)
p u al is p s
(12)
where t:. corresponds to that instant of time at
which Eq. (11) is satisfied. Small-amplitude
motion has been assumed. The static angular
pressure gradient is to be evaluated at the location
of the suction peak corresponding to the steady
condition a = a.. The unsteady inception angle 4;,,
for a given reduced frequency K is obtained from
Eqs. (1) and (12).
cos
Oo. = 6 ar (Gi, = Gh.) COSY)
s )
= a] sing \j/l - ( (a, # 0) (13)
As a consequence of Eq. (12), no singularity is
expected inside the square root. Due to the
372
unsteady effect the inception angle ajy is generally
different from AEC Let Aa be
(14)
which can be used to measure the magnitude of the
unsteady effect. From Eq. (13), it follows that
Ney S (SG) SOE, 1)
is fo) &
2
Sagres (15)
= Ci sino 1- a
a 76
(a, # 0)
For the case where the phase angle > is small at
the location of inception, we have
Qa.
1
SO ae es oa) a 7
a,&
(16)
Although a small phase angle, $¢, approximation is
not required, it is useful to make this approxi-
mation for the sake of discussing the implications
of Eq. (15). The first term on the right-hand side
represents the effect of the ratio of dynamic to
static angular pressure gradients &(K) on unsteady
cavitation inception. The second term represents
the effect of phase angle, amplitude of oscillation,
and the ratio of pressure gradients on cavitation
6.0
Sins a FROM EQ (15)
<%= 6.0 DEG
FROM EQ (15)
m = 2.8 DEG
CAVITATION-INCEPTION ANGLES a (DEG)
RUNS 1205 TO 1208
3.25 + 2.8 Sin wt
9.75 m/s,o = 1.35
4.0
ab) 1.0 1.5 2.0
REDUCED FREQUENCY, K
FIGURE 8. Measured cavitation-inception angles for
test runs 1205 to 1208, a) 2.8 deg.
inception. For example, as seen
phase angles of dynamic pressure
leading edge lag behind the foil
for values of K less than 1.0 at
to this phase lag,
tation inception is further delayed.
in Figure 4 the
response at the
angle (negative 6)
X/C 0.033. Due
the occurrence of unsteady cavi-
Contributing
TABLE 3 - EXPERIMENTAL RESULTS ON UNSTEADY
CAVITATION-INCEPTION ANGLES, &
Run Vv f K
No n/s HZ x10
1205 9.75 4.0 2.4 - 307
1206 9.75 B55) 2.4 ~423
1207 C675) U5) 2.4 Belt
1208 9.75 10.0 2.4 768
1301 11.49 4.0 2.8 264
1302 11.49 EGE) 2.8 - 362
1303 11.49 is) 2.8 494
1304 11.49 10.0 2.8 659
1305 11.49 15.0 2.8 - 988
1306 11.49 25.0 2.8 1.646
1307 14.78 4.0 Sjo7/ 205
1308 14.78 Dyed) 7) - 282
1309 14.78 7.5 3.7 384
1310 14.78 10.0 S\6i7/ 512
1401 11.49 4.0 2.8 264
1402 11.49 5.5 2.8 -362
1403 11.49 7.5 2.8 ~494
1404 11.49 10.0 2.8 659
1405 11.49 15.0 2.8 987
1406 11.49 25.0 2.8 1.646
1407 14.78 4.0 3.7 205
1408 14.78 5398) 3.7 282
1409 14.78 708) 3.7 384
1410 14.78 10.0 S}57/ 513
1501 16.42 4.0 4.1 185
1502 16.42 DoS) 4.1 255
1503 16.42 Hoe) 4.1 347
1504 16.42 10.0 4.1 - 462
1505 16.42 15.0 4.1 694
1506 16.42 25.9 4.1 al galey/
iu
a a
e ea is iu
Deg Deg Deg
1235) 2.8 4.3 5.28
35 2.8 4.3 5.28
35 AGEy 8) 5.28
S5 2.8 4.3 5.28
alga ks} “95 35 3.94
aq ks} 395 315 3.94
abe als} 395. 3.5 3.94
LS 7995) Sop) 3.94
3 195) S\o5) 3.94
ib abs} 195 35 3.71 ~3.94
Algal7d 1.00 35 Sey 19.4
alent? 1.00 3.5 Sorat
ial 1.00 S}G5) 3.94
ye 1.00 365 3.94
iL Gals} HI55 35) 3.90 ~4.20
bats} 1.55 355 3.90 ~4.20
13 Te55 35) 4.20
alGals) 1.55 S}o5) 4.20
1.13 1.55 375 4.20
algal} 1.55 35) 3.60 ~ 3.90
1.14 55) 3.5 3.90
1.14 155, 3.5 4.20
1.14 LG 5s) S355) 4.20
1.14 IG S35) 35 3.90
LS +95 S}58) 3.71
1.15 095 355) So 7/4
ilo als) +95 S}55) SJov/l
algal) -95 e}55) 3.94
1.15 295 Si5) Slozal
nals) 95 &}55) Sef 394
TABLE 4- THEORETICAL CALCULATION OF Aw AND &
1205 TO 1208 AT x/c
K é E Aa
DEG DEG
0.05 -7.41 91 42
0.1 -10.38 .83 64
0.3 -10.53 .70 .86
0.5 -8.08 65 .87
(57/5) -5.48 62 85
1.0 -3.58 61 83
1,5} -1.02 .60 .74
2.0 +1513 .60 66
to the inception delay is the oscillation amplitude
a]. It is noted that the effect of oscillation
amplitude on inception angle is strongly coupled
with the phase angle. Thus, there will be no effect
of q,; on inception if there is no phase shift. This
is a consequence of the small oscillation amplitude
assumption. As the reduced frequency K approaches
zero, &>1 and 9-0, and the steady-state inception
angle (Aa*0) is recovered.
Experimental Results
The range of Reynolds numbers covered in the cavita-
tion tests was 2.4 to 4.1 x 10®. Because it is
shown in Acosta and Parkin (1975) and Huang and
Peterson (1976) that the existence of laminar
separation may trigger premature cavitation in
model tests, the boundary layer characteristics on
the foil under stationary conditions were calculated.
Within the Reynolds number range of the test program,
the occurrence of laminar separation around the
leading edge was not predicted. Flow visualization
with dye injection supported this conclusion. The
unsteady effect of foil oscillations on the boundary
layer characteristics was not included in the
calculation.
In order to simulate prototype viscous effects
as closely as possible, the model was tested at
high tunnel speeds (11.5 to 16.4 m/s). Fora
given body shape the laminar boundary layer thick-
ness based on chord length decreases approximately
as (Rn)-%. The effect of surface roughness on flow
characteristics becomes more important at higher
Reynolds number. This roughness effect was found
in the present model tests with cavitation appearing
prematurely in a few "weak" spots even though the
surface was highly polished. This caused some
difficulty in determining accurate values of
373
4y FOR TEST SERIES
= 0.018
NOTE:
R
fT
4.30 DEG.
R
a
3.25 DEG.
2.80 DEG.
R
a
cavitation inception angle. The relative importance
of this uncertainty was minimized by applying the
same cavitation inception criteria to both the
steady and unsteady test results.
Six series of oscillating foil tests were carried
out. The test conditions and the test results are
given in Table 3. Only 30 pictures were taken to
cover one and 1/5 cycles of oscillating motion,
and thus the angle at which inception occurred can
only be related to two successive pictures. There-
fore, in some cases, the inception angle is given
in terms of a small range of angles instead of a
single value.
The test results from runs 1205 to 1208 are
shown in Figure 8. In these cases, the foil was
oscillated around a mean angle G) = 3.25° with a
4.5 o = 3.25 + 1.55 Sin wt
RANGE OF FOIL ANGLES IN
4-0 TWO SUCCESSIVE PICTURES
EQ(15), % = 1.55 DEG
ZX RUNS 1401 TO 1406, o= 1.13
CAVITATION-INCEPTION ANGLES, a |, (DEG)
O RUNS 1407 TO 1410, w= 1.14
T r
oy 1.0 Wo 2.0
REDUCED FREQUENCY, K
FIGURE 9. Measured cavitation-inception angles for
runs 1401 to 1410.
TABLE 5 - THEORETICAL CALCULATION OF Aw AND @ FOR TEST SERIES
1401 TO 1410 AT x/c = 0.018"
K 6 rd Ao wy NOTE:
DEG. DEG. DEG.
0.05 =F) .91 .23 Be O,, = 3-5 DEG.
0.1 -10.38 .83 134 3.84
0.3 -10.53 .70 142 3.92 @ = 3,25 Wz8,
0.5 -8.08 .65 .40 3.90
0.75 -5.48 62 .36 3.86 CA =nEINSSEDEGE
1.0 -3.58 61 .33 3.83
a5 =1.02 -60 .26 3.76
2.0 +1.13 .60 .20 3.70
374
TABLE 6 - THEORETICAL CALCULATION OF Aw AND ae
FOR TEST SERIES 1301 TO 1306
Iso Tomis06 nt x! Gio Ole
NOTE:
K é g Aa Mu
DEG DEG DEG
0.05 -7.41 ht 14 3.64 a, , = 3.5 DEG
0.1 -10.38 - 83 oil 3.71
0.3 -10.53 -70 +26 3.76 &% = 3.25 DEG
0.5 -8.08 -65 -25 3.75
0.75 -5.48 -62 23, SJoV/s) &% = .95 DEG
1.0 -3.58 - 61 22 3.72
ale) -1.02 - 60 .18 3.68
2.0 Cra bo ale} - 60 ols} 3.65
4.5 ‘ @ = Q,5. As seen in Figure 7, the steady suction
@ = 3.25 + .95 Sin wt 5
S peak occurs at a location near X/C = 0.018. The
a predicted results based on Eq. (15) for the unsteady
2 RANGE OF FOIL ANGLES IN cavitation inception are given in Table 4 and plotted
J TWO SUCCESSIVE PICTURES in Figure 8 along with the experimental data. The
a 4.0 phase angle ¢, and the ratio of dynamic to static
2 angular pressure gradients, €, used in the predic-
z tion were calculated with Geising's computer program.
E Reasonably good agreement between theoretical calcu-
S BUSI) o Gh Sh0 Use lations and experimental measurements is observed.
2 6 ate Ree The test results from runs 1401 to 1406 and runs
= Gite ees ere 1407 to 1410 are given in Table 3 and plotted in
= El seen one nas Figure 9. In these cases, the foil was oscillated
Ss around a, = 3.25° with a pitch amplitude of a) =
1.55° and cavitation number of o = 1.14. The
3 1.0 1.5 2.0 measured cavitation inception angle at the stationary
REDUCED FREQUENCY, K
FIGURE 10. Measured cavitation-inception angles for
runs 1301 to 1310 and 1501 to 1506.
pitch amplitude of a, = 2.8° and cavitation number
fo} 1.35. The measured cavitation inception angle
at the stationary condition was aj, = 4.3°. Within
the range of reduced frequency 0.3 < K s 0.77, the
measured unsteady cavitation inception angles were
5, = 5-28°. That is, a significant delay of
cavitation inception was observed at nonzero
reduced frequencies. The unsteady inception angles
computed. from Eq. (15) will now be examined. A
previous discussion indicates that the suction
pressure peak with the foil in oscillation is
located at essentially the same X/C position as the
suction peak corresponding to the steady condition
TABLE 7 - THEORETICAL CALCULATION OF Aw AND @&
AT x/c =
K é g
DEG
0.05 -7.41 91
0.1 -10.38 83
0.3 -10.53 .70
0.5 -8.08 .65
0.75 -5.48 62
1.0 -3.58 61
ilo) -1.02 . 60
2.0 casa li} .60
0.018
DEG
1.06
condition is ajg = 3.5°. The measured unsteady
inception angles vary from ajy = 3.9 to 4.2° between
K = 0.2 to 1.0 and oj = 3.6 to 3.9° at K = 1.65.
The theoretical results obtained from Eq. (15) are
given in Table 5 and plotted in Figure 9. Once
again, a Significant delay in cavitation inception
is observed experimentally and predicted theoret-
ically at nonzero reduced frequencies. The agree-
ment is fair. Part of the discrepancy between theory
and experiment may be due to the lack of accurate
resolution in measuring foil angles, since only
30 pictures were taken to simulate 1 and 1/5 cycles
of foil oscillation. The phase angle ¢$ is seen to
change the sign from negative to positive values
at K above 1.5. Consequently, at high values of
reduced frequencies the amount of reduction in
cavitation inception delay is reduced. This trend
is observed experimentally and predicted theoret-
ically.
The test results from runs 1301 to 1306, 1307
Lu
NOTE:
a, = 4.3 DEG
s
1
a
> = 3.25 DEG
(ogy = 6.0 DEG
Peres = 5 m/s
8 fe) o = 2.5
(DEG)
O
lu
a =3+6 Sin wt
CAVITATION-INCEPTION ANGLES, a.
oy 1.0 1.5 2.0
REDUCED FREQUENCY, K
FIGURE 11. Measured cavitation-inception angles by
Miyata (1972).
to 1310 and 1501 to 1506 are given in Table 3 and
plotted in Figure 10. The foil was oscillated
around a, = 3.25° with a pitch amplitude of a) =
0.95° and cavitation number o = 1.12 to 1.15. The
measured cavitation inception angle at the stationary
condition is aj, = 3.5°. The measured maximum
steady inception angles are aj, = 3.70 to 3.93°.
Once again, a Significant delay in cavitation
inception at nonzero reduced frequencies is mea-
sured. The theoretical calculations based on Eq. (15)
are given in Table 6 and plotted in Figure 10. The
agreement is reasonably good.
In order to provide an insight into the effect
of a] on cavitation delay, a theoretical example
is computed in Table 7 and plotted in Figure 8.
The foil is assumed to pitch around a, = 3.25° with
an amplitude of a] = 6.0°. The stationary cavita-
tion inception angle is assumed to be dj, = 4.3°.
It is seen in Figure 8 that a significant delay in
cavitation inception can be expected if the pitch
amplitude is increased. This trend is also observed
experimentally by comparing Figures 9 and 10.
A two-dimensional foil undergoing pitch oscil-
lations around an axis located at mid-chord was
tested by Miyata et al. (1972). Two of the typical
test results are produced in Figure 11 for com-
parison. For the data shown the foil was oscillated
with a pitch amplitude of a] = 6.0°. As expected
(See Figure 8) a significant increase in the angle
of cavitation inception is noticed for 0 < K < 1.2.
For the second set of data shown in Figure 11, the
foil was oscillated with a pitch amplitude of
a] = 3.0°. A similarity between Figure 8 and
Figure 11 is noticed. Although the foil shapes and
the locations of pitch axes are different between
Miyata's experiments and ours, the effect of
unsteadiness on cavitation inception is similar for
two model tests. A similar trend is also noticed
in Radhi's experiments (1975).
In the review papers by Acosta and Parkin (1975)
and Huang and Peterson (1976), one is clearly
reminded that even under steady conditions the
cavitation inception process is extremely complex.
375
The theoretical prediction of cavitation inception
angle under steady conditions is still very difficult.
However, if the steady-state inception angle Ais is
known from model tests, the effect of unsteadiness
on cavitation inception may be estimated reasonably
well by Eq. (15). Further investigations are
needed to explore discrepancies between theory and
experiment and the applicability of Eq. (15) to
different foil shapes and for pitch axis different
from the ones examined here.
5. LEADING EDGE SHEET CAVITY INSTABILITY
Wu (1972) has provided a very useful review of the
physics of cavity and wake flows which may help to
explain the observations of the present experiment.
The essence of his description, applicable to the
partial cavity condition, is that the free shear
layer enveloping the cavity is unstable. The cavity
occupies a portion of what can be referred to as
the wake bubble or near wake, physically delineated
in steady flow by a dividing streamline that is
characterized by a constant or nearly constant
pressure. For the condition where the cavity within
the near wake is unsteady, the region is, strictly
speaking, not defined by a streamline but by a
material line which is difficult to observe experi-
mentally. Because of this difficulty, we will
initially assume that a quasisteady approximation
is valid. When the cavity is just beyond the
inception condition, its surface should be smooth
as would be expected with a laminar shear layer.
As the cavity grows in length the free shear layer
would tend to become unstable. Transition from a
laminar to turbulent shear layer initially takes
place at the downstream end of the near-wake. A
further extension of the cavity length causes
transition to gradually move upstream along the
free shear layer and the far-wake becomes irregular.
This is comparable to the bursting of a short laminar
Separation bubble in a single phase fluid. With a
continued increase in cavity length, transition
can begin at the leading edge of the cavity.
In applying here the general features of the
near-wake outlined by Wu (1972) no assumption is
made as to whether the cavity occupies all of the
near-wake region since the detailed physics of the
region downstream of the cavity trailing edge are
uncertain. One possibility is that the roll-up of
the shear layer into vortices is completed at the
near-wake closure where the vortices break away.
If this occurs, it is reasonable to expect a
periodicity in this shedding process.
The variation in foil pressure at the P; location
(see Figure 1) can give a useful insight into what
is happening both downstream of the cavity and
within the cavity when the foil is oscillaing with
a pitch amplitude (a1) of 1.55°. Figure 12 shows
an oscillograph record of the pressure variation
P, for a cavity that reaches its maximum length
downstream of the gage location. (A) is the
region where the foil surface is fully wetted and
the pressure appears to follow the variation
expected as the angle of attack, a, is varied. At
point (B), the cavity begins to cover the gage and
in this example the pressure drops from the fully
wetted pressure of 31.7 kPa to the cavity pressure
in 0.003 seconds. The cavity pressure remains
constant, except for several pressure spikes (C) of
376
FIGURE 12. Sample oscillograph record for
the variation in foil surface pressure with
foil angle at K = 0.26, Van) RS am/S\,
Dey =i/6%2) kPa) =) 325°) Pal oo san Ot.
millisecond duration, until the trailing edge of
the cavity recedes past the P, gage (point D).
The absolute magnitude of the cavity pressure
could not be accurately determined from these
experiments since the in situ pressure gages were
not calibrated for the condition of a gas/liquid
interface at the entrance to Helmholtz-type chamber
over each gage. As shown in Figure 12, point B,
the growing cavity does not appear to produce large
foil surface pressure fluctuations at its downstream
edge. However, when the cavity recedes, (ie., point
D) then the foil surface pressure fluctuations can
be comparable to the magnitude of the dynamic
pressure.
Based on photographic records it appears that
when the cavity is expanding, its trailing edge is
disturbed as one would expect if the shear layer
were unstable at that location. Beginning at the
cavity trailing edge and then moving forward, the
cavity surface becomes highly disturbed, irregular,
and bubbles are introduced into the shear layer,
just as one would expect when transition in the
shear layer moves forward. The cavity pressure,
as measured by the gages Pj, P2 and P3, remains
constant throughout this change in the surface of
the cavity.
During the early stages of sheet cavity growth,
when only the cavity trailing edge appears disturbed,
small regions of bubbles are shed from the sheet
cavity trailing edge. This shedding process becomes
more accentuated as the sheet cavity length increases
and more of its surface becomes disturbed. High
speed movies taken at 9,300 frames per second
clearly show the highly turbulent characteristics
originating at the trailing edge of the sheet
cavity and progressively moving upstream.
The sequence of vapor shedding from the cavity
trailing edge, as determined by high speed movies
taken at 9,300 frames per second, is as in the
sketches of Figure 13. The photographs of Figure
14 demonstrate a phase in the vapor bubble shedding
process from the sheet cavity as sketched in 13c
with two regions of shed vapor downstream. It
should be noted that since the foil surface is
very smooth, a reflection of the shed vapor is
seen in the side views. Therefore a dashed line
PRESSURE GAGE Ps
PRESSURE GAGE Po
PRESSURE GAGE P)
FOIL ANGLE
" FOIL ANGLE MAX.
ai neeanimttamenen CAMERA PULSE TRACE
has been added to Figure 14 to indicate the
separation of the vapor and its reflected image.
This shedding process is periodic and for the
example shown in Figure 14 the shedding frequency
at a given spanwise location is nominally 700 hertz.
The view shown in Figure 14 covers nominally the
center third of the foil span. Visual observations
with strobescopic lighting indicate that the leading
edge sheet cavitation, for nonzero values of K,
typically consists of a series of 3 dimensional
cavities across the span.
In Figure 15 the top view shows a depression in
the cavity surface (a) just above P); and a rise in
cavity height (b) just downstream of the depression.
At this instant a pressure "Spike" is detected by
P, (see for example C in Figure 13). This condition
precedes the shedding of a small region of vapor
bubbles upstream of the sheet cavity trailing edge
and significantly deforms the cavity trailing edge
shape. It is the forerunner of the condition that
will be referred to in this paper as "cloud" cavita-
tion. It is interesting to note that after
correlation: of over 600 photographs of the leading
edge sheet cavitation with the pressure gage signal,
the pressure "spike" always occurs when a depression
in the cavity surface exists over the pressure gage.
The converse, however, was not observed, ie., the
"spike" can occur when no depression was discernable
in the photographs. These "spikes" can occur without
any significant gross change in the observed
character of the sheet cavity surface in the general
—
Lo
CAVITY FLOW
+——
(a) FOIL LEADING (b)
EDGE
(c) x (4)
FIGURE 13. Sequence of vapor shedding from the cavity
trailing edge.
<—TOP VIEW
<— TOP VIEW
vicinity of a pressure gage.
"spikes" can occur during the life of the sheet
cavity it appears improbable that they are due to
the interaction of a postulated reentrant jet with
Since numerous pressure
the sheet cavity surfaces [Knapp et al. (1970)].
These "spikes" frequently have amplitudes which are
comparable to the dynamic pressure and certainly
exceed the estimated static pressure in the free
shear layer over the pressure gage location. Quite
possibly, these pressure "spikes" are due to the
free shear layer itself since they only occur when
the cavity surface indicates a turbulent shear
layer is present. When the reduced frequency is
high, for example at K = 1.65, the fully wetted
pressure variation leads the foil angle by 68° and
then no pressure "spikes" are produced at the pressure
gage location as can be seen in Figure 16. At
these high reduced frequencies the periodic shedding
t SIDE VIEW
4 ste VIEW
7/7
ROW
FIGURE 14. Progressive shed-
ding of vapor from sheet cavity
trailing edge, K = 0.26,
V_= 11.5 m/s, P_ = 76.2 kPa,
© (o) come
a = 3.25 + 1.55 sin wt.
—— FLOW
FIGURE 15. Cavity surface de-
pression producing pressure
"SPIKE" P, gage location,
1S = Oo dain Ni, > aiboS m/s,
Po =, 76-2 ry 6 = SoOSo, +
1.55 sin wt.
from the sheet cavity trailing edge downstream of
the pressure gage is still observed.
The last aspect of the leading edge sheet cavity
instability to be described in this paper is that
which will be called cloud cavitation. The three
principle features of cloud cavitation for K 2 0
are as follows:
(1) A large surface area of the sheet cavity
becomes highly distorted and undergoes a
significant increase of overall cavity
height in the distorted region, (Figure 17).
(2) Once this distorted region begins to
separate from the main part of the sheet
cavity, the upstream portion of the sheet
cavity develops a smooth surface and a
reduced thickness (Figures 18 and 19).
(3) The trailing edge of the smooth surfaced
WwW
~~
ss)
FIGURE 16. Sample oscillo-
graph record for the variation
in foil surface pressure with
foil angle at K = 1.65,
Vina ellis> m/s, P_ = 76.2 kPa,
a) colt
ON= 35255 1 S5iisiniwwts. ERT
FIGURE 17. Initial stage in
the process of cloud cavitation
formation, K = 0.51, woe 14.8
m/s, mS 124.1 kPa, a = 3.25
+ 0.95 sin wt.
region then moves downstream, becomes
unstable at its trailing edge, and quickly
develops the characteristic appearance of
the leading edge sheet cavity elsewhere
along the span (see feature a in Figure 20);
or, the trailing edge of the smooth portion
of the sheet cavity moves upstream to the
foil leading edge and the cavity disappears
(Figure 21). In Figure 21 a dye trace
injected at the foil leading edge can be seen.
When the foil is stationary (K=0) cloud cavita-
tion shedding can be very periodic as can be seen
in Figure 22 which shows the oscillograph trace of
the pressure gage response. The frequency of
shedding for the condition illustrated in Figure
ponvonslisa cbintasnsitauanoanwanshonanatbia
PRESSURE GAGE P3
PRESSURE GAGE Po
PRESSURE GAGE Py
FOIL ANGLE
~=— FOIL ANGLE MAX.
CAMERA PULSE TRACE
se cecbetaonetta
{ SIDE VIEW
*—TOP VIEW
22 is 42 Hz based on the response of the pressure
gage P}. Figure 23 shows a photograph of the type
of cavitation that produced the time pressure history
of Figure 22. In Figure 23, (a) is a cloud just in
the process of being shed, (b) is a cloud previously
shed at a nearby spanwise location, and (c) is a
cloud shed earlier at the same location as (a).
The cavities did not shed in the manner of the two- -
dimensional separation which typically occurs in
sharp leading edge foils [Song (1969),Besch (1969),
Wade and Acosta (1965)]. Instead, cavity shedding
was highly three-dimensional and more or less
independent of the sheet cavity instability occur-
ring several cavity lengths away along the foil
Span. However, it appears that for the trailing
edge shedding and the cloud cavitation (at least
| FLOW
TOP VIEW
—=TOP VIEW
sa
for K = 0) shedding occurrence alternated between
several spanwise locations. This is clearly seen
in Figure 23.
Several other aspects of the cavity shedding
process were apparent. The shed vapor had an
initial gross rotation with the same direction as
occurs in the free shear layer. This was evident
from the high speed movies viewing the cavitation
along the span (ie., a side view), and can also be
inferred from the pulse camera photographs taken
from the same view. The gross volume of the shed
vapor had relatively little dispersion prior to its
collapse but frequently developed within it regions
of apparent bubble coalesence prior to collapse,
as can be seen in Figures 14 and 24.
On the basis of the previously described defini-
tion of cloud cavitation, its occurrence was
determined from available photographs. The presence
of cloud cavitation as a function of the ratio of
SIDE VIEW
{ SIDE VIEW
379
———— [PIL
FIGURE 18. Cloud cavity sepa-
ration from leading edge sheet
cavity (example 1), K = 0.99,
We S dled m/s, P= 76.2 kPa,
a = 3.25 + 1.55 sin wt.
—— FLOW
FIGURE 19. Cloud cavity sepa-
ration from leading edge sheet
cavity (example 2), K = 0,
WV = deo8 m/s, P= 124.1 kPa,
Oe—isey25eR
maximum sheet cavity length, %m, to chord length,
C, and reduced frequency K is shown in Figure 25.
The data used to define the condition for the
occurrence of cloud cavitation were all taken at
nominally the same value of o. Figure 25 shows
that for a given (2m/c) value, cloud cavitation
can occur at nonzero K values whereas none would
be apparent for K = 0. For example, if test
conditions were adjusted such that 2m/e = O26, Ete
0.3 < K < 0.4, then one could conclude as did Ito
(1976) that there was a "critical" reduced frequency
associated with the onset of cloud cavitation.
Figure 25 also shows two curves representative
of the influence of the value of a, on cloud cavita-
tion. It is readily apparent from the data in
Figure 25 that the conditions for cloud cavitation
cannot be simulated by quasi-steady experiments.
As shown in Figure 25, cavity length is strongly
dependent on K. If the angle of a stationary foil
380
FIGURE 20. Final stage in cloud
shedding process, K = 0.21,
VEX = 14.8 m/s, Pi = 124.1 kPa,
a = 3.25 + 0.95 sin wt.
FIGURE 21. Desinent condition
for leading edge sheet cavity;
K = 0.49, V| = 11.5 m/s,
P_ = 76.2 kPa, 0 = 3.25 + 1.55
sin wt.
was set to the maximum angle the oscillating foil
attained (4.2° for a; = 0.95 in Figure 25), the
maximum cavity length could be as much as a factor
of two larger than for finite values of K (eg.,
K = 1.2).
The data plotted in Figure 26 show that within
the accuracy of the experiments, a variation in
velocity from 11.5 to 16.4 m/s produced no signifi-
cant change in the results shown in Figure 25 other
than that expected for the small variation in o
that occurred between tests. It appears that the
parameters of K, 0, and aj, are sufficient to
correlate all of the present data with the presence
of cloud cavitation.
6. CONCLUSIONS
In order to improve the physical understanding of
the cavitation inception process and the formation
—— FLOW
eee ee ee ee
La tne
i} SIDE VIEW
LOCATION OF DYE INJECTION
<—TOP VIEW f s10e VIEW
of cloud cavitation on marine propellers, a large
two-dimensional hydrofoil was tested in the DINSRDC
36-inch Water Tunnel under pitching motion. The
foil was instrumented with pressure transducers to
measure the unsteady surface pressure due to foil
oscillation, and photos were taken to correlate
cavitation inception and cavity patterns.
Prior to the occurrence of cavitation on an
oscillating foil, the foil is in a fully wetted
condition. Knowledge of the pressure distribution
on a fully wetted foil can be expected to provide
useful information for prediction of unsteady cavi-
tation. Fully wetted, time dependent, experimental
pressure distributions were compared with results
from Giesing's method for calculating unsteady
potential flow. Good correlation between the
prediction and the experimental measurements was
obtained for both dynamic pressure amplitudes and
phase angles within the range of reduced frequencies
investigated (K = 0.23 to 2.30). This good corre-
Nera Ni eaten td Nt
ee eee
?
i \| i | \ \
} | \ \ i\ i f | i\ j\ r é % A h\
Ls Ne Ned Nal Newel Sot Nga Singer mene! “enamel Neto Nin OES emf Site etna Sana
Besa |
}
PAAR arte nnengnatn th RAFAL GR gee ac nN ty ANETTA NG
| FLOW
TOP VIEW
lation supports McCroskey's conclusion that unsteady
viscous effects on fully wetted oscillating airfoils
are less important than unsteady potential flow
effects, if the boundary layer does not interact
significantly with the main flow.
Six series of oscillating foil experiments were
carried out in this test program to study the
leading edge sheet cavity growth and collapse.
A simplified mathematical model was developed to
explain experimental results for leading edge sheet
cavitation inception. The mathematical model
utilizes Giesing's method for calculating the
unsteady potential flow. A significant delay in
unsteady cavitation inception was both predicted
and measured. A further delay in cavitation
inception was also observed and predicted with
increasing pitch amplitude. It is shown that
unsteady cavitation inception is a function of:
ett aiat
i cathnpinyyremrea FOIL ANGLE (K=0)
SIDE VIEW
381
PRESSURE GAGE Ps
“| PRESSURE GAGE P,
\eotem.| PRESSURE GAGE Py
FIGURE 22. Surface pressure
fluctuations for K = 0,
We datos m/s, P= 76.2 kPa,
| CAMERA PULSE TRACE GS 3,250,
a— FLOW
FIGURE 23. Alternate spanwise
cloud cavitation shedding for
i= Op Wy = UitsS m/s, i = 1962
kPa’, O) — 3/25).
(1) the ratio of dynamic to static angular
pressure gradients
(dc_/da) / (dc _/da)
iS) u Pp s
and,
(2) the phase shift between the foil angle
and the dynamic pressure response.
Due to the phase lag in pressure response a signifi-
cant delay in unsteady cavitation inception is
predicted theoretically and observed experimentally.
Additionally, the angle at which cavitation inception
occurs increases with increasing pitch amplitude.
This effect results from a change in the phase angle.
It is well known tha even in a steady condition
the cavitation inception process is extremely complex.
The theoretical prediction is still very difficult.
FIGURE 24. Apparent coales-
cence of vapor bubbles within
cloud cavity; K = 0.28,
We = Wot m/s, Po = 124.1 kPa,
a = 3.259 + 1.55° sin wt.
a = 3.25 +a Sin wt
Vq = 11.49 m/s 6= 1.13
(°) = FOIL ANGLE AT K = 0
SEVERE CLOUD CAVITATION
DURING CAVITY LIFE
O O
CAVITY LENGTH 2m/,
MARGINAL OR NO
CLOUD CAVITATION
FIGURE 25.
<—e—T0P VIEW 4 SIDE VIEW
= 1.55 DEG
CLOUD CAVITATION
AT og ONLY
= 0.95 DEG
1.0 1.5 2.0 25)
REDUCED FREQUENCY, K
Variation in cloud cavitation with reduced frequency K and pitch amplitude Os
@ (DEG) Px (kPa) Vo (m/s)
@ 0.95 76.3 11.49
Oo 0.95 124.3 14.78
ro) 0.95 158.8 16.42
\v] 1.00 165.7 16.42
O 1.55 76.3 11.49
a 1.55 127.7 14.78
5 0
FOIL ANGLE AT K = 0
CAVITY LENGTH {m/¢
75 1.0
REDUCED FREQUENCY, K
Nevertheless, if the inception angle djg is known
from the steady model tests, the unsteady effect
on cavitation inception, to the first order, may
be estimated by the present method. Since the
present tests were carried out with only one foil
shape and only one pitch axis location, further
experiments are required, and in particular, the
range of variables should be extended.
Based on photographic observations of the leading
edge sheet cavitation instabilities, it appears
that the free shear layer and near-wake stability
concepts reviewed by Wu (1972) give a reasonable
qualitative description of the physical process.
The inherent instability of the free shear layer
and associated vortex shedding appear to provide
a reasonable model for the breakup of a sheet
cavity. However, the detailed hydrodynamics
associated with the near-wake closure region can
still only be postulated. The commonly held concept
of a reentrant jet, Wu (1972), may provide a reason-
able description applicable to the closure of the
near-wake region during the actual shedding of
vapor. For sheet cavitation extending over only a
portion of the foil chord this reentrant jet may
not actually penetrate the cavity itself but pene-
trate only a locally separated region just down-
stream of the sheet cavity trailing edge. In any
event, the presence of a reentrant jet is not
required to explain the inherent instability and
breakup of the sheet cavity.
For the conditions of the experiments reported
here, where the gross flow is nominally two dimen-
sional, the cavity instability is not coherent to a
significant extent along the foil span. In other
383
FIGURE 26. Influence of Vie
1.5 Por Gyr and K on cavity length
(2m/c) .
words, the cavity instability is highly three-
dimensional and appears to be principally dependent
on conditions in the immediate upstream free shear
layer flow. The most extreme form of cavity insta-
bility is manifest as a large shed cloud of vapor
and thus referred to in the literature as "cloud"
cavitation.
Within the context of the experimental results
reported here, the principle parameters controlling
the formation of cloud cavitation are reduced fre-
quency, K, cavitation number, o and foil oscillation
amplitude, 4}. The maximum cavity length, (2m/c),
is a function of these three parameters. However,
it has been shown that predictions of lm/e at finite
reduced frequencies cannot be based on the cavitation
observations at zero reduced frequency. With o con-
stant, the results show that it is possible to have
no cloud cavitation at finite reduced frequencies -
even though it was present on a stationary foil set
to the maximum unsteady angle. However, if the
steady foil is set to the mean angle of oscillation,
@, and no cloud cavitation is present, then it is
easily shown that at finite reduced frequencies
cloud cavitation will be present. Thus, Ito's con-
clusion that there exists a "critical" reduced fre-
quency for the onset of cloud cavitation appears to
be the result of the specific chosen values of the
parameters, K, 0, and 4).
The implication of the above results is that the
prediction of the occurrence of cloud cavitation
for hydrofoils in waves and propellers in wakes can-
not be based solely on the performance in calm
water or uniform flow.
384
ACKNOWLEDGMENTS
Grateful appreciation is expressed to Mr. G. Kuiper
for his skill in the pressure gage dynamic calibra-
tion, assistance in the test set-up and his helpful
discussions. Grateful appreciation is also due to
Mr. R. Pierce for his excellent work in performing
the data reduction. Finally, the reviews and
constructive comments by Mr. J. McCarthy and
Dr. W. Morgan are greatly appreciated.
The work described in this paper was sponsored
by Naval Sea Systems Command and the General Hydro-
dynamic Research Program at DTNSRDC.
REFERENCES
Morgan, W. B., and F. B. Peterson (1977). Cavita-
tion Inception, A Review-Progress Since 17th
ATTC. Proceedings 18th ATTC, Annapolis, MD.
van Manen, J. D. (1962). Bent Trailing Edges of
Propeller Blades of High Powered Single Screw
Ships. Proceedings of IAHR Symposium, Sendai,
Japan
Tanibayashi, H. (1973). Practical Approach to
Unsteady Problems of Propellers. International
Shipbuilding Progress, 20, No. 226.
Ito, T. (1962). An Experimental Investigation into
the Unsteady Cavitation of Marine Propellers.
Proceedings of IAHR - Symposium on Cavitation
and Hydraulic Machinery, Sendai, Japan.
Tanibayashi, H., and N. Chiba (1968). Unsteady
Cavitation of Oscillating Hydrofoil. Mitsubishi
Heavy Industries Technical Report (in Japanese),
Be WO Ao
Chiba, N., and T. Hoshino (1976). Effect of Un-
steady Cavity on Propeller Induced Hydrodynamic
Pressure, Journal of the Society of Naval
Architects of Japan, 139.
Meijer, M. C. (1959). Some Experiments on Partly
Cavitating Hydrofoils. International Shipbuilding
Progress, 6, No. 60.
Chiba, N. (1975). Behavior of Cavity Collapse as
a Cause of Cavitation Damage of Propeller Blades.
Cavity Flows, ASME Symposium Proceedings, 1975,
ial als}
Miyata, H. (1972). Pressure Characteristics and
Cavitation. M. S. Thesis, Department of Naval
Architecture, Tokyo University.
Miyata, H. et al. (1972). Pressure Characteristics
and Cavitation on an Oscillating Hydrofoil.
Journal of the Society of Naval Architects of
Japan, 132. 10; 107-115.
Radhi, M. H. (1975). Theoretische und Experimen-
telle Untersuchung uber den Kavitationseinsatz
an Schwingenden Tragflugelprofilen. PhD Thesis,
Technischen Universitat Berlin, D83.
Hilten, J. S., et al. (1972). A Simple Sinusoidal
Hydraulic Pressure Calibration. National Bureau
of Standards, Technical Note 720.
Nuttall, A. H. (1971). Spectral Estimation by
Means of Overlapped Fast Fourier Transform Pro-
ceeding of Windowed Data. WNUSC Report 4169.
Abramson, H. N. (1967). Hydroelasticity with
Special Reference to Hydrofoil Craft. NSRDC
Report 2557.
Giesing, J. P. (1968). Two-Dimensional Potential
Flow Theory for Multiple Bodies in Small -
Amplitude Motion. Douglas Aircraft Company,
Report No. DAC-67028.
McCroskey, W. J. (1977). Some Current Research in
Unsteady Fluid Dynamics - the 1976 Freeman
Scholar Lecture. Trans. ASME Journal of Fluid
Engineers, 99, Series 1, No. 1, 8-38.
McCroskey, W. J. (1975). Recent Review in Dynamic
Stall. Proceedings, Unsteady Aerodynamics
Symposium, The University of Arizona, 1-34.
Acosta, A. J., and B. R. Parkin (1975). Cavitation-
Inception - A Selective Review. Journal of Ship
Research, 19, 4; 193-205.
Carta, F. O. (1971). Effect of Unsteady Pressure
Gradient Reduction on Dynamic Stall Delay.
Journal of Aircraft, 8, 10; 839-841.
Huang, T., and F. B. Peterson (1976). Influence of
Viscous Effects on Model/Full Scale Cavitation
Scaling. Journal of Ship Research 20, 215-223.
Wu, T. Y. (1972). Cavity and Wake Flows. Annual
Review of Fluid Mechanics, 4.
Knapp, T. R., et al. (1970). Cavitation. McGraw-
Hill Book Company.
Song, C. S. (1969). Vibration of Cavitating Hydro-
foils. St. Anthony Falls Hydraulic Laboratory,
University of Minnesota, Project Report No. lll.
Besch, P. K. (1969). Flutter and Cavity-Induced
Oscillation of a Two-Degree-of-Freedom Hydrofoil
in Two-Dimensional Cavitating Flow. NSRDC Report
3000.
Wade, R. B., and Acosta, A. J. (1965). Experimental
Observations on the Flow Past a Plano-Convex
Hydrofoil, J. Eng. for Power, ASME, No. 65-FE3,
TOK
Cavitation on Hydrofoils in
Turbulent Shear Flow
H. Murai, A. Ihara,
and Y. Tsurumi
Tohoku University, Sendai, Japan
ABSTRACT
Conditions and positions of inception, locations of
zones, and aspects and behaviors of bubbles and
cavities of cavitations occurring on two hydrofoils
with the profiles of Clark Y 11.7 and 08 in shear
flows and a uniform flow have been observed and
measured, and correlated with measured pressure
distributions on the hydrofoils and turbulence
levels and size distributions of cavitation nuclei
in free streams.
At attack angles small for the profile, traveling
cavitations begin near positions of minimum pressure
and at cavitation numbers about the same as absolute
values of minimum pressure coefficients, irrespective
of flow shears in free streams provided local values
are used. Discrepancies between conditions and
positions of inceptions and pressure coefficients
and their distributions, and sizes of traveling
bubbles depend on qualities of free streams.
On the hydrofoil with the Clark Y 11.7 profile,
a traveling bubble in a zone of rising pressure,
deforms, creating a projection in shear flow, or
two projections in uniform flow, leaves only the
projection and then collapses. On the hydrofoil
with 08 profile, a traveling bubble collapses after
the deformation caused by the instability of the
bubble surface. On both hydrofils, bubbles collaps-
ing symmetrically and asymmetrically, looking like
micro jets forming, can be found.
At attack angles large for the profile, fixed
cavitations occur. Conditions and positions of
inception are similar to those of traveling cavita-
tions. In the boundary layers on both side walls,
fixed cavitations occur at relatively large
cavitation numbers, possibly equal to the absolute
values of local minimum pressure coefficients, and
even develop beyond the boundary layers. Cavitation
zones on the low-speed side are larger than those
on the other side, and those occurring in the
boundary layers of uniform free streams are of an
intermediate size.
385
At attack angles intermediate for the profile,
fixed and traveling cavitations occur at the same
time and tend to become fixed only on the Clark Y
11.7 profile. On the 08 profile, fixed cavitations
at the leading edge and traveling cavitations at
about the mid-chord appear at the same time in shear
flows, but only fixed cavitations occur and develop at
the leading edge in uniform flows.
1. INTRODUCTION
Many researches on the cavitation characteristics
of hydrofoil profiles have been published, and the
appearance, the degree, and the effects on the
hydrodynamic behavior of hydrofoil of the incipient
and developed cavitations occurring on hydrofoils
have been discussed by Numachi (1939, 1954), Daily
(1944, 1949), and Kermeen (1956a, 1956b). Recently,
the effects of the behavior of boundary layers and
the turbulence in the free stream on the inception
and development of cavitations on hydrofoils were
reported by Casey (1974), Numachi (1975), and Blake
et al. (1977). Although they have been concerned
with cavitation occurring on hydrofoils in a free
stream of uniform velocity, actual blades of
hydraulic machines, including ships' propellers,
work mostly in nonuniform flow, and the effect of
nonuniformity might have to be examined as well.
Investigations on cavitation occurring in shear
layers have been made by Daily and Johnson (1956)
in a zone of wall shear turbulence, by Kermeen and
Parkin (1957) in a wake behind a circular plate
and by Rouse et al. (1950) and Rouse (1953) in
submerged jets. But research concerning the cavita-
tion occurring on hydrofoils laid in a free stream
with a shear is not available as far as the authors
are aware.
The present report is intended to clarify the
influence of the spanwise shear, uniform in the
core and the accompaning boundary layers on both
sides of the free stream and its turbulence on the
386
>
ice
2 ALS,
3 Lrgéte ep
o
he eee ae ee ea
a MMM ha hdd Lhd hhh hhd hhhdE
10096
o
Oo
mM
FIGURE 1. High speed water tunnel.
inception and development of cavitation and the
aspect and behavior of cavitation cavities occurring
on two hydrofoil profiles with different cavitation
characteristics.
2. EXPERIMENTAL APPARATUS AND METHODS
High Speed Water Tunnel
The water tunnel used for the experiment is shown
schematically in Figure 1. The tunnel contains
180m? of water. The water is circulated by the
centrifugal pump, P, whose revolution is controllable.
Bubbles generated in the measuring section, the
duct, and the pump mainly disappear in the reservior
T. In the reservoir the water first flows upward
to the free surface at the top of the reservoir,
and then down very slowly through an area of 20m?
to the bottom. Two spaces, one at the entrance
corner of T and the other at the top of the tunnel,
separate bubbles from the water and continuously
remove the separated air. The water sucked up from
the bottom of T turns to the horizontal direction
through corner vanes, and enters the measuring
section through the honey comb, S, made of synthetic-
resin pipes of 26mm diameter, 6mm thick, and 450mm
long. Then it flows through two nozzles, Nl and
N2, which contract the cross section from 2100x1400mm?
to 1500x1000mm2 and to 1200x200mm? , the room for
installing the shear grid, and the nozzle for
contracting the cross section from 1200x200mm? to
610x200mm2. The contraction ratio is 24:1 in all.
The water flowing out of the measuring section flows
through the diffuser and back to the circulating
pump P.
The tunnel pressure is controlled by introducing
compressed air to the top of the reservoir or by
U/Uc
3
0,02 &,
Uc 0.01 =
oa
© 8.23 m/s Q A
| 4 5.60 m/s =
07 | SIL 1 ati
00 601 0.2 03 #04 O05 0.6 Or . OG 09 1.0
y/h
FIGURE 2. Velocity distribution at no grid condition.
lowering the free surface led from the top of the
tunnel, the maximum and minimum pressures being
48x10° Pa and -0.8x10° Pa. The flow velocity at
the measuring section is controlled from the measur-
ing station by controlling the speed of the
circulating pump P.
Measuring Section
The measuring section has a cross section 200mm
wide and 610mm high and its total length is 3000mm.
The first upstream 1000mm has two plexiglass windows
in each side, and upper and lower wall. In this
experiment, the hydrofoil is installed through two
downstream-Side windows in both side walls. Figure
2 shows the spanwise distributions of the velocity
and the static pressure at the position of the
mid-chord of the hydrofoil in the case of no grid.
The velocity profile is almost uniform except in
the 10% the boundary layers on both side walls.
The static pressure, expressed as the difference
from that at the side wall, is constant within the
accuracy of this experiment.
Hydrofoils
Two hydrofoils have been prepared for the experiment,
each of which has 100mm chord and 700mm span. ‘Two
profiles have been selected; one is Clark Y 11.7
and the other 08, dimensions of which are shown in
Table 1. The former is selected for the purpose of
examining the influence of the behavior of the
boundary layer on the hydrofoil surfaces on the
inception and development of cavitation and the
aspects of cavitation bubbles or cavities, because
it has a round nose and a surface pressure distri-
bution rising toward the trailing edge. The latter
is selected as a typical profile among ones designed
by Numachi (1952) for high-speed flows, and has a
sharp leading edge and comparatively good cavitation
characteristics for its simple shape.
The hydrofoil of the Clark Y 11.7 profile has
14 and 13 piezometer holes of 0.4mm diameter on the
suction and pressure surfaces respectively, and one
of the 08 profile has 13 and 13 piezometer holes,
“Table 1 Profile Forms of Hydrofoils
@ilaals Ye Ayal g 7/ Og
x Y x Y
Upper Lower Upper Lower
0.0 3590) | S550 0.0 OQclS O56
Le2S (Sed 598 LS5A5 O35 1 OO
2S GsSQ do@7/ Bo) O57 O50
by 50) T3990 O93 5) 50) 1247) O20
ted Bois) © Moss} 10.0 Aol OO
10.0 Veo 0) 5c7) WSs 35695 O.0
USO) above Gite} (0) Gal) 20.0 S502 0.0
2050 alt 3G 0.03 30.0 502 Oo)
SJO6@! dhalo7o Oo 50.0 3400 OO
Ayo ei; alak 6A@) | OS @ 70.0 G502 60
SOcoO) dO6S2 . 0.) 80.0 S507, Wo
60.0 Vols O50 90.0 259 OO
70.0 3) DO lO) O70 oty oO)
80.0 5222. ©. 7 > O57 Oo
TORO 27.80) ORO OWI57S Oo35 OO
100.0 OZ O10 100.0 OclG OO. 16
Table 2 Positions of Piezometer Holes
Culaels Ye Ibi 47 Og
Upper Lower Upper Lower
X & X % X & X %
1 (0) al SON 4 2} 50)
2 S}50) alS 350 2 G60 as 6.0
3 Go@) ike 6.1 3 i@,@ Le wO>o
4 WO ca aly —al@ Gal 4 S50 aby abs 5
5 IL Ey 5 ab iL} | LA Og} 5 BOO Ae 20.60
6 PX0) Gal 19 2) Gal 6 30.0 19 30.0
7 30.0 20 $3) Gal 7 40.0 20 40.0
8 40.5 21 39) 59) Soo0 2 5050
©) S§@oal AA 30.10) J 690.0 22° O60 50
EOS GlO\=2 2s) (50), 10 70.0 23 70.0
dat 70.4 24 69°58 iil 8050 24 s050
12 80539 25 V9 58 2 8550 25 85.0
13 B60 26 84.8 13 GOO AS SO, O
{
6789
2345
10
Wi2i3
aS
ribo
14116 118 19 20 21 22 23 24/26
17 2)
1S
as are shown in Table 2. The holes are inclined to
the direction of the free streams as to have no
influence on the pressures measurements of each other.
Pressures are measured by using a mercury-water
manometer.
For measurements of pressure distributions, the
hydrofoil is shifted spanwise so as to allow the
piezometer holes to cover the whole 200mm span.
For observations of cavitations, the part of hydro-
foil having no piezometer hole is used.
Shear Grids
In order to examine the influence of shear flow,
the free stream at the measuring section has been
made to have the simplest shear, that is, uniform
shear. The grids for creating uniform shear flow
are composed of straight rods arranged perpendicular
to the free stream and the hydrofoil span with non-
uniform spacings calculated by using the theory of
Owen and Zienkiewicz (1957). The spaces near both
side walls were modified according to Liverey and
Turner, (1964) and Adachi and Kato (1973) and are
shown in Table 3. In order to make two different
free streams having the same shear but different
turbulence, two grids were made, composed of rods
with different diameters, 20mm for No. 1 and 15mm
for No. 2.
TABLE 3 Rod Spacings of Shear Grids
Grid No. l
Rod Number it 2 3 4
distance from low-speed
side wall (mm) AWoil SQLS) alosjo%4 al Sjs}3}
Grid No. 2
Rod Number 1 2 3 4 5
distance from low-speed
side wall (mm) 16 47.4 81.5 118.4 161.2
387
The shear grid is installed at a position 1500mm
upstream from the mid-chord of hydrofoil, where
the cross section of the duct is about twice as
great as that of the measuring section so as to
keep the grid free from cavitation.
Measurement of Velocity and Static Pressure at the
Measuring Section
Spanwise distributions of the velocity and the static
pressure are measured at the position of the mid-
chord of the hydrofoil in the absence of the
hydrofoil, by using a Prandtl-type Pitot tube of
3mm diameter. They corresponded to the difference
of static pressures at the inlet and exit of the
second nozzle, N2, and the static pressures at the
exit of the nozzle and the position 530mm upstream
and 170mm below the position of the mid-chord of
hydrofoil.
It has been pointed out by Lighthill (1957) that
total pressures measured by using a Pitot-tube in
a shear flow exhibit larger values than real ones
due to displacement effects of a Pitot-tube. The
displacement thickness of the boundary layer on
the Pitot-tube used in this experiment, having a
ratio of outer to inner diameters of 0.6, is
calculated as about 0.54mm by use of the empirical
equation presented by Yound and Mass (1936) and
Macmillan (1956). The error in this experiment
caused by the displacement thickness is the order
of 0.08mm/s for a shear factor of 0.15 in the core
of the shear flow so that it can be neglected,
except in the boundary layers. There the shear
factor, on which the error is proportional, is
considerably large, especially near both side walls.
The static pressure at the measuring section is
limited due to the following two reasons: at the
upper limit, by the strength of the differential
piezometer used for detecting the velocity at the
measuring section; and at the lower limit by the
need to prevent the shear grid from cavitating.
The prescribed velocities at the measuring section
are determined so as to keep the static pressure
at the measuring section within the above-written
limits for obtaining the inception and development
of cavitation corresponding to the angles of attack
of the hydrofoils, as shown in Table 4.
Measurement of Turbulence
Spanwise distributions of the components of turbu-
lent velocity in the directions parallel to the
free stream and perpendicular to the free stream
and the hydrofoil span are measured at the position
of the mid-chord of hydrofoil (in the absence of it)
by using the Laser-Doppler velocimeter, DISA 55L
Mark II. Each component of turbulent velocity is
TABLE 4 Velocity and Pressure at the Test Section
on Cavitation Experiments
a (rad) Velocity (m/s) Pressure (105 Pa)
0.0 iil ©) -0.64 ~ -0.45
0.052 10.0 HOS —0)53)5)
0.105 9.0 —=0)5 (60) ~ 0), ALab
(0) 5 15) 7/ 8.0 =0).33' + +0)-40
388
Pulse Modulator Pulse Power Amplifier
Transmitter
Sweep
Oscillator
X-Y Recorder
Standard
Vessel
Stream
Passage
Receiver
Frequency Analyser’
Esai cir ears Ang gl come
ee Impedance Converter
land Amplifier pe
FIGURE 3. Schematic diagram for nuclei measurements.
detected as an absolute value of the root mean
square.
Observations and Measurements of Cavitation
Cavitation inceptions are seen by the naked eye
under 50Hz, stroboscopic 3yus flash illumination.
An incipient cavitation number is defined by using
the static pressure at which the inception is
detected while reducing the static pressure at a
low rate and the local free stream velocity. How-
ever, in the boundary layers the velocities at
outside edges are taken while the free stream
velocity is kept at the prescribed value. Desin-
ences are too intermittent and indefinite to be
detected definitely in the course of raising the
static pressure.
For the measurements of positions of inception
and the observations of appearances of cavitation
bubbles or cavities, photographs of 3 Us exposure
and high-speed motion pictures of 3000 frames per
second and 2 us exposure for each frame were taken.
For the high-speed photography, the high-speed
camera, FASTAX, was used synchronized with the
high-speed stroboscope made by E. G. and E Co. Ltd.
For the measurements of average locations and shapes
of cavitation regions, photographs of 1/60 s
exposure were used.
U/Uc
2
PUc
2 i eres mt aR 002 &,
Wee Se eV ee = es
ye ea ri 3 ¢ i -2—_¢—_2—¢ iS + — = ool x
in ij | Uc —|0™ &
0 583 m/s =
4946ms
07 + 4 ae i | Brn f
(b) Grid No.2
FIGURE 4. Velocity and static pressure distribution
for shear grids.
Relative Measurement of Cavitation Nuclei
Size distributions of gas nuclei are measured by
using the sound-attenuation method of Schiebe 1969.
The measuring system is shown in Figure 3. The
frequency range of swept pulses was 20kHz~1000 kHz.
Both probes for emission and reception were 25mm
diameters, made of a crystal, and exposed directly
to water. The measurements were relative ones for
comparison between the three cases of no grid and
grids No. 1 and No. 2 because the system has not
yet been calibrated for bubbles with prescribed
definite diameters.
Measurements were carried out at four positions
in the spanwise direction at the mid-chord of
hydrofoil perpendicular to the free stream and the
hydrofoil span.
3. RESULTS OF EXPERIMENT AND DISCUSSIONS
Shear Flow at Measuring Section
The velocity profiles normalized by each velocity
at the mid-span and the distributions of the static
pressure expressed as the difference from one at
the side wall and normalized by each dynamic pressure
at the mid-span for the grids No. 1 and No. 2 are
shown in Figure 4. The flow shear for grid No. 1
is uniform in the free stream core and the non-
dimensional shear factor is 0.15. That for grid
No. 2 is about the same as for grid No. 1 at half
the core of the free stream on the high-speed side
but smaller at the other half. The non-dimensional
shear factor is 0.06. Both have boundary layers of
10% thickness span on both sides. The static
pressure is higher in the free stream core than on
the side walls by about 1% or a little more of the
dynamic pressure at mid-span. Scatters of plots
are within the accuracy of this experiment.
Spanwise Distribution of Turbulence
Root mean squares of two components of turbulent
velocity, one stream-wise and the other perpendic-—
ular to it ahd the hydrofoil span, are measured
in every free stream, and shown in Figure 5
normalized by Uc. The velocity at the mid-span was
kept at 9.86 m/s. When both are expressed as the
turbulence levels based on the local velocity of
free stream, U, for the cases of the two shear grids,
both u'/U and w'/U vary so little in the spanwise
direction that they can be regarded as constant
10.0 =I T T
i No.Grid| O | | Uc=9.86
jo .Gri c=9. m/s
NO. An) oan]
8.0 |— [no.2 [ofan] A
* a
Bea arti
u/Uc , w/Uc (%)
FIGURE 5. Spanwise
distribution of
turbulence.
104 103 104 1o%
lam T Ir = = Tet oT T \aeeenal
10S F ,aa a +
- e} 4
é 5o° & #8 6 ayo
ra
5 4 a
oo F s a 4 a
a
lo* 08 8 Oo aj
Y/h = 0.125 0.375 0.625 0.875 4
kd = 0.65 ray
a
103 Ir Uc = 11.Om/s S © Bo
© No.l ray ry o4
& No.2
ea el yp ss 1 I !
10-4 10% 104 1o3
Ro (cm)
FIGURE 6. Spanwise variation of size distribution of
cavitation nuclei.
within the accuracy of this measurement. u'/U was
6.8 and 6.2% in the case of grids No. 1 and No. 2,
respectively, and w'/U was 3.6% in the case of both
two shear grids. It has been reported by Harris
et al. (1977) that in a shear flow generated by a
shear grid, w' and the other lateral component of
turbulent velocity, say v', are almost the same.
If it is also assumed that v' = w' in this experi-
ment, the resultant turbulence levels were 8.5 and
8.0% in the case of the grids No. 1 and No. 2,
respectively, in the core of the free stream. In
the case of no grid both u'/U and w'/U were 0.1%,
and the turbulence can be regarded as isotropic
at a level of 0.17%, in the core of free stream.
Spanwise Variation of Size Distribution of Cavita-
tion Nuclei
Attenuations of sound pressures were measured at
four positions in the spanwise direction ( 12.5,
37.5, 62.5, and 87.5% span) from the low-speed side
at the position of mid-chord in the absence of the
hydrofoil, and at the cavitation numbers of 2.75
and 0.65. Because the levels of attenuated sound
pressures were not calibrated for micro bubbles
of known sizes, sound pressure levels in the shear
flows at each measuring position were compared with
one in the uniform flow in which any spanwise
variation was not noticed. Frequencies and
differences of sound pressure levels were related
to equivalent radii, and to differences of the
numbers of cavitation nuclei from those in the
uniform flow by using the formulae presented by
Richardson (1947) and Gavrilov (1964).
At a cavitation number of 2.75, any noticeable
difference of size distributions between the shear
flows and the uniform flow was not found. Ata
cavitation number of 0.65, however, remarkable
differences were ncticed as can be seen in Figure
6. Numbers of nuclei with radii smaller than 24m
in both shear flows are considerably larger than
those in uniform flow, and the larger the numbers
of nuclei the smaller the nuclei radii are. Size
distributions in the two shear flows were not so
different from each other in the high-speed sides
of free streams, but in the low-speed sides, the
shear flow made by the grid No. 2 is richer in
nuclei, especially in the range of small radii, than
the other.
389
Cavitation Inception
Spanwise variations of local incipient cavitation
numbers are plotted in Figure 7 for the Clark Y
11.7 profile, and in Figure 8 for the Og profile.
Also, spanwise variations of positions of minimum
pressure for the case of no grid, grid No. 1, and
grid No. 2, are shown.
Clark Y 11.7 Profile
In the case of no grid incipient cavitation
numbers, kdi, are a little smaller than absolute
values of minimum pressure coefficients, |cpmin|'s
over the whole span at the attack angles, a, of
0 and 0.052 rad, and in the core of free stream at
a's of 0.105 and 0.157 rad. Differences between
kdi's and |Cpmin|'s increase as a increases until
06 | ae !
0) 0.2 0.4 0.6 0.8
aS T T
mo]
= i Poe, (b) @= 0 052rad
Ce ae ee ee
ee a ae SS = =
2 e \ ¥N
By Numachi (1947) Sy
0 L SiG |
0 02 04 06 08 1.0
(c) @=0O 105rad =I
— |
= -
2.0 eS]
oS
18 alk eee | at
(0) 0.2 04 0.6 0.8 1.0
kdi
on
[o)
S
a= 0 157rad
Gy
PlOIP\O
|
|
with tip clearance |
Sor |
z
2
z
i]
is
B
eens eee aee,|
FIGURE 7. Spanwise variation of incipient cavitation
numbers for the Clark Y 11.7 profile.
390
= + + + *
go oe eG
(a) a@=0O rad
i pel et ec le a Th
0 0.2 04 06 0.8 1.0
20
z xe
15 A
——
1.0 — =)
05 : : ~|
0 02 04 06 08 1.0
35 a a el
+ (c) a= 0.105 rad
Ae
0) 0.2 0.4 06 08 1.0
Y/h
FIGURE 8. Spanwise variation of incipient cavitation
numbers for the Og profile.
it reaches 0.105 rad, but become smaller at a =
0.157 rad.
At a's not smaller than 0.105 rad, fixed cavita-
tions occur in the boundary layers at positions
very close to both side walls at kd's much greater
than local |Cpmin|'s. At the same time a zone of
cavitation widens spanwise beyond each boundary
layer with the inception so that detection of
inception becomes difficult in the region neighbor-
ing both boundary layers on the side walls. This
is the reason the lack of points between y/h = 0.025
~ 0.3 and 0.7 ~ 0.975. Frequency distributions of
cavitation occurrences analyzed by using high speed
motion pictures for 1 second illustrate those facts,
as can be seen in Figure 9.
In free streams with shears made by the grids
No. 1 and No. 2, kdi's almost equal or are a little
larger than local |Cpmin|'s. They vary spanwise
under the influences of the flow shears in the
core and the boundary layers on both side walls,
and the accompanied secondary flows, except at
a = 0.105 rad, which indicates that these free
streams are rich in cavitation nuclei. At a = 0.105
rad, kdi is a little smaller than |cpmin|, which
can be assumed to be due to cavitations changing
from traveling to fixed, as mentioned in the next
section.
Differences between kdi's and |Cpmin|'s in the
boundary layers are larger than those in the case
of no grid on the low-speed side, but are the
contrary on the high-speed side, due to the
secondary flows induced by the flow shears in the
cores. The above-mentioned effect is most remark-
able at a = 0.105 rad: kdi's in the boundary layer
on the low-speed side in cases of the shear grids
are larger than those not only in the case of no
grid but also |cpmin|'s in the boundary layer,
though only by a little. The mechanism causing
the effect has been examined by measuring spanwise
variations of static pressures on three points near
the leading edge in the boundary layer on the low
speed side at the attack angle of 0.105 rad in the
case of the grid No. 2. It was confirmed that the
detected incipient cavitation number, 2.53, in the
boundary layer lies near the largest absolute value
of the pressure coefficient based on the local
velocities in the zone between 3 and 5 mm from the
side wall. However, measured velocities in the
zone are not very reliable. Symbols A in Figure
7 show kdi's when the hydrofoil has a tip clearance
of about 0.1mm on the high-speed side in the case
of grid No. 1. It was found that effects of a
boundary layer are weakened by tip clearances,
especially at large angles of attack, although
another cavitation occurs at the tip clearance.
08 Profile
At 0 angle of attack, traveling cavitations
occurred and kdi aimost coincide with | Cpmin| in
the case of no grid, but were larger than the latter
in the case of grid No. 1. The difference decreases
spanwise toward the high-speed side, in correspon-
dence with the size distribution of cavitation
nuclei. At angles of attack larger than O rad,
however, fixed cavitations occurred and kdi's were
much larger than measured |Cpmin|'s because of the
lack of a piezometer hole at the position of the
largest |Cpmin|, which is closer to the leading
edge than the closest hole at 3% chord. At a =
0.052 rad, in the case of the grid No. 1, another
cavitation of the traveling type appears around
the position of the measured second lc min | and the
kd almost coincided with the measured Cpmin| . kdi's
in the case of grid No. 1 were smaller than those
in the other case on the high-speed side. The
discrepancey can be surmised as due to the discrep-
ancy between structures of laminar separation
bubbles just behind the leading edge in the two
cases because of the difference between turbulence
levels. At a = 0.105 rad, kdi in the case of the
grid No. 1 was larger than in the case of no grid
in the core of free stream, but was the opposite
in the boundary layer on the high-speed side wall.
In the case of no grid, cavitations with long and
wide zones occurred in the boundary layers on both
sides close to the side walls and the leading edge
as with the Clark Y 11.7 profile.
Location of Incipient and Developed Cavitations
Spanwise variations of positions of cavitation
inception and front and rear edges\of (time) average
zones of developed cavitation are shown in Figures
10 and 11 for the profiles Clark Y 11.7 and 08
respectively. Also are shown spanwise variations
FIGURE 9. Frequency
distribution of cavi-
tation occurrence.
pee I
ee |
(0) ray
| SPiepree nf 8)eukeiee
a
08 + 0 g 4 $4
a 4 a=Orad
06 +
i oT oo a 03 :
= /By Numachi (1947) _| oo i Fae) Aut
aaa aS Se on a= 0 rad
| ze l al
) 02 04 06 08 10
Y/h
a I T T
S
« ° e 5 =) @
os; 8 g EP
04
Re ko oe %
Clark Y 11.7
RO! a = 0.052 rad
| i | |
io) 0.2 0.4 0.6 0.8 1.0
y/h
FIGURE 10 (a) (b). Spanwise variations of
position of inception and front and rear
edges of cavitation zones for the Clark Y
11.7 profile.
1.0 aan Uhm
° = 8 6
_ 08 br 5 z
ae
eS a g a 3 al
Q § 9 o g
04 rn a= 0.105 rad a
—_— kee os LO. =Cpmin 02
No. | a at = =
No.2 2 oF Be (c)
[Ma lo. | @ > with Tip clearance] _| 0! (lees ab delay)
i 6 s # —] a = 0.105 rad
mI ! | | 0
(0) 2 04 06 08 10
y/h
O5 T T 3 T
<< 8 ) a
i 04 b & a =) o a S)
E a
03 mal
a
; i)
a g o
o2 g 4 as g z 4
a=0.157rad (d)
kai [25 | 2.0 | -CPmin Clark Y 11.7
ST RAE i a = 0.157 rad
No 2 s a [o[—--— 0.1
No! eo! | with fip clearance i
AL i a 3
| eS es Sem Shah's
Ce) 02 04 06 08 1.0
y/h
FIGURE 10 (c)(d). Spanwise variations of
position of inception and front and rear
edges of cavitation zones for the Clark Y
11.7 profile.
391
of positions of minimum pressure, in each case
indicated in the figures. The bottom and the
second (at a = 0, 0.052 rad) groups show the
positions of inception or front edges of cavitation
zones and refer to the scales written on the right-
hand side, and the other groups show rear edges of
cavitation zones and refer to the scales written on
the left-hand side. Open symbols correspond to
traveling cavitations and closed and semi-closed
hey T |
a
HOM =
S A
Az a RK
3 e
08 6 ® “4
Q
in )
0.6 ° O =|
° a= Orad
|_kdi [056 [053 |-Comin |
iegid TO Loke. 5
No.1 [AT [AlAlAla|—-—
= =| OB.
ah RN Es Ree ae SL 3
| | | 04
(0) 0.2 04 06 0.8 10
Y/h
(a)
08
a= O rad
1.0
= 4 A
= A 4& S)
es re ) =) e g
0.5 cal
a = A a a
0 2 R a o a i) 4
a= 0.052 rad
[ [_kdi [0.9 0.7 [=Cpmin
[NoGrid | |@| {2 o
[Not | fal lalaAtala|—-—= ae
uae es a nee | os
Za
eg e | e ese e—20
(0) 02 0.4 0.6 0.8 Ke)
Y/h
(b)
O08
a = 0.052 rad
| I |
a= 0.105 rad
0.15 [ kdi [2.5 | 2.0 —
[No Grid e| jo! jo o
(No. 1 al [al [a g
0.10 4
me Hw 4 g e
Ze A a 2
0.05 fr z a 4
0 ) ) a) )
9 he
(0) 0.2 0.4 0.6 08 1.0
Y/h
(c)
08
a = 0.105 rad
FIGURE 11. Spanwise variation of the
position of inception and the front and rear
edges of cavitation zones for the Og profile.
392
symbols to fixed. kd's indicated in the figure are
based on the velocity at the mid-span.
Clark Y 11.7 Profile
In the case of no grid, traveling cavitations oc-
curred a little downstream from positions of minimum
pressure at a = 0 and 0.052 rad. Front edges of
average zones of cavitation move forward beyond posi-
tions of minimum pressure as kd is reduced, uni-
formly in the core of the free stream. At a = 0.105
rad, in the core of the free stream, cavitations,
mainly traveling mixed with fixed, occurred just
downstream from positions of minimum pressure. How-
ever, with a small decrease of kd from the incipient,
the type of cavitation changes to fixed and the
front edges of cavitation zones move backward from
positions of inception and forward with a further
decrease of kd. In the boundary layers on both
side walls, fixed cavitations occurred very close
to the leading edge of the profile and to the side
walls, and front edges of cavitation zones move
little as kd is reduced. At a = 0.157 rad, fixed
cavitations occurred just downstream from positions
of minimum pressure and front edges of cavitation
zones moved forward just a little and never ex-
ceeded positions of minimum pressure, in the core
of free stream. In the boundary layers on both
side walls, fixed cavitations occurred just down-
stream from the leading edge of the profile and al-
most attached to the side walls, and front edges of
cavitation zones moved little as kd was reduced.
At all attack angles, lines of rear edges of
cavitation zones have shapes similar to the velocity
profile at kd's a little smaller than the incipient.
But rear edges move backward with a further decrease
of kd to be almost uniform in the spanwise direction.
In cases of grids No. 1 and No. 2, positions of
inception are closer to positions of minimum
pressure than in the case of no grid, in correspon-
dence with size distributions of cavitation nuclei:
Front edges of cavitation zones move forward beyond
positions of minimum pressure in the cores of free
streams at a = 0, 0.052, and 0.105 rad. Ata=0,
0.052, and 0.105 rad, incipient cavitations are of
the traveling type, but at a = 0.105 rad, in the
cores of the free stream, cavitations sometimes
change their type from traveling to fixed as kd is
reduced, and in those cases front edges of zones
of fixed cavitations move backward from the inception
position. In the boundary layer on the low-speed
side wall a fixed cavitation occurred very close
to the leading edge of the profile and to the side
wall, but no inception of cavitation of any type
can be detected in the boundary layer on the other
side wall, in the range of kd in this experiment.
At a = 0.157 rad, fixed cavitations occurred at
positions of minimum pressure, including the boundary
layers on both side walls, and front edges of
cavitation zones move little.
At kd's a little smaller than the incipient,
lengths of cavitations are larger on the high-speed
Side than on the other side at a = O and 0.052 rad.
At a = 0.105 and 0.157 rad, however, they are larger
near the wall on the low-speed side than on zones
more distant from the wall. Rear edges of cavita-
tion zones have a tendency to be uniform in the
spanwise direction at all attack angles as cavita-
tions develop.
Much difference between the two grids in the loca-
tions and movements of cavitation zones cannot be
found.
08 Profile
At O angle of attack, positions of cavitation
inception and movements of front and rear edges of
cavitation zones with a decrease of kd, compared
with positions of minimum pressure, are quite
similar to those of the Clark Y 11.7 profile in
the cases of no grid and grid No. 1. However, at
a's larger than 0, fixed cavitations always occurred
at the leading edge over the whole span, irrespective
of the existence of the shear grid. Front edges of
cavitation zones never moved from the leading edge
as kd's were reduced. Lengths of cavitation zones
do not grow much, owing to the steep negative-
pressure zones just behind the leading edge, until
kd's are reduced to about the second Cpmin|'s.
But in the case of no grid, once kd's increase,
they develop suddenly beyond positions of minimum
pressure and tend to be uniform in the spanwise
direction as can be seen in Figure 13(b) at a =
0.052 and kd = 0.7. In the case of grid No. l,
however, lengths of the fixed cavitation do not
grow enough to reach positions of minimum pressure.
Instead cavitations of the traveling type appear
around positions of the second minimum pressure, as
can be seen in Figure 13(b) at a = 0.052 and kd =
0.7 and as shown in Figure 11(b) by the symbols A.
The length of the cavitation zone is about the same
as that in the case of no grid in the free stream
core but smaller than that in the boundary layers
on both sides, at the beginning of development.
At a = 0.105 rad, the length of the cavitation zone
is much larger than that in the case of no grid at
the beginning of development, but becomes about the
same as the others with a further decrease of kd.
Aspect and Behavior of Cavitation Bubbles and
Cavities
Figures 12 and 13 show several examples among the
3us-exposure photographs and an example of high-
speed motion pictures of cavitations taken at the
inception and each stage of development occurring
on the Clark Y 11.7 and 0g profiles, respectively.
Cavitation numbers indicated in the figure on the
left hand side are based on the velocity at the
mid-span.
Clark Y 11.7 Profile
At a = O and 0.052 rad, incipient cavitations are
of the traveling type in all cases, and in general,
the bubble radius and number of bubbles in the case
of no grid were the largest and the smallest,
respectively, of the three cases, followed by the
case of grid No. 1, which agrees with the size
distributions of cavitation nuclei given previously.
Each bubble is circular when observed perpendicular
to the hydrofoil surface, but as the cavitation
number is reduced, two, in the case of no grid, or
one, in both cases of two shear grids, horn-like
projections are projected behind each bubble from
the downstream or both sides. The groups of plots
lying second from the bottom in Figures 10 (a) (b)
show positions of the upstream tips of the projec—
393
No Grid Grid No.4 Grid No.2
. (a) Oo Oar
No. Grid
FIGURE 12 (a) (b). Cavitation on the hydrofoil of the Clark Y 11.7 profile.
Ww
©
Kd
v/s)
Kd
2.5
Z.25
2.0
FIGURE 12 (c) (d).
No. Grid Grid No.1
(c) @= 0.105 rad
(d) = (Sz) Aol!
Cavitation on the hydrofoil of the Clark Y 11.7 profile.
200mm
Kidii=n2 ai
tions, which seem to be little affected by either
kd or the shear of the free stream. The projections,
in the case of no grid, are supposed to be generated
in cores of trailing vortices and adhere to the
hydrofoil surface, because velocities of the bubbles
exceed those of surrounding water in regions down-
stream from positions of minimum pressure. It can
be seen in high speed motion pictures shown in
Figure 12(f) that the main body of bubbles, having
generated projections, decay, leave behind them
projections of two string-like bubbles, and then
collapse. In cases of the two shear grids, bubbles
are inclined upward toward the high-speed side due
to the secondary flow caused by the flow shears.
Trailing vortices on the low-speed side reach the
hydrofoil surface easier than those on the high-
speed side. Bubbles which generate projections be-
come fewer as kd is reduced in the case of the
shear grids. Several bubbles can be found which
seem to collapse and generate micro jets.
At a = 0.105 rad, cavitations of both types,
traveling and fixed, appear, though the former are
395
FIGURE 12 (e). Behavior of
fixed cavitation on the hydro-
foil of the Clark Y 11.7 pro-
file, a = 0.157 rad, flow up
to down, 12 ms between frames,
2us exposure.
fewer than the latter. Front edges of fixed cavi-
tation zones are round compared with tips of the
above-mentioned projections. At a = 0.157 rad, only
fixed cavitations occur. A cycle of formation of
the break off of a fixed cavity is shown in high
speed motion pictures in Figure 12(e). At IEILASKE p
a clear bubble is generated, like those observed in
our laboratory on the surface of an axisymmetrical
body with a hemispherical nose. The bubble develops
in both streamwise and spanwise directions. The
middle part of the spanwise breadth of the bubble
becomes bubbly, then wavy, and after the development
of the middle part breaks off in pieces of micro-
bubble clouds which are transported downstream al-
though a few remaining small parts grow and
disappear.
08 Profile
At O angle of attack tiny bubbles of traveling
cavitation can be found at a kd a little smaller
396
than kdi, in both cases of no grid and the grid No.
1. Little difference between sizes of the bubbles
can be noticed, although bubbles can hardly be
found on the low-speed side in the case of the grid
No. 1. As cavitation numbers are reduced, however,
the bubbles grow larger and are fewer in the case
of no grid, due to the difference in size of
cavitation nuclei as stated above. Bubbles deforming
to generate projections like those on the Clark Y
11.7 profile can barely be found. Instead, cavities
collapsing to clusters of small bubbles appear.
The discrepancy of the collapse aspect between the
two hydrofoils can be considered to be caused by
the difference of pressure distributions. Cavita-
tions of the above type become more than traveling
bubbles with the decrease of kd, in the case of
the grid No. l.
At a = 0.052 rad, only fixed cavitations occurred
at the leading edge in cases of both no grid and
grid No. 1. The fixed cavitations grown without
changing the front edges of cavitation zones from
the leading edge and develop their lengths slowly
until the kd's are reduced to about the second
Cpmin|'s, being about equal to each other and
existing at the mid-chord in both cases. Nonuniform-
ity of lengths can be found in the case of grid
No. 1. When kd's reach the second |Cpmin|'s, how-
FIGURE 12 (f). Behavior of
traveling cavitation on the
hydrofoil of the Clark Y 11.7
profile, a = 0 rad, flow up to
down, 0.3 ms between frames,
2us exposure.
ever, a remarkable difference in the aspects of
cavitations between the two cases occurs in spite of
only a small difference in the measured pressure dis-
tribution. In the case of no grid, fixed cavitation
develops beyond the position of minimum pressure,
whereas in the case of grid No. 1, the rear edge
of the zone of fixed cavitation does not reach the
position of minimum pressure. Instead, another
cavitation of the traveling type appears around the
position of minimum pressure, and bubbles of the
traveling cavitation are found more on the high-
speed side. The mechanism of this difference can
be surmised as follows: a free shear layer on an
interface between cavity and water may be laminar
near the point of inception in either case, but
the distance necessary for its transition in the
case of no grid is larger than in the case of grid
No. 1 because of the difference of the turbulence
level in the free stream between the two cases,
and the distance necessary for a cavity surface to
reattach the hydrofoil surface might be the same.
The fact that the cavity surfaces in Figure 13(b)
at kd = 0.7 are clear in the case of no grid but
wavy in the other case may show this. Furthermore,
the effect of rolling up the cavity surface caused
by the secondary flow may be expected in shear flow.
At a = 0.105 rad, only fixed cavitations can be
Kd
0.6
0.56
054%
(a) @ = 0 rad
found in both cases. Even in the case of the grid
No. 1, much uniformity of cavitation zones can be
found, although some tail wisps of cavitation can
be found in the case of no grid, e. g., ones
gathered in cores of streamwise vortices.
4. CONCLUDING REMARKS
Conditions and positions of inception, locations of
zones, and the aspect and behavior of bubbles and
cavities of cavitations occurring on two hydrofoils
with the profiles of Clark Y 11.7 and 0g in shear
flows made by shear grids and a uniform flow have
been observed and measured. They have been corre-
lated with measured pressure distributions on the
hydrofoils and the qualities of free streams, i.e.
turbulence levels and size distributions of cavita-
tion nuclei in free streams. The main conclusions
deduced from the results may be summarized as
follows.
At attack angles small for the profile, when
pressure distributions have gradual chordwise
changes, traveling cavitations incept near positions
of minimum pressure and at cavitation numbers about
equal to absolute values of minimum pressure coeffi-
cients, irrespective of flow shears in free streams,
provided local values influenced by flow shears are
-used. Discrepancies between conditions and posi-
tions of inceptions, and pressure coefficients and
their distributions depend on the free stream quali-
ties. The sizes of traveling bubbles depends on the
size distribution of cavitation nuclei.
On the hydrofoil with the Clark Y 11.7 profile,
having a relatively large positive pressure gradient,
a traveling bubble in a zone of rising pressure
deforms, creating a projection in shear flow, or
two projections in uniform flow, leaves only the
397
FIGURE 13 (a). Cavitation on the
hydrofoil of the Og profile.
projection and then collapses. On the hydrofoil
with the 08 profile having gradual pressure gradient,
a traveling bubble collapses after the deformation
caused by the instability of bubble surface. On
both hydrofoils, bubbles collapsing symmetrically
and asymmetrically, looking like micro jets forming
can be found.
At attack angles larger for the profile, when
the pressure distribution declines steeply followed
by a relatively large positive pressure gradient,
fixed cavitations occur. Conditions and positions
of inception are similar to those of traveling
cavitations, although discrepancies of them from
pressure coefficients and their distributions are
less than those of traveling cavitations. In the
boundary layers on both side walls, fixed cavitations
occur at relatively large cavitation numbers,
possibly equal to absolute values of local minimum
pressure coefficients. They develop in both stream-
wise and spanwise directions even far enough beyond
the boundary layers to affect cavitation inceptions
in zones neighboring the boundary layers. Cavita-
tion zones on the low-speed side are larger than
those on the high-speed side. Fixed cavitations
of this kind occur in the boundary layers on both
sides of uniform free streams also.
At attack angles intermediate for the profile,
fixed and traveling cavitations occur at the same
time and tend to become fixed only on the Clark Y
11.7 profile. On the 08 profile, fixed cavitations
at the leading edge and traveling cavitations at
about the mid-chord appear at the same time in shear
flows, but only fixed cavitations occur and develop
at the leading edge in uniform flows. Discrepancies
of conditions and positions of inception from
pressure coefficients and their distributions are
the largest of the three cases mentioned on the
Clark Y 11.7 profile, but about the same as above
mentioned two cases, on the O08 profile.
FIGURE 13 (b) (c). Cavitation on
the hydrofoil of the Og profile.
(c)
Q>= 0.057 rad
a=0.105 rad
ACKNOWLEDGMENT
The authors wish to express their thanks to Mr. S.
Onuma, the technician of Institute of High Speed
Mechanics for his assistance in the experiment.
NOMENCLATURE
Cp: pressure coefficient
|Cpmin | absolute value of minimum pressure
coefficient
fp: number of total occurrences of
cavitation
fy: number of local occurrences of
cavitation at position y
h: width of measuring section
kd: cavitation number
kdi: incipient cavitation number
1: chord length of hydrofoil
n(R,): number of bubbles at radius R,
p: static pressure at hydrofoil surface
or in free stream
Py: static pressure at side wall of
measuring section
Rg: bubble radius
U: local free stream velocity
Uc: velocity at mid span of hydrofoil
installed in measuring section
RMS values of turbulence velocity
components parallel to free stream,
parallel to hydrofoil span and
perpendicular to u' and v', respec-
tively
X, Y; X, y : co-ordinate system fixed in hydrofoil;
the X(x) axis is parallel and the
Y(y) axis is perpendicular to the
chord of the hydrofoil
a attack angle in radian
fe) water density
r chordwise distance from leading edge
of hydrofoil to rear edge of cavita-
tion zone
r
o ? Chordwise distance from leading edge
of hydrofoil to inception point or
front edge of cavitation zone
REFERENCES
Adachi, T., and E. Kato (1973). An experimental
study on the turbulent linear shear flow. J.
Japan Soc. Aeronautical & Space Sci. 21, 573.
Blake, W. K., M. J. Wolpert, and F. E. Gieb (1977).
Cavitation noise and inception as influenced by
boundary-layer development on a hydrofoil. J.
Fluid Mech. 80, 617.
Casey, M. Y. (1974). The inception of attached
cavitation from laminar separation bubbles on
hydrofoils, Proc. Confe. Cavitation held at
Edinburgh, 1.
Daily, J. W. (1944). Force and cavitation charac-
teristics of the NACA 4412 Hydrofoil, Calif.
Inst. of Tech. Hydrody. Lab. ND 19.
399
Daily, J. W. (1949). Cavitation characteristics
and infinite-aspect ratio characteristics of a
hydrofoil Section. Trans. ASME 71, 269.
Daily, J. W., and Johnson (1956). Turbulence and
boundary-layer effects on cavitation inception
from gas nuclei. Trans. ASME, 78, 1695.
Gavrilov, L. €. (1964). On the size distribution
of gas bubbles in water. Soviet Phy.-Acoust.
US, Ake
Harris, V. G., J. A. H. Graham, and S. Corrsin (1977).
Further experiments in nearly homogeneous tur-
bulent shear flow. J. Fluid Mech. 81, 657.
Kermeen, P. W. (1956). Water tunnel tests of NACA
4412 and Walchner profile 7 hydrofoils in non-
cavitating and cavitating flow. Calif. Inst. of
Tech. Hydrodyn. Lab. 47-5.
Kermeen, P. W. (1956). Water tunnel tests of NACA
66-012 hydrofoil in non-cavitating and cavitating
flows. Calif. Inst. of Tech. Hydrodyn. Lab.
47-7.
Kermeen, R. W., and B. R. Parkin (1957). Incipient
cavitation and wake flow behind sharp-edged disks.
Calif. Inst. Tech. Engr. Div. 85-4.
Lighthill, M. G. (1957). Contribution to the theory
of the Pitot-tube displacement effect. J. Fluid
Mech. 2, 493.
Liversy, J. L., and J. T. Turner (1964). The
generation of symmetrical duct velocity profiles
of high uniform shear. J. Fluid Mech. 20, 201.
Macmillan, F. A. (1956). Experiments on pitot tubes
in shear flow. Aero. Res. Counc., London 18235.
Numachi, F., and T. Kurokawa (1939). Uber den
Einfluss des Luftgehaltes auf die Kavitationsent-
stehung am Tragfltigel. Tech. Rep. Tohoku Imp.
Univ. 13, 236. Werft-Reederei-Hafen, Bd.XX.
Numachi, F. (1954). Summary Report on the research
of cavitation phenomena obtained hitherto by our
Institute. Rep. Inst. High Speed Mech. Tohoku
inulin C7 ALES).
Numachi, F. (1975). Effect of turbulence in free
stream on cavitation incipience of hydrofoil.
J. Fluid Engr. Trans. ASME 97, 180.
Numachi, F., M. Nakamura, and I. Chida (1952).
Cavitation tests on hydrofoil profiles of simple
form for blade elements. Memoirs Inst. High
Speed Mech. Tohoku Univ. 6, 113.
Owen, P. R., and H. K. Zienkiewiez (1957). The
production of uniform shear flow in a wind
tunnel. J. Fluid Mech. 2, 521.
Rechardson, J. M. (1947). J. Acoust. Soc. Amer.
ID, G66.
Rouse, H. and A. H. Abdul-Fetouh (1950). Charac-
teristics of irrotational flow through axially
symmetric orifices. J. Applied Mech., Trans.
ASME 17, 421.
Rouse, H. (1953). Cavitation in the mixing zone
of a submerged jet. La Houille Blanche, Janu-
Feb, 9.
Schiebe, F. R. (1969). The influence of gas nuclei
size distribution on transient cavitation near
inception. Univ. Minnesota Proj. Rep. 107.
Young, A. D., and J. N. Maas (1936). Behavior of
Pitot tube in a transverse total-pressure
gradient. Aero. Res. Counc., London, Rep. and
Mem. 1770.
Scale Effects on Propeller
Cavitation Inception
G. Kuiper
Netherlands Ship Model Basin
Wageningen, The Netherlands
ABSTRACT
The boundary layer of four propeller models in
uniform flow is investigated and related with cavita-
tion inception. Laminar separation is found to be
an important phenomenon on model propellers. The
radius where laminar separation starts is found to
be a limit for the radial extent of cavitation.
No inception takes place in regions of laminar flow.
The effect of nuclei in the flow is investigated
using electrolysis. Nuclei seem to be important
for cavitation inception when laminar separation
occurs, but they do not initiate sheet cavitation,
when the boundary layer flow is laminar. When the
boundary layer on the blades is tripped to turbu-
lence by roughness at the leading edge it is shown
that this changes the cavitation by restoring cavita-
tion inception at the vapour pressure. The effect
of electrolysis on cavitation becomes very small
when the propeller blades are roughened. Calcu-
lations of the pressure distribution and the laminar
boundary layer were made and related with test
results.
1. INTRODUCTION
When cavitation patters, observed on full scale
ship propellers, are compared with observations on
model scale, differences are often found [e.g.,
Bindel (1969), Okamoto et al. (1975)]. These
differences are caused by two main factors:
correct scaling of the incoming flow of the
propeller, including propeller-hull interaction,
and incorrect scaling of cavitation.
Considerable efforts have been made to improve
the simulation of the incoming flow by testing the
cavitating propeller model behind the ship model
in a large cavitation tunnel or in a depressurized
towing tank, or by correcting the measured model
wake to simulate the full scale wake in a cavitation
tunnel [Sasajima and Tanaka (1966), Hoekstra (1975)].
in-
400
In this paper the problem of proper scaling of
cavitation will be investigated.
Scaling rules for cavitating propellers can be
formulated using dimensional analysis when the
relevant parameters are known.
This results in the
following well-known dimensionless quantities:
the advance ratio
the cavitation index
the Froude number By
the Reynolds
V =
where
A
h =
Vv =
ee 1
= oD (1)
-p +
ager ogh
Fe (2)
on2D2
2
ans D (3)
g
2
number Re. = Das (4)
N v
advance velocity of the propeller
number of propeller revolutions
propeller diameter
pressure at some reference level
vapour pressure
density of water
acceleration due to gravity
vertical distance from reference level
kinematic viscosity
When these dimensionless parameters are kept the
same for model and prototype,
the cavitation
behaviour of a propeller is independent of size,
provided that no additional parameters play a role
in the cavitation process.
The choice of the cavitation index as a parameter
implies the assumption that inception occurs when
the local pressure is equal to the vapour pressure.
When the inception pressure deviates from the vapour
pressure these deviations are called "Scale effects
on cavitation inception".
Two scaling problems do arise now. First it is
impossible to maintain the Froude number and the
Reynolds number at the same time. The Reynolds
number is abandoned and is lowered on model scale
by a factor of 3/2, where A is the scale ratio.
Even if the Froude number is not maintained it is
practically impossible to obtain the full scale
Reynolds number on model scale. The second scaling
problem is that nuclei play a role in cavitation
inception. Both problems manifest themselves as
scale effects.
Pure water can withstand very high tensions and
nuclei are necessary to generate inception of
cavitation. Nuclei are mostly considered to be gas
pockets in the fluid, possibly trapped in small
crevices of hydrophobic particles. For a review
see Holl (1970). In a cavitation tunnel, however,
the flow will also contain free air bubbles which
come out of solution at the pump, at sharp corners,
or at the cavitating propeller in the test section.
Resorbers are used to bring the free gas back into
solution, or the tunnel can be prepressurized.
When no large nuclei are present, however, scale
effects on cavitation become larger [Hill and
Wislicenus (1961)]. Inception of cavitation becomes
related to the pressure at which the largest gas
bubbles become unstable and start to expand, and
this pressure is lower than the vapour pressure
when the nuclei are small [Daily and Johnson (1956)].
In a towing tank there are very few nuclei since
they will rise to the surface or to go into solution.
Therefore Noordzij (1976) created additional nuclei
in the NSMB Depressurized Towing Tank by electrolysis
and showed the "stabilizing" influence of nuclei on
propeller cavitation behind a ship model. A similar
effect was reached by Albrecht and Bjorheden (1975)
who injected additional nuclei into the water of
their free surface cavitation tunnel after the low
pressure in the test section had deaerated the
water so much that nuclei were no longer formed in
the tunnel.
It is very difficult to control the nuclei content
of the incoming flow [Schiebe (1969)]. When the
nuclei are large enough, the inception pressure
will be close to the vapour pressure. However,
when the nuclei are too large they can lead to
"gaseous cavitation" [Holl (1970)] with inception
above the vapour pressure, or they can be removed
from the region of lowest pressures by the pressure
gradient in the flow, as was theoretically shown by
Johnson and Hsieh (1966).
Variation of the Reynolds number leads to viscous
effects on cavitation inception. Arakeri and Acosta
(1973) and Casey (1974) showed the effect of the
boundary layer on cavitation inception. Laminar
separation was shown to be especially important.
Arakeri and Acosta (1973) visualized the boundary
layer by a schlieren technique and they tentatively
related the cavitation index at inception and the
pressure coefficient at laminar separation or at
transition. Increased pressure fluctuations in
the reattachment region of a laminar separation
401
bubble and in the transition region were measured
by Arakeri (1975) and by Huang and Hannan (1975).
Van der Meulen (1976) also observed the inception
process on headforms by means of holography. He
showed that suppression of laminar separation by
polymers also could suppress cavitation inception.
The relation between the inception pressure and
the pressure at laminar separation or transition
was not always confirmed. In a recent case study
[Kuiper (1978)], it was shown that viscous effects
were responsible for a delay in cavitation inception
on a propeller model. Additional nuclei had no
effect in this case, but it was not yet clear if
nuclei did interact with the boundary layer to
create cavitation inception.
In this study, scale effects on cavitation on
three propellers with different characteristics
were investigated. When a propeller operates in
a wake, scaling problems of the incoming flow and
of cavitation cannot be separated. Therefore the
propellers were tested in uniform axial flow. The
tests were carried out mainly in the Depressurized
Towing Tank. A description of this facility is
given by Kuiper (1974). The advantages of this
tank for the research on scale effects on cavitation
inception are the, supposedly, very low and constant
turbulence level and nuclei content, the uniform
inflow of the propeller, and the absence of wall
effects. Both advance speed and propeller revolu-
tions can be controlled very accurately. The range
of Reynolds numbers which can be tested is lower
than in a cavitation tunnel (maximum carriage speed
is 4 m/sec.) but is not smaller.
The aim of the present study is to gain insight
into the occurrence of scale effects on cavitating
propellers and to develop means to improve the
correlation with full scale observations. Paint
tests were carried out to visualize the boundary
layer flow on the propeller blades. Methods to
calculate the pressure distribution on the blades
are discussed and the calculated pressure distri-
butions are used for the interpretation of the
results of the paint tests and the cavitation
observations. The nuclei content is varied by
using electrolysis, and roughness at the leading
edge of the propeller blades is applied to make the
boundary layer on the blades turbulent, thus simu-
lating a higher Reynolds number. The relation
between the boundary layer on the blades and
cavitation inception is shown and the effect of
leading edge roughness and electrolysis is investi-
gated.
2. TEST PROGRAM
Propellers and Test Conditions
Four propellers were investigated in uniform flow.
Propeller A is the propeller which was investigated
behind a model in a case study by Kuiper (1978).
This propeller showed viscous scale effects on
Cavitation inception but was insensitive for
electrolysis (Figure 1). Behind the model, this
propeller operated in a nozzle. In this study it
was tested without a nozzle.
Propeller B is the propeller which was tested by
Noordzij (1976) behind a model. This propeller
was very strongly influenced by electrolysis.
Without electrolysis the sheet cavitation varied
per revolution, (Figure 2). With electrolysis the
propeller an electrolysis grid was mounted, as
shown in Figure 5. The wires had a diameter of
0.2 mm and a current of 0.2A was used to generate
nuclei. The propeller shaft was at 0.4 meter below
the water level and the lowest wire at 0.5 meter.
Therefore the effect of electrolysis could only
be observed in the upper half of the propeller disk.
-The propeller boundary layer. Two ways of
affecting the boundary layer were used. First,
sandroughness at the leading edge was used to trip
the boundary layer to turbulence. Second, the
9 FULL SCALE OBSERVATIONS
FIGURE 1. Viscous effects on cavitation inception on
propeller A behing the model.
cavitation pattern was present and identical at
every revolution. This "stabilizing" effect of
nuclei is important because it affects the induced
pressure fluctuations on the hull.
Propeller C had a very distinct collapse of the
cavity when the blades left the wake peak, as can WITHOUT ELECTROLYSIS
be seen in Figure 3. This irregular collapse of
the cavity was thought to be caused by viscous
effects and it can also strongly influence the
pressure fluctuations on the hull.
Propeller D was not tested in cavitating con-
ditions. It was used only for boundary layer
visualization. This propeller is an example of a
smaller propeller model used behind models with a
maximum length of 7 meters. This propeller was
made of a copper-nickel-aluminium alloy (CUNIAL).
Propellers A, B and C were of aluminium.
The most important geometrical characteristics
of the four propellers are given in Figure 4. The
complete description, necessary for the calculations,
is given in the Appendix. Most tests were done in
the NSMB Depressurized Towing Tank. To obtain
uniform inflow the propellers were mounted on a
right-angle drive unit, which was kept afloat by a
catamaran-type vessel, as shown in Figure 5. Only
a few comparative tests were done in a cavitation
tunnel.
The following parameters were varied:
-The propeller loading. Two advance ratio's were
used, namely 70% and 40% of the pitch ratio at
r/R=0.7. (Slip ratio's of 30% and 60% respectively).
The slip ratio of 30% corresponds to a loading which
is about normal behind the ship, the slip ratio
of 60% corresponds to an overloaded condition, as
occurs when the blades are in a wake peak. Propeller
A was also investigated at an intermediate loading WITH ELECTROLYSIS
: : of
with a slip of 40% FIGURE 2. Effect of electrolysis on propeller B
-The nuclei content. At 1 meter in front of the behind the model.
FIGURE 3.
C behind the model.
Irregular collapse of cavitation on propeller
propeller Reynolds number was varied with a factor
of about three.
-The cavitation index. Three values of the
cavitation index were used: Oyq=1.5, 2.0, and 2.5.
The reference level of the cavitation index was
always taken at the propeller tip in the top position.
In this paper most cavitation observations will be
shown at Oyp=l.5. At higher revolutions a lower
cavitation index was possible: Oymp=0.5 in the
towing tank and oyp=1.0 in the cavitation tunnel.
Paint Observations
To visualize the character of the boundary layer
at the propeller blades a surface oil flow technique
was used [Maltby, ed. (1962)]. This technique was
adapted for use in water on propellers by Meyne
(1972) and Sasajima (1975). It is particularly
useful on rotating bodies because the difference
in friction coefficient between laminar and turbu-
lent boundary layer flow, in combination with the
centrifugal force acting on the paint, creates a
clear difference in the direction of the paint-
streaks in laminar and turbulent regions.
The paint, used in our paint tests, consisted
of lead-oxide, diluted with linseed oil and coloured
with red "Dayglo" pigment. This mixture produced
a finely detailed pattern of streaks on the metal
surface of the propeller. When the propeller
blades were painted yellow with a thin layer of
zinc-chromate primer, as is done with the cavita-
tion observations to improve contrast and to avoid
reflections, no streaks were formed. Consequently
the flow visualization tests were done with the
propellers not painted.
The viscosity of the paint was controlled by
the amount of linseed oil and was chosen such that
the formation of the pattern took about one full
run in the towing tank. At least 500 revolutions
were always available to form the patter. To
403
1,047
1.0
—07
o5
bo -0.2
5 BLADES
D = 0.3268m
Ag/Ao = 0.820
Co7/D = 0.368
PROPELLER A tic ©) = 0.042
TR
1.0
0.72
053
0.24
TR
4 BLADES
D = 0300m
Ac /Ag = 0.630
Co7/D = 0.430
t/c(07) = 0.022
Ac lAg = 0.824
Co7/D = 0.307
PROPELLER D t/c (07) = 0.050
FIGURE 4. Geometry of propellers.
reach the desired condition took about 100
revolutions, most of them very close to the final
condition. Paint tests were also done in the
cavitation tunnel. The pictures obtained there
were more profuse, especially at high tunnel veloc-
ities, because of the relatively long time it took
to reach a stable condition. For runs longer than
404
ELECTROLYSIS GRID
-CAMERA
a
DETAIL ELECTROLYSIS GRID
~.
0.15 |
0.35m
050m
FIGURE 5. Test equipment for open-water tests.
a few minutes the viscosity of the lead-oxide is
too low and the blades are cleaned by the flow.
The paint is put on the propeller blades at the
leading edge to about 10% of the chord. The layer
must be rather thick to provide enough paint to
cover the whole blade. Some pictures were taken
with UV light using the fluoriscent properties of
the pigment. The bulk of the pictures of the paint
tests was taken in colour photography with natural
light. This gave good colour prints, but unfortu-
nately the contrast in monochrome paper turned out
to be rather poor.
Roughness at the Leading Edge
To trip the boundary layer to turbulence the leading
edge of the propeller blades was covered with
carborundum. The leading edge of the propeller
blade is wetted with watery thin varnish to about
0.5 mm from the leading edge. This is done by
touching the leading edge with a pad wetted with
varnish. The softness of the pad determines the
length of the wetted area from the leading edge.
Then carborundum is put on the wetted area by
spreading the grains on a felt cloth and by wiping
the wetted leading edge with that cloth. Two grains
sizes were used: 30 wm (31-37) and 60 um (53-62).
Microscopic inspection afterwards is necessary. An
example is given in Figure 6.
3. CALCULATION OF PRESSURE DISTRIBUTION
The analysis of boundary layer phenomena and of
cavitation on propeller blades becomes very specu-
lative when the pressure distribution is not known.
No firm experimental verification of calculations
of the pressure distribution is available yet, only
the total thrust and torque give some evidence of
the value of calculations. The calculations are
always potential flow calculations and the effect
of viscosity on the propeller sections cannot yet
be derived with suitable accuracy. Since the
propeller thrust is least sensitive to viscous
effects this quantity gives the most reliable
verification of calculations. When the propeller
geometry and the nominal inflow are known two
approaches are available to obtain the distribution
of propeller loading, viz. the lifting line theory
and the lifting surface theory. Hereafter, both
approaches will be considered with models going
back to the work of Lerbs (1952) for the former
and Sparenberg (1960) for the latter theory.
Lifting Line Calculations
The lifting line theory concentrates the loading of
a propeller section at one point. Using the induc-
tion factor method [Wrench (1957)], a relation
between the hydrodynamic pitch angle, 8;, and the
circulation, I, at each section is found.
B. (i) = ae] ie (a) I (5)
When a given propeller is analysed 8; and I are
unknown. To find them a second relation is necessary,
which is derived from two-dimensional profile
characteristics. The lift coefficient
de"
Cy -( =) (ata,) (6)
where a, is the zero lift angle of the propeller
section. Since the angle of attack a is taken from
oS (BS (2) (7)
Pp i
where 8, is the known geometrical pitch angle, a
second relation is formulated between Cy, (or IT)
and §;, in which dC;/da, is assumed to be known.
When the two-dimensional value for dc; /da, based
on the geometry of the propeller section, is used
the results are rather drastically wrong. This is
caused mainly by the finite length of the propeller
section, which creates a distribution of induced
velocities affecting camber and angle of attack.
1mm
60 Lm 30 Lm
CARBORUNDUM CARBORUNDUM
FIGURE 6. Microscopic picture of leading edge
roughness.
MEASURED 6
Rey = 2.310
CALCULATED L. SURF
—— — — — (AUeVIYATH) (b, (HIN:
No
fo} 01 02 03 0.4 loks) 06 07 08 o9 10
PROPELLER A
MEASURED
Rey = 2.9x10
——-—-— CALCULATED L. SURF.
———— CALCULATED L. LINE
f°) 01 0.2 0.3 04 Os 06 07 08 09 1.0
PROPELLER C
FIGURE 7.
So Eq. 6 has to be corrected to obtain a three-
dimensional lift curve. At one point of the lift
curve, at the ideal angle of attack, results of
systematic lifting surface calculations are avail-
able [Morgan et al. (1968)] and they can be expressed
as correction factors on camber, Ko, and the angle
of attack, Ky. Van Oossanen (1974) used these
correction factors to define the three-dimensional
lift curve over the whole range of angles of attack
instead of at the ideal angle of attack only. He
wrote
fol
(<2) Ry Saue ea
da Zl Gls Ko (8)
3a ‘ (9)
MEASURED
Rey = 1.6210)
————— ——— CALCULATED EL SURE:
Nu) Ka
fo) 02 04 06 08 10 1.2 14 1
PROPELLER B
MEASURED 6
Rey = 15x10
CALCULATED L. SURF.
—— —— CALCULATED LLINE
fe) 0.1 02 03 04 o5 06 O07 08 09 1.0
PROPELLER D
Propeller open-water characteristics.
where a; is the ideal angle of attack of the
propeller section. Substitution of these three-
dimensional values in Eq. 6 makes it possible to
solve the set of Eqs. 5-7, resulting in a radial
distribution of 8, Qos and Cy,.
In Figure 7 the calculated open-water character-
istics using this approach are compared with experi-
ments. The agreement between measurements and
calculations is acceptable. Propeller B could not
be calculated since the regression formula's for
K, and K, in the program were restricted to a
maximum pitch ratio of 1.4.
Viscosity is taken into account by assuming a
viscous lift slope
ac 3
L t
(<2)yan = 0.947-0.76 (£)
where t= max. thickness of propeller section
c= chord length of propeller section
The drag is calculated using the characteristics
of the equivalent profiles of the NSMB B-series
propellers.
Lifting Surface Calculations
The lifting surface theory calculates the induced
velocities over the propeller blades, in chordwise
and radial direction, thus including the effects
of finite aspect ratio of the blades. The draw-
back is that the theory is linearized, which
restricts the validity to lightly loaded propellers.
Van Gent (1977) has shown in his thesis how
heavily loaded propellers can be treated with a
linearized theory since the vorticity in the wake
induces an additional axial velocity component in
the propeller plane, keeping the angles of attack
of the propeller sections small.
The boundary conditions on the propeller blades
are fulfilled at a number of chordwise and spanwise
points. In our calculations four chordwise and
ten radial points per blade were chosen. The pitch
of the vortex sheet in the wake was taken rather
arbitrarily as the pitch at 0.7D.
A very approximate description of the viscous
effects is used. The drag force of the propeller
sections is split into two parts: a drag force
as a result of losses in the suction peak at the
leading edge and a drag force due to friction. The
latter is calculated using a friction coefficient
of 0.0080, irrespective of the Reynolds number.
The first drag force is taken as half the theoretical
suction force. The same correction is also applied
to the sectional lift, which is obtained from chord-
wise integration of the lift distribution. In the
calculation of the induced velocities the geometrical
pitch angle is reduced by 3/4 degree to simulate
viscous effects on the zero lift angle.
The open-water diagrams as calculated with the
lifting surface theory as described by Van Gent
(1977) are shown in Figure 7 together with experi-
mental results and lifting line calculations. The
general agreement with measurements is as good as
the lifting line calculations. This makes clear
that the linearized lifting surface theory can
indeed produce reliable open-water characteristics
up to high propeller loadings. At very low advance
ratio's the calculations deviate from the measure-
ments but this might well be caused by an erroneous
estimate of the viscous effects.
Calculation of the Pressure Distribution
Lifting line as well as lifting surface calculations
give the radial distribution of the lift coefficient,
of the angle of attack, and of the induced camber
(or camber distribution) which can be translated
into a zero lift angle. In Figure 8 these results
are compared for propeller A at 40% slip. The
lifting line calculation gives a higher loading at
the tip and a lower loading at inner radii, compared
with the lifting surface calculation. This is
characteristic for all four propellers in all
conditions. The total thrust does not differ very
much. Large differences, however, are found for
the angle of attack and for the zero lift angle.
LIFT. SURFACE
ET NES
AroT= AincineNce * %o
Q, =ZERO LIFT ANGLE
FIGURE 8. Radial distribution of lift coefficient and
angle of attack on propeller A at 40% slip.
Since these values will be used in the calculation
of the pressure distribution this discrepancy needs
further attention.
The source of the discrepancy is the choice of
Eqs. 8 and 9, used in the lifting line calculation.
The reduction of the slope of the lift curve with
the lifting surface correction factor for the camber,
Ko (Eq. 8), is an empirical one, first suggested
by Lerbs (1951) when he analyzed the lift slopes
of his "equivalent profiles". The physical meaning
of this correction is not clear, but it still can
lead to correct results for thrust and torque,
since the lift slope for the equivalent profiles
was derived using a lifting line theory and experi-
mental values of thrust and torque. Therefore, this
correction for the lift slope, used in combination
with the same lifting line theory, should give
results for thrust and torque not too far from the
experimental results. The definition of the three
dimensional zero lift angle (Eq. 9) is another
empirical relation, bringing the calculated open
water characteristics in line with experiments.
However, this does not necessarily mean that the
three dimensional angle of incidence and zero lift
angle have a physical meaning and can be used for
the calculation of the pressure distribution.
Therefore, the results of the lifting surface cal-
culations are used in the following to calculate
the pressure distribution.
To calculate the pressure distribution on the
blades, the effect of propeller thickness has to
be calculated and the leading edge singularity of
the lift distribution has to be dealt with. Tsakonas
et al. (1976) calculated the pressure distribution
on the propeller blades using a singularity distri-
bution for the thickness, in combination with a
linearized lifting surface theory. These calcula-
tions, however, remain linearized, producing an
infinite velocity at the leading edge, which was
removed by the Lighthill correction for thin air-
foils [Lighthill (1951)]. In our study, three-
dimensional effects on the pressure distribution
are neglected. Interaction effects between thickness
PROPELLER A
PROPELLER C
Tp =0.95
and loading, which occur due to the non-planar
surface of the propeller blades are taken into
account by a correction factor [Morgan et al. (1968)].
This makes it possible to apply conformal mapping
to calculate the pressure distribution. An approx-
imation of the original theory of Theodorsen (1932),
known as Goldstein's third approximation [Goldstein
(1948)] was used. The determination of the "effec-
tive geometry" was done using a camber line, derived
from the calculated induced velocities of the lifting
surface calculation. This can be done because the
problem is linearized. The calculated induced
camberline and the geometrical thickness distribution
were combined in the NACA-manner to obtain the
geometry of the effective profile. The pressure
distribution on the propeller section was then cal-
culated using the induced angle of attack from the
407
PROPELLER B
FIGURE 9. Calculated pressure distribution on the
suction side at 30% slip.
lifting surface calculation. The lift coefficient,
which is found from the lifting surface calculation,
is maintained using the method of Pinkerton (1934).
This is necessary because the potential flow lift
coefficient of the effective profile is slightly
lower at inner radii, where the sections become
thicker. The differences are of the order of 0.02.
In Figures 9 and 10 the calculated pressure
distributions at the suction side are given for
propellers A, B, and C.
4. RESULTS OF PAINT TESTS
In Figure 11 the paint patterns are shown for pro-
pellers A, B, and C at 30% slip and at Reynolds
numbers typical for testing behind 12 meter models.
408
PROPELLER A
PROPELLER C
These pictures were taken with UV-illumination.
At the leading edge the paint is removed, due to
high local velocities. The streaks are formed
gradually, either in a nearly tangential direction
(the turbulent region) or pointed outwards (the
laminar region). The transition from laminar to
turbulent boundary layer flow is shown by a change
in direction of the streaks.
Laminar boundary layer flow occurs in all cases
near the leading edge. Transition in chordwise
direction to turbulent boundary layer flow occurs
gradually, but a transition region can be distin-
guished and at the trailing edge the boundary layer
is turbulent. When the paint streaks are nearly
in the radial direction the flow is separated. At
inner radii the boundary layer if often close to
separation. Laminar separation was clearly present
PROPELLER B
FIGURE 10. Calculated pressure distribution on the
suction side at 60% slip.
on propeller D, as is shown in Figure 12. At 60%
slip the radius where laminar separation is replaced
by natural transition can be seen by the sharp
corner in the paint streaks.
At the suction side near the tip a turbulent
region exists immediately from the leading edge
(Figure 11). An increase in propeller loading
showed a radial increase of the turbulent region at
outer radii, as illustrated in Figure 12. The
change in radial direction of the laminar region
near the leading edge to the turbulent region at
outer radii on the suction side is abrupt and
nearly discontinuous, as sketched in detail in
Figure 13. The laminar region is cut off and the
region of natural transition at inner radii does
not reach the leading edge. We will designate the
radius where this discontinuity occurs, the critical
SUCTION SIDE
PROPELLER C , Rey= 0.66 x 10°
radius of the propeller. Such a critical radius
can also be observed from the paint pattern of
Sasajima (1975) and of Meyne (1972). This critical
radius turned out to be very important for cavita-
tion inception and could be discerned in all cases.
No photographs are shown because of the bad contrast
of the monochrome prints. (Figure 16).
On propeller B at 60% slip a separation bubble
at the leading edge was observed, connected with a
409
PRESSURE SIDE
FIGURE 1l.
30% slip.
Paint patterns at
stagnation region near the tip on the suction side,
which indicated the position of the tip vortex. In
the direction of the hub the laminar separation
bubble extended exactly until the critical radius.
This lead us to the hypothesis that laminar sepa-
ration near the leading edge was the cause of the
discontinuous character of the paint streaks at the
critical radius. To verify the hypothesis of laminar
separation at the critical radius, boundary layer
410
30 °%e SLIP
CRITICAL
RADIUS
LAM. SEPARATION
NEAR MIDCHORD
60 °%. SLIP
Rey = 0.47 x10°
FIGURE 12. Variation of the critical radius with
propeller loading on propeller D (suction side).
calculations were made, using the pressure distri-
butions as calculated in Section 3. The laminar
boundary layer was calculated with Thwaites' method
[Thwaites (1949)]. Laminar separation was predicted
using Curle and Skan's (1957) criteron. This cal-
culation method does not take into account the
delaying effect of rotation on laminar separation,
but since laminar separation occurs very close to
the leading edge the effect of rotation on the
development of the boundary layer will still be
small. The correlation between the calculated and
the observed critical radius is given in Figure 14,
and this correlation is quite good. The critical
radius at all conditions and the variation of the
critical radius between the propeller blades can
FIGURE 13. Discontinuity of paint streaks at the
critical radius.
1.0
" B
Cc
fa) 10)
w
a
=>)
3)
<
wi
=
O5
fe)
CALCULATED
FIGURE 14. Correlation of calculated radius of
laminar separation and measured critical radius.
also be found from Figure 14. As can be seen, the
variation of the critical radius per blade in one
condition can be considerable, showing the sensi-
tivity of laminar separation to the manufacturing
accuracy. The critical radius per blade, however,
reproduced remarkably.
The position of laminar separation is independent
of the Reynolds number. So another check on the
hypothesis of laminar separation at the critical
radius is the independence of the critical radius
from the Reynolds number. Propellers A and C were
therefore tested with about twice the original
number of revolutions. Propeller A was also inves-
tigated in a cavitation tunnel: the highest
Reynolds number in the towing tank was repeated
and another condition with about three times the
original Reynolds number was tested. The paint
tests in the cavitation tunnel were less accurate
since turbulent spots occurred, which caused a
wedge shaped tangential streak through the laminar
pattern. This was strongest at the higher Reynolds
numbers.
Figure 15 gives the critical radius as a function
of Reynolds number for the blades available for
comparison. There is a slight trend for the critical
radius to decrease with increasing Reynolds number,
but this is only very slight. The critical radius
is strongly dependent on the propeller loading and
a slight increase of the propeller loading with
increasing Reynolds number might cause the decrease
of the critical radius. For comparison the obser-
vations of Sasajima are also drawn in Figure 15.
He observes a larger shift of the critical radius
with Reynolds number, but his results from the
tank show no variation with Reynolds number. The
variations found in the cavitation tunnel might
well be caused by variations in propeller loading
or by wall effects. The conclusion seems justified
that the critical radius is independent of the
Reynolds number, at least until natural transition
occurs close to the minimum pressure point. In
that case a critical radius no longer exists.
@ PROPELLER A’ SLIP =0.3 TUNNEL
© " ” ” ” TANK
x ” " n =0.6 '
q " B n =03 "
1.0 © SASAJIMA (1975) TANK
% °
0.9
{e) os 1.0 15 2.0 4G 25 FIGURE 15. Effect of Reynolds number on the
Rey x 10 critical radius.
It is important to note that in Figure 12 at 60%
slip the radius where laminar separation occurs
near midchord is not the critical radius, although
in this case the difference between both is small.
With increasing Reynolds number, however, the region
of laminar separation near midchord will decrease,
while the critical radius will remain unchanged.
The distance between the sharp corner in the paint
streaks of Figure 12b and the critical radius will
therefore increase with increasing Reynolds number.
An increase of Reynolds number causes a shift
in the chordwise position of the transition region
at radii inside the critical radius, as is illus-
trated in Figure 16. This was also observed on
the pressure side. In Figure 17 the chordwise
position of the transition region is given at
r/R=0.7 as a function of the sectional Reynolds
number, which is related to the entrance velocity
and the chordlength of the propeller section at
that radius. The transition region is averaged in
Figure 17. This makes clear that a complete turbu-
lent boundary layer at a radius of 0.7R requires
sectional Reynolds numbers of about 5x106. At the
suction side, turbulent flow at this radius also
occurs when the loading is increased, i.e. the
critical radius is smaller than 0.7.
Empirical criteria for transition of the boundary
layer to turbulence have been given as a relation
between the Reynolds numbers based on the length
from the stagnation point, Re,, and based on the
momentum thickness, Reg. [Michel (1951), Smith
(1956) ]. Van Oossanen used the Smith line
Rey = 0.73 x10"
Rey = 1.56 x 10°
= 1 1740Re, aac
SO ears (10)
as a criterion. When the relation between Reg and
Re, Over the chord was calculated, both on the
suction side and on the pressure side, this relation
was so closely parallel to the criterion of Eq. 10
that no reliable intersection was possible. When
there is a strong negative pressure peak at the
leading edge the relation between Reg and Re, is
such that Eq. 10 always predicts transition very $$ GRIMIENG RASIUS
close to the leading edge. When the pressure £22 777 7~7~— TRANSITION
distribution was nearly shockfree, the prediction
was erroneous. FIGURE 16. Effect of Reynolds number on the transi-
To calculate the transition region, calculation tion region. Propeller A at 30% slip.
On propeller A and on propeller B at 30% slip
the radial extent of the cavitation is clearly
restricted by the observed critical radius. Some—
PRESSURE SIDE times there is a small difference between the
FF SUP =0.6 critical radius and the inception radius, which is
probably caused by a change in the pressure distri-
bution by the cavitation.
The calculated ideal inception radii at 60% slip
should be considered with caustion. They are close
to the hub and the influence of the hub is not
taken into account in the calculations. For example
on propeller B at 60% slip the inception radius is
larger than calculated. In that case the critical
radius is smaller than the inception radius and
does not cause any viscous effects on cavitation.
The distance between the ideal inception radius
and the critical radius on propeller C is small,
so the scale effects due to the critical radius
will be small too.
We can conclude that no cavitation occurred in
regions of laminar flow near the leading edge. The
radial extent of cavitation can be seriously
5x10° 10° 5 x10° restricted by the critical radius. Since the crit-
Re (0.7) ical radius is connected with laminar separation
this means that variation of the Reynolds number
does not remove this restriction until very high
Reynolds numbers. From Figure 17 the sectional
Reynolds number at r/R=0.7 has to exceed 5x106,
whereas a value of 3x105 is mostly considered
enough to avoid Reynolds effects on thrust and torque.
O PROPELLER C- SUCTION SIDE SLIP=0.3
a cr PRESSURE SIDE co BOOKS
O PROPELLER A SUCTION SLIP =0.3
e@
a
FIGURE 17. Chordwise position of natural transition
inside the critical radius.
of the stability of the laminar boundary layer
might give better results [Smith and Camberoni (1956) ].
Since transition occurs far from the leading edge,
the effect of rotation can be important. When the
calculation scheme of Arakeri (1973) is used it is
possible to take the effect of rotation into account
using Meyne's (1972) results. This was beyond the
scope of this paper. j
Variation of Reynolds Number
Propellers A and C were tested at a higher Reynolds
number in the towing tank, while propeller A was
Rey = 0.73x10" Rey =0.51x10° Rey =0.66x10°
5. CAVITATION OBSERVATIONS
The cavitation on propellers A, B, and C is sketched
in Figure 18 for both slip ratio's. The cavitation
index at the blade tip in top position, Oymp (Eq. 2)
was always 1.5. The Reynolds numbers Rey, were
about 5x105. At 30% slip the condition is not far
from inception and a cavitating tip vortex is
present in nearly all cases. However, in some cases
at low Reynolds numbers, propellers A and C were ob-
served without any cavitation. This was not due to
intermittent cavitation during one test, but oc-
curred when tests were repeated with time-intervals
of some weeks. During one test the observations
were quite consistent, indicating that the varia-
tions are caused by factors which are still not
under enough control, e.g., air content, nuclei
content, turbulence.
30% SLIP
Correlation with Paint Test
Of interest is the correlation of the radial extent
of the cavity with the observed critical radius,
found from the paint test. In Figure 18 the
observed position of the critical radius is indicated,
as well as the calculated ideal inception radius, PROP. A PROP. B PROP. C
which is the radius where the minimum pressure on
the blades equals the vapor pressure. Also indicated
is the cavitation, observed when the leading edge
was roughened, as will be discussed in the next
section. FIGURE 18. Cavitation observations at Chom = 1.5.
——— WITH ROUGHNESS
= OBSERVED CRITICAL RADIUS
— CALCULATED IDEAL INCEPTION RADIUS
also tested in the cavitation tunnel at two Reynolds
numbers. No differences in cavitation pattern due
to variation of the Reynolds number were observed
in the towing tank. Notably the radial extent of
the cavity was unchanged, which confirmed that the
critical radius restricted cavitation inception
independent of the Reynolds number. The results
of propeller A at 30% slip are shown in Figure 19.
In this figure the observations of the tests in
the cavitation tunnel are also shown. These show
some differences requiring further attention. The
cavity in the cavitation tunnel at Rey=1.56x106 is
somewhat larger than in the towing tank, but the
difference is not significant and is probably
caused by a slight difference in propeller loading.
(The tunnel condition was taken at a’K,-value
derived from the open water measurements. The flow
velocity was not measured). Remarkable are the
spots of cavitation at Rey=l.56x106 which increased
in number when time increased!
At Rey=2-72*106 there is a sheet outside r/R=0.9,
the same as at Rey=1.56x106. The spots however,
have increased in number and they coalesce at some
distance from the leading edge, forming a cavity
until about r/R=0.8 with isolated spots until r/R=0.7,
which is the ideal inception radius. The increase
of the number of spots with time was not observed
in this situation, but the time to reach a stable
condition was much longer than at lower Reynolds
numbers.
TANK
Rey, =1.36x10°
a. TANK b.
Rey =0.73x10°
TUNNEL
Rey =2.72x10°
c. TUNNEL d.
Rey = 1.56x10°
FIGURE 19. Effect of Reynolds number on propeller A
at 30% slip with On = 1.5.
413
The occurrence of cavitating spots in the laminar
region agrees with the observation of turbulent
streaks in the paint tests in the cavitation tunnel
at higher Reynolds numbers. Therefore, it is
conjectured that, in the tunnel, tiny particles
were deposited on the leading edge of the propeller,
thus creating turbulent streaks. The number of
these streaks may increase with time, and these
turbulent streaks cause spots of cavitation.
Another possible effect is that the propeller
is not hydrodynamically smooth. With increasing
Reynolds number the boundary layer becomes thinner
and more sensitive to local roughness. In this
case the streaks would always be in the same position.
Not enough observations were made to verify this,
but the strongly reduced occurrence of turbulent
spots in the towing tank points to the flow as the
origin of the disturbances. The occurrence of
these streaks was also apparent in the tank when
the pressure was drastically lowered, as is shown
in Figure 20. It is of course very important to
recognize these cavitating spots since they indicate
a region of laminar boundary layer flow anda
possible restriction of the radial extent and the
volume of the cavity.
The effect of Reynolds number on cavitation in
the region from the critical radius to the tip is
small. In nearly all cases cavitation took place
in this region at low Reynolds numbers. In some
cases no cavitation was present in this region at
a low Reynolds number, as shown in Figure 21. A
paint test is included to show the critical radius.
At a higher Reynolds number, cavitation was present
until the critical radius. The ideal inception
radius in this case is at r/R=0.7. A similar effect
was sometimes seen at propeller C and can be
explained by the fact that the reattachment region,
where inception is assumed to occur, shifts to
lower pressure regions with increasing Reynolds
number. Calculations of such an effect are given
by Huang and Peterson (1977). It is not certain,
however, that the Reynolds number is the only
variable since application of electrolysis also
caused inception at low Reynolds numbers. Apparently
the nuclei distribution becomes more critical with
lower Reynolds numbers.
Observations with Oyp = 0.5
Laminar boundary layer flow was seen to prevent
sheet cavitation at the leading edge. To see if
there is some threshold for inception the cavitation
index was drastically lowered to opy=0-5. This
was only possible at high Reynolds numbers. In
Figure 22 propeller A is shown at 30% slip, a
condition comparable with Figure 19b, but at a low
cavitation index. It is clear that even in this
extreme condition no cavitation occurred in the
laminar flow region.
A comparison of the local cavitation index with
the pressure coefficients as given in Figure 9
shows that, e.g., at r/R=0.8, the minimum pressure
coefficient is 0.54 while the cavitation index at
that radius is 0.08 to 0.012, depending on the
position of the blade. The cavitation index at this
radius is lower than the pressure coefficient over
most of the propeller section. When turbulent spots
appeared inside the critical radius these spots
were supercavitating, as is also shown in Figure 20.
Bubble cavitation can be expected near midchord
Ont =1.5
Oyt =0-5
Rey = 1.29 x 10°
FIGURE 20. Turbulent streaks inside the critical
radius at higher Reynolds numbers. Propeller C at
30% slip.
at inner radii, where the minimum pressure exists
near midchord. At propeller A at r/R=0.6 the cavita-
tion index is between 0.13 and 0.20, at Oyp-O-5,
while the minimum pressure coefficient is 0.26.
As can be seen in Figure 22 no bubble cavitation
occurred. The cavitating spot at midchord is a
dent in the propeller surface and illustrates the
low local pressure. Similar observations were made
with propeller C at Oyp=0.5. No threshold for sheet
cavitation could be established and no bubble
cavitation occurred near midchord at inner radii.
Both phenomena are suspected to be caused by a lack
of nuclei. So electrolysis was applied, as will be
discussed in the next section.
6. VARIATION OF NUCLEI CONTENT BY ELECTROLYSIS
Some measurements in the NSMB Depressurized Towing
Tank with the scattered light method indicated
that the nuclei content of this tank was nearly
independent of the pressure. The density of small
nuclei (17 um) was 1.2x107 m7~3 and that of the
largest available nuclei (45 um) was 1.2x10° Wo
[This corresponds with nuclei number densities, as
defined by Gates (1977) of 9x10!! ana 2.4x101°
respectively]. A description of the measuring
technique which was used is given by Keller (1974).
A comparison with similar measurements in the NSMB
large cavitation tunnel [Arndt and Keller (1976) J
shows that the nuclei content is lower than that
in the cavitation tunnel at the lowest air content
by a factor of about 5. The nuclei content in the
cavitation tunnel was very much dependent on the
total air content of the water, showing variations
of a factor of 10 between high (12.5 ppm) and low
(6.3 ppm) air content. This dependency was absent
PAINT OBSERVATION
Rey= 0.73x10°
CAVITATION OBSERVATION
Rey=0.73x 10°
Ont= 1.71
CAVITATION OBSERVATION
Rey = 1.56 x10°
Gyt=2-09
Effect of Reynolds number on_ cavitation
FIGURE 21.
inception outside the critical radius. Propeller A
at 40% slip.
Rey = 1.56 x10°
Guz = 0.5
FIGURE 22. Cavitation at very low cavitation index.
Propeller A at 30% slip.
in the Depressurized Towing Tank. So when cavita-
tion observations in the tank are compared with
observations in the tunnel, we can assume that the
nuclei content in the tunnel is always larger than
that in the tank by at least a factor of 10. Perhaps
most important, however, is that in the tank nuclei
greater than 60 wm are absent.
The nuclei content in the tank has been varied
using electrolysis, as described in Section 2. The
nuclei size distribution from the wires of 0.2 mm
diameter has not been measured. Exploratory
photographic observations showed that the bubbles
coming from the wires are in the range of 50 to
100 um under comparable conditions.
The influence of the wires on the propeller
boundary layer was checked by a paint test on
propeller A at 30% slip. The paint patterns with
and without wires were identical. So we assume that
the turbulence, coming from the wires, did not
affect the propeller boundary layer. This assumption
should be treated with some caution, because Gates
(1977) showed widely different effects of flow
turbulence on two headforms, both with laminar
separation.
Gates also showed that large amounts of nuclei
can influence the boundary layer. Notably the
laminar separation bubble on his hemispherical
headform was removed. To see if this was also the
case in our tests a paint test was carried out with
propeller A at 30% slip. The cavitation index was
just above inception, so cavitation was avoided.
To correct for the higher pressure in this condition
the current through the electrolysis wires was
increased to produce the same volume of gas per
second as in the cavitating condition. No effect
on the paint pattern could be observed. Especially
the critical radius remained unchanged. So we
assume that the nuclei had no disturbing effect on
the boundary layer. As to the effect of electrolysis
415
on the cavitation pattern, three regions on the
suction side of the propeller blades can be
distinguished:
a. At radii larger than the critical radius,
where, at least near the critical radius,
laminar separation takes place.
b. At radii smaller than the critical radius
having a negative pressure peak at the leading
edge.
c. At radii smaller than the critical radius
having a pressure distribution which is
nearly shockfree.
At radii larger than the critical radius no effect
of electrolysis on sheet cavitation could be seen
in those cases where it was present. In the few
cases where no cavitation was present in this
region application of electrolysis restored inception.
An example of absence of cavitation, apparently due
to a lack of nuclei, is shown in Figure 23, where
blade 3 of propeller C at 60% slip showed consider-
able cavitation , while blade 4 was free of sheet
cavitation during the whole run (9 photographs in
3 different blade positions).
Absence of cavitation in regions of laminar
separation, however, is an exception in the steady
case. A possible explanation is that the water is
never completely without nuclei and sooner or later
a nucleus will expand in the separated region and
cause inception. After inception cavitation seems
to be more or less self-sustaining. This agrees
with the observation of Gates (1977) that inception
on a hemispherical body appeared to be insensitive
to freestream nuclei content as long as laminar
separation took place. The situation is different,
however, in the unsteady case, when a blade passes
a wake peak. Only a very restricted time is avail-
able for inception at every propeller revolution
and a high frequency of encounters with nuclei is
necessary to obtain inception at every revolution.
This can explain why the "stabilizing" effect of
electrolysis is more pronounced behind a ship model
than in the open-water tests of the current test
program.
At higher Reynolds numbers absence of cavitation
in regions of laminar separation was not observed.
Apart from viscous effects this can also be caused
by an increase in encounter frequency of nuclei,
since an increase in Reynolds number of the same
propeller models always implied an increase in
propeller revolutions.
At radii smaller than the critical radius elec-
trolysis surprisingly had no effect at all. No
cavitation was initiated in the minimum pressure
peak, although the pressure was far below the vapor
pressure. Even the cavitation pattern at very low
cavitation index, as shown in Figure 22, was
unchanged. It is not clear yet why the nuclei do
not expand. Possibly nuclei do not reach the
minimum pressure region due to a screening effect
as described by Johnson and Hsieh (1966). Ina
situation as shown in Figure 23, however, nuclei
promoted cavitation inception and were not pushed
away. This is only possible when the critical
size of nuclei in a laminar flow region is different
from the critical size in the reattachment region
of a laminar separation bubble.
The third region which has to be considered is
the region where the pressure distribution is
nearly shockfree and has its minimum pressure near
midchord. When the pressure is low in these regions
bubble cavitation can be expected. A situation
416
Rey = 0.66 x 10°
Ont Sikhs
FIGURE 23. Inconsistency of cavitation inception
outside the critical radius at low nuclei content.
Propeller C at 60% slip.
like this is shown in Figure 22, but none or only
a few transient bubbles were seen.
Electrolysis sometimes restores bubble cavitation
in this region, but in many cases it does not.
This inconsistency could even be found on the same
propeller in virtually the same condition when
tested repeatedly with long time intervals. In
one case an abundant amount of large bubbles was
visible without causing bubble cavitation, while
an amount of invisibly small nuclei did cause
bubble cavitation in the same condition. In Figure
19d it was seen that in the cavitation tunnel
cavitating spots at the leading edge were formed at
high Reynolds numbers. When the cavitation index
was lowered, bubble cavitation occurred in the
wake of these spots, while at radii in between of
the spots no bubble cavitation was observed. When
the cavitation index was lowered to about Oyp=0.5
the spots were connected with intense bubble cavita-
tion, as shown in Figure 24. It can be seen that
the bubble cavitation is related to the spots at
the leading edge. Apparently the stream nuclei,
which were abundant in the tunnel at this low
cavitation index, did not create bubble cavitation,
while nuclei, generated by a cavitating spot
created intense bubble cavitation. The possible
relation between pressure distribution, boundary
layer, and nuclei distribution must be studied to
analyse these phenomena.
7. VARIATION OF THE BOUNDARY LAYER BY ROUGHNESS
AT THE LEADING EDGE
In all tests, at least one of the propeller blades
was roughened at the leading edge, as described in
Section 2. With paint tests, it was verified that
the laminar regions were changed into turbulent
ones. Although the grain size of 30 wm and 60 um
is larger in comparison with the boundary layer
thickness, there was a lower limit in the region
which had to be covered with carborundum to cause
turbulent flow. For thin sections an evenly dis-
tributed layer of carborundum of say 0.5% of the
chord was necessary to trip the boundary layer.
There was little difference between the effect of
30 um and 60 um carborundum. At thick sections
to be effective roughness was necessary until about
the minimum pressure point. At the pressure side
the boundary layer remained increasingly laminar
when the loading increased. At 70% slip the
the pressure side of the roughened blades was
completely laminar near the leading edge.
Attention, given until now to the propeller
boundary layer, was focussed on the effect on
torque and thrust. Calculation methods to account
for Reynolds effects on open-water characteristics
are based on the assumption of turbulent boundary
layer flow on the propeller model [Lerbs (1951) ]
or on an empirical value in between fully turbulent
and fully laminar, as compiled by Lindgren (1972).
From the paint tests however, we saw that the
turbulent region at the suction side strongly
depends on the propeller loading. The difference
between the dimensionless thrust and torque coeffi-
cients, therefore, will not only depend on the
Reynolds number, but also on the propeller loading.
In order to eliminate the dependency of thrust
and torque coefficients on the Reynolds number,
turbulence stimulators have been used. Sasajima
(1975) used studs, Yasaki and Tsuda (1972) and
Tsuda et al. (1977) used trip wires at some distance
from the leading edge. Apart from changing the
boundary layer, these devices also have considerable
resistance of their own. Effects both on thrust
and torque are difficult to separate. The influence
of roughness at the leading edge on thrust and
6
Rey = 2.72 x10
Oyy = 0.5
FIGURE 24. Bubble cavitation in the wake of spots at
the leading edge. Propeller A at 30% slip in the cavi-
tation tunnel.
{e) 0.2 0.4 06 08 1.0 1.2 1.4
ADVANCE COEFFICIENT J
FIGURE 25. Effect of leading-edge roughness on torque
and thrust coefficients.
torque coefficients is given in Figure 25. These
measurements were carried out with a special dyna-
mometer inside the propeller hub to assure that the
differences were not insignificant due to inaccuracy
of the measurements. The accuracy in Figure 25 is
still only about + 0.005.
Using Lindgren (1972), the value of AK, between
fully turbulent and fully laminar boundary layer
flow on the propeller is 0.0035. The actual
influence of the roughness at the leading edge is
smaller, so that we can conclude that the resistance
due to the carborundum was very small. An analysis
of the effect of roughness at the leading edge on
the performance of the propeller is beyond the
scope of this paper.
The effect of leading edge roughness on cavitation
is sketched in Figure 18. The radial extent of the
cavitation is increased in those cases where the
critical radius was a limit for cavitation. The
risk of scale effects on cavitation inception due
to laminar boundary layer flow is largest at low
propeller loadings, when the risk of laminar
separation is smallest. But it still can be
considerable at high loadings, as is shown in Figure
26, where propeller A is shown with and without
roughness at 60% slip.
Application of roughness at the leading edge is
expected to cause two problems. First the geometry
of the leading edge may be altered, having a pro-
found influence on the minimum pressure peak.
Secondly, the local inception index may be changed
due to roughness. The effect on the shape of the
leading edge can only be minimized by using small
grain sizes. However, to obtain a turbulent boundary
layer the current 30 wm grainsize was about the
minimum and no differences in cavitation behavior
were observed between blades roughened with 30 um
and 60 wm roughness. The effect of surface irreg-
ularities on cavitation inception can be large, as
was shown by Holl (1965). Moreover, Holl points
out that "the most disastrous place to locate
surface roughness is at the point of minimum
pressure of a parent body". This is exactly what
cannot be avoided at the rather sharp leading edge
of thin propeller sections. The situation very
close to the leading edge, however, is different
from the situation of an isolated roughness at a
417
surface, as studied, e.g., by Holl (1965) and
Benson (1966). Application of their results is
also difficult, because the ratio between grainsize
and boundary layer thickness without roughness,
which is required for the calculations, varies
rapidly in this region. The boundary layer thickness
on the smooth blades near the end of the roughness
was about 30 um in all conditions, when no separation
took place. At the position of laminar separation
the boundary layer thickness was only a few um.
Thus, the ratio of grainsize to boundary layer
thickness easily varies by a factor of ten. Appli-
cation of inception calculations on distributed
roughness [e.g., Arndt and Ippen (1968)] seems
more appropriate, but this is difficult, because a
friction coefficient is required for the calculations,
as well as an "equivalent sandroughness". Both
are strongly interrelated [Bohn (1972) ] and espe-
cially near the leading edge these quantities are
difficult to estimate.
The roughness elements do form a massive distur-
bance of the boundary layer and an increase in the
ES
SMOOTH
60 [Lm CARBORUNDUM
6
Rey = 0.73 x10
Ont = 1-5
FIGURE 26. Effect of leading edge roughness on cavi-
tation. Propeller A at 60% slip.
Rey = 0.73 x 10°
6
Rey =1.36 x 10
60 [Lm CARBORUNDUM
Ont = 15
FIGURE 27. Effect of Reynolds number on spot cavita-
tion at roughness elements. Propeller A at 30% slip.
cavitation inception pressure to a value greater
than the vapor pressure is possible, which would
create additional scale effects on cavitation
inception. To estimate the importance of a possible
increase in cavitation inception index, the ideal
inception radius is also given in Figure 18. This
is the radius where cavitation should start when
the calculations of the pressure distribution were
correct and when no scale effects would occur. As
a
SMOOTH
60 [Lm CARBORUNDUM
0.73 x10®
Gur = 2-5
a
o
z
u
FIGURE 28. Effect of roughness near inception. Pro-
peller A at 30% slip.
can be seen, no cavitation occurs inside the ideal
inception radius, indicating that the pressure at
inception with roughness is not far from the vapor
pressure. Assuming that full scale inception takes
place near the minimum pressure point at the vapor
pressure [oj=-Cp (minimum)], application of sand-
roughness can effectively simulate this situation
at much lower Reynolds numbers. Further experiments
are necessary to find out the precise effect of
leading edge roughness on the flow and on the
boundary layer. Holographic methods, as applied by
van der Meulen (1976) in studying the effects of
polymers can be attractive for these experiments.
When the effect of roughness at the leading edge
is studied three regions on the model propeller can
again be distinguished. At radii larger than the
critical radius, where inception on the smooth
blades takes place due to laminar separation, the
cavitation behavior is unaffected by roughness.
Cavitation was always present on the roughened
blades. It is unknown if the sensitivity to nuclei
in the unsteady case increases, as is suspected on
the smooth blade. Experiences with several other
propellers behind a model indicate that this is
not the case and that nuclei have very little effect
when roughness is applied.
In the laminar region, at radii smaller than the
critical radius, roughness at the leading edge has
its major effect, as described above. In some
cases, however, problems appeared in the form of
streaky cavitation as shown in Figure 27a. When
the pressure on the blade sections was constant,
as was the case for propeller A at r/R=0.7 and for
propeller C at r/R=0.8, both at 30% slip (Figure 9),
and when the Reynolds number was low, cavitating
streaks were formed behind the roughness elements.
In Figure 27b the same blade in the same condition
at a higher Reynolds number is shown. Here a smooth
cavity is seen. The roughness elements apparently
suffer from laminar separation at low Reynolds
number and cavitation occurs in the separated regions
behind the roughness. The length of the spots is
strongly dependent on the cavitation index, as is
shown in Figure 28b, where the same situation as
in Figure 27a is shown at a somewhat higher cavita-
tion index. The spots disappeared and the propeller
is near inception. Figure 28 also shows that in-
ception of the sheet at the leading edge is not far
from the vapor pressure, because the ideal inception
radius in this case was 0.78. When roughness was
applied, electrolysis had no further effect at
radii smaller than the critical radius.
In the region with shockfree nressure distribution,
bubble cavitation was seen to be promoted in some
cases by roughness at the leading edge. The influ-
ence of roughness, however, was inconsistent again
SMOOTH 60 [Lm CARBORUNDUM
Rey = 2.72x10°
Gut = 1.0
“FIGURE 29. Effect of leading edge roughness on bubble
cavitation. Propeller A at 30% slip in the cavitation
tunnel.
419
in this region, as it was with electrolysis. When
there was cavitation at the leading edge due to
the roughness, again bubble cavitation appeared at
midchord, as is illustrated in Figure 29, where
nuclei generated by cavitation at the leading edge
created bubble cavitation at midchord. The cavita-
tion index at 0.7R in Figure 29 is 0.18 and the
minimum pressure coefficient from Figure 9 is 0.20,
so the situation with roughness seems to be the
situation without scale effects on cavitation
inception. Nuclei in the flow, however, did not
create bubble cavitation.
8.
CONCLUSIONS
The results of the present test program can be
summarized as follows:
ib5
On the suction side of a model propeller a
critical radius can exist outside of which
the boundary layer is turbulent from the
leading edge. This critical radius is due
to laminar separation, as was seen from some
observations, from calculations (Figure 14),
and from the Reynolds independency of the
critical radius. (Figure 15).
To obtain natural transition near the leading
edge on a propeller model, high Reynolds
numbers (Reyy>2 510°) are required.
The critical radius is a limit for the radial
extent of sheet cavitation from the leading
edge. An increase of nuclei by electrolysis
is ineffective in the laminar region (Figure 22).
Outside the critical radius, cavitation is not
inhibited (the inception pressure was not
accurately determined), but a lack of nuclei
at low Reynolds number seems to decrease the
frequency of inception (Figure 23). In the
unsteady case the nuclei content of the water
is probably important in this region.
Roughness at the leading edge can effectively
remove the critical radius, thus simulating a
higher Reynolds number. Inception of cavitation
at the roughness elements occurs close to the
vapor pressure, which is assumed to be also
the case on the prototype.
When the pressure distribution is very flat
and the Reynolds number is low, the roughness
elements can induce spots of cavitation. The
length of these spots is strongly dependent on
the cavitation index and is different from the
cavity length at high Reynolds numbers. This
is probably due to laminar separation at the
roughness elements (Figure 27).
The inception of bubble cavitation near mid-
chord at inner radii is not consistent. There
seems to be an interaction between the pressure
distribution, the nuclei distribution, and
even the boundary layer. When cavitation at
the leading edge is present, bubble cavitation
occurs near midchord when the pressure is below
or near the vapor pressure in that region.
Lifting line and lifting surface calculations
can adequately predict the open-water character—
istics of a propeller. For the calculation of
the pressure distribution, however, lifting
surface calculations are necessary. The corre-
lation between calculations and the results of
paint tests and cavitation observations is good.
420
From the previous investigations in uniform flow
some tentative explanations can be given of the
scale effects on cavitation as shown in Figures
1-3. The explanations can only be tentative since
the unsteady pressure distribution on the propellers
in the wake is not known. Propeller A in Figure 1
apparently had a critical radius at r/R=0.9 in
this blade position, which was removed by roughness
at the leading edge. Also, behind the model in
some situations no cavitation at all occurred in
the wake peak, which is expected to be due to a
lack of nuclei (as seen in Figure 21).
The lack of nuclei is more apparent at propeller
B. The critical radius is expected to be near the
hub, but the low encounter frequency with nuclei
of sufficient size makes cavitation inception more
or less random. The irregular collapse of the
cavity on propeller C is apparently due to a strong
change in the pressure distribution, due to a sharp
wake peak. The critical radius at the position of
Figure 3 is near r/R=0.9 but the cavity at inner
radii is still collapsing. This phenomenon could
also be seen on high speed films, where the sheet
cavity was seen to detach from the leading edge
and collapse while moving with the flow. Some
cavitating spots can be seen at r/R=0.8 on propeller
(Cr
ACKNOWLEDGMENT
Part of this program was supported by the Dutch
Ministry of Economic Affairs under the ICOSTE-
program.
REFERENCES
Acosta, A. J., and B. R. Parkin (1975). Cavitation
inception - a selective review. J. of Ship
Research, 19, 193.
Albrecht, K., and O. Bjorheden (1975). Cavitation
testing of propellers in a free surface tunnel
utilizing micro air bubble control. J. of Fluids
IDK BASIC
Arakeri, V. H. (1973). Viscous effects in inception
and development of cavitation on axi-symmetric
bodies. Thesis, Calif. Inst. of Techn. Report
No. Eng. 183-1.
Arakeri, V. H., and A. J. Acosta (1973). Viscous
effects in the inception of cavitation on axi-
symmetric bodies. J. of Fluids Eng., 519.
Arakeri, V. H. (1975). A note on the transition
observations on an axi-symmetric body and some
related fluctuating wall pressure measurements.
Trans A~SMsk., 82
Arndt, R. E. A., and A. T. Ippen (1968). Rough
surface effects on cavitation inception.
AJS.M.E., J. Of Basic’ Eng., 249)
Arndt, R. E. A., and A. P. Keller (1976). Free
gas content effects on cavitation inception and
noise in a free shear flow. Proc. of IAHR-
Symposium, Paris, France, 3.
Benson, B. W. (1966). Cavitation inception on
three dimensional roughness elements. David
Taylor Model Basin, Rep. 2104.
Bindel, S. G. (1969). Comparison between model and
ship cavitation. 12th ITIC, Cavitation Committee,
App. III, 365.
Bohn, J. C. (1972). The influence of surface
irregularities on cavitation: a collation and
analysis of new and existing data with application
to design problems. Thesis. Penn State Univ.,
Ordnance Res. Lab., Tech. Mem., File No. 72-223.
Casey, M. V. (1974). The inception of attached
cavitation from laminar separation bubbles on
hydrofoils. Inst. of Mech. Eng., Conference on
cavitation. Edinburgh, Scotland.
Curle, S. C., and S. W. Skan (1957). Approximate
methods for predicting separation properties of
laminar boundary layers. Aeron, Quart, 8, 257.
Daily, J. W., and V. E. Johnson Jr. (1956). Tur-
bulence and boundary layer effect on cavitation
inception from gas nuclei. Trans. A.S.M.E.,
78, 1965.
Gates, E. M. (1977). The influence of freestream
turbulence, freestream nuclei populations and
a drag-reducing polymer on cavitation inception
on two axi-symmetric bodies. Calif. Inst. of
Techn. Report No. Eng. 183-2.
Gent, W. van (1977). On the use of lifting surface
theory for moderately and heavily loaded ship
propellers. Thesis, Netherlands Ship Model
Basin, Publ. No. 536.
Goldstein, S. (1948). Low-drag and suction airfoils.
J. of the Aeron Sciences, 15.
Hoekstra, M. (1975). Prediction of full scale
wake characteristics based on model wake survey.
Inst. Shipbuilding Progress, 22.
Holl, J. W., and G. F. Wislicenus (1961). Scale
effects on cavitation. A.S.M.E. J. of Basic
imneteis pn Chip Sec
Holl, J. W. (1965). The estimation of the effect
of surface irregularities on the inception of
cavitation. A.S.M.E. Symp. on Cavitation in
Fluid Machinery, G. M. Wood et al. (eds), 3.
Holl, J. W. (1970). Nuclei and cavitation. Trans.
WNASctleltan Co. Cis EACHIC Iino, SeiL.
Huang, T. T., and D. E. Hannan (1975). Pressure
fluctuations in the regions of flow transition.
David Taylor Model Basin, Report No. 4728.
Huang, T. T., and F. B. Peterson (1976). Viscous
effects on model/full scale cavitation scaling.
Wig: (ere Riehl INAS6 5 BOR 2do
Johnson, V. E., and T. Hsieh (1966). The influence
of trajectories of gas nuclei on cavitation
inception. Proc. 6th Symp. on Naval Hydrody-
eee, MEG to Ie Cap To
Keller, A. P. (1974). Investigations concerning
scale effects of the inception of cavitation.
Proc. Conference on Cavitation, Inst. of Mech.
Eng., Edinburgh, Scotland.
Kuiper, G. (1974). Cavitation testing of marine
propellers in the NSMB Depressurized Towing
Tank. Proc. Conference on Cavitation, Inst. of
Mech. Eng., Edinburgh, Scotland.
Kuiper, G. (1978). Cavitation scale effects, A
case study. Int. Shipbuilding Progress, 25.
Lerbs, H. W. (1951). On the effects of scale and
roughness on free running propellers. J. of
the Am. Soc. of Naval Arch., 63, 58.
Lerbs, H. W. (1952). Moderately loaded propellers
with a finite number of blades and an arbitrary
distribution of circulation. Trans. SNAME, 60.
Lighthill, M. J. (1951). A new approach to thin
aerofoil theory. Aeron. Quart, 3.
Lindgren, H. (1972). Ship model correlation method
based on theoretical considerations. Proc. 13th
ITTC, Report of Performance Committee, App. 2,
181.
Maltby, R. L., ed. (1962). Flow visualization in
wind tunnels using indicators. AGARDograph 70.
Meulen, J. H. J. van der, (1976). A holographic
study of cavitation on axi-symmetric bodies
and the influence of polymer additives. Thesis,
Netherlands Ship Model Basin, Publ. No. 509.
Meyne, K. (1972). Untersuchung der Propellergrenz-
schichtstrémung und der Einfluss der Reibung
auf die Propellerkenngréssen. Jahrbuch der
Schiffbautechnischen Gesellschaft 66, 317.
Michel, R. (1951). Etude de la transition sur les
profils d'ailestablissement d'um point de tran-
sition et calcul de la trainee de profil incom-
pressible. ONERA report 1/1578A.
Morgan, Wm. B., V. Silovic, and S. B. Denny (1968).
Propeller lifting surface corrections. Trans.
SNAME, 76.
Noordzij, L. (1976). Some experiments on cavita-
tion inception with propellers in the NSMB
Depressurized Towing Tank. Intern. Shipbuilding
Progress, 23.
Okamoto, H., K. Okada, Y. Saito, and T. Takahei
(1975). Cavitation study of ducted propellers
on large ships. Trans. SNAME, 83.
Oossanen, P. van (1974).
and cavitation characteristics of propellers
including effects of non-uniform flow and
viscosity. Thesis, Neth. Ship Model Basin,
Publ. No. 457.
Pinkerton, R. M. (1934). Calculated and measured
pressure distributions over the midspan section
of the NACA 4412 airfoil. NACA Rep. No. 569.
Sasajima, H., and I. Tanaka (1966). On the
estimation of wake of ships. Proc. llth ITTC,
Tokyo.
Sasajima, T. (1975). A study on the propeller
surface flow in open and behind conditions.
ieareier, Iiekolag Tne, Sipe Tale
Schiebe, F. R. (1969). The influence of gas nuclei
size distribution on transient cavitation near
inception. Univ. of Minnesota, St. Anth. Falls
Hydr. Lab., Proj. Report No. 107.
Smith, A. M. O., and Nathalie Gamberoni (1956).
Transition, pressure gradient and stability
theory. Douglas Aircraft Co. Rep. 26388.
Smith, A. M. O. (1957). Transition, pressure
gradient and stability theory. Ix Congres
International de Mechanique Appliquée, IV.
Bruxelles, Belgium.
Sparenberg, J. A. (1962).
surface theory to ship screws.
Acad. of Sciences. Series B, 5.
Application of lifting
Royal Netherlands
Calculation of performance
421
Theodorsen, Th. (1932). Theory of wing sections of
arbitrary shape. NACA Report No. 411.
Thwaites, B. (1949). Approximate calculation of
the laminar boundary layer. Aeron Quart, 1,
245.
Tsakonas, S., W. R. Jacobs, and M. R. Ali (1976).
Propeller blade pressure distribution due to
loading and thickness effects. Stevens Inst. of
Techn., Report S.T.T.-D.C.-76-1869.
Tsuda, T., S. Konishi, and S. Watanabe (1977). On
the application of the low pitch and high
revolution propeller to the self propulsion test.
ITTC Performance Committee.
Wrench, J. W. (1957). The calculation of propeller
induction factors. David Taylor Model Basin,
Rep. No. 1116.
APPENDIX
The geometry of the four propellers, used in this
study and shown in Figure 4, is given in this
appendix. The output is from a propeller data
base and is not dimensionless but in mm on model
scale. Propellers A and C were stored in the data
base on a different model scale than actually used
in the tests, but this has no further impact. Cal-
culations were made directly from this data-base.
At each radius, R, the pitch, P, is given,
together with the distance to the generator line of
the trailing edge, TE, the leading edge, LE, and
the position of maximum thickness, TM. The positive
direction is from the generator line to the leading
edge.
The geometry of the propeller section is given
by the thickness and the distance of the face of
the propeller section above the pitch line. The
ordinates of the section geometry are given as
percentages of the distance between point of max-
imum thickness and leading edge (positive) or
trailing edge (negative). The origin therefore
always is at the point of maximum thickness of the
profile.
The profile thickness at leading and trailing
edge is finite in this appendix. The radii at the
leading edge were determined by generating a spline
through the profile contour or by interpolating in
the transformed plane after conformal mapping.
Both interpolating techniques gave nearly the same
results and were very close to the actual propeller
geometries.
tv 3a16vL
‘WW NI SSSNMOIHL 310d
(WW 8'>Oocge SVM WVe9Oed LS3L
SHL NI GASN TAGOW 3HL AO YSALAWVIO YsATTsdOud SHL AN)
‘WWW OF BZL = YSALAWVIG
(WG8ZlO0 340 YSaLaWVIG V IV) VY Y¥3711ed0ud
422
‘eV Savi
OL-
os-
WwW NI
O6- S6- OOL- Ade
SSSNWDIHL 3Ws0¥8d
WW €’SlLd = YALAWVIC
‘G wamadOud
423
EV alagvl
Ede be Shu eves GZo°
Sp ve 6c° ve 2° ve ae jon O
BOR GPCR Y= YR SEO et BOO = po ORR As? “oa 2 a50 * jx59 ec fee sue Bape nae.
2u* 20%= ga* a DI oemocn 6e* Zp vs° BO) eee 00: OS a OG Uy Vee ence reyes 8Gg°
20 pa°= pd I 63 2¢ cp 2s a9 177 94 tio eGGee SOG. Che hee CU hoe oAg?
BUF RUE OO NU Ge REGS ODE GRR PR YP RP YG GaP RD ao a0 eoz°
COrmaeec 6208 Gere z= atte aa%e ge° gt* “2° 2c? tee ze° gue ir BBR PO Garo ang?
68 poe t9 8 49 = 99° £9 © bo'e F9°® gG°= BG°= Op°e Gr°e 6p°= pGr= form Byte Zore Ho! aes*
ty’ p6°s ac*t= za°t= curze cerze go®ze od°2= getz= Z6°z™ Cotze deize CL°z= C9%e= Co®ze CEoze 92°20 tye qe=
OOlL- G/k2
‘WW NI SSSANHOIHL 31sOu’d
“(WU O'OOE SVM WVe9OUd LS3L
SHL NI GaSN TAaGOW AHL 4O YSLAWVIC YSaTTAdOud SHL GN)
WW OO'Ore = YALAWVIG
(W720 JO YSLEWVIG V IV) “D3 YAaTIadOdd
424
eeceeeeee
FOMNN-— —
14
Gy
Te
ra
26
iw4
96
iw
‘yV Alavl
os-
WW AI
O6- S6- OOL- GA/e2
SSSNYDIHL 31sdOu’d
WwW O'?lLe = YSALAWVIG
‘dG = Y¥371ad0ud
425
426
Discussion
SHIN TAMIYA, HIROHARU KATO, and YOSHIAKI KODAMA
SETTLING SECTION —-NUCLE! GENERATION SECTION
TEST SECTION (80x80x10G0)
4
=F
EED TANK
— FILTER TANK “ORIFICE
FIGURE 1. General arrangement.
The discussers appreciate the excellent re-
search work on cavitation inception done at NSMB*.
At the University of Tokyo the discussers also per-
formed similar experiments using both hemispherical
and ITTC headforms tested in our newly built cav-
itation tunnel. This tunnel has a filtering tank
containing 60 cartridge type filters, which contin-
uously remove air nuclei and solid particles larger
than ca. 1 um from water (Figure 1). |.
Figures 2 through 7 show the effect of elec-
trolysis on cavitation inception. The photograph
in Figure 3 was taken a few seconds after that in
Figure 2. The flow conditions are exactly the same
for Figures 2 and 3; the only difference is the
presence of hydrogen bubble nuclei. The photographs
in Figures 4 and 5, as well as 6 and 7, were also
taken under the same conditions.
In the discussers' experiments the cavitation
caused by electrolysis nuclei generates only bubble PICUREV Qe muitehoutl hydrogenabubblles\vasichsunver
type cavitation. Even when sheet cavitation exists, 3 ='Osehh.
the cavitation bubbles caused by the electrolysis
nuclei seem to break up the sheet cavity.
*Netherlands Ship Model Basin FIGURE 3. With hydrogen bubbles V = 6.8 m/s,
#Statens Skeppsprovningsanstalt Ci = Machi,
FIGURE 4. Without hydrogen bubbles V = 6.8 m/s,
o = 0.71.
FIGURE 5. With hydrogen bubbles V = 6.8 m/s,
o=0.71.
FIGURE 6. Without hydrogen bubbles V = 6.8 m/s,
o = 0.60.
O. RUTGERSSON
I would like to congratulate the author of
this interesting paper. As a complement to the data
presented I think that some results obtained at
SSPA# when testing high-speed propellers could be
of some interest. A propeller of the supercavi-
tating type was tested with three different gases
in the water. Also, two different conditions of
the blade surface were used, smooth polished and
painted with a thin spray paint giving the surface
some roughness.
In Figure 1 the propeller characteristics from
these tests for homogenous flow at the cavitation
number, 0 = 0.6, are shown. In the partially cav-
itating region (J > 1.0) there is a very pronounced
influence due to gas content for the polished pro-
peller. For the painted propeller no such influ-
FIGURE 7. With hydrogen bubbles V = 6.8 m/s,
o = 0.60.
ence was found. Cavitation pictures at the advance
ratio, J = 1.1, give the explanation for these
differences. Figure 2 shows the cavitation at the
lowest gas content (a/a, = 0.2) for the polished
propeller. The cavitation pattern is divided into
two parts. The first part is a sheet starting at
the leading edge. The second part is an unstable
sheet of bubble cavitation at the aft part of the
blade. Tests at higher gas contents (Figure 3,
a/a, = 0.4) show that the aft part cavitation now
has a larger extension. The painted propeller
(Figure 4) shows a rather different pattern for the
aft part cavitation (the leading edge sheet is al-
most uninfluenced by gas content and roughness) .
The aft part cavitation now consists of a thin sheet
of very small bubbles. The sheet also has a rela-
428
tively larger extension on the painted propeller
than on the polished propeller. Obviously it is
the changes in this aftpart cavitation that cause
the changes in propeller characteristics.
Full scale tests have also been conducted with
this propeller design. In Figure 5 the full scale
cavitation pattern corresponding to the model tests
is shown. This cavitation pattern is very similar
to that of the painted model propeller.
In the author's Figure 29 bubble cavitation
is shown very similar to that in tests with the
painted propeller at SSPA. The author concludes
08 a/Os
0.4 Painted
Ky —-—-— 0.2 Not painted No
0.4 Not painted —
10Kq ———— 0.6 Not painted
n
° 06
0.4
0.2
all
0.6 0.8 - 1.0 1.2 AIS
ADVANCE RATIO
FIGURE 1. Propeller characteristics at 6 - uU.6.
FIGURE 2. J = 1.1 a/a = 0.2 polished blade.*
that this cavitation is inconsistent. Based on our
experience with full scale cavitation, however, I
think that the pattern shown could very well repre-
sent a full-scale case.
The influence of nuclei content and blade
roughness on the cavitation pattern is found to be
rather similar in the tests at NSMB and SSPA. The
main difference is the necessary amount of rough-
ness. This difference is possibly due to the dif-
ference in Reynolds number, about 10 times as high
in the tests at SSPA as in those carried out at
NSMB.
FIGURE 3. J = 1.1 a/a = 0.4 polished blade.®
FIGURE 4. J =1.1a/a = 0.4 painted blade.
FIGURE 5. J = 1.1 6 = 0.65 full scale.
429
Author’s Reply
G. KUIPER
Both the hemisphere and the ITTC body are
known to exhibit laminar separation in the zenge
of Reynolds numbers (estimated at about 2 x 10°)
used in the experiments of Tamiya et al. as was
already shown by Arakeri and Acosta (1973). They
now point to an apparent discrepancy between the
results as described in my paper and their obser-
vations: on the propellers nuclei were found to
generate sheet cavitation in the very few cases
where it was not yet present, and the nuclei never
changed the appearance of the cavity.
First of all, the cavitation patterns, both
with and without electrolysis, on the headforms of
the discussers show a remarkable resemblance to
various patterns shown on the ITTC bodies at other
facilities [Lindgren and Johnsson (1966) and also
reproduced by Gates and Acosta in their paper on
this program] illustrating that the nuclei content
was at least one of the factors causing the varia-
tion in type of cavitation observed at different
facilities.
From the observations of the discussers it can
be concluded that the nuclei, generated by elec-
trolysis, removed the laminar separation bubble in
the same manner as shown very clearly by Gates and
Acosta in their symposium paper. This phenomenon
was found when there were many large free stream
bubbles in the flow, as can also be observed in the
pictures of the discussers. In our case, however,
we verified with a paint test that electrolysis did
not remove the laminar separation bubble by veri-
fying that the critical radius was unchanged.
The observations of the discussers show that
an overdose of nuclei can change the situation
considerably. Gates and Acosta assume that the
free stream bubbles do trip the boundary layer.
Another possibility, however, is that the dynamic
behavior of the bubbles near the minimum pressure
region changes the pressure distribution on the
body, specifically by decreasing the low pressure
peak. This would also remove laminar separation,
leaving the boundary layer laminar over a longer
distance. In fact the nuclei do not only affect
cavitation inception but they change the free
stream conditions, making a correct comparison of
the inception phenomena impossible.
Rutgersson, in his discussion, gives an illus-
tration of a possible effect of nuclei and roughness
on bubble cavitation. With the pictures alone,
only some assumptions can be made as to what hap-
pened on this propeller, but I will make an attempt
to give an explanation.
Although the Reynolds number was rather high
it looks like the boundary layer within r/R = 0.8
is laminar over a large portion of the chord, while
the minimum pressure region is near midchord
(Figure 2). An increase of the nuclei content
leads to occasional cavitating spots, starting at
the low pressure region (Figure 3). On the painted
blade, however, the boundary layer seems to be
turbulent and bubble cavitation starts there, near
the minimum pressure region (Figure 4). :
If my tentative description is correct there
is a difference between the discussers' case and
Figure 29 (and also Figure 24) from my paper, since
there the boundary layer in the region of low pres-
sure was turbulent, and still no bubble cavitation
occurred. Only when cavitation, generated by rough-
ness, at the leading edge took place, a separate
region of bubble cavitation also appeared.
Whatever may be the case, it must be kept in
mind that these descriptions of phenomena do not
explain them, because it is not clear to me why
there should be any interaction between the bound-~
ary layer and the free stream nuclei and which
parameters would control this. I think more sys-
tematic research is necessary to be able to
simulate bubble cavitation on model propellers
in a reproducible way.
I agree with the suggestion of the author that
the increased amount of bubble cavitation, as shown
many times by roughened propeller models, may well
be representative for full-scale cases. Bubble
cavitation seems to be inhibited on scale models
very easily. When bubble cavitation does occur on
scale models the situation is so bad that invari-
ably erosion problems do occur on full-scale.
Ironically a better simulation of bubble cavitation
may not make the interpretation easier.
In general both discussions have made it clear
again that it is impossible to make general state-
ments about the effect of nuclei or roughness. To
make any interpretation and to avoid confusion the
test conditions must be given as complete as pos-
sible. Finally, I thank the discussers for their
discussions and for their kind attention to my
paper.
yi
uy
“
Session VI
CAVITATION
BLAINE R. PARKIN
Session Chairman
Pennsylvania State University
State College, Pennsylvania
1
F c* et
meats . a
<
s <
» z
i
=
A Holographic Study of the Influence of
Boundary Layer and Surface Characteristics
on Incipient and Developed Cavitation on
Axisymmetric Bodies
J. H. J. van der Meulen
Netherlands Ship Model Basin
Wageningen, The Netherlands
ABSTRACT
This paper describes an experimental investigation
of boundary layer flow and cavitation phenomena on
three axisymmetric bodies. The bodies possess
different boundary layer or surface characteristics.
The importance of these features for incipient and
developed cavitation are studied by using in-line
holography. A good correlation is found between
observations and calculations of laminar flow
separation and subsequent transition to turbulence
of the separated shear layer. The influence of
polymer additives on laminar flow separation is
studied in detail. The results of this study explain
the effect of cavitation suppression by polymer
additives on certain bodies.
1. INTRODUCTION
Axisymmetric bodies have often been used to study
the inception of cavitation. These studies were
usually made by systematically varying the parameters
related to the liquid flow (velocity, turbulence,
air content, pressure history) or to the body (size,
surface roughness, wettability). Although a con-
siderable knowledge of cavitation was obtained in
this way, a complete understanding of many cavitation
phenomena was still lacking. A breakthrough was
achieved by Acosta (1974) who emphasized the need
for a thorough understanding of the basic fluid
mechanics of the liquid flow surrounding the bodies
in which cavitation takes place. This statement
was based on an earlier study by Arakeri and Acosta
(1973) in which the boundary layer flow was visual-
ized by the employment of the schlieren method.
Cavitation inception could be correlated with the
occurrence of laminar flow separation. Unawareness
of this important flow phenomenon had obscured the
results of comparative cavitation studies with
axisymmetric bodies, made in the past.
In general, it can be stated that cavitation
433
inception on a body is affected by nuclei, viscous,
and surface effects. The present study deals with
the two latter effects. The use of holography, a
three-dimensional imaging technique, enabled a new
approach. The employment of this method for the
observation of cavitation inception phenomena has
been reported before by Van der Meulen and Ooster-
veld (1974). In the present study an extended
version of the method has been used by which boundary
layer flow phenomena also could be observed. Viscous
effects were studied by comparing two axisymmetric
bodies, a hemispherical nose having laminar flow
separation and a blunt nose not having it. Surface
effects were studied by comparing two hemispherical
noses, one made of stainless steel, the other made
of Teflon.
The phenomenon of turbulent-flow friction reduc-—
tion by polymer additives of high molecular weight
has been known for about thirty years. In recent
years an increased interest has been shown on the
effect of polymer additives on cavitation. In the
present work the influence of polymer additives on
the flow about the test bodies is studied and
related with the influence on cavitation.
2. EXPERIMENTAL METHODS AND PROCEDURE
Description of Test Facility
The facility used is the high speed recirculating
water tunnel of the Netherlands Ship Model Basin.
Originally, the maximum speed in the 40 mm circular
test section was 65 m/s and the maximum allowable
tunnel pressure 35 kg/cm?. A detailed description
of this tunnel and its air content regulation system
is given by Van der Meulen (1971, 1972). For the
present study a new test section was made. It has
a 50 mm square cross section with rounded corners
(radius 10 mm), to limit the influence of the walls.
The models, having a diameter of 10 mm, occupy 3.25
percent of the cross-sectional area of the test
434
MZ
[_] purse -pamper
ae
POLYMER INJECTION LINE
DEAERATION LINE
CENTRIFUGAL Cc |
[| [ {eeu
—
| {COOLING WATER
c
|
eo;
FIGURE 1. Schematic diagram of high fa Stass aeanee
speed cavitation tunnel with polymer
injection system.
section. Injection of polymer solutions from the
nose of the models was made by a Hughes Centurion-
100 pump unit. The unit consists of a drive mech-
anism fitted with two pump heads. A pulse-damper
was used to minimize flow variations. Further
details are given by Van der Meulen (1974b). A
schematic diagram of the tunnel with the polymer
injection system is shown in Figure 1.
To measure the influence of polymer additives
on the friction factor and the surface tension of
the solutions, a turbulent-flow rheometer and a
surface-tensionmeter have been used. Details on
these measuring devices are given by Van der Meulen
(1974a, 1976b) .
Test Models
According to Arakeri and Acosta (1973), most
axisymmetric models used in cavitation inception
studies, such as the hemispherical nose and the
ITTC standard headform, exhibit laminar boundary
layer separation. It means that the laminar boundary
layer is unable to overcome the adverse pressure
gradient and the flow separates from the wall.
Schiebe (1972) introduced a standard series of
axisymmetric models which, theoretically, should
not exhibit boundary layer separation. To distin-
guish between these two classes of axisymmetric
models, a hemispherical nose and a blunt nose,
selected from Schiebe's standard series, were used
in the present investigations. Both models were
made of stainless steel (SST). In addition, a
third model (hemispherical nose) was used, made of
Teflon. The contour of the blunt nose is derived
from the combination of a normal source disk and
a uniform flow. Schiebe (1972) calculated the
dimensionless coordinates and pressure coefficients
for a series of models in the range, Co din = Oods
(point source) - 1.0. From this series a blunt
nose with a minimum pressure coefficient of 0.75
was selected.
The diameter, D, of the cylindrical part of the
hemispherical nose is 10.00 mm. Theoretically,
the diameter of the blunt nose increases smoothly
TUIUILILZ/7
Oc MOTOR
mtd
100i
y
i pk {
WATER SUPPLY
ROTARY PUMP.
VACUUM PUMP
to an asymptotic value, D, with increasing axial
distance, x.
However,
This value was set at 10.00 mm.
for the manufacture of the blunt nose a
minor deviation from the theoretical contour had
to be permitted.
HEMISPHERICAL NOSE
FIGURE 2.
(dimensions in mm).
Thus,
the actual contour coincides
BLUNT NOSE
Cross sections of stainless steel models
435
1.0 T T T T T 1
0.9 irrotational
Se irrotational flow with walls
o/X 5
yo” =
08 / \ e Re=2.1 x10
2)
N
fo}
a
Pressure Coefficient , Cp
fe}
h
(eo)
WwW
0.2
0.1
0.4 0.6 0.8 10 1.2 1.4
Surface Coordinate over Diameter, s/D
with the theoretical contour over a distance, x/D
= 0-1.6, and next changes smoothly into a circular
cylinder with a diameter of 9.88 mm. The cross
sections of the SST models are shown in Figure 2.
For the Teflon hemispherical nose the dimensions
are the same as for the SST hemispherical nose.
However, the Teflon model was not made of solid
Teflon but consisted of a Teflon nose slipped on
a SST core. Extreme care has to be exercised in
manufacturing models for cavitation studies. An
accurate similarity of the model contour is essential,
but a smooth surface is even more critical. The
drastic effects of surface roughness, in particular
isolated irregularities, on cavitation inception
have been demonstrated by Holl (1960) and Arndt
and Ippen (1968). The present models were made by
Instrumentum TNO in Delft. The models were inspected
by an optical comparator (magnification 50x). For
the SST hemispherical nose the maximum deviation
from the true contour was within 5 um, for the
Teflon hemispherical nose within 10 pm. For the
blunt nose, the maximum deviation for x/D < 0.3
was within a few microns and for x/D > 0.3 within
10 um. The mean surface roughness height for the
SST models was 0.05 um; for the Teflon model this
value was considerably higher.
Computations of the pressure coefficient for the
hemispherical nose and the blunt nose were made at
the National Aerospace Laboratory NLR, The Nether-
lands. The velocity potential for irrotational
flow along the model contour was computed with the
variational finite element method according to
flow without walls _|
FIGURE 3. Computed pressure co-
efficient as a function of surface
coordinate over diameter for hemi-
spherical nose. Data points ob-
tained from measurements by Rouse
and McNown (1948) at Re = 2.1 x
10° are included.
20
1.6
18
Labrujére and Van der Vooren (1974). This method
“is suitable for axisymmetric flows. The relation
between the pressure coefficient, Cp, and the
velocity potential, $, is given by
2
CK) |W se) (1)
where s is the streamwise distance along the model
contour and V, the free stream velocity. The pres-
sure coefficient was computed in the absence of
tunnel walls and with tunnel walls. In the latter
case, it was necessary to substitute the square
cross section with rounded corners by a circular
one (diameter 55.44 mm), having the same cross-
sectional area. For the hemispherical nose, the
results are plotted in Figure 3. Also given are
data points obtained from measurements by Rouse
and McNown (1948) at a Reynolds number of 2.1 x 10°.
The computed Cp-values are claimed to be accurate
within 0.1 percent. The Cp-value for irrotational
flow in the absence of tunnel walls is 0.7746 at
s/D = 0.6825 (y = 78.2°). With tunnel walls the
Cpmin Value at the same location is 0.8367. For
the blunt nose, the results are plotted in Figure
4. The computed Cp-values are accurate within 1
percent. The Cp-value for irrotational flow in
the absence of tunnel walls is 0.750, which is
consistent with the accurate computations by Schiebe
(1972). With tunnel walls the Cpy;j,-value is 0.802.
Tabulated values of Cp are presented by Van der
Meulen (1976b) and Labrujére (1976).
436
Pressure Coefficient, Cp
FIGURE 4. Computed pressure co-
efficient as a function of surface
coordinate over diameter for blunt ie) 02
nose.
Holographic Method
In the present work, in-line holography has been
used to study cavitation and flow phenomena about
the test models. The method consists of making
photographic records containing detailed information
on the cavitation and flow patterns. Holography
has become one of the most important areas of modern
optics since the invention of the laser as a new
light source. Holography is usually described as
a method for storing wavefronts on a record from
which the wavefronts may later be reconstructed.
The record, formed in photosensitive material, is
called a hologram. In forming holograms two sets
of light waves are involved: the reference waves
and the subject waves. In the present case of in-
line holography only one set of waves is used
basically. The undeflected light waves from this
set of waves act as reference waves, the light
waves deflected by the subject act as subject waves.
A schematic diagram of the applied optical system
is shown in Figure 5. The light source is the
Korad K-1QH pulsed ruby laser of the Institute of
Applied Physics TNO-TH. To improve the resolution
of the system, the red light from the ruby laser
is converted to ultraviolet light, with a wavelength
of 0.347 um, in a KDP-crystal. The pulse duration
is 25 nanoseconds and the maximum energy 4 mJ in the
TEMg99 mode. A telescopic system (Ly and L3) is used
to obtain a laser beam with a diameter of 30 mm.
A mirror reflects the beam into the test section of
the tunnel. In the walls of the plexiglass test
section, two optical glass windows are inserted.
irrotational flow without walls
irrotational flow with walls
06 08 1.0 1.2 14 1.6 18 20
Surface Coordinate over Diameter, s/D
The location of the body in the test section is
such that the nose is illuminated by the laser beam
over a length of about 20 mm, and the body contour
is imaged on the hologram. A shutter is placed on
the first window. The camera containing the holo-
graphic plate is located close to the second window.
Agfa-Gevaert Scientia Plates 8E56 and 8E75 with a
resolution up to 3000 lines/mm were used as recording
material. The ruby laser could also be used as a
multiple switched laser. Two or three pulses with
PULSED RUBY ral
LASER v
|
KDP-CRYSTAL| UV-FILTER }
Us
GLASS WINDOW
TUNNEL WALLS,
\
GLASS WINDOW
cog
{ SSEREENS;|
ISS =<
CAMERA /HOLOGRAPHIC PLATE
FIGURE 5. Schematic diagram of optical system for
making holograms of cavitation or flow phenomena in
test section of tunnel.
Lo POLAROID FILTER
Sea f alee a
[MICROSCOPE
STAGE WITH
HOLOGRAM/
FIGURE 6. Schematic diagram of reconstruction
set-up.
pulse separations of 50 or 100 usec could be
generated. This enabled multiple imaging of moving
cavities on one hologram.
Reconstruction of the holograms was made with a
continuous-wave He-Ne gas laser (\ = 0.633 ym). A
schematic diagram of the reconstruction set-up is
shown in Figure 6. The diameter of the laser beam
is enlarged by the lenses, L} and Lz. The intensity
of the light can be adjusted by a polaroid filter.
The hologram is placed on a stage, fitted with
guides so that the hologram can be moved in two
orthogonal directions. The movement of the stage
is measured on vernier scales. The reconstructed
image is studied with a microscope with a magnifi-
Cation between 40x and 200x.
Flow Visualization Technique
A new technique had to be developed to visualize
the boundary layer flow about the axisymmetric
models. A description of the several methods in-
vestigated is given by Van der Meulen (1976b). The
ultimate method consisted of injecting a sodium
chloride solution into the boundary layer from a
hole located at the stagnation point of the model.
The diameter of the hole is 0.08 mm. The sodium
chloride solution has a slightly different index
of refraction from the surrounding fluid. The
light emitted from the pulse laser will be deflected
and the deflections are recorded in the hologram.
Optimum conditions for flow visualization are given
by Van der Meulen and Raterink (1977). In the
present study, the ratio of the injection velocity,
Vi, to the velocity in the test section, Vo, was
usually between 0.1 and 0.2. The sodium chloride
concentration was 2 percent. At first, the fluid
was injected with a hypodermic syringe, but later
on, a plunger with a constant motion was used.
Procedure
The tests performed in the high speed tunnel com-
prised flow visualization tests, cavitation tests
437
and cavitation inception measurements. Essentially,
the flow visualization and cavitation tests consisted
of making holograms at prescribed conditions. Prior
to each series of tests the model was cleaned and
the tunnel refilled. To adjust the air content,
the water was passed through the deaeration circuit
for a period of 1% h at a constant pressure in the
deaeration tank. All tests were made at a constant
air content, a, of about 5 cm?/2 (1 cm? of air per
liter of water at STP corresponds to 1.325 ppm by
weight). For each test the temperature of the
tunnel water was measured to obtain the dynamic
viscosity and the vapor pressure. The average
value of the water temperature was 20°C. The flow
visualization tests covered a velocity range of 2
to 30 m/s. For the cavitation tests, the velocity
ranged from 10 to 20 m/s. The effect of polymer
additives on cavitation and cavitation inception
was investigated by injecting a 500 ppm Polyox WSR-
301 solution from the nose of the models. Polymer
injection was provided by the Hughes Centurion-100
pump unit. The holograms were made at the instant
of maximum injection rate. The injection rate was
such that the average value of Vi/Vo was 0.17. For
the cavitation inception measurements, the velocity
ranged from 10 to 24 m/s. Inception (or desinence)
was observed visually.
3. BOUNDARY LAYER STUDIES
Newtonian Flow
The holograms exhibited a distinct occurrence of
laminar boundary layer separation on the hemispher-
ical nose. The location of separation could be
obtained quite accurately from the holograms. At
this location the interference pattern usually
showed a V-shape. This is shown in the photograph
presented in Figure 7. This photograph also shows
the laminar separation bubble itself and the
subsequent transition to turbulence and reattachment
of the separated shear layer. In the transition
region, the flow is still visualized by the sodium
chloride, but further downstream, where the turbu-
lence becomes more developed, mixing of the sodium
chloride prevents any further observations. The
determination of the length and the maximum height
of the laminar separation bubble from the holograms
was somewhat complicated by the fact that the height
of the bubble may show a maximum, as illustrated by
case A in Figure 8, or that the outer flow line
shows an inflexion point, as illustrated by case B
in Figure 8. The location of separation for the
FIGURE 7. Photograph showing laminar separation bubble and subsequent transition to turbulence on SST
hemispherical nose. The flow is from left to right. At the position of separation the interference pattern
shows a "V". YW, = 4 m/s.
438
A
maximum
Inflexion point
FIGURE 8.
on hemispherical nose (schematically) and definitions
of length and maximum height of bubble.
Observed shapes of laminar separation bubble
hemispherical nose is given in Figure 9. In this
figure the angular location of separation, yg, is
plotted against the Reynolds number. Results on
the length, L, the height, H, and the length to
height ratio, L/H, of the separation bubble are
presented in Figures 10, 11 and 12. Each data
point refers to one hologram (values for the upper
and lower side of the model are averaged). Most
data points refer to the SST hemispherical nose,
a few refer to the Teflon hemispherical nose.
The present observations are in agreement with
those obtained earlier by Arakeri (1973) and Arakeri
and Acosta (1973). From Figure 9 it follows that
the boundary layer separation angle is independent
on the Reynolds number, which is consistent with
theory (Schlichting, 1965). For the SST hemispher-
ical nose the average value of yg is 85.43°. This
value is claimed to be quite accurate. To compare
this experimental value with the theoretically
predicted one, laminar boundary layer calculations
were made using the method derived by Thwaites
(1949). With this method the parameter m is cal-
culated, where m is defined as
a2 dau
ae oi wv) ds (2)
and where 9 is the momentum thickness, U the velocity
at the edge of the boundary layer, v the kinematic
viscosity, and s the distance along the surface.
iv)
» °
o 90 T T 7, T T T T T T
Te)
[= st)
4 Yo
5) 86° NE
ow
c 4
©
fo}
o ° °
w 86 a
“ ¢ 20 @ ON 6) P.O, © fe)
@ Ors @ oo, 8 oe |
2 8
= 84h
a o SST
3 e@ Teflon +
3 82° rn ! et —————E eee
o 02 04 06 o8 1 2 3eee4
Reynolds Number x107>
FIGURE 9. Boundary layer separation angle, Yc,
as a function of Reynolds number for hemispherical
nose.
o SST
04 @ Teflon J
Length of Separation Bubble to Diameter, i/o
02 04 06 O8 10 20 30 40
Reynolds Number x1072
FIGURE 10. Length of separation bubble to diameter,
L/D, as a function of Reynolds number for hemispheri-
cal nose. The solid lines refer to theoretical pre-
dictions.
Laminar boundary layer separation is said to occur
for m = 0.09. The computations of yg were made
with the accurate pressure distributions obtained
earlier (Figure 3). For the actual case (with
tunnel walls) yg was found to be 85.57°, and thus,
in excellent agreement with the experimental value.
The theoretical value of yg is hardly affected by
the presence of the tunnel walls, since in the
absence of tunnel walls we found yg = 85.53°.
Arakeri (1973) found experimental and theoretical
values of 87°. However, his computations were
based on the experimental pressure distribution
data by Rouse and McNown (1948), as shown in Figure 3.
The length and the height of the separation
bubbles decrease gradually with increasing Reynolds
number. The variations in length and height for a
given Reynolds number are partly due to the different
0.05 STs T T Tiana Vemerl T T
‘ ° o SST
0.04 @ Teflon 4
fo)
003 =|
Height of Separation Bubble to Diameter, H/o
002 fo) 4
8 9°
O_o
001 e@ @o =
0 &
YU ©
°
°
fo) [nee ee
02 04 OG O8 2 3 4
-5
Reynolds Number x 10° ~
FIGURE 11. Height of separation bubble to diameter,
H/D, as a function of Reynolds number for hemispheri-
cal nose.
20 - <j T sr * T T T
oS o-
o> ° ©
aes e
& © o oO ‘ e °
» 8 @ 00 ° o@
tm = 10 0° 7
=a fo} fe) ° °
O) 0 QO
ag) {s
2
os
~ 0
(&
= x 4
Do
th o SST
So @ Teflon
02 04 06 o8 1 2 3 4
Reynolds Number x107°
FIGURE 12. Length to height ratio of separation bubble,
L/H, as a function of Reynolds number for hemispherical
nose.
appearances of the separation bubbles near transition,
as illustrated in Figure 8. The length to height
ratio of the separation bubbles (Figure 12) is not
very dependent on the Reynolds number. For the SST
hemispherical nose an average value of 10.8 is found.
To compare experimental values of L with theoret-
ical ones, it is necessary to calculate the location
of transition on the separated shear layer. Recently,
Van Ingen (1975) presented a calculation method for
the laminar part of separation bubbles in which
also the location of transition is predicted. The
method is based on a solution of the Navier-Stokes
equations, valid near the separation point. A
relation is found for the separation streamline
leaving the wall at an angle, 6. By using constant
values of B, Oar and Msepr to be obtained experi-
mentally, a formula is derived to calculate the
length of the separation bubble. It is assumed
that the separation streamline is straight and
that the angle 6 is given by
tan 6 = aL (3)
Re,
sep
where REQsep is the Reynolds number based on the
momentum thickness at separation, given by
Re = (Gao) : (4)
TABLE 1. Separation Streamline Angle 6 For SST
Hemispherical Nose, Derived from Holograms.
Re x 107° Regsep 8 B
0.21 56 14.6° 14.6
0.36 74 9.0° 11.8
0.62 97 710° 11.9
0.97 121 52° it
1.35 144 4.4° Tied
1.41 146 4.2° 10.8
1.87 169 A390 WD7
2.40 191 3.9° 13.0
3.40 228 2.8° Died
439
The amplification factor, o is defined as
a’
a
a
neutral
where a/apeytra, 18 the ratio of the amplitude of
a disturbance to its amplitude at neutral stability.
Meo is the value of m at separation (Msep = 0.09).
According to Dobbinga et al. (1972), B is usually
between 15 and 20, but lower values are also found.
To obtain B for the present case, the separation
streamline angle, &, was derived from a series of
holograms. The results are presented in Table 1.
The average value of B is 12.0.
With Van Ingen's method, the location of transi-
tion has been calculated for oa = 7 and go, = 8,
using Msep = 0.09 and B= 12. The results are
plotted in Figure 10. It is found that most experi-
mental data points lie between the two theoretical
curves. The best fit would be obtained for og = 7.5.
It should be noted that the present experimental
data refer to the beginning of transition. Ina
recent paper, Van Ingen (1976) attempted to corre-
late the amplification factor with the turbulence
level, Tu. For og = 7.5, predicting the beginning
of transition, we find Tu = 0.15%. Although the
turbulence level in the high speed tunnel has not
been measured, it is possible to obtain an approx-
imate value (without considering noise aspects) .
Arakeri (1975a) measured the location of transition
on a 1.5 caliber ogive in the axisymmetric test
section of the CIT high speed water tunnel. The
turbulence level in this tunnel was 0.2%. Recently,
Arakeri (1977) performed similar measurements in
the NSMB high speed water tunnel. The agreement
between the transition data indicates that the
turbulence level in both tunnels was approximately
the same. Hence, the turbulence level in the NSMB
tunnel may have been close to 0.2%, which is con-
sistent with the value derived earlier. The above
considerations on the turbulence level are, however,
not confirmed by the measurements of Gates (1977),
who found that the turbulence level had no effect
whatsoever on the location of transition on a
hemispherical nose.
As shown in Figures 9 through 12, the appearance
of the laminar separation bubble on the Teflon
hemispherical nose is the same as for the SST
hemispherical nose. From Figure 10 it is found
that the higher surface roughness of the Teflon
body has no effect on transition. Apparently, the
amplification of disturbances mainly occurs down-
stream of separation.
The blunt nose exhibited a laminar boundary layer
with normal transition to turbulence. Laminar flow
separation did not occur. A photograph showing
transition is presented in Figure 13. A plot of
the transition data is given in Figure 14. Since
the outflow of the sodium chloride solution from
the nose of the model was in some cases quite
unstable, the determination of the precise location
of transition provided some difficulties, but an
upper or lower bound could still be indicated. [In
Figure 14 these data points are marked with an
arrow. When the arrow is pointing upward the data
point is considered to be the lower bound; when the
arrow is pointing downward the data point is con-
sidered to be the upper bound. Silberman et al.
(1973) made laminar boundary layer calculations for
a series of blunt noses having CPpin Values ranging
FIGURE 13. Photograph showing transition (T)
from 0.333 to 1.0. The calculations showed that
none of the blunt noses were subjected to laminar
separation. The present observations are in
agreement with these theoretical predictions.
Non-Newtonian Flow
The influence of polymer additives on the boundary
layer flow about the models was investigated by
injecting a 500 ppm (parts per million by weight)
Polyox WSR-301 solution from the nose of the models.
To visualize the flow, the injection fluid contained
2 percent sodium chloride. For the SST hemispher-
ical nose, the holograms showed that laminar flow
separation was no longer present. An example is
given by the photograph presented in Figure 15. At
or shortly downstream from the location where
Newtonian flow separation occurred, transition from
laminar to turbulent boundary layer flow is observed.
From the holograms made in the velocity range 4 to
20 m/s, it could be derived that transition to
turbulence occurred close to the location of
Newtonian flow separation. It was difficult, however,
to indicate the precise location of transition.
24
20
over Diameter, St/D
52
@
Streamwise Distance to Boundary Layer Transition
fo}
rs
FIGURE 14. Streamwise distance to boundary
layer transition over diameter, S,/D, asa 03
function of Reynolds number for blunt nose.
from laminar to turbulent boundary layer on blunt nose
(Sp/D = 1.68). The flow is from left to right, W. = 8 m/s.
Another important observation was that the sodium
chloride was not completely mixed in the turbulent
region, but was still able to show the existence of
waves and streaks further downstream, till the end
of the hologram. An example of this phenomenon
has been given by Van der Meulen (1976b). For the
Teflon hemispherical nose it was found that the
influence of polymer additives on laminar flow
separation was the same as for the SST hemispher-
ical nose. Although the observations made with the
blunt nose were somewhat obscured by the irregular
outflow from the nose of the model, the main con-
clusion to be derived from the holograms is that
the polymer causes early transition to turbulence.
The approximate locations of transition are plotted
in Figure 14.
The polymer concentration used during the above
observations is rather high when compared to the
most effective concentration for turbulent-flow
friction reduction. From Figure 16, where the
friction factor, f, for flow through a circular
tube is given as a function of the Reynolds number,
it can be derived that a Polyox WSR-301 concentration
of about 20 ppm gives a maximum friction reduction.
Additional holograms for the SST hemispherical nose
O 2percent NaCl injection
@ 2percent NaCl +500ppm Polyox WSR-301 injection
06 O7 O8 1.0 15 20 30
04 foe)
Reynolds Number x 1075
ee Imm =|
FIGURE 15. Photograph showing boundary layer flow about SST hemispherical nose when a solution of 500 ppm
Polyox WSR-301 is injected. The flow is from left to right. We
were made at polymer concentrations of 100, 50,
and 20 ppm. The injection rate was such that Vj/Vo
= 0.2. The phenomena observed at these lower
concentrations were the same as those found at 500
ppm. Recently, Gates (1977) studied the influence
of polymer additives on laminar flow separation at
low injection rates, and was able to find inter-
mediate stages of separation suppression.
The study on the influence of polymer additives
on laminar flow separation has been limited so far
to the case where the polymer is present only in
a thin layer adjacent to the body (the "inner part"
of the boundary layer). To study the influence of
polymer additives present in the "outer part" of
the boundary layer, additional tests with the SST
hemispherical nose were made in which the tunnel
was filled with a 50 ppm Polyox WSR-301 solution.
To prevent polymer degradation, the water speed in
the test section was set at a low value of 4 m/s.
Three different solutions were injected: a solution
0.04 ey al LU T
ae Water "|
0.03 ae °885es +
Peg, d
fo 0° 2°00 00 ©0000 Soo go0a, |
©
® eo © © 00 0 00 | QrdQp000DmpDLD
oO iy
02 =|
Ga So “ov, Op 20 0B °° |
% 7 VOOo 00 0 Go 00 4
@ ee a vv 4
os 4 gv ea Ea Vo =|
ve a v ]
8 P chaos dt 8 Cowie |
0 Foy o*," V ,dd4an |
0 4) Bal
o we? 6 Pag,
av) esta! a We |
5 FOS? ol — vy
6) A.
c 0.01
F hee
eO01ppm 010 ppm Cash 4
202 v 20
@e@05 @50 =
f= 84 o1 4100
e °02 B 200 a]
a5 v 500
6 10 15 20 25 30
Reynolds Number x1072
FIGURE 16. Friction factor of Polyox WSR-301 solu-
tions in water as a function of Reynolds number,
according to Van der Meulen (1974a).
= 4nm/s.
of 2 percent NaCl, a solution of 2 percent Nacl +
50 ppm Polyox and a solution of 2 percent NaCl +
500 ppm Polyox. The injection velocity Vj was
0.8 m/s (Vi/Vo = 0-2). Photographs showing the
boundary layer flow are presented in Figure 17.
For comparison a photograph is included showing
the influence of polymer injection when the tunnel
is filled with water (Figure 17a). When a 2 per-
cent NaCl solution is injected (Figure 17b), the
boundary layer first shows a tendency to become
unstable but further downstream the instabilities
are suppressed and the boundary layer is laminar
again. When a 2 percent NaCl + 50 ppm Polyox
solution is injected (Figure 17c), the boundary
layer first shows a slight tendency to become
unstable, but further downstream the boundary layer
is laminar. When a 2 percent NaCl + 500 ppm Polyox
solution is injected (Figure 17d), the boundary
layer remains completely laminar, till the end of
the hologram. The conclusions to be derived from
these observations are that the presence of the
polymer in the "inner part" of the boundary layer
leads to destabilization, whereas the presence of
the polymer in the "outer part" of the boundary
layer leads to stabilization, and the latter effect
is predominant. In all cases considered (Figure
17), laminar flow separation is suppressed.
An explanation of the various phenomena observed
can, as yet, not be given. Apparently, some of the
phenomena are in agreement with those reported
elsewhere, others may not have been observed before.
This is mainly due to the fact that numerous studies
have been made on drag reduction in turbulent flow,
but only a few were made on the influence of polymer
additives on laminar flow. In studying laminar
flow around circular cylinders, James and Acosta
(1970) found that the streamline patterns with
dilute polymer solutions were significantly different
from those with Newtonian fluids because of visco-
elastic effects. These effects may also play a
dominant role in eliminating flow separation in
those cases that the boundary layer remains laminar.
In those cases where the boundary layer becomes
turbulent due to the presence of the polymer in the
"inner part" of the boundary layer, it is still
questionable whether flow separation is eliminated
by early turbulence by viscoelastic effects, or by
a combination of these. The occurrence of early
turbulence as found in the present study and reported
before [Van der Meulen (1976a, 1976b), Gates (1977) ]
is consistent with the findings of others. According
to Lumley (1973), polymer solutions producing drag
(a)
> ; hee (d)
1mm Ress
FIGURE 17. Photographs showing boundary layer flow about SST hemispherical nose. The flow is from right to
left. V_ = 4 m/s. (a) Injection of 50 ppm Polyox in water.
(b) Injection of water in 50 ppm Polyox. (c) Injec-
tion of°50 ppm Polyox in 50 ppm Polyox. (d) Injection of 500 ppm Polyox in 50 ppm Polyox.
reduction display a positive Weissenberg effect for
which destabilization is predicted analytically.
Destabilization is also predicted by the numerical
analysis of Kiimmerer (1976) on the stability of
boundary layers in an idealized viscoelastic fluid.
Experiments by Forame et al. (1972) and Paterson
and Abernathy (1972) also suggest destabilization.
On the other hand, Castro and Squire (1967) and
White and McEligot (1970) found that polymer solu-
tions in water cause a delay in transition to
turbulence. According to Lumley (1973), drag-
reducing polymers tend to increase the thickness
of the viscous sublayer. Experimental evidence
08 T T T T T T T T
) ° O. 2
° @ e
e eo f® ee
© O6r oe @ ee -
« 5 °o (Oo
: 0 @ 00
3
e }
c &
oc 4 A &
= 4 4 4 4
i] a 4 a & a a a
5 04- 5 : =|
” inception 3
=43cm-/1
o e pieces S S, /
| & inception, 500ppm Polyox WSR —301 injection 3 al]
a=47¢em>/1
& desinence,500ppm Polyox WSR -301 injection
02 . 1 4 L Jt 1 1 4
08 12 16 20 24
Reynolds Number x 10-2
IGU Cavitation inception and desinence number
as a function of Reynolds number for SST hemispherical
ose with and without polymer injection.
for this phenomenon has been provided by Rudd (1972),
who measured velocity profiles in a polymer solution
by using a laser dopplermeter. By examining the
expansion behavior of isolated polymer molecules
in a flow field, Lumley (1973) postulated a mech-
anism which predicted a decreased intensity of
small-scale turbulence in the buffer layer and which
also predicted that, in the maximum drag reduction
regime, the turbulence should consist primarily of
larger eddies. The present observations of waves
and streaks along the surfaces of the models seem
in agreement with the above predictions. They also
agree with the observations made by Hoyt et al.
(1974) on the structure of jets of polymer solution
discharged in air.
4. CAVITATION STUDIES
Inception
Cavitation inception data for the SST hemispherical
nose are plotted in Figure 18. Inception was
measured by gradually lowering the pressure until
the first appearance of cavitation was observed.
Desinence was measured by starting from developed
cavitation and gradually raising the pressure until
cavitation just disappeared. The type of cavitation
mostly observed at inception was sheet cavitation.
Also plotted in Figure 18 are cavitation inception
data when a 500 ppm Polyox WSR-301 solution was
injected from the nose of the model. The type of
cavitation observed in this case was travelling
bubble cavitation. Cavitation inception data for
the Teflon hemispherical nose are plotted in Figure
Llama hae aaa Ta T T T oo ]
O Inception
Aa=44c mA |
e @ desinence |
: |
© ee
b é ° é |
> 2 }
6 ee e
rs) e e el
E |
3 |
2 °o 4
5 ie)
S © ro) OO
= °
5 O68 ©) 4
C ° 9 6 |
(S) ° |
Os °
1
O04 1 1 1 i 1 1 1 i
os 12 16 20 24
Reynolds Number x 10-2
FIGURE 19. Cavitation inception and desinence number
as a function of Reynolds number for Teflon hem-
ispherical nose.
19. The type of cavitation observed at inception
was spot cavitation. The spots were usually located
between the pressure minimum (y = 78°) and the
transition of hemisphere and cylinder (y = 90°).
The most striking differences between the inception
data for both models are: (a) the inception data
for the Teflon model are much higher than for the
SST model and (b) the Teflon model exhibits a strong
cavitation hysteresis [Holl and Treaster (1966) ]
whereas the SST model exhibits no hysteresis. Such
observations have been reported before by Reed
(1969), Gupta (1969), and Van der Meulen (1971).
Since the viscous flow behavior of the Teflon model
is the same as for the SST model (see Section 3),
the above differences can only be explained by
surface effects. Teflon is a porous material and
has a high contact angle. Both properties are
essential features of the Harvey nucleus [Harvey
et al. (1944) ]. Hence, the Teflon surface acts as
a host for surface nuclei, from which (gaseous)
cavitation is initiated. The mechanism most probably
08
b
co O6
© a
€ inception (or desinence)
=)
z
6
2
oO
2
204
oO
oO Cp (Irrotational flow)
T
4
02 er mes | ES 1 “=|
10 14 18 22
Reynolds Number x1072
FIGURE 20. Comparison of cavitation inception (or
desinence) number with pressure coefficient at sepa-
ration, Cp_, and at transition, Cpr for SST hem-
ispherical’ nose.
443
involved with inception on the SST hemispherical
nose has been described by Arakeri (1973). Inception
takes place in the transition and reattachment
region of the separation bubbles, where high pressure
fluctuations occur [Arakeri (1975a)]. The nuclei
may either originate from the surface (Arakeri)
or from the stream where they become trapped in.
the strong vortices occurring in the reattachment
region.
When o; (or og) for the SST hemispherical nose
is to be compared with the pressure coefficient,
several problems arise. The most obvious pressure
to compare 0; with would be the pressure coefficient
at transition, Cp,, since the onset of cavitation
takes place at the location of transition. Accord-
ing to Arakeri (1973), however, the important
pressure coefficient to compare 0; with would be
the pressure coefficient at separation, Cpg- This
opinion is probably based on the assumption that
the pressure within the separation bubble is con-
stant (and thus Cpg = Cp) but, according to Van
Ingen (1975), this is a good approximation only
at low values of Re. A mean curve of the present
inception (and desinence) data is plotted in Figure
20. Also plotted are Cp, = 0.76 and Cp, for
irrotational flow, derived from Figures 3 and 10
(with og, = 7.5). The real (or viscous) values of
Cin are unknown and should be obtained from pressure
measurements. It can be estimated that the real
values of Cp,, are considerably larger than those
for irrotational flow, but still smaller than Cpe:
Thus it would seem that oj; (or oq) can be correlated
with the real value of Coe eEnnthatecase waktacan
be argued that the peak pressure fluctuations,
measured by Arakeri (1975a), are creating the
negative pressures necessary to overcome the sta-
bilizing pressure in stream nuclei, caused by the
surface tension.
Cavitation inception data for the blunt nose are
plotted in Figure 21. Also plotted are inception
data with polymer injection. At inception, a
region of travelling bubbles was observed. The
approximate location of this region was x/D = 0.2
- 1.0. In Section 4, a further analysis will be
given of the type of cavitation occurring. The
inception data show that the o,;-and og-values are
almost identical and nearly constant (Sia = 0.46,
in the absence of polymers). When oj is to be
compared with a suitable pressure coefficient, the
ean T T T T T T T
b
¢ 0.6F a)
F
€ 4
2 e A a A |
ee = a
2 oo8 e ag A aA &
& ° @o ® On Sie A @ )
3 ° ° o Me
5 Os), © inception 7
ry) F X= 4.8 cm3/1
@ desinence
4 inception, 500ppm Polyox WSR-—301 injection J]
- aly ob a=51 em3/L
& desinence, 500ppm Polyox WSR -—301 injection
St Sa ete FE ak wi shee St Nl
08 U3 1.6 20 2.4
Reynolds Number x 10-2
FIGURE 21. Cavitation inception and desinence number
as a function of Reynolds number for blunt nose with
and without polymer injection.
444
06 = aaa =e T T
|
04 “/_inception (or desinence) =i}
b
C
©
Qa
E
=)
z
c
°
~ O02
i)
=
>
%
16)
Oo
10
Reynolds Number x 107°
FIGURE 22. Comparison of cavitation inception (or
desinence) number with pressure coefficient at tran-
sition, Cpe for blunt nose.
best choice would seem the pressure coefficient at
the location of cavitation inception. However,
this location can not precisely be indicated. For
bodies with attached boundary layers, Arakeri (1973)
suggested correlating 6; with the pressure coeffi-
cient at transition, Cp,. For a 1.5 caliber ogive
a close correlation was found between measured
values of og and computed values of Cp,. The same
comparison can be made for the blunt nose. In
Figure 22, o5 q and Cp,, derived from Figures 4 and
14, are plottéd agesinee the Reynolds number. In
this case it may be assumed that the real (or viscous)
values of Cp,, are the same as those for irrotational
flow. It is evident from Figure 22 that oj (or
Og) cannot be correlated with Cp,,. The location
where Cp, = 0.46 (= 53a) is well in the laminar
region of the boundary layer for the Reynolds numbers
considered.
The influence of polymer additives on cavitation
inception is a rather new phenomenon. Darner (1970)
investigated the addition of polymers to water on
Ellis
reported on the effect of polymer
acoustically induced cavitation inception.
et al. (1970)
Surface Tension S, dyne/cm
lo) 100 200 300 400 500 600
Polyox WSR - 301 Concentration , ppm
FIGURE 23. Surface tension as a function of Polyox
WSRT301 concentration in water, as measured in surface
tensionmeter.
solutions on flow-generated cavitation inception.
The effect of the polymer was to suppress cavitation
inception. An explanation for the effect could,
as yet, not be given. Ting and Ellis (1974) studied
the growth of individual gas bubbles in dilute
polymer solutions but concluded that the polymers
hardly affected bubble growth. From Figure 23 it
is found that the surface tension is slightly
reduced by small additions of Polyox WSR-301, but
according to Hoyt (1973) this effect should cause
earlier cavitation instead of cavitation suppression.
From Figure 18, a considerable effect on oj and
Og is found when a 500 ppm Polyox solutuion is
injected from the nose of the SST hemispherical
model. For Re above 1.2 x 10°, the reduction
amounts to 30 percent. For the mean value of oj
and 0g we have Siva = 0.445. The o;- and og-values
are independent of Re. From Figure 21 it is found
that oj; and og are hardly affected by the injection
of a 500 ppm Polyox solution from the nose of the
blunt model. For Re above 1.2 * 10°, the mean
value of oj and og in the absence of polymers is
i,q = 0-45. Hence, inception on the SST hemispher-
ical nose with polymer injection takes place at the
same cavitation number as inception on the blunt
nose in the absence of polymers.
As found in Section 3, the influence of the
polymer is to suppress the laminar boundary layer
separation on the hemispherical nose. Hence, the
strong pressure fluctuations, occurring at the
position of transition and reattachment of the
separated shear layer [Arakeri (1975a) ] and being
the principal mechanism for cavitation inception,
are eliminated and cavitation will start at a much
lower cavitation number. The flow visualization
studies described in Section 3 do not only explain
the suppression of cavitation inception by polymer
injection, but also by having a polymer ocean
[Ellis et al. (1970)]. Earlier studies by Van der
Meulen (1973, 1974b) showed that polymer injection
had hardly any effect on cavitation inception on a
Teflon hemispherical nose. The reason for this
finding is clear now, since cavitation inception on
a Teflon hemispherical nose is related to surface
effects and not to viscous effects.
Appearance on Hemispherical Models
The appearance of cavitation on the SST hemispher-
ical nose is closely related to the occurrence of
laminar boundary layer separation. Arakeri (1973)
showed that cavitation bubbles are first observed
at the location of transition and reattachment of
the separated shear layer. This type of cavitation
is usually called bubble cavitation. An example
is shown in Figure 24a. The larger bubbles at the
location of transition are preceded by smaller ones
which, according to Arakeri (1973), are travelling
upstream with the reverse flow in the separated
region. With a reduction in o, the larger bubbles
create a single cavity as shown in Figure 24b.
With a further reduction in o, the cavity is filling
the separated region, and a smooth attached cavity
is observed (Figure 24c). This type of cavitation
is usually called sheet cavitation. When o is
further reduced, the length and the height of the
cavity extend, but the first part of the cavity
remains smooth (Figure 24d, e). By analyzing
double exposure holograms made of developed cavita-
tion, it could be established that the first smooth
part of the cavity is stable.
Smm
The appearance of cavitation on the Teflon
hemispherical nose is closely related to the presence
of weak spots on the surface. Discrete cavities
originate from points located on the hemisphere.
The cavities develop cone-shaped in the downstream
direction. The first part of the cavity surface
is smooth; the cavity leaves the wall at a very
small angle. Some of these features can be observed
on the photographs presented in Figure 25. The
cavitation separation angle Ycg for both hemispher-
ical models is plotted in Figure 26. For the Teflon
model it is found that the cavities start upstream
of the minimum pressure point (Yes < YPmin)! when
0 is sufficiently low. For the SST model it is
found that the cavities always start downstream
of the minimum pressure point (Yoo < Wien Jo. Yes
is both a function of o and Re. For a given Re,
Yes decreases with decreasing o and for a given
5, Yes decreases with increasing Re. These tenden-
cies for the SST model are in agreement with the
observations by Arakeri (1975b).
The shape of the cavity nose on the SST model
has been analyzed further. A schematic drawing of
the geometry of the cavity nose is presented in
Figure 27. From a detailed study of the holograms
it could be established that the cavity nose was
circularly shaped. It was found that the nose
angle 8 varied between 70° and 120°, but was
independent of o or Re. An average value of 90°
was obtained from 28 cavity noses. Since the cavity
nose is immersed in the separation bubble and the
flow comes to a standstill near the cavity nose,
445
FIGURE 24. Photographs showing progressive
development of cavitation on SST hem-
ispherical nose. The flow is from left to
right. V, = 13.2 m/s. (a) o = 0.60;
(b) o = 0.59; (c) o = 0.56; (d) o = 0.47;
(e) o = 0.39.
it is to be expected that the nose angle equals
the contact angle for the present liguid-gas-solid
system. This is confirmed by the fact that, accord-
ing to Adamson (1966), the contact angle for a
water-air-steel system is 70°-90°. The nose radius
ry was independent of o but, as shown in Figure 28,
the radius decreases with increasing Re. The length
of the sheet cavity (the smooth part preceding the
developed cavity) is more or less independent of
oO but decreases with increasing Re. In Table 2,
mean values of Lo¢/D are compared with corresponding
values of L/D, obtained from Figure 10 (with
5, = 7-5). From this table it can be concluded
that transition to turbulence on the cavity surface
is closely related to transition to turbulence on
the fully wetted separated shear layer. The shape
of the developed cavity is determined by the total
length to maximum height ratio of the cavity,
L¢/Her (in most cases the cavity reached its maxi-
mum height close to the trailing edge of the cavity).
Values of this ratio are given in Figure 29. The
mean value of Lc/He is 10.2. Since the mean value
of the length to height ratio of the separation
bubble is 10.8, it may be concluded that the shape
of the developed cavity appearing on the SST hemis-—
pherical nose is strongly governed by the shape of
the separation bubble.
With polymer injection, the cavities on the SST
hemispherical nose are either attached or may show
the appearance of travelling bubbles, resembling
the type of cavitation observed on the blunt nose.
Details are given by Van der Meulen (1976b).
446
FIGURE 25. Photographs showing progres-
sive development of cavitation on Teflon
hemispherical nose. The flow is from left
to right. Ve = 13.2 m/s. (a) o = 0.96;
(b) o = 0.63; (c) o = 0.40.
10 T T T T
Teflon Pp Pp
— Re=134x 10°
O9fF : p P 7
SST S
Oo Re =094 x102 Sp. =78.2
os @ Re =127 x102 aes ea 4
5
b d Re=154 x102
is a Re =2.03 x 105 ve)
rs
E O7P 4
=)
2 -
c p p A
) L 4
. 0.6 aN
= @® wo
S pP pP ° °
oO
6) m 44 00 J
0.57 ey 6
foe}
ay A
0.4 p p 00 J
p Pp
03 ——— He 1
65° 70° 75— 80° 85° 90° 95
Cavitation Separation Angle Ocs
FIGURE 26. Cavitation separation angle, Ycs, as a
function of cavitation number and Reynolds number for
hemispherical nose.
water
3 VN cavity me
ip \
Sas ELLIE 7
TL, Wate TA VSAM TSE Ut lof HA 7
model
JRE 7 schematic diagram of cavity nose on SST
hemispherical model.
fa)
”
=
°
a
xo
&
vo o
ome
2.9
a
>
SS (5
> ©
o >
oO 0
FIGURE 28.
a function
model.
0.606
re) fo)
O0.004F- fo) |
0.002} © 4
oO LL}
08 1.2 1.6 20 24
Reynolds Number x 107°
Cavity nose radius over diameter, r/D, as
of Reynolds number for SST hemispherical
065 T T = T T T =
a 5
o Re=094 x10 a a
= 5
e@ Re=1.27 x10 aura |
O60
[ a Re=154x 10° iy &
A Re = 2.03 x 10°
eo e fe}
sal 4
055 ° °
6
&
ve) ray
E osob ° a A 4
=}
Zz
{=
3 e e
S
o
= O45 fo) ° 5]
>
oO
oO
Aa
is 8
O40-r (o) ©) |
e e
035 : ss = =
(0) 2 4 6 8 10 12 14
Length to Height Ratio of Cavity, Lc /He
FIGURE 29. Total length to maximum height ratio of
cavity, Lo/HAe as a function of cavitation number and
Reynolds n
Ser for SST hemispherical nose.
TABLE 2. Length of Sheet Cavity Over Diameter,
Lsc/D, and length of Separation Bubble Over
Diameter, L/D, for SST Hemispherical Nose.
Re x 107° Sc L/D
0.94 0.156 0.124
1627 0.124 0.096
1.54 0.070 0.084
DNOS 0.074 0.068
Appearance on Blunt Model
The type of cavitation occurring on the blunt nose
is typically travelling bubble cavitation. An
example is shown in Figure 30a (o = 0.33). When
o is reduced, a single transient cavity may develop,
as shown in Figure 30b (o = 0.28). The transient
character of the cavities occurring on the blunt
nose is clearly observed in the photographs taken
from multiple exposure holograms. Figure 31 shows
a photograph taken from a hologram, where three
pulses were generated by the ruby laser with pulse
separations of 50 usec and 100 usec respectively.
The flow is from right to left. The picture shows
the growth of a cavity near the nose of the model.
The cavity is attached to the model and its shape
is a spherical segment. The cavity grows (its
radius increases) and, at the same time, travels
along the surface with a velocity slightly below
that of the surrounding fluid. When the cavity
reaches a certain height, its shape becomes more
like an attached bubble, as shown in Figure 32.
In this figure, the flow is from left to right.
The attached bubble hardly grows, travels along
the surface, and finally collapses.
The streamwise distance to cavitation separation
on the blunt nose obtained from a series of holograms
taken at various values of o and Re, is plotted in
Figure 33. Also plotted are data points where no
cavitation was observed in the hologram on either
one or both sides of the model. It is found that
| x/p =0247
| x/o=0267
FIGURE 30. Photographs showing cavitation on blunt nose.
(2) @ = Oo33p (Gey) @ = Wolo
447
the streamwise distance to cavitation separation
decreases with increasing Re (apart from the scatter,
typical for travelling bubble cavitation). For
Re = 2.08 x 10°, cavitation separation is located
at a short distance from the pressure minimum
[(s/D) pis, = 0-371.
The Observations of the cavity growth as
represented in Figure 31, enables a comparison with
theory. Plesset (1949) analyzed experimental
observations by Knapp and Hollander (1948) and
compared the growth and collapse of bubbles on a
1.5 caliber ogive with the equation of motion for
a bubble. The agreement was quite satisfactory.
Recently, Persson (1975) introduced some refinements
in the comparison. The present analysis is based
on the so-called Rayleigh-Plesset equation according
to Hsieh (1965). For a vapor bubble, the motion
of the bubble wall is given by the equation
0 ae 25 _ 4uk
OPIS teh) (6)
where p is the liquid density, R the instantaneous
bubble radius, Py, the vapor pressure, P the instan-
taneous ambient pressure, S the surface tension,
and uw the dynamic viscosity. The dots indicate
differentiation with respect to time t. The
multiple exposure hologram (Figure 31) provided
data on Ro(to), Ri (tot50us), and R2 (to+150us),
whereas P(t) could be derived from Figures 31 and
4. Equation (6) was solved numerically to obtain
a theoretical value of Ro. The results of the
computations are given in Table 3. To compare the
significance of the right-hand side terms of Eq.
(6), numerical values of these terms are presented
in Table 4. The main conclusion to be derived from
Table 3 is that the experimentally observed growth
of the cavity on the blunt nose is fairly well
represented by the Rayleigh-Plesset equation of
motion. This is mainly due to the fact that the
blunt nose does not exhibit laminar flow separation
and viscous effects seem to be small.
The appearance of developed cavitation on the
blunt nose with polymer injection was essentially
the same as that without polymers. Details are
given by Van der Meulen (1976b).
f= 5mm
(b)
The flow is from left to right. We = 218) m/si.
448
FIGURE 31. Photograph of multiple exposure hologram showing three stages of cavity growth near nose of blunt
model. The pulse separations are At) = 50 usec and Atp= 100 usec. The flow is from right to left. Ye = 10 m/s;
1 = 0.31. The radii of the growing cavity are indicated on the lower figure.
x/D=0543 = Imm =|
FIGURE 32. Photograph of multiple exposure hologram showing three stages of travelling bubble along blunt nose.
The time separations are: At, = 50 c and At2 = 100 usec. The flow is from left to right. VA = 10 m/s;
0.50 + T a T =T T T T
o Re =094x10°
(s/D)p =037 @ Re =1.22 x10°
045+ min is 4|
a Re=155 x10
A a
A Re = 2.08x10°
fe}
oaol J
b
e i Ir
3 6 a
L a 4
5 O35+ Q
e A Ae
5 A A
2} fe) fe)
= o30} e a a 44 e |
>
3 fats
c
©
o25s- 2 4
5S
oO
8
fe}
c
Oro —— —— fin 1 r i 1 —_
fo) 02 04 06 08 10 12 14 16
Streamwise Distance to Cavitation Separation
over Diameter, S¢/D
FIGURE 33. Streamwise distance to cavitation separa-
tion over diameter, s_/D, as a function of cavitation
number and Reynolds number for blunt nose. Also plotted
are some data points where no cavitation was observed
on one or both sides of the model.
5. CONCLUSIONS
The application of in-line holography and injection
of a 2 percent sodium chloride solution from the
nose of the axisymmetric bodies are useful methods
to visualize the boundary layer and to obtain
detailed information on boundary layer phenomena
and cavitation patterns.
Laminar boundary layer separation and transition
to turbulence of the separated shear layer on the
hemispherical nose can be predicted quite accurately
by existing approximate calculation methods.
Cavitation on axisymmetric bodies may be strongly
influenced by boundary layer effects. For the SST
hemispherical nose, inception and appearance of
cavitation are both related to the location and
appearance of the separation bubble. For the
blunt nose, however, cavitation is apparently more
related to nuclei effects than to viscous effects.
The type of cavitation occuring in this case is
travelling bubble cavitation. The growth of a
cavity on the blunt nose is adequately described
by the Rayleigh-Plesset equation of motion for a
cavitation bubble.
TABLE 3. Theoretical (R) and Experimental (R
Values of Bubble Radius for Cavity Growth on
Blunt Nose (Figure 31).
exp)
R R R R
D exp
t m/sec m/sec mm mm
ee -5710 259)3} 0.84 0.84
te so 50 we -5290 2.66 0.98 0.98
-4540 Qos Le 23} 1.28
449
TABLE 4. Influence of Vapor Pressure, Py, Liquid
Pressure, P, Surface Tension Pressure, 2 S/R, and
Viscosity Pressure, 4y R/R, on Cavity Growth on
Blunt Nose (Figure 31).
Py P DESVAR 4u R/R
t N/m? N/m? N/m? N/m?
om 1940 -6320 170 15
t + 50 us 1940 -3590 150 12
t + 150 us 1940 + 330 120 8
Surface effects on the Teflon hemispherical nose
play a dominant role in both inception and appearance
of cavitation.
The presence of polymers in the "inner part" of
the boundary layer on the SST hemispherical nose
leads to destabilization, whereas the presence of
the polymer in the "outer part" of the boundary
layer leads to stabilization, and the latter effect
is predominant. For all cases considered, laminar
boundary layer separation is suppressed.
Since the influence of polymer additives is to
suppress laminar boundary layer separation on the
hemispherical nose, the strong pressure fluctuations,
occurring at the position of transition and reattach-
ment of the separated shear layer and being the
principal mechanism for cavitation inception, are
eliminated and cavitation will start at much lower
pressures. As a consequence, the cavitation charac-
teristics of the SST hemispherical nose with polymer
injection are approximately the same as those of
the blunt nose without polymer injection.
NOTATION
B Constant in Equation (3)
Cp Pressure coefficient
CPmin Minimum pressure coefficient
CPs Pressure coefficient at separation
Cpm Pressure coefficient at transition
D Model diameter
H Height of separation bubble
He Height of cavity
L Length of separation bubble
Le Length of cavity
Lsc Length of sheet cavity
P Static pressure
Po Free stream static pressure
Pmin Minimum static pressure
2 Vapor pressure
R Bubble radius
Re Reynolds number, VoD/v
REQ sap Equation (4)
iS Surface tension
Tu Turbulence level
U Velocity at edge of boundary layer
Vo Free stream velocity
Vi Injection velocity
a Amplitude of disturbance
Aneutral Amplitude of disturbance at neutral
stability
if Friction factor
m Equation (2)
ie Nose radius of cavity
450
s Surface coordinate
Sc Streamwise distance to cavitation
separation
Sp Streamwise distance to boundary layer
transition
x Axial coordinate
a Air content
8 Nose angle of cavity
Y Angular coordinate
Ys Boundary layer separation angle
Yes Cavitation separation angle
6 Angle at which separation streamline
leaves wall
Momentum thickness
Dynamic viscosity
Kinematic viscosity
Liquid density
Cavitation number, (Po-Py) /40Vo2
Incipient cavitation number
Desinent cavitation number
Amplification factor
Velocity potential
ab
ar pre ne we ee
REFERENCES
Acosta, A. J. (1974). Cavitation and fluid machin-
ery. Cavitation Conf., Edinburgh, Scotland.
Adamson, A. W. (1966). Physical chemistry of
surfaces. Interscience Publishers, New York.
Arakeri, V. H. (1973). Viscous effects in inception
and development of cavitation on axisymmetric
bodies. Cal. Inst. Tech. Rep. E-183-1.
Arakeri, V. H. (1975a). A note on the transition
observations on an axisymmetric body and some
related fluctuating wall pressure measurements.
J. Fluids Engng., Trans. A.S.M.E. 97, 82.
Arakeri, V. H. (1975b). Viscous effects on the
position of cavitation separation from smooth
bodies. J. Fluid Mech. 68, 779.
Arakeri, V. H. (1977). Experimental investigation
of laminar boundary layer stability with surface
heating and cooling. N.S.M.B. Rep. 01054-2-CT.
Arakeri, V. H., and A. J. Acosta (1973). Viscous
effects in the inception of cavitation on
axisymmetric bodies. J. Fluids Engng., Trans.
Me Soicitie G5, Sale)
Arndt, R. E. A., and A. T. Ippen (1968). Rough
surface effects on cavitation inception. J.
Basic Engng., Trans. A.S.M.E. 90, 249.
Castro, W., and W. Squire (1967). The effect of
polymer additives on transition in pipe flow.
Appl. Scientific Res. 18, 81.
Darner, C. L. (1970). Sonic cavitation in water.
N. L. R. Rep. 7131.
Dobbinga, E., J. L. van Ingen, and J. W. Kooi (1972).
Some research on two-dimensional laminar separa-
tion bubbles. AGARD CP-102, Lisbon.
Ellis, A. T., J. G. Waugh, and R. Y. Ting (1970).
Cavitation suppression and stress effects in
high-speed flows of water with dilute macro-
molecule additives. J. Basic Engng., Trans.
AOS-MoE. 927 405).
Forame, P. C., R. J. Hansen, and R. C. Little (1972).
Observations of early turbulence in the pipe
flow of drag reducing polymer solutions.
A LeCiiek . JOULNAL LG m2 Sis
Gates, E. M. (1977). The influence of freestream
turbulence freestream nuclei populations and a
drag-reducing polymer on cavitations inception
on two axisymmetric bodies. Cal. Inst. Tech.
Rep. E-183-2.
Gupta, S. K. (1969). The influence of porosity
and contact angle on incipient and desinent
cavitation. M.S. thesis, Pennsylvania State
University.
Harvey, E. N., D. K. Barnes, W. D. McElroy, A. H.
Whiteley, D. C. Pease, and K. W. Cooper (1944).
Bubble formation in animals. J. Cellular and
Comparative Physiology 24, 1.
Holl, J. W. (1960). The inception of cavitation
on isolated surface irregularities. J. Basic
Engng., Trans. A.S.M.E. 82, 169°.
Holl, J. W., and A. L. Treaster (1966). Cavitation
hysteresis. J. Basic Engng., Trans. A.S.M.E.
88, 199.
Hoyt, J. W. (1973). Jet cavitation in polymer
solutions. A.S.M.E. Cavitation and Polyphase
Flow Forum, 44.
Hoyt, J. W., J. J. Taylor, and C. D. Runge (1974).
The structure of jets of water and polymer
solution in air. J. Fluid Mech. 63, 635.
Hsieh, D. Y. (1965). Some analytical aspects of
bubble dynamics. J. Basic Engng., Trans.
A-S.M.E. 87, 991:
James, D. F., and A. J. Acosta (1970). The laminar
flow of dilute polymer solutions around circular
cylinders. J. Fluid Mech. 42, 269.
Knapp, R. T., and A. Hollander (1948). Laboratory
investigations of the mechanism of cavitation.
TRANS) VAvsS). Mare Oya Os
Kiimmerer, H. (1976). Numerische Untersuchungen
zur Stabilitat laminarer Grenzschichten in
viskoelastischen Fliissigkeiten. Rheol. Acta 15,
579
Labrujére, Th.E. (1976). Berekening van de
wandinvloed voor twee lichamen in een water-
tunnel. National Aerospace Laboratory Rep.
NLR TR 76017 L.
Labrujére, Th.E., and J. van der Vooren (1974).
Finite element calculation of axisymmetric sub-
critical compressible flow. National Aerospace
Laboratory Rep. TR 74162 U.
Lumley, J. L. (1973). Drag reduction in turbulent
flow by polymer additives. Macromolecular
Reviews 7, 263.
Paterson, R. W., and F. H. Abernathy (1972). Trans-
ition to turbulence in pipe flow for water and
dilute. solutions of pclyethylene oxide. J. Fluid
Mech. 51, 177.
Persson, B. O. (1975). Reconsideration of a problem
in cavitation bubble dynamics. J. Appl. Mech.,
GRINS NoSoNGite CA Ts
Plesset, M. S. (1949). The dynamics of cavitation
bubbles. J. Appl. Mech., Trans. A.S-M.E. 16,
277.
Reed, R. L. (1969). The influence of surface
characteristics and pressure history on the
inception of cavitation. M. S. thesis, Pennsyl-
vania State University.
Rouse, H., and J. S. McNown (1948). Cavitation
and pressure distribution; head forms at zero
angle of yaw. State University of Iowa, Eng.
BUaOlyaees aie
Rudd, M. J. (1972). Velocity measurements made
with a laser dopplermeter on the turbulent pipe
flow of a dilute polymer solution. J. Fluid
Mech. 51, 673.
Schiebe, F. R. (1972). Measurément of the cavitation
susceptibility of water using standard bodies.
St. Anthony Falls Hydraulic Lab. Rep. 118.
Schlichting, H. (1965). Grenzschicht-Theorie,
Verlag G. Braun, Karlsruhe.
Silberman, E., F. R. Schiebe, and E. Mrosla (1973).
The use of standard bodies to measure the
cavitation strength of water. St. Anthony Falls
Hydraulic Lab. Rep. 141.
Thwaites, B. (1949). Approximate calculation of
the laminar boundary layer. Aero. Quart. l,
245.
Ting, R. Y., and A. T. Ellis (1974). Bubble growth
in dilute polymer solutions. Phys. of Fluids
17, 1461.
Van der Meulen, J. H. J. (1971).
hemispherical nosed teflon bodies.
Symposium, Leningrad.
Van der Meulen, J. H. J. (1972). Incipient and
desinent cavitation on hemispherical nosed bodies.
Int. Shipb. Progress 19, 21.
Van der Meulen, J. H. J. (1973). Cavitation supp-
ression by polymer injection. A.S.M.E. Cavita-
tion and Polyphase Flow Forum, 48.
Van der Meulen, J. H. J. (1974a). Friction reduction
and degradation in turbulent flow of dilute
polymer solutions. Appl. Scientific Res. 29,
161.
Van der Meulen, J. H. J. (1974b).
polymer injection on cavitation.
Conf., Edinburgh, Scotland.
Van der Meulen, J. H. J. (1976a)
Cavitation on
IUTAM
The influence of
Cavitation
Holographic study
451
of polymer effect on cavitation. A.S.M.E.
Cavitation and Polyphase Flow Forum, 4.
Van der Meulen, J. H. J. (1976b). A holographic
study of cavitation on axisymmetric bodies and
the influence of polymer additives. Ph.D. thesis,
Enschede.
Van der Meulen, J. H. J., and N. B. Oosterveld
(1974). A holographic study of cavitation
inception on a hemispherical nosed body. Symp.
High Powered Propulsion of Large Ships, Wagen-
ingen.
Van der Meulen, J. H. J., and H. J. Raterink (1977).
Flow visualization of boundary layers in water
by in-line holography. Int. Symp. on Flow
Visualization, Tokyo.
Van Ingen, J. L. (1975). On the calculation of
laminar separation bubbles in two-dimensional
incompressible flow. AGARD, CP-168, Gdttingen.
Van Ingen, J. L. (1976). ‘Transition, pressure
gradient, suction, separation and stability
theory. Workshop Rand Corporation, Santa Monica.
White, W. D., and D. M. McEligot (1970). Transition
of mixtures of polymers in a dilute aqueous
solution. J. Basic Engng., Trans. A.S.M.E. 92,
411.
Mechanism and Scaling of
Cavitation Erosion
Hiroharu Kato
University of Tokyo
Toshio Maeda
Mitsubishi Heavy Industries Ltd.
Atsushi Magaino
University of Tokyo
Tokyo, Japan
ABSTRACT
Recently cavitation erosion has been primarily
treated experimentally. However a need exists for
both a theoretical cavitation erosion model and
more quantitative erosion test methods. As a
contribution to the state of the art, the authors
have summarized their research at the University
of Tokyo using the soft surface erosion test method
(the aluminum erosion test).
Two test series were completed, the first using
the NACA 16021 foil section and the second using
the NACA 0015 foil section. Two-dimensional erosion
tests were systematically made at various velocities
and cavitation numbers to obtain a correspondence
between the erosion and the hydrodynamic character-
istics of the cavitation pattern. It was found that
the estimation of the cavity length and its fluctua-
tion are important factors in the prediction of the
cavitation erosion.
The results of these tests are used to illustrate
the effectiveness of Mean Depth of Deformation Rate,
MDDR, aS a Cavitation Erosion Index. These test
results also served as a background for extending
the cavitation erosion scaling theory, previously
proposed by Kato, to include differences in the
cavitation number.
After determining two empirical constants, the
resulting predicted MDDR Cavitation Index was shown
to be in good agreement with both Thiruvengadam's
(1971) and the authors' test results.
In addition to this basic research, two additional
studies are summarized. The first is a comparative
test of the aluminum erosion test and the paint
test and the second is a study in the influence of
air injection in reducing the cavitation erosion
intensity. The test results obtained from the paint
and aluminum tests were found to be in good agreement
and for routine cavitation erosion checks, the paint
test should be adequate. It was found that small,
air injection rates reduced the cavitation erosion
intensity dramatically and large injection rates
452
did not result in substantial reduction of the
cavitation erosion intensity.
1. INTRODUCTION
Erosion is one of the largest problems caused by
cavitation. Cavitation tests of model propellers
have been made for the purpose of predicting cavita-
tion erosion, especially for low-speed merchant
ships. However, the prediction was mainly based
on the observer's "feeling" of the cavitation
pattern on the propeller blade. Recently a new
testing method, i.e., paint test, was developed at
several laboratories [Sasajima (1972) and Lindgren
and Bjdrne (1974) ]. In this test the erosion inten-
sity is judged by the area of paint peeled off.
At the University of Tokyo in the authors'
laboratory, erosion tests of soft aluminum test
pieces have been made for several years [Sato et
al. (1974) and Sato (1976)]. The main purpose for
developing the soft aluminum method are:
(1) Development of a quantitative prediction
method for cavitation erosion.
(2) Obtain a deeper insight into the mechanism
of cavitation erosion by the observation
of eroded metal surface.
(3) Establishment of cavitation erosion scaling
laws.
The test piece is usually made of pure aluminum,
which is easy to obtain, has stable quality, good
machinability, and is relatively cheap. Its
mechanical properties can be roughly established
by hardness and tensile tests. The erosion resist-
ance of pure aluminum is very low and its surface
is roughed by cavitation attack within one half
hour of test exposure which is similar to the testing
time of the paint test. The increase in roughness
is a first indication of erosion [e.g., Young and
Johnston (1969) ]. It can be measured by a roughness
tester and the quantitative erosion intensity can
be obtained with sufficient accuracy.
Micro-appearances of the eroded surface such as
the pit shape, can also be qualified by examination
of roughness records and microscopic pictures of
the surface.
The erosion intensity has been evaluated by
mean depth of penetration (MDP) GoGo 5 Hammitt
(1969) ] or energy absorbed by the material eroded
l@aGio p Thiruvengadam (1966) ]. In addition one of
the authors recently proposed a new concept of
erosion intensity, mean depth of deformation (MDD)
which functions as a bridge between surface rough-
ness, SR, and MDP [Kato (1975) ]. Thus MDD corres-
ponds to SR at the initial stage and MDP at the
final stage of erosion.
This paper discusses the experimental results
of two-dimensional aluminum foil sections (pure
and aluminum alloy), various considerations of the
erosion mechanism in connection with the hydrodynamic
characteristics of the foil section along with the
modeling and scaling of erosion, and summarizes
experiments using an air injection system which the
authors found very effective in cavitation erosion
preventation. Nomenclature is shown at the end of
this paper.
2. FOIL SECTION EROSION TEST
High Speed Cavitation Tunnel at University of Tokyo
Erosion tests of two-dimensional foil sections
were made using a high speed cavitation tunnel at
University of Tokyo. The test sections of this
tunnel can be changed according to the experiment.
For the present test two test sections were used.
One was the rectangular high speed section with
cross section dimensions of 100mm x 10mm. Test
Series I was carried out using this section in
1976. Since the side wall effect was so large that
the two-dimensionality of the flow was almost lost
near the trailing edge of the foil section, it was
concluded that the 10mm width was too narrow. There-
fore the test section was modified to a 80mm x 15mm
cross section prior to starting Test Series II in
1977. The maximum velocity of the section was
about 50m/s.
The second test section was the rectangular low-
speed section used only in Test Series II (1977).
It has cross section dimensions of 120mm x 25mm
and a maximum velocity of 35m/s.
Foil Section
Two foil sections (NACA 16021 and NACA 0015) were
tested. The NACA 16021 foil section used in Test
Series I (1976), was the same foil section used in
Kohl's experiment [Kohl (1968) ]. Kohl made his
tests at an attack angle of a = 0°. Since this
foil section has no camber, when it is set at a =
O°, the inception point of cavity appears around
60% chord. Thus, testing at a = 0° was not suitable
for cavitation erosion tests, so the authors chose
a test condition of a = 4°. Since its chord and
Span are 40mm and 10mm respectively, the aspect
ratio A = 0.25, was so small that the spanwise
pattern of the cavity was not uniform. The cavity
closed at midspan appearing as a kind of streak
cavitation. Another disadvantage of using the NACA
16021 section is its chordwise pressure distribution
which is the "roof-top" type. The cavity length
453
drastically changes with only slight changes in the
cavitation number. While this characteristic is
desirable in practical applications, it was found
to be undesirable in the present study since erosion
would occur only in a narrow range of cavitation
numbers which makes the experiment difficult.
Therefore prior to starting Test Series II in
1977, two major improvements were made. From wind
tunnel tests the minimum aspect ratio necessary to
maintain two-dimensional flow was found to be about
4 = 0.4 and an aspect ratio, A = 0.5, was chosen
for Test Series II. The smaller foil was designed
with a 30mm chord and a 15mm span and the larger
foil section was designed with a 50mm chord and
a 25mm span.
The second improvement was to change the foil
section, from the NACA 16021 to the older NACA 0015,
which has a chordwise pressure distribution of the
"triangular" type. The experimental chordwise
pressure distribution of this foil is compared in
Figure 1 with the calculated pressure values. It
can be seen that the agreement between the experi-
ment and calculation is satisfactory.
Test Condition
In Test Series I (NACA 16021) the following items
were tested:
(1) Relationships between the mean depth of
deformation (MDD), mean depth of penetration
(MDP), and surface roughness (SR).
(2) Effect of cavitation number, velocity, and
the water's air content on the erosion
intensity.
(3) Comparison between the results obtained by
the soft aluminum erosion test and paint
EeSice
(4) Influence of air injection on erosion pre-
vention.
-2.0
—— THEoRY
EXPERIMENT
S55)
- @= U4 pec,
a=
Cp 2 DEG.
@= 0 pec.
-1.0 $ = - 4 Dec.
-0.5
0
0.5
1.0
FIGURE 1. Comparison of suction side Cp for NACA
0015 foil section.
454
In Test Series I, the size and material of the foil
section were not changed. The material was pure
aluminum, JIS H2102-2 (AL > 99.5%).
In Test Series II (NACA 0015) the following items
were tested.
(1) Effect of cavitation number, velocity, and
chord length, and the hydrodynamic character-
istics of the cavity flow, on the erosion
intensity.
(2) Effect of material properties on the erosion
intensity.
(3) Comparison of the soft aluminum and paint
test results.
The test conditions are summarized in Table 1.
The attack angle was a = 4° throughout Test Series
I and II. In Test Series I, the air content was
a/adg = 0.5, while in Test Series II it was initially
0.2 and increased gradually during the experiment
to a value of 0.4 by the end of the experiment.
In Test Series I-D, before the test began, air
bubbles were injected into the cavitation tunnel
to control air bubble content of the water. Then,
the erosion test was completed to study the effect
of air content on erosion.
Experiments with air injection from the foil
surface were also carried out to study the positive
Table 1
utilization of erosion prevention effect of air
bubbles.
At the start of the tests, the water temperature
was about 25°C which increased during the high speed
tests, reaching a maximum temperature of 50°C.
In addition to the erosion tests, measurements
of the hydrodynamic characteristics such as cavity
length, pressure distribution etc., were completed
using a similar foil section made of stainless steel.
Material and Heat Treatment
In Test Series I the foil section material was pure
aluminum (JIS H2102, 99.5%), while in Test Series
II pure aluminum and two kinds of aluminum alloy,
JIS H4163-2 (AA 5056) and JIS H4163-5 (AA 6063)
were used. These materials were selected for their
low erosion resistance, good corrosion resistance,
and good machinability. The foil sections tested
were machined by a NC-milling machine and the surface
was smoothed by a buffing machine. The foils'
surface roughness was found to be less than 1 um
in the virgin state.
Since the foil surface was work-hardened, a thin
layer of the foil surface had a large degree of
Experimental Conditions
Series I
ie
Cav. No. & Flow Vel.| Duration
Material JIS*
Attack Angle
Flow Velocity
Cavitation
Number
Exposure Time
Air Content
Material
JIS (AA)*
Chord 30 mm, Span 15 mm
HA4LE3—5 Hh163—2
(5056)
(6063)
Flow Velocity
Cavitation
Number
Air Content
4v8 ppm (a/as** =0.200.4)
* JIS Japanese Industrial Standards
AA : The Aluminum Association
**¥ Qo : Saturated Air Content at 25°, 1 ata.
455
Table 2 Chemical Composition and Mechanical Properties
Pure Aluminum
JIS H2102-2
Aluminum Alloy
JIS H4163-5| JIS H4163-2
(AA 6063) (AA 5056)
v0.10
Chemical
Composition
Stress
Tensile
2
Mechanical Strength (kg/mm
Vickers
Properties
Hardness (kg/mm?
Young's
Modulus
(kg/mm?
hardness, requiring heat treatment to remove this
work hardened layer. Following the Japanese
Industrial Standards, pure aluminum and aluminum
alloy H4163-2 were annealed for 1 hour at 400°C
and foils made of aluminum alloy H4163-5, were
annealed for 1 hour at 205°C.
The surface hardness before and after annealing
are shown in Figure 2. This test was made using
a micro Vickers hardness tester. The tensile test
results are shown in Figure 3 and summarized with
the composition of the materials in Table 2.
Surface Roughness (SR), Mean Depth of Penetration
(MDP), and Mean Depth of Deformation (MDD)
For this study a NACA 16021 foil section was
tested for 9 hours to find the relation among SR,
MDP, and MDD. The result is shown in Figure 4.
When a ductile material such as aluminum is exposed
to cavitation, small pits detected by an increase
60
Pure Acuminum (H2102-2)
Non-ANNEALED Pas
7
ANNEALED
Hv (KG/MM)
8
9?
°
|
°
MEAN RoucHNess (gem)
0 200 400 60(
Static Loap (G)
FIGURE 2. Result of Vickers hardness test [pure
aluminum (H2102-2)].
in SR are formed at the first stage of erosion.
At this stage there is no weight loss. This initial
period is called the incubation period where after
an initial increase, the SR value asymptotically
approaches a larger value.
It is well known that MDP remains zero during
the incubation period. The time rate of MDP/(MDPR)
increases to the maximum (acceleration period) then
decreases gradually (deceleration period). Asa
measure of erosion intensity the value of MDD, pro-
30
25
ind
oOo
rH
al
Stress (KG/MM”)
0 5 10 15 20 25
STRAIN (%)
FIGURE 3. Comparison of tensile test result.
456
i
Oo
WN
Oo
10
ho
oO
x
iS)
Wwe
o——o—°-
Mean ROUGHNESS 5
joo
f=)
We1GHT Loss (mG)
Mean DEPTH oF DEFORMATION (gem)
we
0 — @—«@ 0
0 2 4 € 8 19
Time (HR)
FIGURE 4. Extended duration cavitation test [NACA
Pure Al (H2102-2), C = 40 mm, a = 4 deg.].
posed by one of the authors, seems to be more
suitable than MDP. The advantages of using MDD
are that it increases almost linearly over a wide
range of exposure time as well as the fact that MDD
corresponds to SR in the incubation period and to
MDP after long exposure.
In the present tests, SR was measured to shorten
the testing time. Usually the test was completed
within 1 hour so the SR value coincides with MDD.
The degree of erosion after a long exposure can be
estimated using the measured SR.
3. HYDRODYNAMIC CHARACTERISTICS OF CAVITATION ON
NACA 0015 FOIL SECTION
Cavity Length
Because erosion occurs at the collapsing point of
the cavities namely the end of the cavity, it is
important to know the cavity length for predicting
cavitation erosion.
tests, the cavity length and pressure distribution
along the back surface of the NACA 0015 foil were
PROBABILITY (2)
Therefore, prior to the erosion
measured. At the test condition 50 photographs
were taken to measure the cavity length.
The results are shown in Figure 5. As seen in
the figure, above o > 0.8 the distribution of cavity
length is characterized by a peak, but below o < 0.8
the fluctuation becomes so large that there is no
characteristic peak. For the supercavitation
condition (o = 0.45) the fluctuation is reduced and
a characteristic peak can again be observed. The
mean value of cavity length and its standard devia-
tion are shown Figures 6 and 7. The cavity length
increases linearly with smaller cavitation number,
and the standard deviation begins to increase
rapidly about o = 0.85 as clearly seen in the figure.
It is well known that the cavity length of a
partially cavitated foil can not be determined
theoretically by linear cavity models. The cavity
length predicted by a closed type cavity model is
usually longer than the observed length. If we
adopt a open type cavity model., the situation
becomes reversed and the predicted cavity length
becomes shorter than the observed length. Conse-
quently a half-closed type model is usually adopted,
but this model requires the opening of the cavity
end to be determined experimentally.
In this study the cavity length was calculated
using the half-closed type model by Nishiyama and
Ito (1977). This method is based on linear theory
using singularities (source and vortex) distributed
on the cavitated foil. The calculated results are
shown in Figure 7 where the opening de was system-
atically changed. The contour of de = O coincides
with the closed cavity model. The circles in this
figure represent the "mean" value of the observed
cavity length. Using this mean value, the opening
6e can be calculated showing that de increases
with smaller values of o (see Figure 8).
Pressure Distribution and Cavity Shape
The theoretical pressure distribution and cavity
shape for the back side of NACA 0015 foil section
are shown in Figure 9 along with the corresponding
experimental result. Here the Nishiyama-Ito's half-
closed model was used with the de values taken
0
0
PROBABILITY (2)
FIGURE 5. Fluctuation of cavity length (NACA
0015, «a = 4 deg., V = 35.9 m/s).
100
REGS eso
&
Ena
=
rd
S
nig nO Le
50 100 150 0 50 100 150
Cavity LenctH (&CHoRD)
PROBABILITY (4)
50 100 150
Cavity LENGTH (%CHoRD)
=
oO
© V = 35 M/s
@ V= 25 m/s
Ww
oO
4 V=15 m/s
ip]
oO
=
oO
STANDARD DEVIATION (ZCHoRD)
oO 0.6 0.8 1.0
CavITATION NUMBER
FIGURE 6. Standard deviation of measured cavity length
(NACA 0015, a = 4 deg.).
from Figure 8. The pressure distribution diverges
to a positive infinite value at the end of cavity
because of singularity at this point. This singu-
larity makes the agreement between theoretical and
experimental results very poor.
The cavity shape is also compared in Figure 9.
The observed leading edge of the cavity is about
10% chord position. Whereas, in the theory the
leading edge of the cavity begins at the leading
edge of the foil. This appears to be one of the
reasons why the calculated cavity thickness is
much thicker than the experimental thickness even
though the cavities have similar profiles.
4. EROSION TEST
Cavity Length and Position of Erosion
The roughness increment on the foil was measured
for various exposure times. Spanwise roughness
measurements were made over the entire chord at
intervals corresponding to 5% the chord length.
120
THEORY
100 —-— EXPERIMENT
0..08C
30 0.06C
0. 04C
60
457
Two examples of the roughness distribution are
shown in Figure 10. Arrow marks in this figure
indicate the position of cavity end and the standard
deviation of its fluctuation.
The figure clearly shows that the peak of erosion
appears slightly downstream of the cavity end, and
the erosion distribution agrees well with the cavity
fluctuation. Namely, there is an obvious peak in
the region of o > 0.8, but in the region of o < 0.8
the surface roughness distribution spreads over a
wider range. This result indicates that the esti-
mation of cavity length and its degree of fluctuation
are important factors in the prediction of erosion
intensity.
Effect of Hydrodynamic Factors on Erosion
Cavitation Number
The mean increment of surface roughness, SR, and
its time rate of change can be determined from the
roughness distribtuion shown in Figure 10. It
corresponds to the mean depth of deformation rate
(MDDR) because the test was finished within the
incubation period. While Thiruvengadam has proposed
adopting the rate of energy absorbed by the eroded
material, which can be calculated by multiplying
MDP by the energy absorbing capacity of the material
per unit volume, the present research uses MDDR as
a measure of erosion intensity in order to find
which property is responsible for cavitation erosion.
It is known that the erosion intensity, MDDR,
has a peak at the certain cavitation number. The
change of measured MDDR to cavitation number is
shown in Figure 11, where plots (a) and (b) refer
to the NACA 0015 foil tests while plot (c) refers
to the NACA 16021 foil tests. The test result of
Kohl and Thiruvengadam are also presented in plot
(a) [Kohl (1968) and Thiruvengadam (1971) ]. As
mentioned earlier, while the same foil section
(NACA 16021) was tested in Test Series I, a different
attack angle was used.
There are several differences in the results
obtained in the NACA 0015 foil tests and the NACA
= 35 m/s
= 25 m/s
= 15 m/s
o.czc _8,-0
40
Cavity LENGTH (ZCHORD)
20
0.4 0.6 0.8 1,0
CaviITATION NuMBER
FIGURE 7. Comparison of calcu-
lated and observed mean cavity
length (NACA 0015, a = 4 deg.).
1.2 1.4
a
i:
So
3S -1.0
ss
o
Ce
4.0 gl
“0.60 0.80 1.0 1,20 HEORY
CavITATION NUMBER
EXPERIMENT
: , : ; LE.
FIGURE 8. Derived relationship between Se and cavita- Ghern
tion number for NACA 0015 foil, a = 4 deg.
FIGURE 9. Comparison of NACA 0015 foil calculated
cavity shape and Cp distribution with experiments at
a= 4 deg.
STANDARD
DEVIATION
J PEND one
30 MIN Cavity °°
°
STANDARD
<—— DEVIATION
it oF Cavity
—?P
100
MDD (pom)
FIGURE 10. Illustration of MDD
(Mean Depth of Deformation) data
: 0 20 40 60 80 100 0 20 49 60 80 100
koi NITES, (IGM OR EOS Ipsithas L.E, CHorD Position (CHORD) T,£, LE, CHORD Position (%CHORD) T,E,
aluminum (H2102-2), a = 4 deg.,
V = 35 m/s]. (a) C = 50 mm (6) C = 30 mm
459
SON Z102—2
— e— H4163-2 (5056)
—4— H4163-5 (6063)
°
S
=
=
a =—_
= cs
= <
2 st
a
a1
(=
—
i=)
=
0.6 0.8 1.0 ney?
Cavitation NuMBER
(a) NACA 0015, H2102-2, 0.6
a= 4deg., V = 35 m/s
1.0 oe
CAVITATION NUMBER
(b) NACA 0015, C=30 mm
a=4deg., V = 45 m/s
41,7 m/s
32.6 m/s
are
MDDR (pe m/min)
2
Peak Erosion INTENSITY (WATT/M’ )
0.6 0.25 0.30
0.2 0.4
CAVITATION NuMBER
(c) NACA 16021, H2102-2,
C = 40 mm, a= 4 deg.
16021 foil tests. First, the width of the peak of
NACA 16021 is narrower than the NACA 0015 peak.
This is caused by the difference of pressure distri-
bution between the two foil sections. The NACA
16021's distribution is flat, resulting in a larger
change of cavity length with small changes in
cavitation number. In contrast, the NACA 0015
section has a triangular pressure distribution so
the difference between the inception cavitation
number and supercavitation number is large. Since
erosion occurs only when the cavity bubbles collapse
on the foil surface, it seems quite reasonable that
NACA 0015 has a much wider peak than the NACA 16021.
Here the authors would like to point out that due
to side wall effects the measured pressure distri-
bution of the NACA 16021 foil and the peak value
of o = 0.4 can not be obtained directly by a two-
dimensional calculation.
0.35
CavITATION NUMBER
(d) NACA 16021, 1100F-A1
(Thiruvengadam, 1971)
0.40 0.45
FIGURE 11. Summary of MDDR erosion
index and 'test results.
Another difference between these two results is
the value of the maximum MDDR. It is much larger
for the NACA 16021 foil when compared with the
NACA 0015 foil results, even though the chord length
and test velocity are not that much different. The
main reason lies in the difference of cavity pattern.
With the NACA 16021 section, the cavity inception
is concentrated at the mid-span position and the
cavity was a streak type. Correspondingly, the
erosion pattern was a streak type, where a narrow
and deep eroded groove was formed along the middle
of foil. A picture of this groove taken by a
scanning electron microscope is reproduced in Figure
12. Streak cavitation can induce severe erosion in
comparison to sheet cavitation erosion which occurred
in the NACA 0015 foil tests. The difference in the
cavity patterns seems to cause this large difference
in MDDR.
460
FLOW
DIRECTION —>
FIGURE 12. Scanning electron
microscope photographs of eroded
surface (NACA 16021, H2102-2,
Cc = 40 mm, a = 4 deg., V = 41.7
m/s, o = 0.450).
Referring to Figure 11 (b) in the 3 test series
where only the material of the foil was changed,
the position of maximum MDDR changes. This seems
irrational because the flow condition is not changed
by the material. The reason of this shift is the
occurrence of the foil's bent trailing edge. Ona
full scale propeller, cavitation erosion is some-
times accompanied by a bent trailing edge. The
same thing happened in the present test. The: foil
section made of pure aluminum is much weaker than
those made from an aluminum alloy, and it was bent
more at the trailing edge causing the shift of
peak MDDR to the larger cavitation number.
An example of a bent trailing edge is shown by
the profile view in Figure 13. The amount of bend
is large at the corner of the trailing edge, which
exaggerates considerably the shape shown in this
figure. The bent trailing edge was observed on
every NACA 0015 foil sections when the erosion
occurred. On the contrary, it hardly appeared on
“ Poe
Eropep REGION
NACA 16021 foil section because of its thicker
trailing edge.
Velocity
It is well known that the erosion intensity is af-
fected very much by the mean velocity since Knapp's
suggestion of 6th power law [Knapp et al. (1970) ].
The effect of velocity on the peak value of MDDR
is shown in Figure 14. Usually the exponent obtained
‘experimentally, has a large spread falling somewhat
between 3 and 9. In the present tests with the
NACA 16021 foil the exponent, n, was 9 and for the
NACA 0015 foil tests the exponent, n, was 6.
Chord Length
The chord length of a foil also has a large ef-
fect on the erosion intensity. This is very
(a) BEFORE EXPERIMENT
(b) AFTER EXPERIMENT.
FIGURE 13. Impression of bent trailing edge.
important for marine propellers where the scale
ratio between a full scale propeller and its model
is large. Sometimes this ratio exceeds 30. As
mentioned above, while the effect of the velocity
difference is very large, we can still make a model
test with the same tip speed as full scale by
increasing the revolution of the model propeller.
However it is very difficult to reduce the scale
ratio of chord length.
Experimental verifications on this problem are
also very poor. Thiruvengadam (1971) made his
erosion tests using two chord lengths, 1.5 and 3 in.
His result shows that the erosion intensity increases
proportional to the chord length. The result
obtained in the present test is shown in Figure 15.
In the present tests the erosion intensity increases
proportional to the square of chord length. The
effects of hydrodynamic factors such as cavitation
x10"?
15
MDDR ( pem/min)
ine)
0,2
20 40 70 100
VELocITY (m/s)
FIGURE 14. MDDR vs. velocity
(NACA 0015 : H2102-2, C = 30 mm,
a = 4 deg.) (NACA 16021 : H2102-2,
C = 40 mm, a = 4 deg.).
461
MDDR ( pem/min)
0,3
20 40
CHorD LENGTH (mM)
70 100
FIGURE 15. MDDR vs. chord length
(NACA 0015, H2102-2, a = 4 deg.,
V = 35 m/s).
number, velocity, and chord length can be explained
universally by a model of erosion mechanism. The
details of this model will be given in Section 5.
Air Content
The effect of air content was examined using the
NACA 16021 foil section results. The air content
was controlled as follows. As a pretreatment, the
water was degassed to about 8ppm in a vacuum chamber
and introduced into the cavitation tunnel. Then a
certain amount of air was injected into the tunnel
through an injection port before the test. In this
case the ratio of gaseous air to total air content
is much greater than found in ordinary water where
the amount of air is an order of parts per million
of total air content [Ahmed and Hammitt (1969) ].
With increase of air content the value of MDDR
decreases as seen in Figure 16. This tendency
agrees with the test results of SSPA [Lindgren and
Bjarne (1974) ] and those of Stinebring et al.
[Stinebring et al. (1977)]. The reason is attributed
to the damping effect of air in a collapsing cavity
bubble, attenuation effect of tiny air bubbles to
shock wave, or a combination of both.
Material Properties
The effects of material properties on erosion are
usually tested by accelerating devices such as
vibrators, rotating discs, water jets etc. Summa-
rizing these results, Heymann has made the chart
shown in Figure 17 where the hardness of the
material was taken as a factor governing the erosion
[Heymann (1969)]. As seen in the figure the slope
differs according to the material group, namely
the slope of the steel group is steeper than that
of aluminum and copper and brass group. This implies
that the erosion resistance cannot be fully repre-
sented by hardness alone. Thus other material
properties such as strain energy absorbed to material
(engineering strain energy) [Thiruvengadam (1966) J,
ultimate resilience [Hobbs (1966) ], or their com-
MDDR ( pen/min)
0 20 40 60
Air Content (PPM)
FIGURE 16. Effect of air content on MDDR
(NACA 16021, H2102-2, C = 40 mm, a = 4 deg.,
V = 41.7 m/s, o = 0.443).
bination [e.g. Hammitt et al. (1969) ], have also
been proposed by several researchers.
The present test results are also compared with
those material properties, i.e., hardness, engi-
neering strain energy, and ultimate resilience.
Hardness seems to give the best representation as
seen in Figure 18. This will be discussed in Section
5 dealing with modeling the erosion mechanism.
5. THEORETICAL CONSIDERATIONS
Review of Erosion Scaling Theory
Thiruvengadam has made several theoretical. consid-
erations on scaling of erosion. In 1971, he
introduced a scaling formula [Thiruvengadam (1971) ].
He assumed a statistical distribution of air nuclei
and derived the efficiency of erosion, 6, as,
_ o& 1 -2.67
6 = 5 (As) €XD F(A),
where 6, 0, Ao, and W are nondimensional nuclei
size, cavitation number, degree of cavitation, and
Weber number respectively. Equation (1) is very
attractive because it has no empirical constants.
However the calculated values are quite different
from the experimental values. While 9 should be
the order of 10° by the calculation, the 8 obtained
from model tests typically has an order of 10710,
This discrepancy comes from the assumption that the
total energy of the cavity bubbles generates the
erosion. The theory shows that when the cavitation
number is reduced, the efficiency, 8, increases
from the point of cavitation inception to a maximum
and then decreases to zero when cavitation number
reaches zero. This tendency agrees qualitatively
with experiments. It is expected, since the actual
cavity becomes a supercavity at a certain cavitation
number causing the erosion intensity to decrease
greatly and in a practical sense reach zero.
One of the authors has proposed a model of erosion
mechanism in which the discharged energy of the
collapsing bubble is assumed to be distributed
statistically as:
a6), === - —
) =a € exp € (2)
where f is the distribution function of energy
density, €, reached on the material surface. Then
a scaling law for cavitation erosion was derived
using an empirical formula for the erosion resis-
tance of materials. A comparison with only the
peak erosion intensity taken from Thiruvengadam's
tests showed good agreement [Kato (1975)].
Consideration on Effect of Cavitation Number
As mentioned before, MDDR has a peak value of a
certain cavitation number. This is due to a
combination of the following two reasons. There is
an increase in the collapsing cavity volume as the
cavitation number decreases which causes increased
erosion. On the other hand, the decrease of cavita-
tion number causes an increase in the cavity length
so the eroded area shifts towards the trailing edge
of a foil. Also when the cavity length exceeds the
chord length, the cavity does not collapse on the
foil surface, causing no cavitation erosion.
Usually the cavity length fluctuates and the erosion
intensity will change continuously with the cavita-
tion number. Although there seems to be a consider-
able decrease in the collapsing pressure of cavity
decreasing cavitation number, the control factor
of erosion intensity is the change of cavity length
as mentioned above.
The decrease of erosion intensity at the right
hand side of the MDDR peak in Figure 19 is caused
by the lack of cavity and by too long a cavity on
left hand side. By increasing the cavitation number,
the cavity becomes intermittent, and if the cavity
is stabilized by roughing the leading edge, the
MDDR peak shifts to a higher cavitation number
where the peak value is increased. This was verified
in the authors' experiments as shown in Figure 20.
100
0.1
0,06
0), 5010)
10 Hv (k6/MM?)
(b) CopPER AND Brass
NorRMALIZED EROSION RESISTANCE (NE )
0.01
100 1000 15 100 300
Hv (kG/MM*) Hv (kG/MM-)
(a) STEEL (c) ALUMINUM ALLOY
FIGURE 17. Vickers hardness vs. erosion
resistance [Heymann (1969)].
V/MDDR (r1N/ pe)
1/MDDR (MIN/ gm)
20 40
Hv (Ke/MM")
70 100 ] 2 ih 10
S.(kG/mm" )
(a) Vickers Hardness (b) Engineering Strain Energy
300
200
/MDDR (mIN/ pe m)
20
10
0.007 0.01 0.02 0.04 0,070.1 0.2
UR (k6/Mm*)
(c) Ultimate Resilience
FIGURE 18. MDDR vs. various mechanical
properties of material (NACA 0015, C = 30 mm,
a = 4 deg., V = 45 m/s).
Modelling of Cavitation Erosion and Scaling Factors
As mentioned above, one of the authors developed
a model of the cavitation erosion mechanism. How-
ever it is limited to only constant cavitation
numbers and the effects of material properties were
derived empirically from accelerated tests. In the
present paper, this model is developed further to
treat differences in the cavitation number. The
effect of the material's mechanical properties is
also studied and a simple model is introduced.
The total energy of collapsing bubbles per unit
is given as:
E. = n(p-pv)O , (3)
probability of bubble collapses on a
foil’ surface,
where nN
P-Py : pressure difference at the collapse
point,
Q : volumetric flow rate of cavitation
bubbles.
Equation (3) can be modified:
463
GovERNING Factor
<—
Cavity LENGTH
—_
Cavity VoLUME
EROSION INTENSITY ——»
(MDDR)
CavITATION NuMBER —»>
FIGURE 19. Illustration of MDDR peak characteristic
(test data given in Figure ll).
E, * 1(Po-Py) 5 BV
« nopv? eS) es (4)
where 6 displacement thickness of cavitation
bubbles,
B : foil] span,
de : cavity thickness at the cavity end,
V : velocity,
L : reference length.
Assuming that a cavity bubble grows according to
Knapp's similarity law, the volume,V, is:
Vera (wg = ye (5)
where T = ay where A is the cavity length.
The pressure difference, Ap, is assumed as
x10°?
3.0
© SmooTH SURFACE
@ RouGHEeD aT LeaDING EDGE
2.0
1.0
MDDR ( pem/miNn)
0.3 0.4 0.5
CavITATION NUMBER
FIGURE 20. Impression of effect of roughened leading
edge [NACA 16021, tested by Ozaki and Kiuchi (1975)].
464
Ap © Py-Pmin
Ga (ot Gam) OF - (6)
Combining Eqs. (5) and (6), the following equation
is derived.
3
v « 3 [-(o + Cryin) | (7)
The number of cavity bubbles per unit time is then
given as,
de V
va @e
7 (8)
= 3
r3n [-( © + CPmin) ]2
where \ as the nondimensional cavity length, X=
A/D Se is the nondimensional cavity thickness at
the end, and de = Se/L.
Here, we make the same assumption as in the
previous paper [Kato (1975) ] on the statistical
energy distribution of cavity bubbles. The distri-
bution is given as,
n = cE exp (-aE) 9 (9)
where n is the number of bubbles per unit time
whose energy is between E and E + dE. Total number,
N, and total energy of bubbles, E;, are given as
follows:
foe)
c
N= a ndE = a2 (10)
0
Constants a and c can be decided by combining Eqs.
(4), (8), and (10).
1
ES
pp
Il
No
nov2L2r3[-(o + CPmin) |
oom (alah)
ki de
Se Ras Se 9
n2p2v3L7A902[-(0 + CPrmin) ]”
where Kj} and K} are constants independent of the
chord length, velocity and cavitation number. From
Eq. (9), the distribution function of energy density,
f, is derived as a function of energy density, €.
The detailed discussion of this point is given in
the previous paper [Kato (1975) ].
Substituting Eqs. (9) and (11) into the relation
£(E) coon (ele) i, (12)
the final expression for f is
f = C € exp (-Ae) f (13)
where A=
= x)
n2p2v3L32\ 202 [-(o0 + Cp.) ]2
min
In the present case the chord length is taken as a
suitable reference length, L.
Equation (13) is similar to Eq. (2), but it is
extended to include differences in the cavitation
numbers.
The next problem is the modelling of deformation
of a material surface caused by the attack of
collapsing bubbles. For the present tests, hardness
seems the best property to express the erosion
resistance of a material. However it was found to
be insufficient as seen in Figures 17 and 18.
The methods of hardness testing can be divided
into two types. One is the measurement of a dent
size caused by the static load of a sphere or a
pyramid on the material surface. The other method
is the measurement of absorbed energy from dropping
a certain test body on the surface. The Vickers
hardness test made in the present study belongs to
the first type.
When a pyramidal dent whose depth is d, is
formed by a static load F (Figure 21), the energy
used to the deformation is
iy Gaiotel 9B (14)
The hardness has the following relation by its
definition.
(15)
The increase of surface roughness (SR) by the single
dent is given as
wv . ae
Se SS 16
SRES aia) rayne (16)
where Y and S are the volume of the dent and refer-
ence area, respectively.
Combining equations (14) ~ (16),
i
a p
Gos aera vie STE 17
oe EL S H ee
Diamond PYRAMID
FIGURE 21. Model of Vickers hardness test method.
where e
atecraieulo
is the energy density absorbed by the
s plastic deformation.
If e is small
enough, the deformation is within the elastic limit
and no permanent dent will be formed. When e
exceeds a certain limit, e,, the plastic deformation
of surface occurs and a pernament dent is formed.
Then the following relation is derived:
a) = 0 fore < ee
Qa. = © 2°
Pp @ for e > Se 9
(18)
The above mentioned argument is valid for the actual
case of erosion where many cavity bubbles collapse
in a certain period if e is substituted tO}, Er, en
these equations.
Then,
—e =0 for € <e
P
€ =e-e LOnReescme * (19)
p c
and
oO
MDDR « — € £(e)de
Vv
Eo
ao
ay (e-e)c (-ae)d 20
a EQ e exp €)de (20)
Vv
we
Integrating Eq. (20),
K ov3 €
MDDR = —~ —— g (a)( 2+ —1"_)
Loy dele oV LF (o)
ic
ex | = FI (21)
OV2LF(o)
where
= 3
F(o) = naso [-(o + Cp in) J?
G(o) = node o
Here F and G are functions of cavitation number,
where G is proportional to the total energy of the
cavity reaching to the surface, and F is related
to the individual energy of each cavity bubble.
The probability of the bubble collapse on the
foil surface, n, is calculated using the estimated
mean position of collapse and its fluctuation. In
the case of the NACA 0015 foil section, the position
was estimated as 1.3 A from Figure 10 and the
fluctuation is assumed to be the same as the cavity's
fluctuations. The thickness at the end of cavity
is taken from Figure 8. The value of F andG for
NACA 0015 section were calculated at a = 4°. The
results are shown in Figure 22.
While the critical value of energy density, Er
should be expressed by the mechanical properties of
465
material such as yield strength, Young's modulus
etc., at the present stage, for lack of data we
assume the following relation,
Yield strength (22)
aC y o oy
and determine the power, n, from the erosion experi-
ments.
Comparison with Test Result
The results of this theoretical model are compared
with the erosion test of NACA 0015 section in
Figure 23 where the two constants, K, and Ky, in
Eq. (21) were determined using two different test
points. In this figure those points are shown by
dashed marks. The value of the power, n, was taken
as n = 1/4 from the experimental results. The
agreement between this theory and the test results
is satisfactory.
The theory was also compared with Thiruvengadam's
test result [Thiruvengadam (1971)]. In this case,
no data about the cavity was measured, so only the
peak value of erosion intensity was used in this
comparison with the present theory. The agreement
is almost perfect as seen in Figure 24 where one
set of data was used to determine two constants.
Photos in Figure 23 (b) also show the paint test
results discussed in the next section.
Paint Test and Soft Aluminum Erosion Test
Recently the paint test has been routinely used at
several research laboratories to predict erosion
intensity, in contrast to the present research using
the soft aluminum erosion test to predict erosion.
Both of these two test methods have merits and
demerits. The soft aluminum erosion test is some-
what troublesome and the surface of the material
5 O26 0.7 0.8 0.9 1.0 Itai
CAVITATION NUMBER
FIGURE 22. Derived F and G values for NACA 0015 foil
section at a = 4 deg.
$/W Gp = A ‘WwW Og = 0 ‘(Z-ZOLZH) WnuIWN|y and (4)
YdadWNN NOTLVLIAW)
al OT 60 3'0 Z'0
s}[nsey qsey juted
(NIN/Wm™) YCaW
“(“bep 7 = 2
“STOO WOWN) S3INsexr [TeqUueUTrsedxs
UQTM XO8puT UOTSOZe YddW PezOTp
-o01d Jo uostzeduod *ez Tena
YBEWNW NOLLVLIAV)
weil Olea nO 8'0 Z'0 Oi) 0)
wy
S
(NIW/WTT) YW
YIGWNN NOILVLIAV)
els OT 6'0 8'0- £0 oH Sh)
0
S
3
>
=
9 S'0 =
J
oT
7-01
UBAWNN NOLLVLIAV)
iCall O'T 6'0 8°0 E0950 S'0
0
0'T
Ss
o
=
zr
=
=
072
S/W Gp = A ‘‘bapp=0
‘wl O€ = 9 ‘(9S0S
‘C-EQLPH) AON WnuiWNiy (Pp)
s/W Gp = A ‘bapp=0
‘WwW Of = 9 ‘(£909
"G-EQLb~H) AOly WnuiwWNn|y (9)
s/W GE =A
“(Z-ZOLZH) Wnuiuinjy aing (2)
466
nN
f=)
a
(=)
oO
MNS
S)
JS
9,2 THIRUVENGADAM
2
Peak Erosion INTENSITY , IE (waTT/M’ )
o 1.5 IN.FoIL
© 3,0 In.FoIL
0.1
100 (F/s) 200 300
a a
40. 60 80
V (m/s)
FIGURE 24. Comparison with
Thiruvengadam's data (NACA
16021).
is destroyed, as a matter of course, after a long
exposure to cavitation. But as mentioned in Section
1, it has the merits of yielding quantitative and
reliable erosion data, a similar appearance of the
full scale eroded surface, etc.
The paint test has just the opposite merits. It
is a cheap and handy method. And although the
conditions under which the paint is removed changes
with very small changes in the paint composition,
test procedure, etc., it appears that by developing
standards, the paint test can be used to represent
relative differences between similar models.
From this discussion of the paint test merits
and demerits, the paint test appears suitable for
daily routine tests of usual propellers. The soft
aluminum test is suitable for making standard com-
parative tests at different research laboratories
as well as for different types of propellers and
for situations where critical erosion predicitions
are required.
It is valuable to make a comparison of these
test methods using the same foil section. After
testing several kinds of paint, a marking paint
"AOTAC" was found to be the best. Figure 23 (b)
shows appearances of the painted surface after 5
min. test. They can be compared with the theory
and the soft aluminum erosion test results shown
in the same figure. The cavitation number of
maximum erosion intensity is slightly different
between the paint test and theory. But the general
tendency agrees well and the paint test seems very
useful especially for a comparative testing.
The position of maximum erosion intensity esti-
mated from the paint test also agrees well with
the chordwise distribution of MDD shown in Figure
10.
6. AIR INJECTION SYSTEM
Tiny air bubbles in the free stream reduce the
erosion intensity by the action of their damping
effect as mentioned in Section 4. To achieve a
positive damping effect an air injection system
467
with air bubbles injected from holes on the foil
surface is sometimes adopted. This system has
been used very effectively to prevent erosion on
the inner surface of a full-scale ducted propeller
(Ooo p Okamoto et al. (1975) and Narita et al.
(1977) ]. However the mechanism of prevention is
not yet fully explained, and the best injection
position and/or the necessary amount of air injection
have not been clarified.
The authors made the air inject test using NACA
16021 foil sections with three air injection holes
of 0.5mm dia. drilled at 10% or 37.5% chord position
(Figure 25). The tests were made at a = 4°,
V = 41.9m/s, and o = 0.438. The previous test
showed that the peak MDDR value falls somewhere
between 40 ~ 45% chord. The injection position
of 10% chord represents the injection near the
leading edge of the section, and that of 37.5% chord
represents the injection which insures effective
coverage of the eroded area. Air was then injected
at 2, 5, and 10 cc(normal)/min. The quantity of
air was so small that separate air bubbles were
found even at the 10 cc/min, and consequently the
air jet column typical at high flow rate was not
observed. As seen in Figure 26, the injection
from 10% chord gives better performance and even as
small a rate of the injection as 2 cc/min results
in drastic decrease in the erosion intensity. With
injection the MDDR value reduced to 1/5 of non-
injection level. Increasing air volume, the value
of MDDR decreases but the effect seems to become
saturated with a larger rate of air injection.
7. CONCLUSIONS
(1) The purpose of the present research was to
find the mechanism of cavitation erosion and its
scaling laws with special reference to the relation-
ship between the appearance of cavitation and the
erosion intensity.
(2) Detailed observations of the cavity pattern
were made on a two-dimensional foil section (NACA
0015). Then erosion tests, using the same foil
section of pure aluminum and aluminum alloy, were
made to measure the increase of surface roughness.
The erosion intensity was also compared with the
observed cavity pattern and other hydrodynamic
0.1C or 0.375C
ERoDED REGION
(0.45 - 0.5C)
FIGURE 25.
Location of air injection.
468
©0 cc/MIN
Inuection Position 0,375C
MDDR (ypm/mIN)
oD Ce/MIN
wn
o
Invection Position 9.1C
ye
2 cc/MIN
‘4.
ae cc/MIN
4
10 cc/MIN
0 5.0 10.0 |
INJECTION RaTE (A/V - S) x10
FIGURE 26. Effect of air injection on MDDR erosion
index [NACA 16021, pure Al (H2102-2), C = 40 mn,
a = 4 deg., V = 41.9 m/s, o = 0.438).
factors such as cavitation number, water velocity,
etc.
(3) Modelling of cavitation erosion has been made
assuming a statistical distribution of cavitation
bubble. Using the model, a theory of erosion scaling
was established which contains two constants given
by the experiment. The erosion scaling of cavita-
tion number, velocity, chord length, and material
can be made by the theory. The theory has been
shown to give good agreement with the authors' and
Thiruvengadam's tests.
(4) Another two-dimensional foil section (NACA
16021) was also tested, but in this case the side
wall effect was so large that the results were not
compared with the theoretical calculations.
(5) The paint test also was made with the same
foil section (NACAOQ0O15). The results of paint
test agreed with that of the aluminum erosion test
although it gives qualitative data.
(6) The effect of air content and air injection
method was also investigated experimentally. The
air injection was found to be very effective in
preventing erosion.
ACKNOWLEDGMENTS
The authors would like to express their acknowledg-
ments to Prof. S. Tamiya and members of the High
Speed Dynamics Laboratory, University of Tokyo, for
their many valuable discussions and help during
the research work. They also wish to thank Mr. T.
Komura, Mr. R. Latorre, and Miss N. Kaneda for
their sincere help during the preparation of the
Paper.
This research work was financially supported by
the Grant in Aid for Developmental Scientific
Research (2), Ministry of Education, Japan (Research
No. 185087) and the authors are grateful for the
support.
NOMENCLATURE
A, a : constants
B : span
ic : constant, chord length
c : constant
Cp : pressure coefficient
d : depth
E : energy
Et : total energy of bubbles
e : energy density
1 : force
£ : energy density distribution function
Ish re: Vickers hardness
Ky, Kj constants
Ko, Ki : constants
L : reference length (chord length)
al : length
MDD : mean depth of deformation
MDDR : mean depth of deformation rate
MDP : mean depth of penetration
MDDR : mean depth of penetration rate
N total number of cavity bubbles
n distribution function of bubble number
p pressure
Q volumetric flow rate
R bubble radius
Ss area
SR surface roughness
T time
Vv velocity
a attack angle, air content
6 thickness
Se cavity thickness at the end
€ energy density rate
n probability
A aspect ratio
nN cavity length
fo) density
oO cavitation number
Sy : yield stress
Vv : volume
SUBSCRIPTS
critical
min > minimum
Pp plastic deformation
Ss : sSaturate
Vv vapor
©0 infinity
— : nondimensional value
REFERENCES
Ahmed, O., and F. G. Hammitt (1969). Determination
of particle population spectra from water tunnel
using Coulter counter. Univ. of Michigan, Cav.
and Multiphase Flow Lab. Rep. 01357-4-I, 08466-7-I.
Reina I Con Mo Ca Ilbetoe, (5 ah allies beS WIE
Mitchell, Jr., and L. P. Solomon (1969). A
statistically verified model for correlating
volume loss due to cavitation or liquid impinge-
ASTM, STP 474, 288.
Toward quantitative pre-
ASTM, STP
ment.
Heymann, F. J. (1969).
diction of liquid impact erosion.
474, 212.
Hobbs, J. M. (1966).
tion erosion test.
Experience with a 20-kc cavita-
ASTM, STP 408, 159.
Kato, H. (1975). A consideration on scaling laws
of cavitation erosion. Int. Ship. Prog., 22,
25)3)305i
Knapp, R. T., J. W. Daily, and F. G. Hammitt (1970).
Cavitation. McGraw-hill Book Co., New York.
Kohl, R. E. (1968). Experimental studies to
establish scaling laws for modelling cavitation
damage. Hydronautics Tech. Rep. 233-12.
Lindgren, H., and E. Bjarne (1974). Studies of
propeller cavitation erosion. Conf. on Cavita-
iesicra, Beaks
Narita, H., Y. Kunitake, K. Holden, and B. Mugaas
(1977). Investigation on the ducted propeller
cavitation and the duct erosion prevention by
the air injection system. Symp. Hydrodynamics
of Ship and Offshore Propulsion Systems, Paper
W/3¥
Nishiyama, T., and J. Ito (1977). Calculation of
469
partially cavitating flow by singularity method
(1st Report: Two-dimensional isolated hydro-
foil). Trans. JSME, 43, 370; 2165, (in Japanese).
Okamoto, H., K. Okada, Y. Saito, and T. Takahei
(1975). Cavitation study of ducted propeller
on large ships. Trans. SNAME, 83.
Ozaki, H., and D. Kiuchi (1976). Cavitation erosion
of two-dimensional foil sections. Graduation
thesis, Dept. Naval Arch., Univ. of Tokyo, (in
Japanese) .
Sasajima, T. (1972). On cavitation erosion test
method by using a painted model propeller.
Mitsubishi Tech. Rep., 9, (in Japanese).
Sato, R. (1976). Study on cavitation erosion.
3rd Lips Prop. Symp., 19.
Sato, R., S. Tamiya, and H. Kato (1974). Study on
cavitation erosion. Selected Papers from J.
Soc. Naval Arch. Japan, 12, 21.
Stimebrng yD. oh Ra Ee AGe Arndt) and di Wel Hole
(1977). Scaling of cavitation damage. J. Hydro-
nauticsy, Lis) oike
Thiruvengadam, A. (1966). The concept of erosion
strength. ASTM, STP 408, 22.
Thiruvengadam, A. (1971). Scaling laws for cavita-
tion erosion. Proc. IUTAM Symp. Lenigrad, 405.
Young, S. G., and J. R. Johnston (1969). Acceler-
ated cavitation damage of steels and superalloys
in sodium and mercury. ASTM, STP 408, 186.
Experimental Investigations
of Cavitation Noise
G6ran Bark and Willem B.
van Berlekom
The Swedish State Shipbuilding Experimental Tank,
Goteborg, Sweden
ABSTRACT
The requirement of low or acceptable noise levels
onboard ships as well as low levels of radiated
noise for special purpose ships can cause large
problems for the naval architect. Low noise levels
onboard ships are required in living quarters and
also in some working spaces. The radiated noise
field is of concern for instance for fishing vessels
and ships with acoustical dynamic positioning systems.
One important source of noise in ships is cavita-
tion and especially cavitating propellers. The
cavitation noise can have a quite varying character.
It may for example sound like a hiss or like sharp
hammer blows. For the naval architect it is impor-
tant to be able to predict and, if possible, to
reduce undesired cavitation noise.
In this paper some of the research and develop-
ment work on cavitation noise at the Swedish State
Shipbuilding Experimental Tank (SSPA) will be
described. This work at SSPA is mainly experimental
and two projects will be described here in detail.
One concerns the relation between cavity dynamics
and cavitation noise. This work was carried out
using an oscillating hydrofoil in the No. 1 SSPA
cavitation tunnel. The other project concerns the
relation between types of cavitation and cavitation
noise. Different types of cavitation were generated
in the tunnel using axisymmetric head forms and
hydrofoils.
A great deal of effort has been made at SSPA to
develop adequate methods for measuring cavitation
noise in cavitation tunnels. A short review of
the measuring techniques now in use is given in an
introductory chapter. Besides the two projects
mentioned above several other projects are, or
have been, carried out at SSPA.
1. REVIEW OF MEASUREMENT TECHNIQUES AT SSPA
Measurements of cavitation noise started at SSPA
as early as 1958. The first tests concerned cavita-
470
ting axisymmetric head forms and were carried out
in the SSPA cavitation tunnel No. 1. The measuring
equipment was a waterfilled box attached to one of
the plexiglass windows of the tunnel. A hydrophone
was lowered into this box and could thus pick up
the noise emanating from the source (propeller etc).
The transmission path from the noise source is
through water, plexiglass, and water to the hydro-
phone. The transmission loss due to the presence
of the plexiglass window is low. The drawbacks to
this arrangement are reflected acoustic waves and
vibrations in the box. The problem with the
reflected waves may partly be overcome by carefully
calibrating, or rather comparing, results from the
hydrophone in a free field and in the box using the
same known noise source. Vibration problems (from
the vibrating tunnel plating) may be cured by using
a pair of rubber bellows between the box and the
window (see Figure 1).
The signal from the noise source is, however,
still distorted as can be seen in Figure 2. This
figure shows the noise from a cavitating propeller,
as measured by the hydrophone in the box and a
hydrophone near the propeller. The differences in
the curves are striking and show that the general
shape is seriously altered by the box. It is in
fact almost impossible to analyse the signal in
time-domain using the hydrophone in the box. Com-
paring results from 1/3 octave band analysis also
shows differences, especially as regards the
frequency dependence. These differences are,
however, not as striking as those for signals in
time-domain.
The arrangements for noise measurements at SSPA
are at present:
1. Flush mounted pressure tranducers on the hull
(Figure 3)
2. Flush mounted pressure transducers on the tunnel
wall
3. Hydrophones in the flow field near the propeller
(Figure 3)
4. Hydrophone in the water-filled box outside the
tunnel
Tunnel No | Test section 700 x700 mm
~— = blade frequency period
pressure
[| |
i\, thy H
sal
/ Why,
ain Yelle yh if 1 YAN
yee AeA eye
NanenleesNynt
,
a?
un |
Leen oe
iia
Hl | |
: Maya ei bial ja il | iM !
|
lyk
time
Hydrophone in external box
Hydrophone box 1(2mm steel
connected to plexiglass
window by rubber bellows
/
Free water surface
Plexigldss
| |
FIGURE 1.
noise measurement.
=
Ny ot WM vii)
VW eaynt "Oy It Yat When
ty
vonege somone HE
FIGURE 2. Pressure signals at different
hydrophone positions.
First arrangement for
471
472
All, 2 pine, AnD, Sale,
are pressure transducers
Arrangement 1 is intended to be the standard
measurement procedure at SSPA and results are easily
compared with full scale measurements using the
same equipment. This arrangement gives essentially
the near field noise from the propeller.
i 5 WL
4
339
|
6
——— fr, =
Le 190
i oa
1 |
|
|
|
7) 7
|
!
BL | le
0 1/2 1
FIGURE 3. Arrangements for noise measurements on complete ship model. (Tunnel No. 2)
@ Non-cavitating propeller
OA Cavitating propeller
Sound pressure level re 1[Pa in 1/3 octave band
1 OF rTpst onlin lin Tne] in Li ict eT sion poo
If it is of interest to know the radiated noise
into the farfield, arrangement 2 can be used.
Arrangement 4 also gives the farfield noise, but
has its problems, as discussed above. Arrangement
2 has less problems with reflected acoustic waves
and vibrations than arrangement 4. The main reason
why arrangement 4 is still used is to compare results
directly with older measurements. Arrangement 3
(Figure 3) has been especially developed for explor-
ing the influence of variation in cavitation and
the effect on the near field noise. Other arrange-
ments of hydrophones have also been used for special
purposes.
Since the main concern in the noise measurements
is cavitation noise, the effect of flow noise due
to the turbulent boundary is of minor importance.
Usually the increase in noise level due to cavita-
tion is quite substantial, as can be seen in Figure
4, which shows a typical example for a propeller in
non-cavitating and cavitating condition.
2. EXPERIMENTS WITH AN OSCILLATING HYDROFOIL
Background to Experiments with Oscillating Hydrofoil
A typical example of the pressure signal froma
cavitating propeller model is shown in Figure 5.
The pressure was measured by a hydrophone near the
180 z “ |
170 -—t-—-
L ho“Ta,
160 [ Fal Toe
. rage eo
| e |
NOE + s ali =
F | ian
yay ett SU LIL
30 50 100 200 500 tk 2k 5k 10k 20k 40k
Hz
FIGURE 4. Noise measurements on propeller-model.
(Tunnel No. 2)
pressure
| 104 Pa |
(a
time (ms)
0 5
propeller. The signal corresponds to a spectrum of
the the type shown in Figure 4 and typically is a
rather slow variation of pressure interrupted by
sharp and fairly infrequent pulses. The pulses
are presumed to be generated during the final cavity
collapse and they provide the main contribution to
pressure levels at high frequencies. The pulses
are often higher than the low frequency variations,
but because of their low repetition frequency and
wide frequency content the spectrum levels at high
frequencies are lower than at low frequencies.
To understand the scaling of cavitation noise
and how different types of cavitation noise are
generated, and perhaps can be reduced, it is
important to study the mechanism generating different
types of noise. A suitable way to obtain such
knowledge is to carry out high speed filming and
synchronous measurement of the cavitation noise.
The first idea was to carry out such measurements
with a propeller model. Because of high tip speed,
small dimensions, and the complicated geometry of
a propeller it was decided to take the first step
by performing such experiments with oscillating
hydrofoils. By suitable oscillation of a hydrofoil
it is possible to generate cavitation with approxi-
mately the same dynamic behavior as obtained from
a propeller operating in a wake. The experiments
with oscillating hydrofoils were supposed to shed
some light on the following questions that originated
from the search for methods of prediction and
reduction of propeller cavitation noise:
1. Which are the characteristic properties of the
pressure pulses from some special types of
cavitation?
2. Are strong pulses generated by an orderly
collapse of the whole cavity (e.g., a sheet
cavity) or do they originate from large or
small parts that separate from the main cavity?
What is the geometry before and during collapse
of cavities generating strong pulses?
3. How is the pressure pulse related to the size
of the cavity? Is there, for example, any
relation between the maximum extension of a
sheet cavity and the final pressure pulse?
4. Is rebound of cavities important for generation
of sharp pulses?
5. What part of the cavitation period is of main
importance for the generation of different
types of noise (slow pressure variations, sharp
pulses, etc.)?
6. Which are the characteristic properties of the
Blade frequency period
473
FIGURE 5. Pressure signal from a cavitating
10 propeller model.
flow field, oscillation frequency, etc., causing
cavitation with violent collapse?
7. To what extent is collapse time determined by
the oscillation frequency of the hydrofoil?
8. To what extent does the cavity behavior seem
predictable by theoretical methods? How
realistic is it to think that a sufficiently
good scaling from model to full scale is
obtained for the most important cavitation
events?
Experimental Set Up
Cavitation Tunnel
The tests were carried out in SSPA cavitation
tunnel No. 1 (the samller one) equipped with test
section No. 1 (500 x 500 mm).
Oscillation Apparatus
The hydrofoil was located horizontally in the test
section and attached to an oscillation apparatus
fixed to the test section wall (Figure 6). The
hydrofoil was supported only at one end and forced
to oscillate (rotate) around an axis fixed spanwise
through the midchord point, i.e., the geometric
angle of attack oscillated around an adjustable
mean value, a9, (Figure 7). The axis was driven
by a connecting rod and an adjustable crankpin.
By setting the crankpin the oscillation angle, a,
could be varied from 0 to 6°. With the hydrofoil
used in these tests the oscillation frequency, foccr
was varied from 0 to 15 Hz. The limits of water
speed, ag, G, and f,,, were set by the strength of
the hydrofoil and the background noise generated by
the apparatus. One part of the background noise
from such an apparatus is knocking in shaft bearings.
To minimize this knocking, adjustable bearings were
used. The motor, which was not dimensioned for this
experiment, could deliver 16 kW at a maximum speed
Cpe 5X0) 16/435
The dynamic angle of attack, experienced by the
leading edge of the hydrofoil, is composed of the
geometric angle and of an angle caused by the motion
of the leading edge. The angle is also affected by
induced velocity. In the following only the geomet-—
ric angle is considered (Figure 7).
The system with connecting rod and crankpin
results in an approximately sinusoidal oscillation
of the geometric angle of attack. This manner of
474
Hydrofoil
Hydrophone strut
FIGURE 6. Experimental set up.
oscillation does not cause a time variation of the
angle of attack that is completely similar to that
of a propeller blade in a wake. The reason for
using this sytem was that, due to its strength,
high oscillation frequencies with large hydrofoils
could be obtained. If similarity with propellers
is most important it is probably better to use
oscillation systems of the types constructed by
Ito (1962) and Tanibayashi and Chiba (1977).
Hydrofoil
In these introductory experiments an existing hydro-
foil, earlier used for studies in two-dimensional
flow, was used. The profile has NACA 16 thickness-
distribution and is typical of a relatively thick
propeller blade at about 0.7 of propeller radius.
The hydrofoil data are
Mean line a = 0.8
Camber ratio = fy/c = 0.0144
Thickness ratio = s/c = 0.0681
Chord length = c = 120 mm
Span = 200 mm
Profile shown in Figure 7.
Noise Measuring Equipment
Two hydrophones (Briiel and Kjaer Type 8103 with
frequency response 0.1 Hz - 140 kHz +2 dB) were
placed in notches in a tube supported by two hydro-
foils in such a way that photographing of cavitation
was permitted (Figure 6). The frequency response
of the hydrophones mounted in this manner was
checked by white noise. No significant change in
the frequency response was detected.
The hydrophone signals were recorded on FM-
channels on a Honeywell 5600-C tape-recorder (0-40
kHz at 60 ips tape speed). Recordings were also
made on direct channels (300 Hz - 300 kHz at 60 ips).
It was then possible to write out the complete signal
(0-40 kHz) by use of tape speed reduction and UV-
recorder.
Simultaneous with the hydrophone signals, a
FIGURE 7.
signal showing the events of maximum angle of attack
was also recorded.
High-Speed Film Equipment
The requirements set up for the filming were that
the film had to be synchronous with the noise
recordings and permit measurements of cavity size
as a function of time. The intention was not to
measure the detailed behavior of small or very fast
events. The minimum duration of the filming was
set to about one second.
These requirements were met by a Stalex VS 1C
camera capable of 3,000 frames/s. This is a 16 mm
rotating prism camera taking rolls of 30 m film.
Lenses with focus lengths of 9.8 and 50 mm were
used. For synchronization the camera could release
a flash at a preset time. The flash trigging
signal was recorded on tape together with hydrophone
signals and the flash was placed within the frame.
Only one flash was released during each filming.
The camera was also equipped with a crystal-controlled
timée-marker, making one light marking every milli-
second on the edge of the film. This, together
with the synchronization flash, made it possible
to identify and follow cavitation behavior on the
film together with the corresponding pressure
Geometric angle of attack = QA~A +A sin 2T t fosc
Oscillating hydrofoil.
behavior recorded on tape. An example of the
recorded signals is shown in Figure 8.
As light sources, two 1,000 Watt spotlights were
used. To get a proper background without reflections
the hydrofoil was painted with a red matte paint.
A test was performed with black and white film
(Kodak 2479 RAR Film). The result was not very
good, the contrast between hydrofoil and cavitation
being too small. Color film (Kodak Vide News Film)
was then used, with very good results.
Evaluation of Films and Pressure Signals
The pressure pulse generated by a cavity is related
to the volume acceleration of the cavity and thus
it is desirable to measure the cavity volume as a
function of time. With complex cavities this is
not very simple. An estimate of the cavity volume
could be obtained if both cavity extent (area) and
thickness were filmed synchronously. This is
possible by the use of optical systems reflecting
the two pictures into the same frame [Lehman (1966) ].
No such attempts were made. Most photographs were
taken in order to measure the cavity area on the
suction side of the hydrofoil. To obtain information
about the cavity thickness some photographs were,
however, taken from the free end of the hydrofoil.
A method of estimating the relative thickness,
synchronous with the cavity area, was to measure
the length of a cavity shadow generated by the
directed light. The method, which was calibrated
by use of spherical bubbles, was rather rough, but
some general information of thickness behavior was
obtained.
The photographs were studied by use of an analysis
projector permitting single-frame projection on a
focusing screen, where the area of the cavities
could be measured by summing up elements in a
pattern. For identification of cavitation events
on the films and noise recordings the synchronization
flash was the primary starting point. To increase
the accuracy of identification of events far from
the flash easily identifiable events, such as
single bubble collapses, were used as reference
points.
Experiments
The experiments with an oscillating hydrofoil
presented in this paper are the first of this kind
carried out at SSPA and they are to be regarded as
introductory in several respects.
Only one hydrofoil was used. The following
flow parameters were held constant during the tests:
Relative gas content (at atmospheric pressure)
of the tunnel water was 25%
Water velocity in test section = U = 5.0 m/s
Cavitation number at the center of test section
ey), = 12
Bag oe TT. eee ys
1 5 w
where 2
Po = surrounding pressure = 11.850 Pa
Py vapor pressure of water (20°C) = 2.338 Pa
po = density of water = 998 kg/m?
The following oscillation parameters were varied
in the experiments (see Figure 7):
ag = mean angle of attack of the hydrofoil
475
& = oscillation angle
1 eye = oscillation frequency
In the figures the reduced frequency k, is used:
TE c
wo _ osc
where
WwW = 2M eos
c = chord length of the hydrofoil
U = water velocity
After some introductory tests the following con-
ditions of hydrofoil oscillation were selected from
high-speed filming:
ao a £ k
osc (co
(o) (o) (Hz)
3 3 3 0.23
Ad Y 7 O53)
ui rT 10 0.75
HW rf 5 at a abs}
3 4 il 0.08
i W 2 OFMES
uw ui 3 0.23
ui 1 4 0.30
" ¥ 7 0.53
wv uv 10 0.75
Me " 14 1.06
4 3 3 0.23
" WY 7 0.53
ui 10 Os 75
" u 15 dL 5 ALg}
4 5 3 OR23)
u oe 7) @553}
Hi i 10 0.75
Results
Primary results are presented as pressure signals
from cavitating and non-cavitating hydrofoils,
measurements of cavity area, and sketches of the
cavitation pattern at various oscillation parameters.
Presentation of Results
In Figures 9 - 14 a survey of pressure signals
and cavitation patterns at various oscillation
conditions is shown. All pressure signals shown
in these and other figures are from the hydrophone
(Hl) near the leading edge of the hydrofoil. For
each condition some oscillation periods are shown.
The length, Toso = 1/fosc, Of an oscillation period
is identified by the markings of maximum angle of
attack, Omax- The figures show primarily cavitating
conditions (cavitation number = 0.76) but in some
cases signals from the corresponding non-cavitating
condition is sketched (without the fine structure,
which is apparatus noise). The pressure scale is
given as a number of Pascal (Pa) per scale unit
(su) defined at the top of the figures. The time
scale is 6.15 ms/scale unit in all signal examples
in Figures 9-13. For one of the oscillation periods
the number of the oscillation period (relative to
the synchronization flash) is shown in a circle,
and for this period some additional data is given
to the right. In the cavitation sketches are shown
the maximum area extent, the maximum chordwise
cavity length, &max, and the cavitation extent at
FIGURE 8.
hydrofoi
ale
NN ee EEEEyeEeEeEeEeEEEEeE~ILE _—————————EEEKEVaaa
»
Max. angle of attack Qmaqx |
HI | |
\ \ \ \ i \
iy ‘( iA \ | 1
ay ) i \ hy 4 A \ fa iy
ae | whet , 4 VW \r\ lg \
yo \ yf wit, ees | vi ee \ Ny r i 1 \ "| i 4 \ mil \ ; | \
Le eh WA \ ‘hy / Whth | Ven yy 7
ne i Av Wis i Mis i (! WW \ Moat \ |
vaniar ( i" t yl Mi Vi i" 5,
Hydrophone signals t
i} \
H2 A I,
ag \\ NN 1A * i H
uo vey WE ; WN Hy 4 V\ fh 4 IVI Ma
‘ my eft lt A ‘i at h ey AL YP OM Yi Wy fy AM H\\ \W\ /\ 1 \
, ine, ste HedN My \ | wel) We \ 7S wh f |
: Mt W , ef
Sea re eae Os cr tating =—Synchronization flash End of film —=
Recorded signals. ee EEE
Freire ar :
[1 1 1 1 4 50 scale units (su) | Max. extent. |Collapse(=max p)
== pressure 25 Pa/su
79 Pa/su
79 Pa/su
FIGURE 9.
c |
: == ——max extent.q
A
if h
Ml
——
max. p before final collapse
cav. starts
{i \
8
c
£
F
a
roy
1S)
c
=
o
2
3
&
o
a
a
x
cS]
€
3°
o
c
iS
a
3
x
5
2
a
a
o
a
bad
cS]
E
Oscillating hydrofoil. Pressure signals and cavitation. O =m3era=u4on
Pressure signals
50 scale units (su)
125 Pa/su
pressure
265 Pa/su
265 Pa/su
FIGURE 10.
(approximately) that moment when maximum pressure
is generated. For rapidly collapsing cavities the
cavitation patterns shown existed 1/3-2/3 milli-
seconds before the sharp pressure pulse. A note
is also made as to whether or not the maximum
pressure increase coincided with the final collapse
(i.e., the complete disappearance of the cavity).
The collapse velocity during the last stage is
indicated by arrows:
> = slow motion of the cavity boundary in
the direction of the arrow
>> = fast motion of the cavity boundary in the
direction of the arrow
>>> = very fast motion of the cavity boundary
in the direction of the arrow
At collapses with more or less spherical symmetry,
arrows are placed opposite each other.
To the right is shown the cavity growth time, Tg:
and the collapse time, T,, for the complete cavity,
measured by use of the time markings on the high-
speed film. The’collapse time is measured from the
time of maximum area extent to that time when the
cavity generated the maximum positive pressure. For
rapidly collapsing cavities this event coincides
with complete disappearance of the cavity. This
was not the case for slowly collapsing cavities;
477
max. p at final co
a
AC)
i
Cc
=
~
5
ro
x
So
E
Oscillating hydrofoil. Pressure signals and cavitation. a = so 6 S 20.
for these cavities the collapse times for complete
disappearance are also given (in parenthesis).
General
The general character of noise and cavitation be-
havior when the frequency of oscillation is varied
is shown in Figures 9-14. The pressure signals
from the cavitating hydrofoil are to be compared
with signals from the non-cavitating hydrofoil
(Figure 15) and with the curve in Figure 16, showing
the schematic behavior of the pressure generated by
a growing and collapsing cavity.
In comparisons of generated pressure from non-
cavitating and cavitating hydrofoils the most
striking difference is often the high and sharp
pulses generated at the cavity collapse. The
generation of such pulses is obtained especially
when fog, exceeds a certain value. Also the pressure
increase corresponding to cavity growth and the
pressure dip generated near maximum cavity extent
are detectable.
The generated pressure pulses were classified
into three main types:
478
e Slow pressure increase at cavity collapse
(normally obtained at fog, = 1-3 Hz)
e Fast pressure increase (fosc = 4-7 Hz)
e Very fast pressure increase, i-.e., sharp
pulses (fos¢ = 7-15 Hz)
Generation of High Frequency Noise
Sharp pulses (i.e., high frequency noise) were
generated in three main ways:
A. By violent collapse of the main cavity (or
a large part of it).
B. By collapse of small spherical bubbles
occurring independently of the main cavity.
The bubbles generated rather strong pulses.
Cc. By collapse of rather small irregular cavities
separating continuously from the main cavity.
Of greatest interest is the generation process
A, which was obtained at high fo,,- The high and
sharp pulses were generated in three somewhat
different ways:
Al. Separation of a rather large part of the
main cavity at an early stage of the
collapse. Thick cavity formations often
separated in this way, especially if the
cavity was long (large %may) and broken up
by disturbances. At the end the collapse
was often very violent and often followed
by a violent rebound. Also the rebounded
cavities (complex in form) cometimes
Pressure signals
50 scale units (su)
+——Tose ———. @)
Omax
‘ee A, ye
y W “hr a uA W 0 aly
Va, ns Te sions area =O" un,
cav. Pane
ressure 156 Pa/s
P -
a
time
492 Pa/su
492 Pa/su
FIGURE 11. Oscillating hydrofoil. Pressure signals and cavitation.
collapsed violently. An example of this
behaviour is shown in Figure 13 for fog, =
7 Hz (oscillation period 5).
A2. Sharp pulses were also generated when a
sheet collapsed towards the leading edge.
The upstream cavity boundary was attached
to the leading edge during the whole collapse.
This process was normal at the conditions
shown in Figures 11 and 12 and especially
in cases where the main cavity was rather
small. In these cases the whole collapse
was orderly and without extensive separa-
tions of cavity parts from the main sheet.
After the collapse was completed a rebound
of small cavities occurred about 10 mm
downstream from the leading edge and not at
the center of collapse as in the case of
more symmetrical collapses. Also in cases
where large cavities separated from the
main cavity the remaining, rather smooth
sheet often collapsed in this way (Figure
10, 10 and 14 Hz, Figure 13, 7 and 10 Hz).
A3. In cases where the smooth sheet attached
to the leading edge was long and narrow it
was also cut off from the leading edge. For
the downstream part, the collapse then be-
came more symmetric and violent and with
a violent rebound (Figure 11, 10 Hz and
Figure 13, 7 Hz). This process often
occurred near the end of collapse.
Spherical bubbles were very effective as genera-
eo cs nn
50
(71)
0.48
Collapse oman nl
3)
max p at final coll max p at final coll, |max p before final coll! 9
max p at final coll
Pressure signals
L111 1 4 50 scale units (su)
156 Pa/su presse
——cav. Starts
131 Pa/su
131 Pa/su
FIGURE 12.
tors of high frequency noise. This is discussed
later in the text together with cavity area measure-
ments.
The generation of high frequency noise by small
irregular cavities, continuously separating from
the main cavity is the only generation process when
fosc = 0. Also at low fos, (about 1-2 Hz) this
process generated pulses. The separation of small
cavities from the main cavity decreased with
increasing fosc-
When the high frequency noise was obtained it was
always generated during the last part of collapse
of the generating cavity (i.e., a bubble could col-
lapse and generate high frequency noise during the
growth of the main cavity). This is not surprising,
but it should be mentioned that at studies of pro-
peller cavitation it has been noticed that the growth
of cavities in some cases also generates rather fast
pressure variations which indicates that high volume
acceleration can also occur during growth.
Generation of Low Frequency Noise
The generation of low frequency noise (vibration
generating pressure disturbances at multiples of
propeller blade frequency) can be identified by
inspection of signals from non-cavitating conditions,
479
[tavitation nde
Max extent. [Collapsetnax fe] [aii
lmax=105 mm @)
cpl| max p before final col
c
£
G
3
2
a
a
5
a
x
cS)
E
3G
eS
=
~
ro]
a
Imax % 60 mm
max p at final coll.
Oscillating hydrofoil. Pressure signals and cavitation. ae = 2 eg,
cavitating conditions, and the schematic pressure
behavior shown in Figure 16. This is especially
easy in cases where cavitation start is marked
(Figure 9, 3 Hz, Figure 10, 7 Hz, Figure 11, 3 Hz,
Figure 12, 7 Hz, Figure 13, 3 Hz) or where a non-
cavitating period is followed by a cavitating one.
In several cases it can be seen that a rather slow
pressure increase is generated during the growth.
When the volume acceleration is directed inwards,
during a period around the maximum cavity volume,
negative pressure is generated (for example Figure 9,
3 Hz). This pressure variation is rather slow and is
an essential part of the low frequency disturbance.
Because of inertia effects in the motion of cavity
walls this part of the motion will probably never
contribute to really high frequencies.
In most of the figures it can be seen that con-
tribution to the low frequency pressure is also
obtained from the collapse. Especially at low foc.
the collapse seems important. The pressure increase
during collapse is due to the outward-directed
volume acceleration existing during the final part
of collapse. This acceleration depends on the
cavity geometry and the velocity of the cavity walls
and it is in principle possible to obtain a collapse
with constant volume velocity (no pressure generation),
as well as a collapse with decreasing volume velocity,
in which case a pressure increase is generated. It
is supposed that both types of collapse can occur
480
Pressure signals
L111 1 4 50 scale units (su)
cay, starts
=!
4)
o
a
w
—t
N
o
L
=!
w
w
oO
—_
a
1310 Pa/su
Cavitation
f
a cls)
| Max. extent. | Collapse («max plz
)
5 =
a é
3 :
S fo}
o a
re]
E
FIGURE 13. Oscillating hydrofoil. Pressure signals and cavitation. Ce 4° @ = 5°.
on propellers, depending on cavity geometry and medium-high frequencies from a propeller
time variation of the surrounding pressure. (5-20 x blade frequency). The pressure
The contribution from collapse obviously exists fluctuation seems related to the dynamics
(see Figure 9, 1 and 2 Hz) but the quantitative of the main cavity, which at this stage was
results especially at fo,, = 3-7 Hz must be used quite orderly.
with much prudence, because of the resonant character 3. During the last part of collapse very sharp
of the signal in these conditions. This is discussed pulses with durations less than 0.1 milli-
in the Appendix. second were generated. At this scale of
time, measurements and detailed observations
of cavity behavior were not possible. Some
Area Measurements of Some Cavities observations indicated, however, that the
sharp pulses sometimes were generated by a
For the condition a9 = a = 3° and rage = la ee rather well-ordered collapse. Figure 17
some results from measurements of cavity area are shows an example of this behavior. The
shown in Figures 17-23. The main cavity includes cavity was in this case attached to the
the sheet and some small bubbles at the downstream leading edge during the whole collapse.
edge, which follow the behavior of the sheet. 4. More complex cases are shown in Figures 18,
Although the cavities in this condition were rather
simple, with no large separations from the sheet,
quite complex events often occurred during the
last 1/2 millisecond of the collapse.
Some comments on the figures will be made:
1. From the shape of the area curves it can be
seen that the growth of cavities was rather
similar in all cases, while there are
differences in the collapses. Compare, for
example, Figures 17 and 20.
2. It is seen that 1-2 milliseconds before
final collapse a slow or moderately fast
pressure increase was obtained. During this
time collapse is fast, but measurable. This
pressure fluctuation corresponds to low or
21, 22, and 23. Several pulses were generated
during a short time and it is impossible to
separate the generating events (collapses
and rebounds of several small cavities).
Typical of these oscillation periods is
that when the downstream cavity wall moves
towards the leading edge, the cavity separates.
into two parts, both attached to the leading
edge. This separation was caused by a growing
disturbance on the cavity surface. The
disturbance grew from the downstream edge
towards the leading edge. (See also Figure
11). During the collapse some bubbles also
separated from the downstream cavity edge
and the disturbed area. These three cavity
p 125 Pa/su A,
Mi
| W I Hu
| | Wn rf
At)
yi”
We)
‘i
i}
{
Nghe PI) stl daimat q
fosc=
i wt
fe A hy:
a p 265 Pa/su |
(=)
w
| B
Withee '
A | ao
ain
p 265 Pa/su A
Nh
é an 1 Ma
0 5 10
FIGURE 14. Pressure signals during collapse. Expanded
signals from Figure 10.
15 t (ms)
groups seldom collapsed exactly simultaneously
or with the same violence. For example, in
the cases shown in Figures 18 and 21 a part
of the cavities was cut off from the leading
edge during the last millisecond of the
collapse. This resulted in violent collapse
of the cut-off parts.
From these examples it is understood that
in a single oscillation period the character
of the pressure signal is very sensitive to
such things as simultaneousness and violence
of separate cavitation events. Over many
periods, normally used in measurements, the
quantities are smoothed out to a mean value,
which often is less sensitive to small dis-
turbances.
5. In some cases small bubbles and irregular
parts separated from the main cavity and
collapsed rather fast. In the case shown
in Figure 22 a group of small bubbles behind
the main cavity (cavity B) collapsed violently,
simultaneously with the main cavity, and it
is impossible to determine which of the
cavities generated the main pulse. Examples
of cavities that seemed rather fast, but
only generated small pulses are shown in
Figure 18 (B) and 19 (C).
6. The most extensive rebounds resulted from
cavities that were cut off from the leading
edge and then collapsed fairly symmetrically.
The cut-off normally occurred during the
last one or two milliseconds and it often
resulted in two cavities, one of which
remained attached to the leading edge. The
481
p 26 Pa/su fosc = 1Hz
aaa 50 su n
| wih m4
\ i i ea ft Ma Ah i
II bint | \ "Ny th
ail ne haiti
i ii My Whi W,
aa hh
Time
82 Pa/su n
if
H Le
" Ae
Maha th N
82 Pa/su n n ices 2 Oh
Ny svt
Se, out o aye
See “tin
82 Pa/su fosc = 4Hz
' i
v Avs ve
131 Pa/su ffosch=/AiZ,
i] 1
ca ‘
131 Pa/su fose = 10 Hz
1 i 1
fs *, oy d 4 ra
rp ee \ye Sf
265 Pa/su flosc = 116) Hz
eae Miu ies clened
FIGURE 15. Pressure signals from non-cavitating hydro-
Poni, GO = 9 Ge a,
(eo)
Cavity volume
4
= Time
Radiated pressure
A
= Time
FIGURE 16. Schematic behavior of cavity volume and
radiated pressure.
482
FIGURE 17. Cavity area and generated pres-—
sure. Oscillation period -1l.
Cavity area (cm?)
70
60 p 131 Pa/scale unit
Cavity area (cm?) p ‘131 Pa/scale unit ag=3°
50 a=?
r fosc=15 Hz
Ir 50 scale units
mh hava,
[ |
[ on ty at
ant a a a a
L 0 5 10 15 20 25 30 35
L t (milliseconds)
20
10 |
|
|
i |
0 L
0 5 10 15 20 25 30
50 scale units
t (milliseconds)
FIGURE 19.
sure. Oscillation period 6.
Cavity area and generated pres-
30
35 40 45
t (milliseconds)
FIGURE 18. Cavity area and gen-
35 40 erated pressure. Oscillation
t (milliseconds) period 4.
Cavity area (cm?) p 131 Pa/scale unit Gp = 3°
507 a=3°
L fosc= 15 Hz
50 scale units
I leg
30+ be +1 J SSS SS SS eee)
0 5 10 15 20 25 30
t (milliseconds)
25 30
t (milliseconds)
Cavity area (cm?) dp =3°
°
50, p 131 Pa/scale unit @=3
f fosc=15 Hz
40+ 50 scale units
30
a frie]
10 15 20 25 30
20 — ees = {VTMIUUER GHEE
Bmax =/22 mm | |
a
| |
A |
25 30
t (milliseconds)
Cavity area (cm?)
707
t P
483
FIGURE 20. Cavity area and generated pres-
sure. Oscillation period 7.
= 48mm
Imax
131 Pa/scale unit
50 scale units
t (milliseconds)
FIGURE 21. Cavity area and gen-
erated pressure. Oscillation OF 5 0
period 12.
- 2 4
Canty aca (Gar) p 131 Pa/scale unit CoP o
60 a =3
fosc= 15 Hz
j |
50 50 scale units |
40
3
t (milliseconds)
a
15 20 25 30 35 40
t (milliseconds)
FIGURE 22. Cavity area and generated pressure.
Oscillation period 13.
484
FIGURE 23. Cavity area and generated pres-
sure. Oscillation period 14.
rebounded cavity (often a group of small
cavities) collapsed after three to four
milliseconds. Compared with the main cavity
the area of the rebounded cavity was small
(Figure 18 cavity C, Figure 22 cavity C and
Figure 23 cavity B). The rebounded cavity
often generated pulses of nearly the same
height as the main cavity.
7. The equipment was not designed to measure
small and fast collapsing cavities such as
small bubbles, but an example.of a diameter
measurement of a bubble is shown in Figure
24. The area (1a2/4) of the same cavity is
plotted in Figure 19 (cavity A), where the
sharp collapse pulses are also visible.
Other examples of bubble collapses are shown
in Figure 17 (time = t = 5 ms), 18 (t = 10),
20 (t = 0, cavity A), and 23 (t = 0). Bubble
collapses are also shown in Figures 9-13.
Diameter (mm)
8
-2 -1 0 1 2
-3
Time (milliseconds )
FIGURE 24. Diameter of a spherical cavity.
in Figure 19.)
(Cavity A
Cavity area (cm?) p
70
60
50+
131 Pa/scale unit (cm2)
50 scale units
Nea
See
vo
0 5 10 15 20 25
t (milliseconds)
-5 0 5 10 15 20 25
t (milliseconds)
The bubbles studied appeared just before or
during the growth of the main cavity and the
pressure pulses were then easy to identify.
The bubbles normally rebounded once or twice.
From the size of the bubbles and the generated
pressure it is obvious that the bubbles are
very effective as sources of high frequency
noise. During the first life cycle, the
bubble surface was smooth, but in the rebound
cycles it became rough as reported by other
authors.
Dimensionless Presentation of Some Results
The pressure generation at collapse is related
to the violence of the collapse and it is then
natural to study the collapse time, Te, for cavities
generating different types of pressure pulses. Tc,
given in Figures 9-13, is measured for the complete
cavity, but in several cases it is only a separted
part of the cavity that generates the main pressure
pulse. Because of this simplification T. is probably
not significant for the generated pressure in all
cases. The intention was, however, to study the
relevance of parameters for the complete cavity.
In Figure 25 dif (its + Tg), (Ty = growth time),
is plotted for the cavities shown in Figures 9-13.
As seen the steepness of the curves tends to
stabilize at a lower value for foo, resulting in
sharp pulses. The growth and collapse are, however,
not generally related to each other and Figure 25
may thus give a distorted picture of T_-behaviour.
In an effort to remove this drawback Wea also
was plotted, where Ti is a hypothetical collapse
time given by the formula for spherical cavities
(Rayleigh 1917):
Te
Te +1g
(A)a) ny a
06 45) 8
a 3 4
(a) E : Z
(_) complete collapse
(0) no( ) collapse to max pressure
0.5
7 slow pressure increase
” fast pressure increase
A very fast pressure increase
@ mean value of 8 samples
o4p
- bubbles
03>
O2-F
By
0.0 05 10 Reduced freq 1.5
L n eS Cee es ees)
0) 5 10 15 fose (Hz) 20
FIGURE 25. Normalized collapse time.
where
Pg = surrounding pressure
Py = vapor pressure
U = undisturbed velocity
pe = density of water
Oo = cavitation number
Of course this formula at best gives a time
proportional to the collapse time of the sheet with
maximum length, 2£m3y- As is shown in Figure 26
the tendency is simlar to that in Figure 25. The
conclusion is that at high f the collapse is
mainly regulated by a surrounding pressure consider-
ably higher than the pressure inside the cavity,
which results in T./Tc' = constant and a violent
collapse of the type predicted by classical theory
[Rayleigh (1917)]. At low f5,, it can be supposed
that during collapse the pressures outside and
inside the cavity are approximately equal. Then a
violent collapse will not occur and T./T,' becomes
considerably larger than for a "free" collapse.
If the cavity is considered as a monopole source
the generated pressure, p, in the far field is
d2v(t - =)
P= G 2 (1)
4tr at2
where
V = cavity volume
r = distance between cavity and hydrophone
c¢ = velocity of sound
t = time
Applying this and classical theory of cavity
collapse it can be shown [Ross (1976) ] that the
generated maximum pressure, Pmax, at certain con-
ditions is given by
max
p = const
max
485
where
Rnax = the maximum radius of a spherical cavity
AP = Po - Py
Py = surrounding pressure
Py = vapour pressure
According to this
" fs, AP
P Y/ "max (2)
would be an appropriate coefficient to study for
different cavities in our case.
+ : :
Pp = maximum pressure increase at collapse
oe = maximum chord-wise extension of the sheet
cavity (for bubbles 25, = diameter)
The parameters are:
The distance r is measured individually for
every collapse.
1 DB
MPSS OU G 2 Op) ma
Inherent in the coefficient above is an assumption
about the collapse dynamics and, as the dynamics
are dependent on cavity type, there is no universal
value for the coefficient (2). For our purpose
the coefficient may be seen as a measure of the
pressure generation efficiency of different types
of cavities. For spherical cavities this coefficient
was used by Harrison (1952) and Blake et al. (1977).
Another treatment which leads to a dimensionless
pressure coefficient is to suppose that a constant
part of the potential energy available for collapse
is radiated as noise [Levkovskii (1968)]. The
dimensionless parameter derived from this assumption
is
6.0r
Te max p
1."
: bn
5.0 jo 9
(ss 4
Oo 4 3
@ 4 5
7 slow pressure increase
7 fast pressure increase
4.07 “A very fast pressure increase
“1m mean value of 8 samples
3.0
2.0
1.0
0.0
0.0 05 1.0 Reduced freq. 1.5
L 4 4 aa Lt
0 5) 10 15 foge (Hz)20
FIGURE 26. Normalized collapse time.
486
ptr [Pam] ky & a co bubble
4000 SNS) 5 a3 Sie ES:
[ ATSS) 4 ptr [a] ¢) 3
Oo 4 3 (yi 4
e4 5 imax®F fA 8
7 slow pressure increase [ @ 4 5
7. fast pressure increase 7 slow pressure increase
A very fast pressure increase 7. fast pressure increase c® bubble
A very fast pressure increase
3000 [
3.0 [
2000
2.0
1000
10F
=A bubbles
0
0.0 05 1.0 Reduced freq. 1.5
o 5 10 1 fosc (Hz) 20 0.0 05 10 Reduced freq. 1.5
+
FIGURE 27. Pressure p at collapse. Different 1
conditions. ; ) 5 10 15 fog¢ (Hz) 20
+
FIGURE 28. Pressure p at collapse. Different
n va be
3 7 IRE conditions.
29/2) ae manera
R ¥v 9 cAP
max
fy pe 93
y por ‘ce max eh max? (5)
p = density of water
c = velocity of sound From the films it was observed that the cavity
Other symbols as above thickness seemed proportional to the length rather
Here it is necessary to know a time At propor- than to the square root of the cavity area and the
tional to the duration of the pressure pulse. following coefficient was obtained in cases where
With the use of At some information about the the area was measured.
real collapse dynamics is introduced and therefore
coefficient (3) may be somewhat more universal than ‘i
(2). Note, however, that for the original use of Pit Te max p
(3) similarity in cavitation was assumed. teal
Of interest for future work is to what extent 3.0 a a a
the final pressure behavior can be described by a Fi a t
measured cavity data. In this case it is more NSB bubbles
natural to think of methods to estimate a2v/at? in @ 6 3
(1). It is then necessary to know V(t) or to assume Oo 5 bs
a relation between a*v/at? and measured parameters, 7 slow pressure increase
nh llapse time and vity size In thi 2.0 7 fast pressuresrincrease
SSS ao. P A ea yo i a s : A very fast pressure increase
paper only the cavity area A(t) is presented. As
a first approximation it will be assumed that V(t)
is proportional to a3/2 ox (see. From the measure-
ments of A(t) attempts were made to estimate a*v/at?
by difference ratios in the conventional manners. (&)
This failed, due to uncertainty in A(t) during the 10- extreme
final collapse. Then as a very rough assumption
Vv
GAY ~
=~ = const —* (4)
at? an
Cc 0 SSE
0.0 05 1.0 Reduced freq. 1.5
was tested.
This is true only at very special circumstances.
The assumption was, however, used and from (1) and
(4) the following dimensionless pressure coefficient FIGURE 29. Pressure p- at collapse. Different
is obtained conditions.
(ft
0 5 10 15 foge (Hz) 20
487
+r [Pa:m] reached the low value region (Figure 26). There
is considerable scatter in generation efficiency.
It must, however, be remembered that the plot is
based on single cavitation events probably not
always typical, the results must only be seen as a
first hint of tendencies. The coefficient (3) gave
tooo} + = a t aoratry results rather similar to those from (2) but with
somewhat smaller dispersion. In Figure 29 it can
be seen that with coefficient (5) the dispersion
of the points was considerably decreased.
In Figures 30-32 results from Figures 17-23 are
plotted. Only the dimensionless coefficients (2)
and (6) are shown and it is seen that both attain
approximately the same values for similar pulses,
but neither of them brings the values of oscillation
periods 6 and 7 into agreement with the others.
100 | (tees a Le The other coefficients give similar results. Also
if the coefficients are based on values of area,
| time, etc. closer to the final collapse, the scatter
is not decreased drastically. The conclusion of
| ical Iai this is that, in the prediction of noise by theory
0 10 20 30 40 50 60 7, or model tests, good similarity in certain cavitation
Maximum cavity area(em’) events is important, and that these important events
are not generally described by such simple parameters
as To and Vpax-
Because it was not possible to estimate d2v/at2
500
_——— a on 1
osc.per -1 7 6 13 4 2 «4
aP 4 5
FIGURE 30. Pressure p from different oscillation
pamiests € 248 Hu, te ago 20: directly from measured values of V(t) functions of
osc the type:
es ge A Q V(t) = const[1 - cos o(t)] [o(t) is a polynomial
Ee c max BA max max” (6) with six variable para-
In Figures 27, 28, and 29 results are shown for meters ]
the different conditions shown in Figures 9-13. were closely matched to nearly the whole collapse.
ptr is shown in Figure 27 only to provide a reference The pressures then calculated by use of these
for the other parameters. functions agreed fairly well with measured values
Figure 28 shows that the generation efficiency in many cases. These simple computations also
increased strongly at a certain fogc~ (or reduced demonstrated how sensitive the generated pressure
frequency). The increase normally coincided with often was to the final behavior of V(t) and it was
generation of very sharp pressure pulses and at easy to realize that parameters of the types dis-
these f5,, the relative collapse time had also cussed above can only be "universal" if they are
applied to fairly similar cavitation events.
ptr
[PAR
wee Bitdlemaxp
30 Amax'max
20) 2.0 . ey
1.0 10 ree a
05
|
|
YS Saal aaa:
i 2 le
0 10 20 30 40 50 60 70
Maximum cavity area (cm?)
tt"
oscper -1 7 6 130 14 124 osc.per -17 6 3 (4 12
bite + : P . +
FIGURE 31. Pressure p from different oscillation FIGURE 32. Pressure p from different oscillation
periods f S15 fe, Oo ses 3%, periods f = 9 fe, @. = fs =o.
osc © osc °
488
3. SUMMARY AND CONCLUSIONS FROM EXPERIMENTS WITH
AN OSCILLATING HYDROFOIL
1. The generation of sharp pulses was dependent
of the oscillation frequency. At low
frequencies no high and sharp pulses were
generated and above a certain frequency very
high pulses were generated.
2. The sharpest and highest pulses were generated
by cavities which separated from the main
cavity and underwent a rather symmetrical
and orderly collapse. Detailed studies
showed, however, that a series of pulses was
often generated, indicating that the collapse
was not always simple at the very end.
3. Very high pulses could also be generated by
cavities that were attached to the leading
edge during the whole collapse.
4. The highest pressure generation efficiency
was observed for spherical bubbles, which
despite their smallness generated rather
strong pulses.
5. The sharp pulses were generated during the
very last part of the collapse.
6. Rebound of cavities was an important process
for generation of sharp pulses. The most
violent rebounds were obtained for separated
cavities.
7. Low frequency noise was generated during the
growth, near the time of maximum cavity
extent and during the rather late stage of
collapse. Because of a disturbing resonance
the importance of collapse was, however,
difficult to determine.
The basis of existing scaling laws for cavitation
noise is mainly [see for example Levkovskii (1968)
and Baiter (1974)]:
1. Ideas from theory and experiment concerning
the dynamics and radiation properties of a
single cavity.
2. Ideas concerning statistical properties of
the pulse-generating events.
The dynamics and radiation depend on cavity
geometry, cavity size, and the surrounding pressure.
Scaling laws based on simple theory deal with model
scale and magnitude of surrounding pressure, while
similarity has to be assumed in cavitation behavior.
It has to be accepted that complete similarity
in cavitation behavior will not occur, but if it is
known which events in the cavitation process are
crucial for generation of important pulses this
will provide an indication of to what extent
similarity is necessary for proper application of
scaling laws.
Of course these introductory experiments cannot
supply the final and complete answer, but the results
indicate that one of the most important factors is
that the separation of a cavity into parts is
correctly scaled, the reason being that these
separations are often the starting points for violent
collapses.
this often begins at an early stage of the collapse,
or is even initiated by disturbances during the
growth of the main cavity.
Parameters that determine tendencies to separation
of cavities have only been studied to a limited
extent, but it is clear that the combination of a
long (chord-wise) cavity and high reduced frequency
causes extensive separation of large parts from the
main sheet. From the plots of collapse times and
pressure generation efficiency, DeC/ AP ene: as
Especially when large parts are separated,
functions of reduced frequency it can be concluded
that within special regions it is important that
the time variations of the surrounding pressure be
properly scaled. Such a scaling may be critical
for the onset of separation of large cavity parts
from the main cavity.
4. NOISE FROM DIFFERENT CAVITATION SOURCES
Introduction
In order to gain more information concerning the
noise emitted from a cavitating source, tests with
four axisymmetric head forms and two hydrofoils
have been carried out in SSPA cavitation tunnel No.
1. The aim of these tests was to obtain well-defined
and unambiguous types of cavitation, as bubble,
sheet, and vortex cavitation. Comparisons of the
noise levels from these different types of cavitation
were made, as well as some investigations of the
effect of free-stream velocity and gas content.
The results reported here will only concern effects
of the type of cavitation.
Test Set-Up
The tests were carried out in SSPA cavitation tunnel
No. 1 test section, 0.5 m x 0.5 m. The noise was
measured using arrangement 4 (hydrophone in water-
filled box), see also Figure 1. In some of the
later tests a flush-mounted hydrophone in the
tunnel wall (arrangement 2) was used as well as a
hydrophone in the flow field. Signals from the
hydrophone(s) were registered by a tape recorder,
but also directly analysed by a 1/3 octave band
analyser and a narrow-band analyser. Main results
given here are from the 1/3 octave band analysis.
Tests were carried out for a water speed 9 m/s,
but with some additional tests at 7.5 m/s and 11
m/s. The gas content of the water at the tests
was 10% and 40%, with some additional tests at
higher gas content.
Test Set-Up
The first series of tests was carried out with
axisymmetric head forms. The reason for this
choice was that cavitation patterns for these bodies
were well-known and well-defined from rather exten-
sive tests [Johnsson (1972)]. The head forms used
are given below, see also Figure 33.
Head form Shape Cavitation Type of
SSPA iden- of nose number for cavita-
tification contour cav inception tion
U1A hemispherical 0.67 sheet
N39 flatt+elliptic 3:1 0.4 bubble
N3 flat+elliptic 6:1 0.42 sheet
N10 flattelliptic 4:1 0.43 sheet
The head forms were attached to a cylinder and
a faired afterbody, which were suspended from the
tunnel roof via a thin wing. The main difficulty
at the tests was the low cavitation numbers needed.
At cavitation numbers below 0.4 fairly extensive
cavitation occurred at the wing-tunnel roof junction
and at other imperfections along the tunnel walls.
This cavitation caused rather excessive background
U1A
Hemispherical
N 39 (flat nose )
Elliptic 3:1
N
Q
s
oo
c=)
N3 (flat nose)
Elliptic 6:1
i
\
(a)
w
Oo
oe!
N10 (flat nose )
Elliptic 4:1
\
—
@ 05
FIGURE 33. Axisymmetric head forms.
noise and made noise measurements almost impossible
at low cavitation numbers. There is also some
question whether such background noise from undesired
cavitation was obtained at higher cavitation numbers
than o = 0.4, when cavitation numbers are increased.
With regard to these findings the results given
here are limited to cavitation numbers o > 0.6 and
only for decreasing pressure.
In Figure No. 34 1/3 octave band noise spectra
for cavitation numbers o = 1 and o = 0.6 are given.
At o = 1.0 no visual cavitation was obtained and
the noise levels are almost the same as for the
empty tunnel (at the same velocity and cavitation
number). At o = 0.6 the cavitation is well developed
for the hemispherical nose, for the other head forms
no cavitation can be visually observed. There are,
however, rather large differences in noise spectra
for the three "non-cavitating" head forms. Thus
head forms N3 and N1O have noise levels 10 to 20
dB above N39, for which the noise level is equal
to non-cavitating or empty tunnel conditions. These
differences cannot be attributed to unwanted cavita-
tion on the wing or tunnel walls. In that case the
noise levels for head form N39 should also have
increased. The conclusion is thus that head forms
N3 and N1O have audible but not visible cavitation.
From the tests with axisymmetric head forms it
can be concluded that the cavitation numbers will
be low, which implies that effects of unwanted
cavitation will increase background noise levels
and violate results for the cavitating head forms.
Tests with Hydrofoils
In order to obtain cavitation at higher cavitation
numbers tests with two wings have been carried out.
Using wings, vortex cavitation can also be obtained.
The problem is here rather to obtain other types of
cavitation without getting vortex cavitation.
489
One of the wings tested has cambered sections
and elliptical planform, and the other has symmetric
sections and trapezoidal planform, see Figure 35.
Wing Angle Cavitation Type of
(SSPA ident- of number for cavitation
ification) attack, a cav inception
Elliptic, =e =2 sheet
cambered
(16-12.12) the De) vortex
LW22 3 vortex
Trapezoidal,
symm rounded (0) 0.5 bubble
tip (K7 Vbl1*) Bo SL 5) vortex
Trapezoidal
symm with
end plate Sse =1.2 sheet
(K7 Vp3*)
(*The wing K7 was tested with rounded tip, Vbl,
and a small end plate, Vp3, see also Figure 35).
For the comparison of noise emitted from different
types of cavitation it is important that these
comparisons be made at the same cavitation number.
One inherent difficulty is that pure bubble cavita-
tion seems to be possible to obtain only at rather
low cavitation numbers compared with the other
cavitation types.
dB re10°° Pa
150 r
Cav. number O=1
140 -
WD No cavitation
19 — T
05 2 5 10 40 f (kHz)
dB re 10° Pa
150
Cav. number O=0.6
140+ Sheet cav. (U1A)
130 IP supra lice =
(N10)
sedli (N3)
Nob
No cav. (N 39)
100 is 1
05s 2 5 10 40 f (kHz)
FIGURE 34. Axisymmetric head forms, cavitation noise
(1/3 octave band). (Free stream velocity 9 m/s,
gas content 10%.)
490
Wing 16-12.12
Elliptic , cambered
Wing K7
Trapezoidal , symmetric
Tip shape:
Rounded K7 Vb1
S——
End plate K7 Vp 3
FIGURE 35. Wings.
Results from the tests are given here for five
cavitation numbers, o = 3, 2.5, 2, 1.5, andl.
The free stream velocity was 9 m/s and the gas
content ratio was 10%. Results are given as faired
curves for the noise levels from 1/3 octave band
analysis.
For cavitation number o = 3 (Figure 36) only
the cambered wing 16-12.12 at a = 172° cavitates
with vortex cavitation. Noise levels for the wings
with no cavitation are of the same order as for the
empty tunnel. The vortex cavitation at a = 172°
gives an increase in noise levels of 15 to 20 dB
compared with non-cavitating conditions.
At o = 2.5 the wing 16-12.12 has vortex cavita-
tion at a = 2° and a = 172°, Figure 36. It is of
interest to note that the vortex cavitation at
a = 2° is not attached to the wing tip but starts
behind the wing. This vortex can only be obtained
when the pressure in the tunnel is increased
(increasing cavitation number). The increase in
noise level due to vortex cavitation here is also
15 to 20 GB.
For the cavitation number o = 2 the wing 16-12.
12 has vortex cavitation at a = 172°, intermittent
vortex cavitation at a = 2° and sheet cavitation
at a = -2°. The vortex cavitation gives an increase
in noise level of the order of 15 dB. The sheet
cavitation at a = -2° increases the noise levels
at higher frequencies (f > 5 kHz), 10 to 15 dB
above the level for vortex cavitation, see Figure
Bike
At o = 1.5 it can be noted that in some cases
no pure types of cavitation can be obtained. Thus,
wing 16-12.12 gives sheet cavitation at a = -2°,
vortex cavitation at a = 2° and vortex and bubble
Cavitation at a = 172°. Results in Figure 37 show
the largest increase of noise levels for sheet
cavitation. Note also the differences between
dB
150
140
130
120
110
100
dB
150
140
130
120
110
100
FIGURE
(Free
re 10° Pa
Cav. number O=3
Vortex cav. (16-12.12 , @=172°)
No cavitation
f(kHz)
re 10° Pa
Ir
Cav. number O=2.5
Vortex cav. (16-12.12,
Fe os 5
_- <— Vortex not Q=172°, a= 2°)
<a attached (A= 2°)
No cavitation
05 10 f (kHz)
36. Wings, cavitation noise (1/3 octave band).
stream velocity 9 m/s, gas content 10%.)
dB re 10° Pa
150
Cav. number O= 2
140
Sheet cav. (16-12.12,@=-2°)
130-
Vortex cav.(16-12.12,
120
@=172°, a=2°)
eile No cavitation
1
100 05 2 5) 10 40 f(kHz)
dB re 10° Pa
150 Cav. number O=1.5
Sheet cav. (16-12.12 ,@=-2°)
10F
_- Vortex and bubble -
= cav. O decreasing.
IO (16 -12.12 , @=172")
120 Vortex cav. (16-12.12,@=172",
O increasing ,@= 2°)
110
No cavitation
108 0.5 2 5 10 40 f (kHz)
FIGURE 37. Wings, cavitation noise (1/3 octave band).
(Free stream velocity 9 m/s, gas content 10%.)
decreasing and increasing cavitation number for
a = 172°. For decreasing 0 small cavitation bubbles
are obtained, which increase the noise level about
15 dB compared with increasing o.
From the results at cavitation number o = 1.0
(see Figure 38) it is obvious that bubble cavitation
gives the largest increase in noise levels from
25 dB at low frequency (500 Hz) to 55 dB at high
frequency (40 kHz). Sheet cavitation gives less
increase but depends on the intensity of the cavita-
tion. Thus for wing 16-12.12, a = -2°, the sheet
cavitation is extensive and gives an increase from
20 dB at low frequencies to 50 dB at high frequencies
compared with non-cavitating condition. For wing
K7 Vp3 the sheet cavitation is concentrated at the
leading edge and an increase in noise level is only
obtained for higher frequencies (> 2 kHz) and the
increase at 40 kHz is of the order of 25 dB. The
differences in noise level for wing K7 Vbl for
increasing and decreasing cavitation numbers can be
attributed to differences in cavitation patterns.
No pure vortex cavitation could be obtained at
cavitation number o = 1.0.
Conclusions from Tests with Head Forms and Hydrofoils
Tests with head forms are less suited as rather low
cavitation numbers are needed. This may cause
problems with high background levels due to undesired
cavitation on tunnel walls etc. Tests with hydrofoils
can be used to obtain effects on noise levels from
different types of cavitation. There may, however,
be some problems in obtaining pure cavitation types.
Vortex cavitation gives an increase in noise
level of about 20 dB. It should be noted that
differences in vortex cavitation can be obtained
for increasing and decreasing pressure, which also
show as differences in noise level. Also a vortex
not attached to the wing causes increases in noise
level. The increase in noise level due to vortex
cavitation seems to be less for lower cavitation
numbers.
Sheet cavitation gives substantially higher
levels than vortex cavitation. The extent of the
sheet has some influence on the noise level. For
a fairly large sheet increases in noise level of
20 dB at 500 Hz to 50 dB at 40 kHz are obtained.
For a small, leading edge sheet the increases in
dB re 10° Pa Cav. number O=1
160 Bubble and vortex
cav. (16 -12.12 ,a=172°)
150
Sheet cav
140 - (16-1212, @=-2°)
130
al Sheet cav.(K7 Vp 3 a=5)
Vortex and sheet cav.
increasing O.
NOP (K7 Vb1 a=5)
No cavitation
1
ge f (kHz)
0.5 2 5 10 40
FIGURE 38. Wings, cavitation noise (1/3 octave band).
(Free stream velocity 9 m/s, gas content 10%.)
491
noise level are obtained for higher frequencies
(£ > 2 kHz) and for 40 kHz the increase is 25 dB.
Bubble cavitation gives the largest increases
in noise level. Levels are for this case 5 to 10
dB above the levels for sheet cavitation.
ACKNOWLEDGMENT
This work is part of the research program at the
Swedish State Shipbuilding Experimental Tank and
the authors are indebted to Dr. Hans Edstrand and
Mr. H. Lindgren for making this study possible.
Part of the work reported here has been carried
out with financial support from The Defence Material
Administration of Sweden.
The authors would also like to express their
sincere thanks to those members of the staff at SSPA
who have taken part in the investigations and the
analysis of the material.
REFERENCES
Baiter, J.-H. (1974). Aspects of Cavitation Noise.
Symposium on High Powered Propulsion of Large
Ships, Part 2, December 1974, Wageningen, The
Netherlands. Publication No. 490, Netherlands
Ship Model Basin, Wageningen, The Netherlands,
pp. XXV 1-39.
Blake, W. K., M. J. Wolpert, and F. E. Geib (1977).
Cavitation Noise and Inception as Influenced by
Boundary-Layer Development on a Hydrofoil. J.
Fluid Mech. 80, 4, pp. 617-640.
Harrison, M. (1952). An Experimental Study of
Single Bubble Cavitation Noise. J. Acoust. Soc.
Am. 24, 5; 776-782.
Itd6, T. (1962). An Experimental Investigation into
the Unsteady Cavitation of Marine Propellers.
Proceedings of IAHR-Symposium, Sendai, Japan,
1962, Cavitation and Hydraulic Machinery edited
by Numachi, F., Institute of High Speed Mechanics,
Tohoku University, Sendai, Japan, 439-459.
Johnsson, C.-A. (1972). Cavitation Inception Tests
on Head Forms and Hydrofoils. Thirteenth Inter-
national Towing Tank Conference. Proceedings
Volume 1 edited by Schuster. S., and M. Schmiechen.
Versuchsanstalt fur Wasserbau und Schiffbau,
Berlin, Germany, 723-744.
Lehman, A. F. (1966). Determination of Cavity
Volumes Forming on a Rotating Blade. Eleventh
International Towing Tank Conference, Tokyo 1966,
Proceedings edited by Kinoshita, M., Yokoo, K.
The Society of Naval Architects of Japan, Tokyo,
Japan, 250-253.
Levkovskii, Y. L. (1968). Modelling of Cavitation
Noise. Sov. Phys.-Acoust. 13, 3; 337-339.
Rayleigh, Lord, (1917). On the Pressure Developed
in a Liquid During the Collapse of a Spherical
Cavity. Phil. Mag. 34, 94-98.
Ross, D. (1976). Mechanics of Underwater Noise.
Pergamon Press Inc., New York, USA, p. 218.
Tanibayashi, H., and N. Chiba (1977). Unsteady
Cavitation of Oscillating Hydrofoil. Mitsubishi
Technical Bulletin 117. Mitsubishi Heavy
Industries, Ltd. Tokyo, Japan.
492
APPENDIX
LOW OSCILLATION FREQUENCIES MAINLY GENERATING
RATHER SLOW PRESSURE PULSES
The following observations were typical for fy, =
1-3 Hz and ag = 3°, & = 4° (Figure 9) but most of
the results are also valid for other angle conditions:
1. The maximum pressure increase is generated
before the sheet cavity has disappeared
completely. At the moment of maximum pressure
increase the collapse Slowed significantly
and the rest of the collapse was very slow.
Due to hysteresis the total collapse time
was sometimes longer than the growth time,
T.. Typical for the collapse from maximum
extent to maximum pressure was Way (ls + Tg)
> 0.4. The sheet cavities were attached to
the leading edge during the whole collapse
and only small parts were separated from the
downstream cavity edge.
Already during growth a large part of the
cavity is disturbed and consists of one part
with a smooth surface and one with thick
irregular cavity formations. From this total
connected cavity, small parts were separated
both during growth and collapse. Only a few
of these parts collapsed violently, which is
also confirmed by the pressure signals which
do not contain many sharp pulses during
growth and first part of collapse.
At very low f,., (1-2 Hz) these contin-
uously occurring collapses of small cavities
were, however, the only source of high-
frequency noise. At these conditions also
most sharp pulses were obtained in the
hydrophone (H2) near the trailing edge.
3 AG Eose = 2 and! 4) Hz the pressure ancrease
often ends with a sharp pulse. The pulse
was, however, not caused by an orderly and
violent collapse of the main cavity, but
instead by small cavities that separated
from the main cavity and then collapsed
separately. It was also observed that these
rather violent collapses of small cavities
mainly occurred during the time when the
pressure was high owing to main cavity col-
lapse.
On a more expanded time scale it can also
be seen that the sharp pulse is superimposed
on a slower pressure increase. If not very
clear, this tendency is still detectable in
the 7 Hz-condition in Figure 14. This figure
shows the pulse (oscillation period 6) in
the 7 Hz-condition shown in Figure 10, but
with the time axis expanded 40 times.
4. The cavitation sketches in Figures 9-13 show
that for f,., $ 4 Hz the cavitation extent
was approximately independent of fosc, but
that at higher fog, the cavity did not develop
to the full size. One reason for this may
be that the time variation of the dynamic
angle of attack is altered with OSG
5. Characteristic of low fosc is also the fact
that collapsing cavities show little tendency
to rebound. Rebound is only obtained in
small bubbles.
to
HIGH OSCILLATION FREQUENCIES MAINLY GENERATING
SHARP PRESSURE PULSES
Below some observations are reported regarding the
conditions ag = 3°, @ = 4° and £,,, = 10 and 14
Hz (Figure 10). Many of the results are also valid
for other similar conditions. Typical observations
are:
1. The sharp pulses are often much higher than
slow pressure variations.
2. The duration of the final part of the sharp
pulses seems (as long as can be determined
in the recording) independent of fosc (Figure
14). For the earlier parts of the cavitation
period the dependency on ERS is more complex
due to different cavity sizes etc.
3. For this condition (ag = 3°, &@ = 4°) the
complete change of cavity dynamics and
pressure generation occurred between fog, =
7 and 10 Hz (Figure 14). At 7 Hz the cavity
mainly collapsed towards the leading edge.
At 10 Hz a large part consisting of thick
formations separated and performed a violent
collapse at the middle of the hydrofoil (B
in Figure 14). This collapse occurred about
1.4 milliseconds later than the collapse of
those two parts (A) of the sheet that were
attached to the leading edge during the
whole collapse. Also these two parts col-
lapsed rather violently, but a small pulse
was generated. The thick separated cavity
(B) consisted of several parts that did not
collapse exactly simultaneously and, thus,
a series of collapse and rebound pulses was
generated. A significant rebound was only
obtained from the separated cavity. The
group of rebounded cavities collapsed rather
slowly, resulting in a small pulse about
3.5 milliseconds after the collapse of the
separated cavity (B'). In some oscillation
periods the separated cavities and those
attached to the leading edge collapsed almost
simultaneously and it also happened that
high pulses were generated at the collapse
of rebounded cavities.
4. The cavitation behaviour at fy., = 14 Hz is
approximately similar to that at 10 Hz
(Figures 10 and 14). The thick formation
(C) separated and collapsed at a later stage.
The first pulse (Figure 14) was generated
by the outer cavity (A) attached to the
leading edge. About 1.4 milliseconds later
the other cavity (B) attached to the leading
edge collapsed. This cavity was complex
and generated a series of pulses. First
about 3.5 milliseconds after the first pulse
the thick formation (C) collapsed, generating
a sharp pulse. No violent collapses were
experienced by rebounded cavities in this
case. The overall impression from these
two conditions with fog, = 10 and 14 Hz is
that normally the separated thick cavities
generated the highest pulses, but that in
some cases pulses of almost equal height
were generated by cavities attached to the
leading edge.
Another behavior of the signal from the cavitating
hydrofoil is a low frequency variation (about 23 Hz)
that seems rather independent of f,,,- Sequences
containing cavitating as well as non-cavitating
periods indicate that the fluctuations were generated
by cavitation (Figure 10, 7 Hz, Figure 11 and 12).
Inspection of the films shows, however, that no
cavitation is visible on the hydrofoil or about
0.5 chord-lengths downstream it (Figure 9, 11, 3 Hz).
493
Here the most probable cause is that the cavitation
started resonance vibrations in some structure.
These vibrations probably cause disturbing errors
in the pressure signal at some f,,,, mainly in the
region 3-7 Hz, and quantitative results from such
conditions must be used with care.
Cavitation Noise Modelling at
Ship Hydrodynamic Laboratories
Gavriel A. Matveyev
and
Alexei S. Gorshkoff
Krylov Ship Research Institute
Leningrad, USSR
ABSTRACT
Theoretical and experimental correlation of visual
and accoustical effects of cavitation are considered.
The Froude similarity is treated critically because
of the pressure effects on the coefficient of cavity
energy transformation into cavitation noise as well
as because of the increase of noise absorption or
cavitation resistance of water. Though in large
cavitation tunnels which have no free surface the
nonstationary boundary conditions can be reproduced
less perfectly, their capability of simulation
at full-scale pressure is regarded as the leading
factor. It is suggested that extrapolation formulae
should take into account the effect of the rate of
pressure increase (or pressure gradient) in the
cavity collapse area. This corresponds to an
increase in the square of acoustic pressure on the
model, compared to the prototype, inversely pro-
portional to the linear scale of modelling.
1. COMPARISON OF VISUAL AND ACOUSTIC EFFECTS OF
CAVITATION
The occurrence of strong visual and noise effects
of cavitation are usually considered to be coinci-
dent. When this coincidence is actually the case,
it provides certain conveniences. The measurement
of noisiness makes it possible to detect cavitation
on structural elements not easily accessible for
inspection. Visual observation of cavitation on
models is used for the prediction of noisiness of
various prototypes. However, the experiments
involving visual and acoustical recording of
cavitation indicate that there may be a considerable
discrepancy between these two manifestations of
cavitation. It is interesting to discover the
nature and the cause of the discrepancy by means
of a mathematical model of an elementary cavitation
process which is described by the well-known
differential equation of a single spherical cavity
growth
494
jw
(1)
i]
i
Ulr
LAS
ue}
fo)
i}
ue)
Q
es
i}
ne}
Here R and R, are the cavity radius and its initial
value, respectively; p and py are the variable
component and the initial value of ambient pressure;
Par ©, Y, and v are vapor pressure, density, and
the surface tension and kinematic tension coefficient,
respectively.
Computations were made by equation (1) for the
negative pressure pulse
1-1 t
INE) SES! 7 ee
(2)
which is characterized by the time scale, T, and
the amplitude, Pm: Such a pulse represents the
region of negative pressure having the length, L,
on the profile with the maximum negative pressure
coefficient, Coms in the flow with velocity, U:
2
it
pus is) T
Linearization of Eq. (1) with respect to 6 = z2-z
for the small-amplitude oscillation frequency gives
(3)
According to (3), oscillatory properties of the
cavity disappear at pp = 1 ata when Ry < 10-© m.
Bearing in mind that the natural period is limited
by the pulse duration, T, computations were made
for
Po = 10" kg/m?; R, = 107°+1073 m; ~ = 1073410-1m;
V = 10+20 m/s; Cc =o
pm
The following value is taken as a measure of the
accoustical effect of an elementary cavitation
process to an accuracy of the potential energy
transformation coefficient for the maximally expanded
cavity: 23 5/43
G_ = 10 Log a (4)
R?
n
Here, Rn is the threshold value of the cavity radius
which, for the sake of convenience and without
limiting the generality of conclusions, is taken as
10-6 m. For large Rp/Ro values this measure differs
only slightly from a simpler measure used in Figure 1
Vy as 5
Gh 302g (5)
The threshold of the visual observation of cavita-
tion is taken as Re = 1073 m, which coincides with
the upper limit to the size of the cavitation nuclei
under study. For the chosen measure of acoustic
effect this threshold corresponds to 90 dB. Since
the resolution of vision is limited by angular
dimension, the measure of the visual effect where
the distance to the object of observation remains
constant is the first order, linear dimension of
the cavity. Hence, when the origins coincide
al
Giese
B 3.7 @)
and the processes below the level of G, = 90 dB are
out of visual observation.
Thus, leaving out of account the actual signal-
noise ratios, the acoustical recording makes it
possible to penetrate much deeper (by 2-3 orders)
into the "microcavitation" region.
Worthy of notice is the qualitative similarity
of the curves shown in Figure 1 to the experimental
curves of cavitation noise increase against velocity
which are given below, as well as by Sturman's data
(1974). It is evident that at an early cavitation
p= 4°)
FIGURE 1. Calculated comparison of visual and acous-
tical effects of elementary cavitation process in a
limited region of negative pressure.
495
stage the predicted levels drop by 20 dB with the
velocity decreasing 10-fold. This stage is usually
regarded as free of cavitation.
With the increase of velocity there comes a stage
which is sometimes referred to as "true" cavitation
and in which the most intensive growth of cavities
and cavitation noise is observed. This stage corre-
sponds to a decrease and loss of static equilibrium
of the cavity.
At the third stage the intensive cavity growth
ceases and asymptotic saturation of the acoustic
effect occurs due to the fact that the size of the
cavity is nearing that of the zone of negative
pressure. The asymptotic values of saturation shown
in Figure 1 correspond to the rough estimation
L
Gre = 15 +15 Hee + 302g a (6)
n
As to the relationship between visual and noise
manifestations of cavitation, Figure 1 allows one
to assert that:
- at sufficiently high levels of ambient noise
the acoustic detection of cavitation may coincide
with the visual detection or takes place even later;
- potentially, at a fairly low level of the
ambient noise, the acoustic manifestation of cavita-
tion must be detected much earlier than the visual
one.
In particular, the acoustic effect of cavitation
can be rather strong (e.g., an increase of noisiness
by several dozens of decibels) in the case of "micro-
scopic" cavitation invisible to the eye.
The indicated values are largely conditional as
the threshold of visual detection may differ under
different conditions. Nevertheless they are close
to those obtained under laboratory conditions.
It is of interest that Figure 1 reveals such a
contradictory phenomenon as vagueness in respect
to cavitation inception. At high levels the curves
for various nuclei coincide, so for a more correct
determination of cavitation inception one should try
to reduce rather than to increase the accuracy of
recording methods. The increase of accuracy, as is
shown in Figure 1, brings about increasing ambiguity
of cavitation inception and expansion of the vague-
ness region to cover an increasing range of veloci-
ties. However, as the accuracy decreases, more and
more small zones of cavitation inception are left
out of control.
The above analysis simplifies the actual processes
and can be at variance with them mainly due to the
fact that the coefficient of cavity potential energy
transformation into acoustic energy is not constant
being a complex function of many parameters [Benia-
minovich et al. (1975)]. Specifically it may have
a much greater value for small cavities as compared
to larger cavities.
2. EXPERIMENTAL STUDY ON MODELS
There is an urgent need for an effective and well-
founded classification of a great variety of forms
and types of cavitation which substantially differ
in the mechanism of nonstationarity giving rise
to noise and having other practical consequences
of cavitation.
The following brief list of the forms and types
of cavitation represents a more or less established
practice with respect to marine propellers [Goncharov
et all. (1977) ])-
496
——-— - bubble cavity
- sheet cavity
Uniform flow
FIGURE 2. Development of noise and visual manifesta-
tion of cavitation at constant pressure vs. velocity,
in- reference to the conditions of cavitation noise
detection.
According to the location of cavitation zones:
- vortex cavitation (in the cores of tip and
axial vortices),
- leading edge cavitation (on the suction side
and pressure side at the leading edge),
- blade-profile cavitation (in the region of
large blade thicknesses) .
—-—-— - bubble cavity
- sheet cavity
Nonuniform flow
FIGURE 3. Development of noise and visual manifesta-
tion of cavitation at constant pressure vs. velocity,
in reference to the conditions of cavitation noise
detection.
- root cavitation (at the blade roots).
According to cavity pattern:
- bubble cavitation (with cavities moving with
the flow through negative or increased pressure
zones),
- sheet cavitation (with cavities which on the
average are motionless in relation to the propeller).
By steadiness (uniformity) of the incoming flow:
- steady cavitation (noise and other effects
result from the inner unsteadiness of the cavity
which is steady on the average),
- unsteady cavitation (noise and other effects
result from the cavity pulsations at an almost
regular frequency, the phenomenon of cavitation
buffeting),
- cavitation in an unsteady flow (noise and other
effects here again result from the pulsations as
well as from the probable disappearance of cavities
with the frequency of flow condition change).
It seems extremely difficult to provide a com-
parative description of noisiness for about three
dozen cavitation types characterized only by the
above-mentioned features. Some guidance is given
by the experimental data presented in Figures 2 to
Bs
In steady-state conditions the bubble cavitation
types are the most noisy (Figure 2). Among cavita-
tion zones of different locations, vortex cavitation
types are the least noisy (Figure 3), whereas
pressure-side, leading-edge cavitation types are
the most noisy (Figure 4).
In an unsteady (non-uniform) flow the relation
between the noisiness of sheet cavitation and that
of the bubble type is different (Figure 5).
The higher noisiness of the pressure-side leading-
edge cavitation is accounted for by the rapid
increase of pressure behind the suction zone (high
gradient), which is typical of these conditions. In
case of the bubble structure of a cavity this rapid
pressure increase is accompanied by the increase
of acceleration during the collapse. In case of
the sheet structure it is accompanied by the
~o- - vortex cavitation V
—e - leading-edge
cavitation
FIGURE 4. Development of noise and visual manifesta-
tion of vortex and leading-edge cavitation appearing
in succession in a uniform flow at constant pressure.
- cavity on the v
pressure-side of
the leading edge
—o
—eo - cavity en the
suction-side of
the leading edge
FIGURE 5. Development of noise and visual manifesta-
tion of cavitation on both pressure- and suction-side
in a uniform flow at constant pressure.
unsteadiness of even small size cavities due to
closure behind the maximum suction zone.
The change in the relative noise intensity of
sheet and bubble cavities depends upon the fact
that in the case of the bubble cavity structure the
unsteadiness varies but slightly, whereas the volume
of sheet cavities begins to severly pulsate. Passing
over to the unsteady flow, we may even observe the
reduction of bubble cavitation noise. This occurs
when one portion of the propeller gets free from
the cavity whereas, on the other portion thereof,
the intensive development of cavitation is not
accompanied by an increase of noise due to a satura-
tion effect.
Individual points on the graphs shown in Figures
2 to 5 indicate moments of the first visual detection
of cavitation. As is seen, in a large cavitation
tunnel where the measurements were made, the above
conclusion that the noise comes ahead of the visual
detection of cavitation is to a variable degree
valid for any type of cavitation.
3. MODEL-PROTOTYPE CORRELATION AND COMPARISON OF
MODEL-TEST RESULTS WITH FULL-SCALE DATA
It is usually assumed [Levkovsky (1968) and
Sturman (1974)] that the fraction of the cavity
potential energy converted into cavitation noise
(coefficient of transformation) is the same for
the model and the full scale ship. Experience con-
firms the validity of the conflicting conclusions
[Beniaminovich et al. (1975)] that are indirectly
confirmed in some works. The coefficient of cavity
energy transformation into noise proves to be
strongly dependent on the absolute pressure, Po-
It is this fact, that was used by Beniaminovich
et al. (1975) for explaining the reduction of the
transformation coefficient by several orders with
497
a decrease of pressure, Por from 1 to 0.4 ata. ie
is also emphasized that at sufficiently low Pp, the
collapse of cavities is not necessarily accompanied
by shock wave generation.
Vacuum noise measurements, when performed in
ship hydrodynamics laboratories engaged in cavitation
research, show inadmissible noise absorption in
the facility water unless measures are taken to
insure additional removal of gaseous nuclei of cavi-
tation from the water. By intensified water degassing
the absorption may be reduced to an acceptable level,
but the resulting growth of cavitation resistance
of water leads to a drastic change of conditions
for inception and development of cavitation [Gorsh-
kof£ and Lodkin (1966)]. In view of the complicated
character of absolute pressure effects on the
coefficient of cavity energy transformation into
noice it appears to be good practice to perform
cavitation noise measurements at a full-scale value
of pressure.
That the Froude similarity will not be fulfilled
under these conditions, can be accepted provided
that adequate means are available for the description
and reproduction of the conditions of flow non-
uniformity behind the hull. This approach, used
in a large cavitation tunnel in combination with
correlation methods recommended by Levkovsky (1968),
Sturman (1974), has shown that overestimated cavita-
tion noise levels are predicted in this case. This
was found to arise from the fact that the coefficient
of cavity energy transformation into noise is
approximately proportional to the rate of pressure
growth leading to the cavity collapse. In modelling
by the Froude method this pressure growth rate
decreases as VL.
In case of large-scale modelling the comparison
of model-test and full-scale data may not have
revealed this discrepancy among other more pro-
nounced ones. One can use pressure gradient instead
of the rate of pressure variation with time. Then,
for Froude similarity, the noise level model-to-full-
scale extrapolator coincides with that used by Sturman
(1974). Not so with modelling at full-scale absolute
pressure. Here the proportionality of the transform-
ation coefficient both to the velocity of pressure
variation with time and to the pressure gradient in-
volve the same extrapolator. Giving up the construc-
tion of dimensionless parameters of which, with a
great number of constants involved, there is ample
freedom of choice, the extrapolator suggested by
Sturman (1974)
3
A. = 2808 (7)
<p*> —SS SS
2
can be substituted by the following:
is the square of the acoustic pressure,
T is the distance to the point of noise
measurement,
No is the number of cavities collapsing in
unit time.
If we assume in the regular way that the similar-
ity of cavity patterns is observed and the noise is
measured at similar points of the flow, then
V w/e
- model-test data
coo - full-seale test data
FIGURE 6. Comparison of noise levels extrapolated
from model with measured full-scale data in a wide
band of frequencies.
and t = L. The table below shows extrapolators for
scaling the square of the acoustic pressure during
cavitation from model to full-size with reference
to the assumptions of constant and variable coeffi-
cients of cavity energy transformation into noise,
nN, and to fit the cases of constant Froude number
and constant absolute pressure.
That the frequencies vary inversely in proportion
to linear dimensions in modelling at a constant
pressure may turn out to be a significant advantage,
so the acoustic wave lengths change in proportion
to linear dimensions of the model and wave inter-
ference patterns remain unchanged. In modelling by
the Froude method wave lengths on the model are
© model-test and full-scale results for the wide
frequency band
@ model-test and full-scale results for the 1/3-
octave noise frequency band of the model - 80 kHz
* instant of visual detection of vortex cavitation
on the model
FIGURE 7. Comparison of noise levels extrapolated
from model with measured full-scale data.
TABLE 1. Cavitation Noise Levels Scaling
Extrapolator
P P
=e ike
nN = const L T
Fo = const i3/2 Be72 Le
Bs = const iL 1/L 1/L
known to be VI, times larger than the model linear
dimensions.
Figure 6 shows the comparison between the model-
test data (solid line) scaled the comparison between
the model-test scale data (dotted line) for the
noise level in a wide band of frequencies. Figure
7 gives a similar comparison with another prototype.
Curves of cavitation noise increase are also compared
in a 1/3-octact band for the model at the frequency
of 80 kHz. In Figure 7 the moment of visual detec-—
tion of cavitation is marked on the general level
curve with an asterisk. Full-scale data are given
here for individual rates of speed.
The scaling extrapolator (8) needs to be verified
under full-scale conditions and is likely to be
refined. However, the need for stability of the
coefficient of cavity energy transformation into
cavitation noise appears to be an indisputable
argument for cavitation noisiness scaling with the
full-scale pressure retained.
CONCLUSION
The two major conclusions can be formulated as
follows:
- Scaling for cavitation noise measurements with
the full-scale pressure retained gives a high value
coefficient of cavity energy transformation into
noise and substantial advantages in respect to:
a) obtaining high levels of cavitation noise;
b) similarity of sound waves to the model.
- Large-scale modelling with the full-scale
pressure retained confirmed the possibility brought
out by the analysis of an elementary cavitation
process of acoustic detection of cavitation long
before the cavity reaches the size that can be
detected visually.
REFERENCES
Beniaminovich, M. B., K. A. Kondratovich, and I. V.
Krutetsky (1975). On experimental determination
of acoustic radiation energy during cavitation
bubble collapse. (in Russian). Symposium on
Physics of Acoustics-Hydrodynamics Phenomena,
Reports, Izd. "Nauka", Moscow.
Goncharov, O. N., A. S. Gorshkoff, and V. J. Vaniu-
khin (1977). Cavitation and its noise radiation
in steady and unsteady flows. 9th All-Union
Symposium on Physics of Acoustics-Hydrodynamics
Phenomena, (in Russian), Reports, Vol. 2, Izd.
Academy of Sciences of the USSR, Moscow.
Gorshkoff, A. S., and A. S. Lodkin (1966). The
inception of cavitation under symmetrical stream-
lining round a body of revolution with blunt nose.
llth ITTC, Tokyo.
499
Levkovsky, Yu. L. (1968). Scaling of cavitation
noise, (in Russian). Akustichesky zhurnal, 13,
3. Moscow.
Sturman, A. M. (1974). Fundamental aspects of the
effect of propeller cavitation on the radiated
noise. Proceedings of Symposium on High-Powered
Propulsion of Large Ships, Wageningen.
Fluid Jets and Fluid Sheets:
A Direct Formulation
P. M. Naghdi
University of California
Berkeley, California
ABSTRACT
The object of this paper is to present an account
of recent developments in the direct formulation of
the theories of fluid jets and fluid sheets based
on one and two-dimensional continuum models origi-
nating in the works of Duhem and E. and’ F. Cosserat.
Following some preliminaries and descriptions of
(three-dimensional) jet-like and sheet-likeé bodies,
the rest of the paper is arranged in two parts,
namely Part A (for fluid jets) and Part B (for fluid
sheets), and can be read independently of each other.
In each part, after providing the main ingredients
of the direct model and a statement of the conserva-
tion laws, appropriate nonlinear differential equa-
tions are derived which include the effects of
gravity and surface tension. Application of these
theories to various one and two-dimensional fluid
flow problems, including water waves, are discussed.
1. INTRODUCTION
Jets and sheets are a class of three-dimensional
bodies whose boundary surfaces have special charac-
teristic features. To this extent they are, respec-
tively, similar to another class, namely that of
rods and shells (or plates), although the nature of
the specified surface (or boundary) conditions in
the two classes may be different. Moreover, the
kinematics of jets and rods are identical, as are
the kinematics of sheets and shells. Indeed, it is
only through their constitutive equations that a
distinction appears between rods and jets on the
one hand, and shells and sheets on the other. It
is natural to inquire as to the possible utility of
methods of approach in the construction of theories
in the class of rods and shells for that of jets
and sheets and vice versa. The main purpose of this
paper is to call attention to the possible utility
of a direct approach for jets and sheets, an approach
500
which has met with considerable success in the case
of rods and shells. The direct approach for fluid
jets is based on a one-dimensional model, called a
Cosserat (or a directed) curve which is defined in
Section 3; and the direct approach for fluid sheets
is based on a two-dimensional model, called a Cos-
serat (or a directed) surface which is defined in
Section 5. It should be emphasized that a Cosserat
curve and a Cosserat surface are not, respectively,
just a one-dimensional curve and a two-dimensional
surface; but are, in fact, endowed with some struc-—
ture in the form of additional primitive kinematical
vector fields.
The concept of 'directed' or 'oriented' media
originated in the work of Duhem (1893) and a first
systematic development of theories of oriented media
in one, two, and three dimensions was carried out by
E. and F. Cosserat (1909). In their work, the Cos-
serats represented the orientation of each point of
their continuum by a set of mutually perpendicular
rigid vectors. The purely kinematical aspects of
oriented bodies characterized by ordinary displace-
ment and the independent deformation of N deformable
vectors in N-dimensional space has been discusssed
by Ericksen and Truesdell (1958), who also intro-
duced the terminology of directors.
A complete general theory of a Cosserat surface
with a single deformable director given by Green
et al. (1965) was developed within the framework of
thermomechanics; and their derivation (Green et al.
1965) is carried out mainly from an appropriate
energy equation, together with invariance require-
ments under superposed rigid body motions. A re-
lated development utilizing three directors at each
point of the surface, in the context of a purely
mechanical theory and with the use of a virtual work
principle, is given by Cohen and DeSilva (1966). A
further development of the basic theory of a Cosserat
surface along with certain general considerations re-
garding the construction of nonlinear constitutive
equations for elastic shells is given by Naghdi
(1972), which also contains additional historical
remarks relevant to oriented continua and to the
theory of thin elastic shells. A parallel develop-
ment in the theory of a Cosserat curve with two
deformable directors begins with a paper of Green
and Laws (1966) whose derivation is carried out
mainly from an appropriate energy equation, together
with invariance requirements under superposed rigid
body motions. A related theory of a directed curve
with three deformable directors at each point of the
curve, in the context of a purely mechanical theory
and with the use of a virtual work principle, is
given by Cohen (1966). A further development of
the basic theory of a Cosserat curve along with
certain general developments regarding the construc-
tion of nonlinear constitutive equations for elastic
rods is given by Green et al. (1974a,b).
In general, two entirely different approaches may
be adopted for the construction of one-dimensional
and two-dimensional theories of mechanics pertain-
ing to certain motions and (three-dimensional) media
responses which are effectively confined, respec-—
tively, to one-dimensional and two-dimensional re-
gions. For example, the theory of slender rods and
that of fluid jets are both one-dimensional theories;
and, similarly, the theory of thin shells and that
of fluid sheets are both two-dimensional theories
in the context of the particular classes of three-
dimensional bodies mentioned earlier.
Of the two approaches just mentioned, one starts
with the three-dimensional equations of the classi-
cal continuum mechanics and by applying approxima-
tion procedures strives to obtain one-dimensional
(in the case of jets and rods) and two-dimensional
(in the case of sheets and shells) field equations
and constitutive equations for the medium under
consideration. In the other approach, the particu-
lar medium response mentioned above is modelled as
a one-dimensional and a two-dimensional directed
continuum, namely a Cosserat curve and a Cosserat
surface introduced earlier; and one then proceeds
to the development of the field equations and the
appropriate constitutive equations. If full inform-
ation is desired regarding the motion and deforma-
tion of the continuum under study in the context of
the classical three-dimensional theory, then there
would be no need to develop a particular one-
dimensional and a two-dimensional theory. In fact,
the aim of one-dimensional and two-dimensional theo-
ries of the type mentioned above is to provide only
practical information in some sense: for example,
in the case of fluid sheets information concerning
quantities which can be regarded as representing
the medium response confined to a surface or its
neighborhood as a consequence of the (three-
dimensional) motion of the body, or the determina-
tion of certain weighted averages of quantities
resulting from the (three-dimensional) motion of
the body. A parallel remark may be made, of course,
in the case of fluid jets. The desire for obtain-
ing limited or partial information if the basic
motivation for the construction of such one-
dimensional and two-dimensional theories as those
for slender rods and thin shells and for fluid flow
problems of jets and sheets.
The nature of difficulties associated with the
development of both the shell theory and the theory
of water waves on the one hand, and that of rods and
jets on the other, from the full three-dimensional
equations is well known and has been elaborated upon
501
on various occasions.* In view of these, it is rea-
sonable to attempt to formulate one-dimensional and
two-dimensional theories of the types described above
by replacing the continuum characterizing the (three-
dimensional) medium in question with an alternative
model which would reflect the main features of the
response of the three-dimensional medium and which
would then permit the formulation of appropriate
one-dimensional and two-dimensional theories by a
direct approach and without the appeal to special
assumptions or approximations generally employed in
the derivation from the three-dimensional equations.
Of course, the introduction of an alternative
model and formulation of one-dimensional and two-
dimensional theories by the direct approach do not
mean that one ignores the nature of the field equa-
tions in the three-dimensional theory. In fact,
some of the developments of the field equations by
direct procedures are materially aided or influenced
by available information which can be obtained from
the three-dimensional theory. For example, the inte-
grated equations of motion from the three-dimensional
equations provide guidelines for a statement of one
and two-dimensional conservation laws in conjunction
with the one and two-dimensional models, and also
provide some insight into the nature of inertia terms
and the kinetic energy in the direct formulation of
the one-dimensional and two-dimensional theories.
Inasmuch as most of the difficulties associated
with the derivation of the one-dimensional and two-
dimensional theories from the three-dimensional equa-
tions occur in the construction of the constitutive
equations, it is in fact here that the direct ap-
proach offers a great deal of appeal. This construc-
tion, as well as the entire development by the
direct approach, is exact in the sense that they
rest on (one-dimensional and two-dimensional) pos-
tulates valid for nonlinear behavior of materials
but clearly they cannot be expected to represent all
the features that could only be predicted by the
relevant full three-dimensional equations. Theories
constructed via a direct approach necessarily sat-
isfy the requirements of invariance under superposed
rigid body motions that arise from physical consider-
tions and, of course, they are also consistent and
fully invariant in the mathematical sense. More-
over, the development by the direct approach is con-
ceptually simple and does not have the difficulties
involving approximations usually made in the devel-
opment of the theory of thin shells and the theory
of water waves (or the theories of slender rods and
jets) from their corresponding three-dimensional
equations.
Following some general background information
and definitions of jet-like and sheet-like bodies
in Section 2, the remainder of the paper is arranged
in two parts which can be read independently of each
other: one part (Part A) is concerned with the
theory of fluid jets and the other (Part B) is de-
voted to the theory of fluid sheets and its applica-
tion to water waves. In our discussion of the
direct formulation of these two topics, considerably
“the nature of these difficulties with particular reference
to shells is discussed by Naghdi (1972, Secs. 1,4,19,20,21).
Some of the difficulties associated with both nonlinear and
linear theories of water waves are noted by Naghdi (1974) and
are also discussed in the first and final sections of the
paper of Green et al. (1974c).
tsee the remarks following Eqs. (26) and (50).
502
more space is devoted to fluid sheets and water
waves. This is partly due to the fact that, in the
context of the direct formulation, the theory of
fluid sheets has to date received more attention
than that of fluid jets. Thus, in Part A (Sections
3-4) , we summarize the basic theory of a Cosserat
curve and briefly discuss a restricted form of the
theory for straight jets which are not necessarily
circular. The resulting system of nonlinear ordi-
nary differential equations includes the effects of
surface tension and gravity and has been derived for
both inviscid and viscous jets. We do not record
these here; but we call attention in Section 4 toa
number of existing solutions, which serve as evidence
of the relevance and applicability of the direct
formulation of the theory of fluid jets.
In Part B (Sections 5-8), after briefly describ-
ing the basic theory of a Cosserat surface in Sec-
tion 5, we present in outline a derivation of a
restricted theory in Section 6, and then obtain a
system of nonlinear partial differential equations
for the propagation of fairly long waves in a homo-
geneous stream of variable depth (Section 7). This
system of differential equations, which includes
the effects of surface tension and gravity, is de-
rived for incompressible inviscid fluids. Some ex-
tensions of these results to nonhomogeneous and
viscous fluids are available but these are not dis-
cussed here. In the final section of the paper we
make a comparison between the differential equations
derived in Section 7 and the systems of equations
for water waves that are often used in the litera-
ture; and, on the basis of compelling physical con-
siderations, argue as to why the system of equations
of the direct formulation should in general be pre-
ferred to others. In Section 8, we also call at-
tention to a number of existing solutions, which
serve as further evidence of the relevance and ap-
plicability of the direct formulation of the theory
of fluid sheets.
In the course of our development, sometimes the
same symbol is utilized in Parts A and B to denote
different quantities; but this should not give rise
to confusion, as the two parts can be read indepen-
dently of each other. Throughout the paper, Latin
indices (subscripts or superscripts) take the values
1, 2, 3, Greek indices take the values l, 2 only,
and the usual convention for summation over a re-
peated index is employed.
2. GENERAL BACKGROUND
In this section, we provide appropriate definitions
for jet-like and sheet-like bodies. To this end,
consider a finite three-dimensional body, 8, ina
Euclidean 3-space, and let convected coordinates,
61 (i = 1, 2, 3), be assigned to each particle (or
material point) of 8. Further, let tr* be the posi-
tion vector, from a fixed origin, of a typical parti-
cle of § in the present configuration at time, t.
Then, a motion of the (three-dimensional) body is
defined by a vector-valued function, £*, which as-
tone use of an asterisk attached to various symbols is for
later convenience. The corresponding symbols without the
asterisks are reserved for different definitions or designa-
tions to be introduced later.
signs position, r*, to_each particle of 8 at each
instant of time, i.e.,
fa (DE O07) (1)
We assume that the vector function, £*,--a 1-
parameter family of configurations with t as the
real parameter--is sufficiently smooth in the sense
that it is differentiable with respect to 6+ and t
as many times as required. In some developments,
it may be more convenient to set 93 = — and adopt
the notation
Fiz Wem ie gel ta oc (2)
Ae
CES ay 7 Gay Chi Gin gee Celera)
56 = z
gt 0 J; = 8; " gt = g”9, E gt 6 gJ = gtJ , (3)
L,
dv = g*aelae7ae? (4)
and further assume seal
Xs
Ga S ig _g Gill SO co (5)
~1=2=3
In (4), g and g are the covariant and the contra-
variant base vectors at time, t, respectively, Si5
is the metric tensor, gtJ is its conjugate, 6+ is
the Kronecker symbol in 3-space and dv the volume
element in the present configuration.
The velocity vector, v*, of a particle of the
three-dimensional body in the present configuration
is defined by
Wi Sat (6)
where a superposed dot denotes material time dif-
ferentiation with respect to t holding 6+ fixed.
The stress ‘vector t across a surface in the present
configuration with outward unit normal y* is given
by
(7)
Vos are 4
au ey a
oct
i]
ico
where
§ F .
Recall that when the particles of a continuum are referred
to a convected coordinate system, the numerical values of
the coordinates associated with each particle remain the
same for all time. Although the use of a convected coordi-
nate system is by no means essential, it is particularly
suited to studies of special bodies (such as sheets, jets,
shells, and rods) and often results in simplification of
intermediate steps in the development of the subject.
Wine choice of positive sign in (5) is for definiteness.
Alternatively, for physically possible motions we only need
to assume that g@ # 0 with the understanding that in any
given motion [g}g293] is either > 0 or < 0. The condition
(5) also requires that 01 be a right-handed coordinate
system.
* ky kG
Vi vg =v “Gy (8)
Wo ENGiter cia = Gea |p
~ 3] ,
and where tik are the contravariant components of
the symmetric stress tensor. In terms of quantities
defined in (5)-(8), the local field equations which
follow from the integral forms of the three-
dimensional conservation laws for mass, linear
momentum and moment of momentum, respectively, are
* 1
eo a
+ =
in Os OS DSI
x lox i
Z o Gy T =0 (9)
where p* is the three-dimensional mass density, £*
is the body force field per unit mass, and a comma
denotes partial differentiation with respect to 6?.
A material line (not necessarily a straight line)
in 8 can be defined by the equations, 6% = 6%(&);
the equation resulting from (1) with 6% = e"(E) rep-
resents the parametric form of this material line in
the current configuration and defines a l1-parameter
family of curves in space, each of which we assume
to be smooth and nonintersecting. We refer to the
space curve, 6% = 0, in the current configuration
by c. Any point of this curve is specified by the
position vector, xr, relative to the same fixed ori-
gin to which r* is referred, where
v= Z(E,t) = £ (0,0,8,t) . (10)
Let a3 denote the tangent vector along the E-curve.
By (10) and (3),
@
>
ag = a3(E,t) = DE = g3(0,0,€,t) (11)
and the unit principal normal, a,, and the unit bi-
normal vector, aj, to c may be introduced as
da,/dE
a; = a, (&,t) = |[BaaDET P
on = (Ee) ek a 12
a2 a2 Tas] l 2 (12)
lL,
la3| = (a33)7 ,
Glee) Seley 2 Ele) 0
[ajaga3] >O , (12)
where the notation Ja3| stands for the magnitude of
a3. The system of base vectors, aj, are oriented
along the Serret-Frenet triad and satisfy the dif-
ferential equations
day ky
Be ~ 'l@ag) Gao ke3
dag i,
Te | Sega) ano
Be3 1 2233
— = ————— : 13
ae 7 259 Dean OCS ane
503
where K and Tt denote, respectively, the curvature
and the torsion of c. In the special case that c
is a plane curve, we may choose aj as the unit
normal to the curve and then ag will be perpendicu-
lar to the plane of a, and a3. TEC isvarstravight
curve, then there is no unique Serret-Frenet triad
and a, may be chosen as any orthogonal triad with
a1,a2 as unit vectors. Equations (13) are not
identical to the formulas of Frenet because the pa-
rameter, €, is not necessarily the arc length of c.
It may be noted here that the convected coordinate,
&, may be chosen to coincide with the arc length in
any one configuration of the material curve, e.g.,
in the present configuration. However, in a general
motion (involving different configurations) the arc
length between any pair of particles changes while
the convected coordinates of each particle must re-
main the same. Therefore, arc length would not
qualify as a convected coordinate.
A material surface in 8 can be defined by the
equation, —€ = &(6%); the equation resulting from
(1) with € = €(6%) represents the parametric form
of this material surface in the current configura-
tion and defines a l-parameter family of surfaces
in space, each of which we assume to be smooth and
nonintersecting. We refer to the surface, € = 0.
in the current configuration by s. Any point of
the surface, s, is specified by the position vector,
r, relative to the same fixed origin to which r* is
referred, where
BS (Ope) = BY Ore) (14)
Let a, denote the base vectors along the 6°-curves
on the surface, s. By (14) and (3)j,
‘asa (0. ,t) =
~O.
=g (6',0,t) , (15)
~O =a
and the unit normal, a3 = a3(0’,t), to s may be
defined by**
Ey eG ee lO) hb. Crs cia le a
Bn E lepeeeal > Os (XE)
In the next four paragraphs we provide appropri-
ate definitions for jet-like and sheet-like bodies
in fairly precise terms.
Definition of a Jet-like Body. A Representation
for the Motion of a Slender Jet.
Consider a space curve c defined by the parametric
equations, e* = 0, over a finite interval, € SESE>.
Let r be the position vector of any point of c and
let aj,a2 and a3 denote its unit principal normal,
unit binormal, and the tangent vector, respectively.
At each point of c, imagine material filaments ly-
ing in the normal plane, i.e., the plane perpendicu-
kk
The use of the same symbols for base vectors of a surface
in (15)-(16) and for the triad of a space curve in (11)-(12)
should not give rise to confusion. The main developments
for jets and sheets are dealt with separately in the rest
of the paper; this permits the use of the same symbol for
different quantities in the case of jets and sheets without
confusion.
504
lar to a3, and forming the normal cross-section'tT,
Qn - The surface swept out by the closed boundary
curve, 0Q,, Of Gp is called the lateral surface.
Such a three-dimensional body is called jet-like if
the dimensions in the plane of the normal cross-
section are small compared to some characteristic
dimension, L(c), of c (see Figure 1), e.g., its
local radius of curvature 1/K, or the length of c
in the case of a straight curve. A jet-like body
is said to be slender if the largest dimension of
Q@y is much smaller than L(c). If a, is independent
of €, the body is said to be of uniform cross-
section, otherwise of variable cross-section. Since
a material curve in the three-dimensional body, 8,
can be defined by the equations, e% = e%(E), it
follows that the equation resulting from (1) with
e% = 9%(&) represents the parametric form of the
material curve in the present configuration and de-
fines a curve, c, in space at time, t, which we as-
sume to be sufficiently smooth and nonintersecting.
Every point of this curve has a position vector
specified by (10). Let the (three-dimensional) jet-
like body in some neighborhood of c be bounded by
material surfaces, § = &), € = &9, (indicated in
Figure 1) and a material surface of the form
me ,O=) =O 5 (17)
which is chosen such that & = constant are curved
sections of the body bounded by closed curves on
this surface with c lying on or within (17). In
the development of a general theory, it is preferable
to leave unspecified the choice of the relation of
the curve, c, to one on the boundary surface (17).
In special cases or in specific applications, how-
ever, it is necessary to fix the relation of c to
the surface (17). @
Suppose now that £* in (1) is a continuous func-
tion of eit and has continuous space derivatives
of order 1 and continuous time derivatives of order
2 in the bounded region lying inside the surface (17)
and between € = €], € = 9. Hence, to any required
degree of approximation f£* may be represented as a
polynomial in el, 62 with coefficients which are con-
tinuously differentiable functions of &, t. Instead
of considering a general representation of this kind,
we restrict attention here to the approximation.
ak (o}
BE + 6 qa p (18)
where r is defined by (10) and a = qd (6 rt) -
Definition of a Sheet-like Body. A Representation
for the Motion of a Thin Sheet.
Consider a two-dimensional surface, s, defined by
the parametric equation, & = 0, over a finite co-
ordinate patch, a' = 6! Sa", Bg" S 62 = p".. Let ig
and a3 denote, respectively, the position vector and
the unit normal to s. At each point of s, imagine
material filaments projecting normally above and
below the surface, s. The surface formed by the
material filaments constructed at the points of the
closed boundary curve of s is called the lateral
surface. Such a three-dimensional body is called
tt , A é :
The normal cross-section of a jet is a portion of the
normal plane to the curve, c, i.e., the intersection of the
body and the normal plane.
FIGURE 1. A jet-like body in the present configuration
showing the line of centroids with position vector r
and the end normal cross-sections & = &;, § = &. Also
shown are the unit principal normal aj), the unit binor-
mal az and the tangent vector a3 to the curve with po-
sition vector r.
a sheet if the dimension of the body along the nor-
mals, called the height and denoted by h, is small.
A sheet is said to be thin if its thickness is much
smaller than a certain characteristic length, L(s),
of the surface, s, for example, the local minimum
radius of curvature of the surface, or the smallest
dimension of s in the case of a plane sheet. If h
is constant, the sheet is said to be of uniform —
thickness, otherwise of variable thickness. Since
a material surface in the three-dimensional body can
be defined by the equation, — = &(6%), it follows
that the equations resulting from (1) and (2) with
— = £(6%) represent the parametric forms of the
material surface in the present and the reference
configurations, respectively. In particular, the
equation, € = 0, defines a surface in space at time,
e
~
FIGURE 2. Sketch of the cross-section (y = const.) of
a sheet of vertical thickness $ showing a wave motion
propagating over a bottom of variable depth. Also shown
is the surface 6? = 0 (with position vector r and height
Y) chosen such that the center mass of the (three-
dimensional) fluid region lies on this surface. The top
and bottom surfaces of height 8 and a are specified by
93 = 1/2 ana 63 = -1/2, respectively.
t, which we assume to be smooth and nonintersecting.
Every point of this surface has a position vector,
x, specified by (14). Let the boundary of the three-
dimensional continuum be specified by the material
surfaces
B= ei@=ne) 9 § = Bald 0-) Ei < Em p (is)
with the surface, & = 0, lying either on one of the
two surfaces (19)1,2 or between them (see, for ex-
ample, Figure 2), and a material surface
f(0!,02) =o , (20)
which is chosen such that € = const. forms closed
smooth curves on the surface (20). As pointed out
previously [Naghdi (1975)], in the development of
a general theory, it is preferable to leave unspeci-
fied the choice of the relation of the surface, s,
(—§ = 0) to the major surfaces, st ands. In spe-
cial cases of the general theory or in specific ap-
plications, however, it is necessary to fix the
relation of s to the surfaces (19)1,2-
Suppose now that r in (1) is a continuous func—
tion of 61,t, and has continuous space derivatives
of order 1 and continuous time derivatives of order
2 in the bounded region, &)S&S&>. Hence, to any
required degree of approximation, oe may be repre-
sented as a polynomial in € with coefficients which
are continuously differentiable functions of 6%,t.
However, instead of considering a general represent-
ation of this kind, we restrict attention here to
the approximation
ce En sei (Bel (21)
where r is defined by (14) and d = d(9%,t).
PART A
In Part A (Sections 3-4), we summarize the basic
theory of a Cosserat (or a directed) curve and then
briefly discuss a restricted form of the theory ap-
propriate for straight fluid jets. Although we are
concerned here mainly with the purely mechanical
theory involving appropriate forms of the conserva-
tion laws for mass, linear momentum, and moment of
momentum, we also include the conservation of energy.
The latter is useful in some applications and sup-
plies motivation for some requirements in the de-
velopment of certain solutions.
3. THE BASIC THEORY OF A COSSERAT CURVE
Having defined a (three-dimensional) jet-like body
in Section 2, we now formally introduce a direct
model for such a body. Thus, a Cosserat (or a
directed) curve, R, comprises a material curve, L,
(embedded in a Euclidean 3-space) and two deformable
directors attached to every point of the curve,
The directors which are not necessarily along the
unit principal normals and the unit binormals of
the curve have, in particular, the property that
they remain unaltered under superposed rigid body
motions. Let the particles of | be identified by
means of the convected coordinate, &, and let the
505
curve occupied by | in the present configuration of
R at time, t, be referred to as 2. Let r and dy
(a = 1,2) denote the position vector of a typical
point of & and the directors at the same point,
respectively, and also designate the tangent vector
to the curve, &, by a3. Then, a motion of the Cos-
serat curve is defined by vector-valued functions
which assign a position, r, and a pair of directors,
dy to each particle of R at each instant of time,
i.e.
xr = valle) ,
cL SCL) 9 MGC ea) = © (22)
and the condition (22)3 ensures that the directors,
dy, are nowhere tangent to % and that d),d2 never
Change their relative orientation with respect to
each other and a3. The velocity and the director
velocities are defined by
He 7m ee (23)
and from (23), and (11) we have
ay =a 4 (24)
where a superposed dot denotes material time dif-
ferentiation with respect to t holding € fixed.
Consider an arbitrary part of the material curve,
L, in the present configuration, bounded by & = &j
and 1G )= 75 (EG seo) jandy let
1.
ds = (a33)*d& , a33 = a3 ° a3 (25)
be the element of the arc length along the curve,
&. It is convenient at this point to define the
following additional quantities: The mass density,
p = p(&,t), of the space curve, %; the contact
force, n = n(&,t), and the contact director couples
is = p"(—,t), each a three-dimensional vector field
in the present configuration; the assigned force,
£ = £(€,t), and the assigned director couples,
ga = g%(£,t), each a three-dimensional vector field
and each per unit mass of the curve, 2; the intrin-
sic (curve) director couples, 1% = mt (Epe) o per unit
length of & which make no contribution to the supply
of momentum; the inertia coefficients, y® = y%(&)
and yB = y%8(z), with y°8 being components of a
symmetric tensor, which are indenpendent of time;
the specific internal energy, €« = €(§,t); the spe-
cific heat supply, r = r(&,t), per unit time; and
the heat flux, h = h(&,t), along 2, in the direction
of increasing §, per unit time. The assigned field,
£, represents the combined effect of (i) the stress
vector on the lateral surface (17) of the jet-like
body denoted by f,, and (ii) an integrated contri-
bution arising from the three-dimensional body force
denoted by fp, e.g., that due to gravity. A parallel
statement holds for the assigned fields, ge Sim-
ilarly, the assigned heat supply, r, represents the
combined effect of (i) heat supply entering the
5 Spor convenience, we adopt the notation for r in (10) and
(18) also for the surface (22);. This permits an easy iden-
tification of the two curves, if desired. The choice of
positive sign in (22)3 is for definiteness. Alternatively,
it will suffice to assume that [d)d a3] # O with the under-
standing that in any given motion the scalar triple product
[djdja3] is either > 0 or < 0.
506
lateral surface (17) of the jet-like body from the
surrounding environment, denoted by r,, and (ii) an
integrated contribution arising from the three-
dimensional heat supply denoted by rp. Thus, we may
write
1 OLs es) sO ett Olam ny i ry= tat COU (26)
The various quantities in (26) are free to be spec-
ified in a manner which depends on the particular
application in mind and, in the context of the the-
ory of a Cosserat curve, the intertia coefficients,
yo, yo8 and the mass density, 9, require constitu-
tive equations. Indeed, fo, 22 and r,, as well as
fb, &£% and Xp, Can be identified with the corre-
sponding expressions in a derivation from the three-
dimensional equations [see, for example, Green et
al. (1974a)]. Likewise, the inertia coefficients,
yo, yos , and the mass density, p, may be identified
with easily accessible results from the three-
dimensional theory.
With the above definitions of the various field
quantities and with reference to the present con-
figuration, the conservation laws for a Cosserat
curve are:
a iy) * -
dt p as = ’
fe A
Bi Eo a Eo Eo
ae o(v +y w)ds = i pfds + [n] '
G1 ei ey
Bo
a ap
ae oly We ae SY Welds
1
£2 ie 2y)
= i) (OH = gs) es * tee,
el 1
£2
ae | Ol wy G <y, Fa xy)
Ey
+ qe x yew las
b2
£2
zs f p(x +£-v+ fa w)ds
€1 = y = ~OL
ls Onypsde By ie) py (27)
where we have used the notation
=)
£
[ (E,t) 1,
Srp) = wep) (28)
The first of (27) is a statement of the conservation
of mass, the second is the conservation of linear
momentum, the third that of the director momentum,
the fourth is the conservation of moment of momentum,
and the fifth represents the conservation of energy.
Under suitable continuity assumptions, the first
four equations in (27) are equivalent to
1 6 r)
h = A(E) = p(a33)? or 6a33 + pag ° = =0, (29)
an
pe + AE =A(v + yw.) , (30)
ap.
WS ae SNES oes) (31)
0€ ~ ~ ~ ~B
om
a3 x n+ Be @ Key SO (32)
dh :
Ue pe 7 er 1 =O , (33)
where
ia a , a
LSS eS = it er liey Gr ’
(o} a aie ap
QRS SV ES (34)
and
ov ow,
MI fees a. @). Seley
ALE = n DE +7 Wo +p DE (35)
is the mechanical power. With the help of (34), the
local form of the moment of momentum equation (31)
can be reduced to
ad
ag Xn 4d" x qe 4 ie = Oy (36)
u dE P
It may be noted that the local field equations
in the mechanical theory of a Cosserat curve have
the same forms as those that can be derived from the
three-dimensional equations; the latter can be de-
rived by suitable integration of (9)],9,3 with re-
spect to 61 and 62 and in terms of certain definitions
for integrated mass density and resultants of stress
[for details, see Green et al. (1974a)]. Moreover,
given the approximation (18), there is a 1-l corre-
spondence between the one-dimensional field equations
that follow from the conservation laws of a Cosserat
curve and those that can be derived from the three-
dimensional equations provided we identify the
director dy in (18) with (22) 9 and adopt the defini-
tions of the resultants mentioned above. A similar
1-1 correspondence can be shown to hold between (33)
and an integrated energy equation derived from the
three-dimensional energy equation.
The above results include the local form of con-
servation of energy derived from (27)5. For the
purely mechanical theory in which the law of con-
servation of energy is excluded, the appropriate
conservation laws are the first four of (27). In
the context of the purely mechanical theory, it is
worth recalling that the rate of work by all contact
and assigned forces acting on the curve, %, and its
end points minus the rate of increase of the kinetic
energy can be reduced to:
E2 j 4 &2
-w,)ds + In - wi
a
3)
A
iq
+
IwD
ig
+
OD
where P is defined by (35).
Before closing this section, we note that the
restriction imposed on the motion of the medium by
the condition of incompressibility reduces to ||||
[see Green (1976) ]
5, [didza3] = 0 (38)
and can alternatively be expressed in the form
eax Yau yaniecy k= (39)
° x ° — i=
~B a3 We Sil a2) 0& t
where 8 is the permutation symbol in 2-spaces. To
complete the theory of a Cosserat curve under the
constraint condition (39), eee assume that each of
the functions, n, a, and p is determined to within
an additive constraint essen ee so that
f =f 9 7 » Sa ep pCO
n=n+n ,
where fi, #%, and p% are determined by constitutive
equations and the functions n(&,t), Tt (Ep) p and
pa(é,t) are the response due to the constraint; the
latter quantities are arbitrary functions of &,t and
do no work. For an incompressible inviscid fluid
jet, which models the properties of the three-
dimensional inviscid fluid at constant temperature,
we introduce the constitutive assumption that n,74,
ig do not depend explicitly on the kinematic quan-
tities, dv/dé, Wor W/E, and are furthermore work-
less, i.e.,
ov ow
a,
110% erent WON A Se eaife)
DO ae a) ge ee is (41)
provided w., dv/d— satisfy the constraint condition
(39). It can then be shown that [Green and Laws
(1968) and Green (1976) ]
p=0, (42)
lll In. general, there are three conditions of incompressibility
in the theory of incompressible directed fluid jets; for a
discussion of these, see Caulk and Naghdi (1978a, Appendix) .
In restricted forms of the theory discussed in the next sec-
tion, two of the three conditions are satisfied identically.
The specification (38) is motivated from an examination of
the incompressibility condition in the three-dimensional
theory when the position vector is approximated by (18).
507
where isya is an arbitrary scalar function of &,t.
For an incompressible viscous jet, the constraint
response,n, mo, po, are determined similarly with the
use of the Constraint condition (39), but constitu-
tive equations are required for n, ister 1p in (40). We
do not record here the results for a viscous jet and
refer the reader to Green (1976) and Caulk and
Naghdi (1978b) .
4. STRAIGHT FLUID JETS. ADDITIONAL REMARKS
We now specialize the results of the previous sec-
tion to straight jets of elliptical cross-section.
In order to display some details of the kinematics
of a straight jet, including the rotation of the
directors in a plane normal to the jet axis, it is
convenient to introduce a fixed system of rectangular
Cartesian coordinates (x,y,z) with the z-axis paral-
lel to the jet. Further, let the unit base vectors
of Ene rectangular Cartesian axes be denoted by
(i,j rk) and introduce, for later convenience, the
eee lowell base vectors
Qy = ab cos © w 3 Sain F
ep ie San Ol ta ICOSMON my §Cae Kner, (43)
where 8 is a smooth function of z and t. We assume
that the directors are so restricted that they de-
scribe an elliptical cross-section of smoothly vary-
ing orientation along the length of the jet and that
at each z = const., the base vectors, e] and eg,
lie along the major and minor axes of the ellipse,
respectively. Then, the angle, 8, called the
sectional orientation, specifies the orientation of
the cross-section as a function of position. With
this background, henceforth we restrict motions of
the directed curve, R, such that in the present con-
figuration at time, t,
SS] B(Eraes oo Ghee 7 ca = wen) (24)
where $)] and ¢9 measure the semiaxes of the ellipti-
cal cross-section. In the case of a circular jet,
¢1 = 62.
The complete theory also requires the specifica-
tion of explicit values for Ary, yOB, £ and £%. In
particular, the values for d,y%,yoB may be obtained
by an appeal to certain results from the three-
dimensional description of the jet. Thus, recall-
ing (18) and the remark made following (17), here
we choose the curve, 6% = 0, as the line of centroids
of the jet-like body and identify this curve with
the curve, 2, in the theory of a Cosserat curve.
This leads to the identification
L eh
\ = p@aq)e = i g°ae!ae?
a
* 4s a
Ay = (Ni tefl) cls) Cl) 9 yp
a
x 1 i 2
pF iE g*e"e"aa ao, (45)
where p* is the three-dimensional mass density in
(9) and the determinant g defined by (3)3 is cal-
culated from the approximation (18). Again, with
508
the use of (18) and the equations of motion (9)2, 3,
the expressions for f and 2% can be identified in
terms of the integrated body force, £*, over the
cross-sectional area, a, and specified pressure and
surface tension over the boundary, da of a [for
details, see, for example, Caulk and Naghdi (1978a)].
We observe that since y® = 0 hy (45)9, the equations
of motion (30) and (31) assume a slightly simpler
form. We do not record here the details of the
system of ordinary differential equations which can
be obtained from (29)-(33) for both inviscid and
linear viscous fluids. They are readily available
in the papers cited: see Green and Laws (1968),
Green (1975, 1976, 1977), and Caulk and Naghdi
(1978a, b).
In the rest of this section, we briefly call
attention to some available evidence of the relevance
and applicability of the direct formulation of the
fluid jets. Available solutions obtained to date
are limited to those for straight jets and among
these most of them deal with jets of circular cross-
section. Some general aspects of compressible
inviscid jets, including a discussion of ideal gas
jets in the context of a thermodynamical theory,
have been studied by Green (1975). Applications to
incompressible circular jets for both inviscid and
viscous fluids are contained in the papers of Green
and Laws (1968) and of Green (1976). Green (1977)
has also studied a steady motion of an incompressible
inviscid fluid jet which does not twist along its
axis. A more detailed analysis of the motion of a
straight elliptical jet of an incompressible inviscid
fluid in which the jet is allowed to twist along its
axis is contained in a recent paper by Caulk and
Naghdi (1978a). This study, which includes the ef-
fects of gravity and surface tension, utilizes the
nonlinear differential equations of Section 3 with
r and dg at time, t, specified in the form (44). A
number of theorems are proved in the paper of Caulk
and Naghdi (1978a) which pertain to the motion of a
twisted elliptical jet and some special solutions
are obtained which illustrate the influence of twist.
Further, a system of linear equations, derived for
small motions superposed on uniform flow of an in-
compressible circular jet, is employed by Caulk and
Naghdi (1978b) to study the instability of some
simple jet motions in the presence of surface ten-
sion, i.e., the so-called capillary instability that
leads to disintegration of the jet. In particular,
they [Caulk and Naghdi (1978b)] consider the breakup
of both inviscid and viscous jets: in the case of
an inviscid jet excellent agreement is obtained with
the three-dimensional results of Rayleigh (1879a,b);
and for a viscous jet, through a comparison with
available three-dimensional numerical results
[Chandrasekhar (1961)], the solution obtained is
shown to be an improvement over an existing approxi-
mate solution of the problem by Weber (1931). A
related study by Bogy (1978), concerning the insta-
bility of an incompressible viscous liquid jet of
circular section, partly overlaps with the work of
Caulk and Naghdi (1978b) on the temporal instability
of a viscous jet, and considers the spatial insta-
bility of a semi-infinite jet formulated as a
boundary-value problem.
PART B
In Part B (Sections 5-8), after briefly describing
the basic theory of a Cosserat (or a directed) sur-
face, we summarize a special case of the theory
which is particularly suited for applications to
problems of fluid sheets and to the propagation of
fairly long water waves. For the sake of simplicity,
we confine attention here to homogeneous fluids; but
note that, as in Green and Naghdi (1977), the deriva-
tion can be modified to allow for variation of mass
density with depth. Although we are concerned mainly
with the purely mechanical theory involving appropri-
ate forms of the conservation laws for mass, linear
momentum, and moment of momentum, we also include
the conservation of energy. The latter easily sup-
plies motivation for some requirements in the devel-
opment of certain solutions.
5. THE BASIC THEORY OF A COSSERAT SURFACE
Having introduced the notion of a (three-dimensional)
sheet-like body in Section 2, we now formally define
a direct model for such a body. Thus, a Cosserat
(or directed) surface, C, comprises a material sur-
face, S, (embedded in a Euclidean 3-space) and a
single deformable vector, called a director, attached
to every point of the surface, S. The directors
which are not necessarily along the unit normals to
the surface have, in particular, the property that
they remain unaltered under superposed rigid body
motions. Let the particles of the material surface
of C be identified by means of a system of convected
coordinates, 0% (a = 1,2), and let the surface oc-
cupied by S in the present configuration of C€ at
time, t, be referred to as Jd. Let ry and d denote
the position vector of a typical point of J and the
director at the same point, respectively, and also
designate the base vectors along the 6%-curves on
Jd by ay: Then, a motion of the Cosserat surface is
defined by vector-valued functions which assign posi-
tion, x, and director, d, to each particle of C at
each instant of time, imens
r= r(0,t) , d= d(0",t) , [ajard] > 0 (46)
and the condition (46), ensures that the director,
d, is nowhere tangent to Jd. The base vectors, ay,
and their reciprocals, ar the unit normal, a3, and
the components of the metric tensors, aap and ars,
at each point of & are defined by
ax a a
furs a | = | 8g SOB See Zoe
30
OS 2. 8 gah
adie aa Mee at ene le feel
5
a = det at, ane [ajaja3] > O , (47)
where 67 is the Kronecker delta in 2-space. The
velocity and the director velocity vectors are de-
fined by
43)
ig
ll
ine
2 Que
9 YS
*For convenience, we adopt the notation for ry in (14) and
(21) also for the surface (46),. This permits an easy
identification of the two surfaces, if desired. The choice
of positive sign in (46)3 is for definiteness. Alterna-
tively, it will suffice to assume that [ajajd] # 0 with the
understanding that in any given motion the scalar triple
product [a,ajd] is either > 0 or < 0.
where a superposed dot denotes differentiation with
respect to t holding 0° fixed.
Let P, bounded by a closed curve, 0P, be a part
of J occupied by an arbitrary material region of
S in the present configuration at time, t, and let
Vow =| vie (49)
be the outward unit normal to 3P. It is convenient
at this point to define certain additional quantities
as follows: The mass density, p = p(0’,t), of the
surface, J, in the present configuration; the con-
tact force, NES N(8Y,t;v), and the contact director
forcel, M = M(6Y,t;v), each per unit length of a
curve in the present configuration; the assigned
force, £ = £(0Y,t), and the assigned director force,
L= R(0Y,t), each per unit mass of the surface,J ;
the intrinsic director force, m, per unit area of
di the inertia coefficients, k = k(@Y) and k = k(6Y),
which are independent of time; the specific internal
energy, € = e(O8Y,t); the heat flux, h = h(6’,t;v)
per unit time and per unit length of a curve, OP;
the specific heat supply, r = r(6Y,t), per unit time;
and the element of area, do, and the line element,
ds, of the surface, J. The assigned field, £, may
be regarded as representing the combined effect of
(i) the stress vector on the major surfaces of the
sheet-like body denoted by f., e.g., that due to the
ambient pressure of the surrounding medium, and (ii)
an integrated contribution arising from the three-
dimensional body force denoted by fy, e.g., that due
to gravity. A parallel statement holds for the as-
signed field, 2. Similarly, the assigned heat sup-
ply, r, may be regarded as representing the combined
effect of (i) heat supply entering the major surfaces
of the sheet-like body from the surrounding environ-
ment, denoted by Yo, and (ii) a contribution arising
from the three-dimensional heat supply, denoted by
Yp- Thus, we may write
7 Sse, ab ae A & = fl) a A ae ES Ge tb ry o (SO)
The various quantities in (50) are free to be speci-
fied in a manner which depends on the particular ap-
plication in mind and, in the context of the theory
of a Cosserat surface, the inertia coefficients, k,
k and the mass density, p, require constitutive equa-
tions. Indeed, forko and Yor as well as fpr ep and
Yp, can be identified with corresponding expressions
in a derivation from the three-dimensional equations
[for details, see Naghdi (1972,1974)]. Likewise, p
and the coefficients,k,k, may be identified with
easily accessible results from the three-dimensional
theory.
In terms of the above definitions, the conserva-
tion laws for a Cosserat surface can, be stated in
fairly general forms. We do not record these here
since they are available elsewhere [Naghdi (1972),
p. 482) or Naghdi (1974)]. Instead, we turn our
attention to the relatively simple theory of the
next section.
It may be noted that the local field equations
in the mechanical theory of a Cosserat surface have
+
The terminology of director couple is also used for M depend-
ing on the physical dimension assumed for the director, d.
Here we choose d to have the physical dimension of length so
that M has the same physical dimension as N. For further
discussion see Naghdi (1972, Ch. C) and Green and Naghdi
(1976).
509
the same forms as those that can be derived from the
three-dimensional field equations () 1 4) Jey Suites
able integration between the limits, &, and €5, and
in terms of certain definitions for integrated mass
density and resultants of stress [for details, see
Naghdi (1972, Sections 11-12) or Naghdi (1974)].
Moreover, given the approximation (21), there is a
1-1 correspondence between the two-dimensional field
equations that follow from the conservation laws of
a Cosserat surface and those that can be derived from
(9)1,2,3 provided we identify the director, d, in
(21) with (46)9 and adopt the definitions of the re-
sultants mentioned above. As similar 1-1 correspon-
dence can be shown to hold between the two-dimensional
energy equation in the theory of a Cosserat surface
and an integrated energy equation derived from the
three-dimensional energy equation.
6. A RESTRICTED THEORY OF A COSSERAT SURFACE
Special cases of the general theory can be obtained
by the introduction of suitable constraints, thereby
resulting in constrained theories. Alternatively,
corresponding special cases can be developed in which
the kinematic and the kinetic variables are suitably
restricted a priori and then restricted theories are
constructed by direct approach. Such special cases
of the general theory have been discussed previously
by Naghdi (1972, Sections 10 and 15) and by Green
and Naghdi (1974) and are of particular interest in
the context of elastic shell theory. We provide here
an outline of a restricted theory developed by Green
and Naghdi (1977) mainly for application to problems
of fluid sheets. The resulting equations can also
be obtained as a constrained case of those given for
directed fluid sheets [Green and Naghdi (1976)], but
it is more convenient tq restrict the kinematic and
the kinetic variables at the outset and construct a
corresponding restricted theory from an appropriate
set of conservation laws in integral form.
Let the director, d, while deforming along its
length, always remain parallel to a fixed direction
specified by a constant unit vector, b. It should
be kept in mind that b is fixed relative to the body
and not relative to the space. Thus, recalling (46) 9
and (48)5, we write
d= $(0,t)b , w=w(0,t)b , w= . (51)
Further, in view of the assumed form of (51), for
the director, it is convenient to decompose M,m and
2 into their components along and perpendicular to
the) unit vector, b, iJe.,
MS MOV ews Bis Oey) 7 SSDS O 2
=n(Ob be s@e) > s °
n=}
I
ow
Il
)
= DOM oa Hs oO ae) eS)
i)
|
2Q0
°
ey
Il
je)
where M,m and % are scalar functions and S,s,c are
vector functions of their arguments. According to
the decomposition (52) ; the vector, M, is resolved
into two parts. One pat is along b and the other
part is the perpendicular projection of M onto the
plane defined by S * b = O which is perpendicular to
510
b. Parallel statements hold for vectors, m and L,
in (52)9,3-
Also, it is convenient to decompose the assigned
fields, f and 2, into two parts, one of which repre-
sents the three-dimensional body force acting on the
continuum which is assumed to be derivable from a
potential function, 2(r,$), and the other which
represents the effect of applied surface loads on
the major surfaces of the fluid sheet. Thus, we
write
dQ 8Q
i i fo ’ oe (— nen Lo) O (53)
With the foregoing definitions of the various
field quantities and with reference to the present
configuration, the conservation laws for a restricted
theory of a Cosserat surface [different from the re-
stricted and constrained theories discussed previ-
ously by Naghdi (1972) and by Green and Naghdi (1974) ]
are:
d
ae do =
s |e (o} OQ 4»
d
—= ne = +
ae exe + kwb) do J ,eta0 lhe N dst;
ae : p (kv + kwb) wdo = bt J (or-mas ae Mds]
+b x if (pc-s) do
Diese
+ J gp80s] :
A
ae px x v + k(x x wh + d x v)]do
12
=[ ple x £ + ax (b x c)]do
5 C
+ [ope x N+ax x s)las ,
aE ple +2 + 4(v + v + 2kv + wh + kw?)]do
x, avant 2
= [ pirte “y+ 2 w)ao
P CLO! ~ (©
+ foe ONY ar Iie) > TNCIS= 5G (54)
In the above equations (54) ; is a statement of con-
servation of mass, (54) 9 the conservation of linear
momentum, (54)3 that of the conservation of the
director momentum, (54), the conservation of moment
of momentum and (54)5 represents the conservation
of energy. It should be noted that the quantities,
M and 2% .¢ no contributions to the moment of momen-
tum equation, and the quantities, ¢ and S, make no
contribution to the equation for conservation of
energy in the present restricted theory.
Under suitable continuity assumptions, the curve
force, N, the director force, M, and the heat flux,
h, can be expressed as S
(er Ch. a
Erie Mabeaicbia aU Doki 0
(55)
where q is the heat flux vector and the fields, N°,
seme, q%, are functions of eY,t. The five conserva-
tion equations in (54) then yield the local equa-
tionst
pa- = y(@)) (56)
a) eyes
(aN) PAE = Wy > Ky) (57)
; z v £
Bw :
Dr). ey = ue + y(kv * b+kw) ,
‘ s
74.0 5 :
(a°s’) Bi Netra ek MR or bx ykv , (58)
Qa Qa
ay x N + dq x (b 1) <5 Gl " x) (6b) Si) Oe 9)
r - div =pe + N° OW + + Mw = 0 (60)
f s@ p = ~, a. ie pe) u
where "div is the surface divergence operator de-
fined by divs q = q,q ° ao and a comma denotes par-
tial differentiation with respect to the surface
coordinates,0". It should be noted that the vector
fields, Sc and s, are workless and do not contribute
to the reduced energy equation (60).
The above results include (60), which is derived
from (54)5. For the purely mechanical theory in
which the law of conservation of energy is excluded,
the appropriate conservation laws are the first four
of (54). In the context of the purely mechanical
theory, it is worth recalling that the rate of work
by all contact and assigned forces acting on P and
on its boundary, dP, minus the rate of increase of
the kinetic energy in P can be reduced to [see
Naghdi (1972,1974)]:
fpolt “vt 22 * w)do + f (N > v +M ° w)ds
pois Yves we ys” s
bgt (vy + v + 2kv + w + kw2)ao = [Pao 5 (Gib)
dt Yp~ ~ ~ ~ = P
where
P= N° OF AY + mw + Mw
is the mechanical power.
Before closing this section, we also note that
the restriction imposed on the motion of the medium
by the condition of incompressibility, in the context
ache restricted theory under discussion, reduces |
to
i line with a remark made at the end of the previous
section, we note that equations (56)-(60) can also be
derived by suitable integration across the thickness of
the sheet, respectively, from the three-dimensional equa-
tions (9)1 2,3 and the three-dimensional energy equation.
am general, there are two conditions of incompressibility
in the theory of incompressible directed fluid sheets; for
a discussion of these, see Naghdi (1974, Section 3). In
our present discussion, since d is assumed to have the form
(51) ,, the second condition is satisfied identically and
the corresponding pressure (arising from the constraint
response) is a part of the response functions for Se and s.
The specification (62) is motivated from an examination of
the incompressibility condition in the three-dimensional
theory when the position vector is approximated by (21).
d
ae [ajagd] = 0 (62)
and can alternatively be expressed in the form
a a
(Glo aa” = (el oa aglow way °o we @ ofGs)
= A! 2
For an incompressible inviscid fluid sheet, which
models the properties of the three-dimensional in-
viscid fluid at constant temperature, we introduce
the constitutive assumption that N*,m,M% do not de-
pend explicitly on the kinematical quantities, Yiar
WW ye and are furthermore workless, i.e.,
lo}
IN
+mw+Mw =O , (64)
, '
provided v g and w satisfy the constraint condition
(63). With the use of (51), it can then be shown
that [see Green and Naghdi (1976, 1977)]
NO = - p!{(d + a3)a" = (d + a°)ag}
So g ap
ne no
Mes wa, om kr SOG (65)
where PS is an arbitrary scalar function of ey,t
and e¢8 is the alternating tensor in 2-space. With
the help of the energy equation (60) and the fact
that the mechanical power vanishes identically for
an incompressible inviscid fluid at constant tempera-
ture, it can be shown that [see the appendix of Green
and Naghdi (1976) ]
7. WATER WAVES OF VARIABLE DEPTH
Within the scope of the restricted theory of the
previous section, we include here an outline of a
derivation of a system of nonlinear differential
equations governing the two-dimensional motion of
incompressible fluids for propagation of fairly long
waves in a stream of water of variable initial depth.
Our developments include the effects of gravity and
surface tension but we assume that the mass density
of the fluid does not vary with depth. However, a
more general derivation for a nonhomogeneous inviscid
fluid in which the mass density is allowed to vary
with depth is given by Green and Naghdi (1977). Let
e€1,e2,e3 be a set of right-handed constant orthonormal
base vectors associated with rectangular Cartesian
axes and choose the unit vector, b, to coincide with
e3.- Then, the position vector, xr, in (46), and the
director, d, in (51), can be represented as
(66)
BS son > Yep Sg GOSS sg
where x,y,,> are functions of 9! ,62,t. The velocity,
v, and the director velocity now take the forms
YS Us 1 Vena Ney op We ue; a (67)
where
511
u=x ,v=y , _=0 ,w= (68)
and we note that the velocity components, u,v,A,w,
may be regarded as functions of either 6) ,02,t or
of x,y,t. From (67) follow the expressions
Vue qt ven + he; " w = wes (69)
and
u POUL act UL Ulta A qv =v, + uv VV '
t y t x y
Noh. “oh. 2 a i wo we & uw + vw , (70)
t x y x
where the subscripts, x,y,t, designate partial dif-
ferentiation with respect to x,y,t, when u,v,A,w are
regarded as functions of x,y,t. With the use of (67)
and (70), the incompressibility condition (64) as-
sumes the simpler form
OGL AP A)! Ey SO) 5 (71)
x y
In order to complete our development, we need to
specify values for the assigned force, £, and the
assigned director force, 2, and to identify the co-
efficients,y,k and k, which, in general, require
constitutive equations. For this purpose we consider
the corresponding fluid sheet in the three-dimensional
theory in which an incompressible homogeneous fluid
under gravity|| ,-g*e3, flows over a bed specified by
the position vector
r* = xe; + yep + a(x,y)e3 (72)
and we specify the surface of the fluid by
r* = xe, + yeo + B(x,y,t)e3 (73)
In (72), a is a given function of x,y but 8 in (73)
depends on x,y,t. At the surface (73) of the stream
there is constant pressure, Por a constant normal
surface tension, T. At the bed the (unknown) pres-
sure, Pr depends on x,y and t. Thus, the normal
pressure, p*, at the top surface (73) is
w=
19) 15)
Sone neg 7
T{(1 + 82)8 - 2888 + (1 + 62)6 3
& WL 28 eV MKS EXammVs ;
(Le BE & Bayeve
x y
(74)
At the bed (72) the normal velocity of the fluid is
zero and the pressure, p* takes the value
p* = p(x,y,t) , (75)
where p is to be determined.
To proceed further, we recall the notation ais ((3})) 5
let the surface, & = 0, defined by (15) coincide with
the surface, J, and consider the three-dimensional
region of space between the surfaces (72) and (73)
occupied by the fluid. Any point in this three-
dimensional region is then specified by
I We use g* (instead of g) for gravity, since the letter, g,
is used for a different quantity in (3), (5) and elsewhere in
the paper.
sae Fert 3he3 = xje]+ yep + (p + 0%p)e3 Pen GG) 7%
where the surfaces, a and $8, in (26) or (72) and (73)
correspond to 93 = Bale 93 = Eo, respectively. Also,
x,y,W and $¢ in (76) are functions of 61,62 and t and
GS Waren, g SW sp Saw o (77)
Next, in order to obtain explicit values of y,k,k,f
and 2 in relation to the top and bottom surfaces of
the fluid, we choose the surface, 93 = 0, so that the
center of mass of the three-dimensional fluid region
under consideration always lies on this surface and
we then identify this surface with the surface,d,
in the theory of Cosserat surface. Without loss in
generality, we may choose &) = -, E> = +5 (see
Figure 2). This leads to the identification:
1 *; ol * 3 ( )
= cakee= Zola) \3 eee x1Y
Y = pa = ij Gel" =.) Sa
5 aanes)
1
eo
Ss 4 x L
k = [ o g*e3ae3 =0 ,
es 7. Be)
% > Sp
p g7(6%)2ae2 = FS - (78)
= 3 (61,67)
where p* is the three-dimensional mass density in
(9) and the determinant g defined by (3)3 is cal-
culated from the approximation (21) so that
1
3
a (x,y)
= ae SS (79
y 3(6!,62) :
Substitution of (78) and the appropriate expressions
for f and QR into (57) to (59) results in the dif-
erential equations of motion
ne 0
p gu = = (Py = DB = pa,
*ov=-p + (p -@)® - pa ,
p> I 1D. = Gi Be Pa,
eta = Rok
PRON ESP eet etd ¢ <7
iL oO =
ao OY SAG = 2.) = ape : é (80)
where
P=plo . (81)
Moreover, Since the bed of the stream is stationary,
from (77) and (70) 3,4 we have
a = ua, + va = V -}i $ =o PAW 6 (82)
The above system of equations is independent of the
remaining equations (58) which involve S%,s. The
fields, $%,s, correspond to appropriate constraint
responses for the restricted motion (51).
ak
In (76) to (78), we have returned to the notation
6° instead of & introduced in (2)
The questions of continuous dependence upon the
initial data and uniqueness for solutions of initial
boundary-value problems for a class of symmetric
flows characterized by a special case of the system
of nonlinear partial differential equations given
by Green et al. (1974c) has been discussed by Green
and Naghdi (1975). A similar procedure may be used
to establish uniqueness for the more general system
of equations (80).
For later reference, we consider here the reduc-—
tion of the system of nonlinear differential equa-
tions (71) and (80) for unidirectional flow in the
absence of surface tension, T. Without loss in
generality, we set the ambient pressure, py = 0,
and consider flows in the x-direction only. Then,
with q = 0, from (71) and (80) we obtain
Qn 3? COBY) =O"
OMS. st
ep dh =p-o*g% ,
pe ow=-spee (83)
We may solve (83)3,, for p and p and obtain the ex-
pressions
= * * °
DSO Oe SP WY)
oF D 5 io
bp=%p ¢ (g+dAt aw 6 (84)
Introduction of (84); 9 into (83);,9 yields a system
of two partial differential equations in u and w but
we do not record these here. A further simplifica-
tion of these equations results for a horizontal bed.
For a horizontal bottom a may be taken to be zero and
(77)1,2 and (68) 3,4 reduce to
a=O , B=o , W=%o ,rA=4W . (85)
8. FURTHER REMARKS
The system of nonlinear differential equations (71)
and (80)),2,3,4, which include the effects of gravity
and surface tension, govern the two-dimensional mo-
tion of incompressible inviscid fluids for the propa-
gation of fairly long waves in a stream of variable
initial depth. They are derived here by a direct
approach as consequences of the conservation laws
(54) subject to the incompressibility condition (64) .
Upon specialization to unidirectional flow, the non-
linear differential equations (71) and (80) reduce
to those for inviscid fluids over a bottom of vari-
able initial depth given by Green and Naghdi (1976a,
Sections 5-6), while the equations for two-
dimensional flow over a horizontal bottom were de-
rived earlier [Green et al. (1974c)].
The differential equations governing the motion
of a viscous fluid sheet are discussed briefly by
Green and Naghdi (1976a, Section 11) and a similar
development can be given within the framework of the
restricted theory of Section 6, but we do not con-
sider this aspect of the subject here. The system
of differential equations obtained in Section 6 is
valid for incompressible, inviscid, and homogeneous
fluids. A more general derivation for propagation
of fairly long waves in a nonhomogeneous stream of
variable initial depth in which the mass density is
allowed to vary with depth is contained in a recent
paper of Green and Naghdi (1977).
In the case of incompressible inviscid fluid
sheets, the nonlinear equations for wave propagation
in water of variable depth can also be derived from
the three-dimensional theory: the procedure involves
the use of the (three-dimensional) equation for con-
servation of energy, the incompressibility condition,
invariance requirements under superposed rigid body
motions, along with a single approximation (21) for
the position vector. Then, by (6) and (21), the ap-
proximation for the (three-dimensional) velocity
field is given by
Vay Oy (86)
where v and w in (86) have the same forms as those
in (67). A derivation of this kind has been carried
out by Green and Naghdi (1976b). It is important,
however, to note that this derivation is limited to
incompressible inviscid fluids which do not require
constitutive equations. tt
It is natural to ask what are the relationship and
advantages (if any) between the above system of equa-
tions and those which are currently employed by other
investigators. To provide a ready comparison, we
list below from Whitham (1974) alternative forms of
equations for water waves moving in the direction of
a fixed x-axis for a stream of initial constant depth,
h. Let the elevation of the stream be h + yn. Then,
for unidirectional flow and in terms of n and the
horizontal velocity, u, we recall from Whitham
(1974, pp. 460-463) the system of equations
nt + f{u(h + nyt =OMine
* L 2 2
us + uu, +g es + c“hn =O© > (87)
and the pair of equations attributed to Boussinesq,
namely
nte(h+tnu =0 ,
x
° x it
ar += =
wi i. 3 Wuleare ORS, (88)
where the notations in (87) and (88) are the same as
those in (70), g* is the acceleration due to gravity
introduced in Section 7 and c@ = g*h. Both systems
of equations (87) and (88) allow for wave propagation
in either direction along the x-axis. For waves moyv-
ing along the positive x-direction only there is the
Korteweg-deVries (1895) equation--hereafter referred
to as the K.dV. equation--i.e.,
De & On =0 (89)
h x
3
+ —
Ne OMe se 6 XXX
2
tt A , E 2
Recall that in the three-dimensional theory of incompress-
ible inviscid fluids the stress vector is specified in terms
of a pressure which is determined by the equations of motion
and the boundary conditions.
513
or an equation due to Benjamin et al.
by
(1972) given
Ss
+
Q
Ee
+
LOH es)
Ab uo) =
yn, 6 ch eee ='0) 3 (90)
‘gh
h
As already remarked by Green and Naghdi (1977),
it may immediately be verified that the set of equa-
tions (88) and (90) only have steady state solutions
if n and u are both constants. Also, although the
K.dV. Eq. (89) admits a solitary wave in which the
velocity at infinity is zero and the stream there is
at its undisturbed height, h, it does not admit a
steady state solution with u constant and n = 0 at
infinity. This fact is related to another property
of (89) which is also shared by (88) and (90): the
three sets of equations (88) to (90) are not invari-
ant in form under a constant superposed rigid body
motion of the whole fluid. To see this, suppose
that a constant superposed rigid body translational
velocity is imposed on the whole fluid so that the
particles at the place, x, are displaced to x? at
time, t , specified by
+
See tb ee a te ue aE (91)
where a and a are constants. The variables that oc-
cur in the differential equations (87)-(90) are n =
n(x, )) and w= wtx,t)). Let ni = nica t) and ut =
ut(xt,tt) be the corresponding scalar quantities de-
fined over the region of space occupied by the fluid
after the imposition of the superposed rigid body
motion (91),. Then, from (68), and (8.6) we obtain
+ + +
Dex) = el (Ge pie) Se
+ —_
U 63 + ae, oe > am a «2 (92)
We expect the elevation, h + n, of the fluid to re-
main unaltered by superposed rigid body motions; and,
since h remains unaltered also, this leads us to re-
quire that
n(x,t) = a’ Gee) = a” (GR eye, oh Gy) 5 (ES)
From (92) and (93), we calculate expressions of the
type
= + =
ue U) re en 2 ike Win, 9
t x x
ae ar 2, +
= +
xtt. Vee ae” etl). fa CW ne te
ae me oS 18 oS 2S
. + + o+ e+
Hh, FM Sy, Fe Hh = i , (94)
te x +
ste x
with similar results for uz,uy and u in terms of ut
and their derivatives. It was noted by Green and
Naghdi (1977) that if the independent variables, x,
t, in (88) to (90) are changed to (91), the equations
for u,n in terms of xt+,tt+ are different from those in
terms of x,t and this was illustrated explicitly with
reference to the K.dV. equation (89). Here, we con-
sider the pair of equations (88) 1 2- After substi-
tuting (92)-(94), they become
514
x
Pe Gap ees hg
Ue Ul at, oe
x Gta
+ +
=— 2a) - at :
i tre Ie ca) oP ar op 2)
BS) Bre) 1S Re “Bie ge
The first of (95) is of the same form as (88), and
hence invariant but clearly the second of (95) dif-
fers from (88). This means that the character of
the solutions of (88), (89) and (90) is substantially
altered by superposing a constant rigid body trans-
lational velocity on the fluid, which is contrary
to what happens if we use the full three-dimensional
equations of motion for an inviscid fluid. On the
other hand, the set of equations (87) is not subject
to this drawback, and the equations do have useful
steady state solutions. It may be argued that be-
cause of the nature of the approximation in obtain-
ing (88) to (90) from the three-dimensional theory
we should not expect these equations to be invariant
under a superposed constant translational velocity,
but this then leaves in doubt which version of any
of the sets (88) to (90) are to be chosen as basic.
The difficulty disappears if we linearize any of
the above sets since the resulting equations are
then invariant under a small superposed constant
translational velocity, as we would expect.
From the above discussion, it might appear that
the equations (87) may be preferable to any of (88)
to (90), but arguments are put forward by Whitham
(1974, p.462) to suggest that the system (88) is to
be preferred to (87). Although considerable use has
been made of some of the equations (87) to (90), it
would seem that they all rest on a somewhat shaky
physical foundation. By contrast, the system of
equations (71) and (80) do not possess the undesir-
able features of the type noted above: they are
properly invariant under superposed rigid body mo-
tions, admit general steady state solutions, and are
free from anomalies mentioned earlier.
For the purpose of providing a more explicit com-
parison with the system of equations (87) to (90),
we specialize the system of equations (83) to that
for a horizontal bottom for which (85) j 2,3,4 hold.
Then, denoting again the elevation of the stream by
h +n, the differential equations (83) 2 can be re-
corded in the form
i) ar (a a) SO) A
beg. = iH (96)
Pep) ube = 3 tt | Be 0
where
2
BF Uae > sy Vaden
11 i42 2
-== + oo
3 > p (Aes se Bh BU Shae i
b=h+tn . (97)
Clearly if R on the right-hand side of (96) can be
neglected, then (96), 2 reduce to those of Boussinesq
given by (88)],2. It should be emphasized, however,
that the nonlinear equations (96); 9 are invariant
under a constant superposed rigid body translation
while (88) 9 are not.#+ Within the scope of the
nonlinear theory, it does not seem reasonable to
neglect the quantity, R, in (96) on the basis of
either physical considerations or mathematical argu-
ments. It may be, however, that in some special
circumstances the solution of (88) is a good approxi-
mation to the solution of (96), but this is a dif-
ferent question than that discussed above. In this
connection, it is worth noting that a solution to a
system of differential equations, which results from
neglecting certain terms in a more general system of
equations, in general, will not be the same as a
solution obtained by approximation from a correspond-
ing solution of the more general system of equations.
We close this section by calling attention to some
available evidence of the relevance and applicability
of the direct formulation for fluid sheets. The sys-
tem of equations (71) and (81), or a special case of
it, has already been employed in some detailed stud-
ies of a number of two-dimensional problems of in-
viscid fluid sheets, as well as in some comparisons
with known previous solutions on the subject. We
mention here some of these studies and refer the
reader to the papers cited for additional informa-
tion: (a) the nonlinear differential equations admit
a solitary wave solution [see Green et al. (1974c)]
which is the same as that attributed by Lamb (1932,
Section 252) to Boussinesq and Rayleigh; (b) this
solitary wave solution, as well as appropriate jump
conditions and certain results derived from the
energy balance for an inviscid fluid sheet at con-
stant temperature [Green and Naghdi (1976a, Appen-
dix)], has been used by Caulk (1976) to discuss the
flow of an inviscid incompressible fluid under a
sluice gate; (c) the steady motion of a class of
two-dimensional flows in a stream of finite depth
in which the bed of the stream may change from one
constant level to another, and the related problem
of hydraulic jumps, both for homogeneous and non-
homogeneous incompressible fluids [Green and Naghdi
(1976a, Section 7) and Green and Naghdi (1977)];
and (d) a class of exact solutions [Green and Naghdi
(1976a, Section 9)] which characterize the main fea-
tures of the time-dependent free surface flows in
the three-dimensional theory of incompressible in-
viscid fluids [Longuet-Higgins (1972)].
ACKNOWLEDGMENT
The results reported here were obtained in the course
of research supported by the U.S. Office of Naval
Research under Contract NO0014-76-C-0474, Project
NR 062-534, with the University of California,
Berkeley.
REFERENCES
Benjamin, T. B., J. L. Bona, and J. M. Mahony (1972).
Model equations for nonlinear dispersive systems. —
Phil. Trans. Royal Soc. Lond. A272, 47.
Bogy, D. B. (1978). Use of one-dimensional Cosserat
theory to study instability in a viscous liquid
jet. Phys. Fluids 21, 190.
Fone difference between the two disappears, of course, upon
linearization.
Caulk Dey Ate (LOG)
under a sluice gate.
FIBIBIES Ye
Caulk, D. A., and P. M. Naghdi (1978a). The influ-
ence of twist on the motion of straight ellipitcal
jets. Arch. Rational Mech. Anal. (to appear) =
Report No. UCB/AM-77-5, Office of Naval Research
Contract NOO014-76-C-0474, Project NR 062-534
(June 1977), Dept. of Mechanical Engineering,
University of Calif., Berkeley.
Caulk, D. A., and P. M. Naghdi (1978b). The onset
of breakup in inviscid and viscous jets, Report
No. UCB/AM-78-3, Office of Naval Research Contract
NO0014-76-C-0474, Project NR 062-534 (July 1978),
Dept. of Mechanical Engineering, University of
Calif., Berkeley.
Chandrasekhar, S. (1961). Hydrodynamic and Hydro-
magnetic Stability. Oxford: Clarendon Press.
Cohen, H. (1966). A nonlinear theory of elastic
diimectedmcurves!) int.) Wi. Engng. SCi- 4,05).
Cohen, H., and C. N. DeSilva (1966). Theory of
directed surfaces. J. Math. Phys. 7, 960.
Cosserat, E. and F. (1909). Théorie des corps
déformables. A. Hermann et Fils, Paris; also
Theory of deformable bodies [transl. from original
1909 edition], NASA TTF-11, 561, Washington, D.C.,
1968.
Duhem, P. (1893).
pression hydrostatique.
3p ausi7/ 5
Ericksen, J. L., and C. Truesdell (1958). Exact
theory of stress and strain in rods and shells.
Arch. Rational Mech. Anal. 1, 295.
Green, A. E. (1975). Compressible fluid jets.
Rational Mech. Anal. 59, 189.
Green, A. E. (1976). On the nonlinear behavior of
fluid jets: Int. TF. Engng. Sei. 14, 49).
Green, A. E. (1977). On the steady motion of jets
with elliptical sections. Acta Mechanica 26,
ALWAYS
Green, A. E., and N. Laws (1966). A general theory
of rods. Proc. Royal Soc. Lond. A293, 145.
Green, A. E., and N. Laws (1968). Ideal fluid jets,
Inte Wi. Engng. Sel. 6), 30
Green, A. E., and P. M. Naghdi (1974). On the
derivation of shell theories by direct approach.
J. Appl. Mech. 41, 173.
Green, A. E., and P. M. Naghdi (1975).
On the problem of fluid flow
WBNS Wis ligleperess enh, gel
Le potentiel thermodynamic et al
Ann. Ecole Norm. 10,
Arch.
Uniqueness
and continuous dependence for water waves. Acta
Mechanica 23, 297.
Green, A. E., and P. M. Naghdi (1976a). Directed
515
fluid sheets, Proc. Royal Soc. Lond. A347, 447.
Green, A. E., and P. M. Naghdi (1976b). A deriva-
tion of equations for wave propagation in water
of variable depth. J. Fluid Mech. 78, 237.
Green, A. E., and P. M. Naghdi (1977). Water waves
in a nonhomogeneous incompressible fluid. J.
Appl. Mech. 44, 523.
Green, A. E., P. M. Naghdi, and W. L. Wainwright
(1965). A general theory of a Cosserat surface.
Arch. Rational Mech. Anal. 20, 287.
Green, A. E., P. M. Naghdi, and M. L. Wenner (1974a).
On the theory of rods. I. Derivations from the
three-dimensional equations. Proc. Royal Soc.
Lond. A337, 451.
Green, A. E., P. M. Naghdi, and M. L. Wenner (1974b).
On the theory of rods. II. Developments by
direct approach. Proc. Royal Soc. Lond. A337,
485.
Green, A. E., N. Laws, and P. M. Naghdi (1974c). On
the theory of water waves. Proc. Royal Soc. Lond.
A338, 43.
Korteweg, D. J., and G. deVries (1895). On the
change of form of long waves advancing in a
rectangular channel, and on a new type of long
stationary waves. Philosophical Magazine (Fifth
Ser.) 39), (422).
Lamb, H. (1932). Hydrodynamics, 6th edn.
University Press.
Longuet-Higgins, M. S. (1972).
time-dependent, free surface flows.
Mech. 55, 529.
Naghdi, P. M. (1972). The theory of shells and
plates. S. Fliigge's Handbuch der Physik, VIa/2,
C. Truesdell, ed., Springer-Verlag, Berlin, 425-
640.
Naghdi, P. M. (1974). Direct formulation of some
two-dimensional theories of mechanics. Proc.
7th U.S. National Congr. Appl. Mech., Amer. Soc.
Mechanical Engineers, New York, N.Y., 3-21.
Naghdi, P. M. (1975). On the formulation of contact
problems of shells and plates. J. Elasticity 5,
S725
Rayleigh, Lord (1879a). On the instability of jets.
Proc. Lond. Math. Soc. 10, 4.
Rayleigh, Lord (1979b). On the capillary phenomena
of jets. Proc. Royal Soc. Lond. 29, 71.
Weber, C. (1931). Zum Zerfall eines Fltissigkeits-
strahles, ZAMM 11, 136.
Whitham, G. B. (1974). Linear and Nonlinear Waves,
John Wiley and Sons.
Cambridge
A class of exact,
J. Fluid
516
Discussion
G. L. CHAHINE
I would like to congratulate the author on his
very fine work and to comment on his conclusion
that the Rayleigh-Plesset equation represents fairly
well the growth of bubbles attached to a wall. As
is well-known, the Rayleigh-Plesset equation relates
the growth and collapse of a spherical bubble, with-
out relative motion with respect to the unbounded
surrounding fluid, for a given variation of pressure
far from it. It then seems really surprising that
such an equation could describe so well the growth
of the bubble on a blunt nose as shown in Figure 31.
None of the requirements for the validity of the
Rayleigh-Plesset equation are fulfilled:
a. the bubble is non-spherical, even if we
agree that the shape in the figure plan
is a portion of a circle,
b. presence of a wall,
c. shear flow around the bubble,
d. yrelative motion between the bubble and the
fluid (as pointed out by the author).
Moreover, the presence of gas inside the
bubble is not taken into account, while the gas
behavior has been shown to be very important
(Chahine (1974, 1976)]. We believe that the good
agreement between experimental results and analyt-
ical computations shown in this paper is mainly
due to:
a. the time of observation is too small com-
pared to the hypothetical lifetime of the
bubble. (For a bubble radius of 1.3 mm
and an external pressure of 5,000 N/m2,
the Rayleigh time is about 0.7 ms and the
lifetime is greater than 1.5 ms; say 10
times the observation time.)
b. in order to integrate numerically the
Rayleigh-Plesset equation one needs two
initial conditions: an initial radius
and an initial growth rate. If R, and R
replace these initial conditions it is
not surprising that the result deduced
for Ry differs only 4% from the experi-
mental result.
Concerning Table 4, the calculated relatively
small effect of surface tension and viscosity is
in good agreement with previous asymptotic studies
{Chahine (1976) and Poritsky (1952) ].
REFERENCES
Chahine, G. L., (1974). Etude Asymptotique et
Experimentale des Oscillations et du Collapse
des Bulles de Cavitation. EWSTA Report 042,
CEDOCAR, MF 50831.
Chahine, G. L., (1976). Etude Asymptotique du
Comportment d'une Bulle de Cavitation dans un
Champ de Pression Variable. Jl. de Mecanique, 16
(2), pp. 287-306.
Poritsky, H., (1952). The Collapse or Growth of
a Spherical Bubble or Cavity in a Viscous Fluid.
Proceeding of the First U.S. Nattonal Congress in
Applted Mechanics, ASME, pp. 813-821.
517
Author’s Reply
J. H. J. van der MEULEN
The author appreciates Dr. Chahine's comments
and would like to point out that the principal aim
of comparing the cavity growth on the blunt nose
with theory was to show that the travelling bubble
type of cavitation is more related to bubble dynam-
ics than to boundary layer phenomena.
The surprising observation (Figure 31) that
the shape of the attached, growing cavity is a
spherical segment is, to a certain extent, consis-—
tent with the observation by Dr. Chahine (1977)
that the growth of the lower part of a bubble below
a free surface is not influenced by the presence
of the free surface.
It seems most unlikely that the presence of
gas originating from a small stream nucleus or from
diffusion may have affected the growth of the cav-
ity during the observation period. Oldenziel (1976)
has shown that such effects can be neglected for
explosive bubble growth.
REFERENCES
Chahine, G. L., (1977). Interaction between an
Oscillating Bubble and a Free Surface. J. Fluids
Engng., Trans. ASME, 99, p. 709.
Oldenziel, D. M., (1976). Gas Transport into a
Cavitation Bubble during an Explosion. JLAHR Symp.
on Two Phase Flow and Cavitation tn Power Genera-
tton Systems. Grenoble, France.
518
Discussion
R. LATORRE
Our lack of understanding of cavitation noise
and its measurement technique is an area of recent
concern and the authors' experiments and discussion
will hopefully aid other researchers with these
problems.
The correlation of cavitation noise and the
observed cavitation is a complicated research topic.
In my dissertation I am studying tip vortex cav-
itation noise and as a contribution to the authors’
paper, I would like to present some illustrative
A
UNIV, TOKYO CAVITATION TUNNEL
2 3 4
1
KEY:
A B
hie) / A FOIL / PROPELLER
. B TIP VORTEX CAVITATION
— om 1 B& K 8103 HYDROPHONE
2 B&K 2626 COND. AMP.
3 RION 1/3 BAND PASS FILTER
ee 4 — RION HIGH SPEED LEVEL
FIGURE 1. Tip vortex cavitation noise measure-
ment.
SHIP RESEARCH INSTITUTE
CAVITATION TUNNEL
RECORDER
SOUND PRESSURE LEVEL, dB IN 1/3 OCTAVE
0.5
1/3 OCTAVE BAND CENTER FREQUENCY
SRI MEASUREMENT
PROPELLER No. 121
J = 35, N= 20 RPS
1 - 6, = 23.06
2-6, = 20.0 3-6,= 19.
1/4 SRI FOIL, 6.5 M/S, 10°
= J
1-6, 8 49
2 = Oy 3
3 hy 0
a,
SOUND PRESSURE LEVEL, dB IN 1/3 OCTAVE
2 5 10 KHz OSL 2 b) 10 KHz
1/3 OCTAVE BAND CENTER FREQUENCY
D) 3.15 KHz BAND CENTER FREQ,
Ww
=>
=
3 10 oB
= 40
a
a ' ' ' \ \ I \
= 30 ha SO 8 Bis RO PY BUS. aay
7 N/
a AIR CONTENT:
ef a 26 %, 2.4 PPM
w UNIV. TOKYO TEST
= 1/4 SRI FOIL
wo 12 M/S, 10° 10 dB
a 0 7 1-6, =3.
FIGURE 2. Tip vortex cavitation noise measure- = NO =1 INTERMITTENT = 2 STEADY = 3
ments of propeller and foil tests comparison of = TIP VORTEX CAVITATION TEST (REF. €)
intermittent and steady tip vortex cavitation 0.5 2 5 10 KHz
noise.
1/3 OCTAVE BAND CENTER FREQUENCY
DEVELOPMENT OF TIP VORTEX CAVITATION NOISE
TRACE OF NOISE SIGNAL AND NOISE ENVELOPE
noise measurements made at the University of Tokyo's
and the Ship Research Institute's (SRI) cavitation
tunnel -
Figure 1 shows the measurement apparatus. The
hydrophone was set in a 50 mm acrylic cup mounted
on the tunnel's observation window and filled with
water. The measurements were made in uniform flow
at constant speed with the section pressure lowered,
using propellers and foils. The propeller was SRI
No. 121 (D = 250 mm, z = 6, area ratio = 0.8,
constant P/D = 0.75). The foil (1/4 SRI Foil) was
a scaled version of Dr. Ukon's (SRI) design using
NACA 4412 wing section and a planform of c(n) =
c,(1-n*)%. The 1/4 SRI Foil had an aspect ratio
of 3, semi-span = 50 mm, and base chord, c, = 40 mn.
The measurements are briefly illustrated in
Figures 2, 3, and 4. In Figure 2 the noise spec-
trum and envelope of tip vortex cavitation noise
is shown for SRI and Tokyo University tests. The
intermittant tip vortex noise appears as spikes in
the spectrum between 2 and 6.3 kHz, as denoted by
"2" in this figure. Using the complete test
record it is possible to construct the envelope
shown in Figure 2D. The shifts in the frequency
TRIGGER SIGNAL : 6.3 KHz BAND CENTER FREQUENCY
519
appear to be a function of both the low pressure
vortex core and the condition of the water.
In an attempt to gain an understanding of the noise
mechanism, additional experiments were performed.
In Figure 3, the intermittant tip vortex noise
signal at 6.3 kHz was used to trigger the camera
shutter to photograph the intermittant tip vortex
cavitation. It appeared that the noise mechanism
is due to the pressure wave caused by the filling
of the low pressure vortex core by dissolved gases.
To test this hypothesis of the tip vortex cav-
itation noise mechanism, air was injected from the
1/4 SRI Foil tip and the noise spectrum measured.
Figure 4 shows the results of the initial tests
illustrating a qualitative agreement in the actual
tip vortex cavitation noise spectrum and the sim-
ulated tip vortex using air injection. At the time
of writing, it has been possible to improve this
technique and duplicate the intermittant "spikes"
in the noise spectrum.
Thus by the experimental results a basis for
understanding the low frequency aspects of tip vor-
tex cavitation noise has become possible.
40 1 o= 3.8
NO. CAVITATION
DGy= 3.8
AIR INJECTION
FROM FOIL TIP
3 Gye Sul
STEADY TIP
VORTEX
CAVITATION
1/4 SRI FOIL
V = 12 M/S5) 105
AIR CONTENT:
24%, 2 PPM
nN WN
=) =)
ex
(ao)
SOUND PRESSURE LEVEL, dB IN 1/3 OCTAVE
%0,8 i 2 5 10 KHz
1/3 OCTAVE BAND CENTER FREQUENCY
1/4 SRI FOIL, 10 M/S, 10°, 6,
AIR CONTENT: 23%, 1.9 PPM
© PHOTO
= 3,36
FIGURE 3. Intermittent tip vortex cavitation
noise signal and photo.
FIGURE 4. Comparison of tip vortex cavitation noise
spectrum trace and simulated tip vortex using air
injection.
Authors’ Reply
GORAN BARK and WILLEM B. van BERLEKOM
It is very interesting to hear of this hypoth-
esis concerning generation of noise by tip vortex
cavitation. We have performed experiments with tip
vortex cavitation at propellers and hydrofoils and
found that intermittant tip vortex cavities were
noisiest. However, we have not performed high speed
filming or other more advanced attempts to study
the real mechanisms involved in the volume fluctu-
ations of the tip vortex cavity. In the case of
bubble cavitation and unsteady sheet cavitation,
which we have studied in more detail, we are of
the opinion that the highest pulses are generated
during the final part of a collapse, which often
is rather symmetrical, and that filling the cavities
with gas is of minor importance as a primary gen-
eration mechanism. However, some results indicate
that this gas decreases the violence of the collapse.
Session VIT
GEOPHYSICAL FLUID DYNAMICS
WALTER H. MUNK
Session Chairman
University of California, San Diego
La Jolla, California
teen ee
The Boussinesq Regime for waves
in a Gradually Varying Channel
John Wilder Miles
University of California
San Diego, California
ABSTRACT
The Boussinesq equations for gravity waves of ampli-
tude a(x) and characteristic length 2£(x) ina
gradually varying channel of breath b(x) and depth
d(x) are derived from Hamilton's principle on the
assumptions that a/d =a << 1, (a/2) 2 = OCC), io (9)
= 0(a3/2b/a) and d“(x) = 0(a3/2) (* = d/dx). The
further assumption of unidirectional propagation
then leads to the Korteweg-deVries equation for a
gradually varying channel. It is shown that the
latter equation admits two integral invariants.
The second-order (in amplitude) invariant measures
energy, as expected, but the first-order invariant
measures mass divided by pid’ ; accordingly, mass
is conserved only if either the first-order invariant
vanishes identically or ba is constant, and only
the former possibility appears to be consistent
with conservation of energy. An approximate solution
for a cnoidal wave, which conserves both energy and
mass, is developed. The corresponding approximation
for a solitary wave (which may be regarded as a
limit of a cnoidal wave) does not conserve mass but
nevertheless provides an approximation to the evolu-
tion of the amplitude, a « bp 2/3q7l, that is in
agreement with experiments for gradual decrease of
depth or increase of breadth but not for decrease
of breadth.
1. INTRODUCTION
The Boussinesq régime for gravity waves of amplitude
a and characteristic length 2 in water of depth d
is characterized by
GCSv“vd<i, Ge GM <« il, BS OG), Gade)
where a and 8 are measures of nonlinearity and
dispersion, respectively, and (lc) refers to the
asymptotic limit a + 0. The assumptions of one-
dimensional wave motion and uniform depth and the
523
neglect of compressibility and viscosity then imply
Boussinesq's equations for the free-surface displace-
ment and the depth-averaged velocity, n(x,t) and
u(x,t). The further assumption of undirectional
propagation permits the elimination of u to obtain
the Korteweg-deVries (KdV) equation for n. The
classical derivations are given by Whitham (1974,
§13.11). An alternative derivation, starting from
the Luke-Whitham variational principle and using &,
the velocity potential at the free surface, and n
as dependent variables also has been given by Whit-
ham (1967).
I consider here the generalization of the
Boussinesq and KdV equations for a channel of grad-
ually varying breadth and depth b(x) and d(x) and
their approximate solution for slowly varying
cnoidal and solitary waves. I begin (in Section 2)
by deriving (what may be called) the Boussinesq chan-
nel equations directly from Hamilton's principle (to
which the Luke-Whitham variational principle is
equivalent in the present context) on the basis of
(1) and the further assumptions (which imply
gradually varying)
3 3
b*(x) = O(a /2b/a) , a7 (x) = O(a /2) (2a,b)
I then (in Section 3) invoke the hypothesis of uni-
directional propagation to obtain the KdV channel
equation, which was developed originally by Shuto
(1974) through a rather more involved procedure.
I then go on to consider cnoidal waves in Section
4 and the solitary wave in Section 5 on the basis
of the stronger assumptions
|b*| << 0 3/2 (pa) la*| << a °72 (3a,b)
A prominent feature of the KdV equation for a
uniform channel is the existence of an infinite
number of integral invariants (Whitham, 1974, §17.6).
The KdV equation for a slowly varying channel admits
only two such invariants, of first and second order
in the amplitude; the latter measures energy, as
524
expected, but the former measures mass only if ba”
= constant. This deficiency is presumbly a conse-
quence of the implicit neglect of the weak reflection
that accompanies the gradual variation of the channel:
the reflection coefficient for energy is second
order in some appropriate measure of the channel
variation and therefore has no cumulative effect,
whereas that for mass is first order and does have
a cumulative effect. The resulting difficulty may
be avoided for a wave that is either periodic or of
compact support simply by choosing a horizontal
reference plane such that the mean value of the
free-surface displacement vanishes identically (see
Section 4), but the problem is more subtle for an
aperiodic disturbance of unlimited extent such as
a solitary wave (see Section 5) and remains unre-
solved.
The primary goal, at least for practical applica-
tions, of the analysis of waves in a gradually
varying channel is the prediction of a as a function
of b and d. Green's law, which neglects both non-
linearity and dispersion, predicts [Lamb (1932,
§185)]
A GI” 5 (4)
It is often used for practical shoaling calculations,
and Shuto (1973) finds that a « d “ holds for
solitary waves on relatively steep slopes for a/d as
large as 2. On the other hand, the joint assumptions
of Boussinesq similarity (a/d « d2/22) and conser-
vation of energy (which is proportional to abo)
imply [Miles (1977a)]
ace pb 2/3q-1 . (5)
Comparison with experiment (see Section 5) suggests
that (5) should be valid for a shoaling-or laterally
diverging channel if |6| < 0.1, where
5 = Ge) 72a) & Sey (6)
but perhaps not for a laterally converging channel.
The present results also have implications for
the approximate treatment of nonlinear wave propa-
gation along the lines initiated by Whitham (1974,
Ch. 8) in his treatment of shock-wave propagation
and since applied to solitary waves [Miles (1977a)].
2. BOUSSINESQ CHANNEL EQUATIONS
The boundary-value problem for gravity waves in an
ideal, homogeneous liquid may be deduced from
Hamilton's principle in the form [Broer (1974),
Miles (1977b) ]
LP 1
6[[Lté n}axat = 0, L = En, - Bune esy - 590°, (7a,b)
where x and y are horizontal and vertical coordinates;
6 (x,t) and n(x,t) are the velocity potential at, and
the displacement of, the free surface; dx is an
element of area in the x space; d(x):is the quiescent
depth; and the velocity potential $(x,y,t) is
determined by
V26= (0) (-d <y <n) , (8)
Re ar NKSIONKy = 0) (C7 el) ) StS GZ =) (9a,b)
The solution of (8) and (9) is given by
§ = & - y¥-(avey - Sy2v2e + 01826) (10)
where 8 is defined by (lb) with d and 2 as scales
of y and x. The corresponding approximation to
the kinetic energy integral, after invoking n = O(ad),
aVE = 0(8%E), B = O(a), (2), and V*(AVB) = VA*VB +
AV2B, is
n
f (Vo) 2dy = (atn) (VE)? - $a3 (026)? + 29- [a3 (v2E) VE]
OGRE) (11)
Substituting (11) into (7), invoking the further
approximation that ¢ is independent of the transverse
coordinate in a channel of slowly varying breadth
b(x) and depth d(x), and integrating across the
channel, we obtain
L 2, 13-2 dL
-= + =a - =
offen, 3 (d+n) ES Go ex 7 GIN |bdxdt = 0. (12)
The corresponding Euler-Lagrange equations,
Ll g3
=. + =
3 (bd See [b(d+n)e), ur bn, 0) (13a)
and
E, + 562 + on = 0 (13b)
t Dex. .
are counterparts of the Boussinesq equations [cf.
Whitham (1967)].
It is worth noting that the approximations to
this point are consistent with conservation of both
mass and energy:
3, [nbax = Op a, [Jptarm ey - aa3e2 + Sgn? bdx = 0,
(14a,b)
where the integrals are over either (-~,~) or a
periodic interval. The integral (14a) follows
directly from the integration of (13a) with respect
to x, subject to appropriate null or periodicity
conditions at the end points. The integral (14b)
may be similarly established or may be inferred
(through Noether's theorem) from the invariance of
the Lagrangian density in (12) under a translation
of t;:it is an exact invariant of (13), but it
would be consistent with the antecedent approxima-
tions to approximate the specific energy in (14b)
by 4 (d&2 + gn7).
3. KORTEWEG-DEVRIES CHANNEL EQUATION
The Korteweg-deVries (KdV) equation for uni-
directional wave propagation in a uniform channel
may be deduced from the Boussinesq equations by
assuming that & and n are slowly varying functions
of t in a reference frame moving with the wave speed,
c. It is expedient in the present context to choose
x, rather than t, as the slow variable (since b and
d are prescribed as slowly varying functions of x)
and to introduce
- = gd alls)
aes) E (c gd) (15)
as a characteristic variable. The direction of
propagation may be reversed by reversing the sign
(ope qo) bhigy, ((alS))) 4
The reduction of (13) on the hypothesis that Ny
= O(ans) yields
Qe a j=
3d Vike ) pe + 3i(cd)) mm. + an, + (Abe)im = 0; (16)
where
A( ) = (d/dx)log( ) (17)
(note that Ac = 4Ad). Equation (16), which appears
to have been derived originally by Shuto (1974),
reduces to the KdV equation if b and d are constant.
The vertically averaged, horizontal velocity is
given by
u = (gn/c) [1+0(a)], (18)
whilst the vertical velocity is O(a %u). The mass,
Momentum, and energy of the wave therefore are given
by
co foe) co
M = pbc [ nas, M = pba [ uds| ="Me,, & = pgbe [n2ds,
— co —-o —o
(lS ay o7C).
within 1+0(a). The limits of integration may be
replaced by +4T for a wave of period T.
Multiplying (16) through by (be) and ben,
respectively, and integrating over -~ < s < ~ on
the assumption that n, Ns, and ngg vanish in the
limits, we obtain the integral invariants
Cc foe}
3 2
I = (bc) nds, J =bc n“ds. (20a,b)
—oco —oO
It follows that E = pgJ is conserved. On the other
hand,
1
ai(aa) ema Ma pn@ses)?
M = (2la,b)
so that, except for special combinations of b and
d, M and M are conserved only if S°.nds = 0. Non-
conservation of momentum is acceptable in consequence
of the horizontal thrust exerted on the fluid by
the bottom and walls of the channel, but non-
conservation of mass is generally unacceptable.
We remark that the neglect of both dispersion
and nonlinearity, as represented by the first and
second terms, respectively, in (16), yields Green's
law, (be) 2n = f(s), where f is an arbitrary function
of the characteristic coordinate, s.
4. SLOWLY VARYING CNOIDAL WAVE
Theory
Kinematical and scaling considerations suggest that
an approximate solution of (16) for a wave of pre-
scribed period
We Ann = (t/g)? (22)
be posited in the form
n(s,x) = a(x)N(6,x), 68 = ws - x(x), (23a,b)
where 8 and x are fast and slow variables, a(x) is
a slowly varying amplitude, and x(x) is a slowly
varying phase shift. It also is expedient to
introduce
525
y¥ (x) = 2(cd/aw) x7 (x), (24a)
such that the phase speed of the wave is given by
. 1
-8,/8, = c/{1-y(a/a] = [g(dtya)]*.
Conservation of mass and energy imply the constraints
(see Section 3)
ne (24b)
<N> = (0), a*beT<n2> = 5, (25a,b)
where < > implies an average over a 2m interval of
8 and J is the integral invariant obtained through
the substitution of (23) into (20b).
A formal, asymptotic development of the descrip-
tion (23) may be obtained by expanding N(8,x) and
y(x) in powers of an appropriate measure of the
slow variation of b and d and invoking (25a) and
the requirement that the period of 6 be 27. The
first approximation, which is obtained by substit-
ing (23) into (16) and then neglecting all
derivatives with respect to the slow variable x,
corresponds to that for a cnoidal wave [Lamb (1932,
§253)]. It may be placed in the form
N en? [ (K/t) 6|m] - <cn2> F <cn2> =
l
[m-1+(E/K)] /m,
(26a,b)
il
y = [2-m-3(E/K)]/m, aL/a% = (16/3)mk2 = U(m), (26c,d)
where en (u|m) is an elliptic cosine of modulus vn
and K and E are complete elliptic integrals in the
notation of Abramowitz and Stegun (1955), and U(m)
is the local Ursell parameter. Substituting (26)
into (25b), we obtain
ah opie 2) = WPaes = Fen), (27a)
where
<n2> = <cnt> - <cn2>2 = [2(2-m) (E/K) - 3(E/K)2
- (1-m)]/(3m?) (27b)
and
E = (4°/3°)x2[2(2-m)EK - 3R2 - (1-m) K?]. (27¢c)
It follows from (27), which determines m(x), that
m is constant if and only if pad/2 = constant, in
which special case (23), (26), and (27) constitute
an exact similarity solution of (16). e
The results (26a) and (27a) provide a parametric
relation between aL/d2 and gL3/2 /pa/2 that may be
graphically represented as a plot of log F vs log U
[see Miles (1978b)]. The case of constant depth is
especially simple in that the plot of log F vs log
U is equivalent to -log b vs log a. The limiting
relations
F > au? JAS Ghia cas 2 yo (283i)
and
ae an”, ie Saar (U + o) (29a,b)
intersect at U =
for U> 150.
The preceding calculation is a generalization of
that of Svendsen and Brink-Kjaer (1972), who consider
the one-dimensional (b = constant) shoaling problem;
however, they replace 8 + wt in (23b) by the
equivalent of [1 - 4y(a/da] (x/c), which is clearly
150 and provide rough approximations
526
in error unless both b and d are constant.
The problem also is attacked by Shuto (1974),
who allows for the variation of both b and d but
arrives at a result (which he integrates numerically)
that appears to be inconsistent with conservation
of energy. However, his result is consistent with
(28) in the limit U + 0 and with (29) in the limit
U + © or, more precisely, with the result obtained
by neglecting only terms of exponentially small
order in (27),
3/2 2
F~ Gu “- 2G) Ute), (30)
which is in error by less than 1% for U > 70. It
therefore appears that Shuto's numerical results
are not significantly in error (on the scale of his
plots) over the entire range of U.
Experiment
Shuto (1974) compares his results with his own
experimental observations and with those of Iwagagi
and Sakai (1969) for shoaling waves periods from
1.2 to 6 seconds on uniform slopes of 1/20 and 1/70.
He concludes that linear surface-wave theory (which
presumably accounts exactly for dispersion) is
superior to his cnoidal-wave results for U < 30 and
conversely for U > 30 and that the latter are good
for a/d as large as 0.8.
5. SLOWLY VARYING SOLITARY WAVE
Theory
The slowly varying solitary wave
*5 1
nN = asech? Cra)” (fee t) , C= [g(dta)]°*,
(3la,b)
Die OVS =.
a= a3 %% cia (31c)
is obtained by letting U + © with KO = 0(1) in (26)
and (27).* There is, however, a new difficulty:
none of the integrals I, M, and M [see (20a) and
(2la,b)], which now are proportional to pl/6g3/4
b2/3q, and p2/3q3/2 respectively, is conserved ex-
cept for special variations of b and d. [The failure
of the condition <N> = 0 in the limit U + ~ is a con-
sequence of the loss of the displacement -a<cn*> ~
a/K, which cancels the mean of acn? (2K8) when inte-
grated over -K < 2K9 < K.] It follows that, except
in the special case ba?/2 = constant for which (31)
is an exact solution of (16) and M and M vary like
a-? and a3/2, respectively, (31) cannot be a uni-
formly valid approximation to the solution of the
KdV channel equation (16); instead, it is the first
term in an inner expansion, which must be matched
to an appropriate outer expansion.
Johnson (1973) obtains the next term in an inner
expansion for b = constant and finds that it can
be matched to an appropriate outer expansion if d
is increasing in the direction of propagation (the
solitary wave may undergo fission if d is decreasing) ;
*The prediction that a «= b72/3q71 appears to be due
originally to Saeki, Takagi, and Ozaki (1971); see
also Shuto (1973, 1974) and Miles (1977a).
however, he does not obtain an explicit description
of the oscillatory tail, nor does he allow for the
possibility of expanding the slowly varying phase
X(x) as well as N(6,x) [see (23)].
Ko and Kuehl (1978) have criticized Johnson for
this latter omission and develop a joint expansion
of (the equivalents of) N and x. They conclude
that the solitary wave ("soliton") experiences an
irreversible energy loss in the sense that it does
not re-establish itself if the channel gradually
reverts to its initial, uniform breadth and depth.
This may be, but the proper form of the inner
expansion is to some extent a matter of expediency,
and the ultimate validity of any particular expansion
can be established (albeit heuristically) only
through matching to a proper outer expansion. Ko
and Kuehl appear to overlook the crucial role of
matching, and, at least in this important respect,
their results must be regarded as incomplete.
Johnson's results are readily generalized to
allow for the variation of both b and d and reveal
that
§ = 2(3a/d) °/2aA(ba9/2) = (3a)~3/2 (2aAb + 947)
(32)
is an appropriate measure of the slow variation of
the channel (this same measure also is appropriate
for a cnoidal wave for U > 100). The Boussinesq
equations (13) and KdV equation (16) are based on
the restriction 6 = O0(1) asa ¥0 [cf. (2)], whereas
(26) and (31) are based on the stronger assumption
|| << 1 [cf. (3)]. Moreover, a consideration of
the special case of linearly increasing breadth and
constant depth [Miles (1978a)] suggests that the
wave ultimately ceases to be solitary and evolves
0.4
0.02
0.01
10
FIGURE 1. Decay of a solitary wave in a linearly ex-
panding channel. The wave is propagating in the posi-
tive-x direction, where x is measured from the virtual
origin at which b = 0, and enters the diverging chan-
nel (from an entry section of uniform width) at
x/d + 10. The amplitudes at the transition station are
a/d = 0.05(x), 0.1(+), 0.2(0), and 0.4(:*). The dashed
lines have slopes of -2/3.
0.8
0.4
0.2
a
id
04
0.05
0.025
Xx /d
FIGURE 2. Growth of a solitary wave in a linearly con-
tracting channel. The wave is propagating in the nega-
tive-x direction (right to left), where x is measured
from the virtual origin at which b = O, and enters the
converging channel (from an entry section of uniform
width) at x/d + 94. The amplitudes at the transition
station are a/d = 0.05(x), 0.1(+), 0.2(0), and 0.4(:);
the corresponding slopes of the dashed lines are
“DoE, “Oo, SO MLE Etvel o)o4i-
into a dispersive wave train for which the first
peak closely approximates a solitary wave in shape
but is followed by successive peaks of only gradually
diminishing amplitude. There remains, however, the
difficulty of nonconservation of mass, and the
general problem of an aperiodic wave (in particular,
an initially solitary wave) in a gradually varying
channel is unresolved at this time.
Experiment
Shuto (1973) compares Green's law, a = ars, and the
present prediction a « qd7l, with the experimental
observations of Camfield and Street (1969) and Ippen
and Kulin (1954) for shoaling of solitary waves on a
uniform slope. He concludes that the range of valid-
ity of the "-l power" law decreases with increasing
slope and that the "4 power" law holds for slopes in
excess of 0.045 and a/d as large as 2.0. A more
precise comparison can be made on the basis of (32),
which reduces to
6 = 9(3a/d)~3/2a° (b = constant) (33)
for a channel of constant breadth. The estimated
critical values of 6, such that a « a7! or a-*% pro-
vide better fits to the data for 6 < 6* or 6 > 6%,
respectively, are 6* = 0.10, 0.10, and 0.09 for
slopes of .01, .02, and .03, anda « a! is typically
within the experimental scatter for 6 < 0.01.
Chang and Melville (unpublished) have recently
measured a(x) in linearly diverging and converging
channels. Their results for a diverging channel
(Figure 1) tend to confirm the prediction a « b~2/3
for initial values (at the transition from a uniform
channel) of 0.05 < a/d < 0.2 [the corresponding
values of 6 = 2(3a)-3?/2(db“/b) are in the range
(0.01, 0.07], although the decay ultimately exceeds
this inviscid prediction--presumably in consequence
of viscous or other dissipation -- and exceeds it
after only a rather brief section for an initial
527
value of a/d = 0.4. Their results for a converging
channel (Figure 2) predict a growth that is roughly
approximated by a = b-9-4, Dissipation in the
converging channel would tend to decrease the magni-
tude of the exponent, but why this decrease should
be so much larger than the corresponding increase
for the diverging channel is not clear at this time
(intuition suggests that reflection could be more
significant in a converging than in a diverging
channel, but neither analytical nor experimental
evidence is available to support this conjecture).
ACKNOWLEDGMENT
This work was partially supported by the Physical
Oceanography Division, National Science Foundation,
NSF Grant OCE74-23791, and by the Office of Naval
Research under Contract NO0014-76-C-0025. Most of
the material in Sections 3 and 4 has been published
elsewhere [Miles (1978b)] in slightly different
form.
REFERENCES
Abramowitz, M., and I. Stegun (1965). Handbook of
Mathematical Functions, Bureau of Standards,
Washington, D. C.
Broer, L. J. F. (1974). On the Hamiltonian theory
of surface waves. Appl. Sci. Res. 30, 430-446.
Camfield, F. E., and R. L. Street (1969). Shoaling
of solitary waves on small slopes. Proc. ASCE,
Waterways and Harbors Div. 95, 1-22.
Ippen, A., and G. Kulin (1954). The shoaling and
breaking of the solitary wave. Proc. 5th Coastal
Engineering Conference, 27-49.
Iwagaki, Y., and T. Sakai (1969). Studies on cnoidal
waves (seventh report) - Experiments on wave
shoaling. Dis. Pre. Res. Inst. Annals, No. 12B,
Kyoto Univ., 569-583 [in Japanese; cited by Shuto
(oT):
Johnson, R. S. (1973). On the asymptotic solution
of the Korteweg-deVries equation with slowly
varying coefficients. J. Fluid Mech. 60, 813-824.
Ko, K., and H. H. Kuehl (1978). Korteweg-deVries
soliton in a slowly varying medium. Phys. Rev.
Lett. 40, 233-236.
Lamb, H. (1932). Hydrodynamics, Cambridge University
Press.
Miles, J. W. (1977a).
a Slowly varying channel.
149-152.
Miles, J. W. (1977b). On Hamilton's principle for
surface waves. J. Fluid Mech. 83, 153-158.
Miles, J. W. (1977c). Diffraction of solitary waves.
ZAMP 28, 889-902.
Miles, J. W. (1978a). An axisymmetric Boussinesq
wave. J. Fluid Mech. 84, 181-192.
Miles, J. W. (1978b). On the Korteweg-deVries
equation for a gradually varying channel. J.
Fluid Mech. (sub judice).
Rayleigh, Lord (1876). On waves, Phil. Mag. 1,
257-279; Papers 1, 251-271.
Saeki, H., K. Takagi, and A. Ozaki (1971). Study
on the transformation of the solitary wave (2).
Proc. 18th Conf. on Coastal Engg. in Japan, 49-
53 [in Japanese; cited by Shuto (1974)].
Note on a solitary wave in
J. Fluid Mech. 80,
Shuto, N. (1973). Shoaling and deformation of non-
linear long waves. Coastal Engineering in Japan
HG, alow.
528
Shuto, N. (1974). Nonlinear waves in a channel of
variable section. Coastal Engineering in Japan
7, dA.
Svendsen, I. A., and 0. Brink-Kjaer (1972). Shoaling
of cnoidal waves. Proc. 13th Coastal Engineering
Conference (Vancouver 1, 365-383.
Variational methods and
Proc. Roy. Soc.
Whitham, G. B. (1967).
applications to water waves.
Lond. A 299, 6-25.
Whitham, G. B. (1974). Linear and nonlinear waves,
Wiley-Interscience, New York.
Study on Wind Waves as a
Strongly Nonlinear Phenomenon
Yoshiaki Toba
Tohoku University
Sendai, Japan
ABSTRACT
Recent studies on wind waves in our laboratory,
from a view point of strong nonlinearity of the
wind waves, are reviewed. The main items are as
follows. (1) It has been shown by experiments and
theoretical analyses that the mechanism of initial
generation of waves by the wind is the instability
of shear flows of two-layer viscous fluids, air and
water. It is a selective amplification of distur-
bances at the frequency of maximum growth rate.
However, the transition of the initial wavelets to
irregular wind waves including turbulence follows
within several seconds [Kawai (1977)]. (2) Flow
visualization studies of the internal flow pattern
of wind waves show that the shearing stress of the
wind is concentrated at the crest and windward face
of individual waves, and a special area is formed
where the surface wind drift, and consequently the
vorticity is concentrated, causing the forced con-
vection or the turbulent mode, which is the origin
of the irregularity of wind waves [e.g., Toba et al.
(1975); Okuda et al. (1977)]. (3) Statistical
investigation of instantaneous individual waves in
a wind-wave tunnel shows clearly the existence of
similarity in the individual waves [Tokuda and Toba
(1978)]. Namely, the energy spectrum, which is
newly defined for the individual waves, is virtually
equivalent to the traditional energy spectrum at
the frequency range from 0.7- to 1.5-times the
frequency of the energy maximum. However, the energy
peaks which usually appear in the traditional spec-
trum at the higher harmonics of these dominant waves
completely disappear. The apparent phase speed of
individual waves, for each wind and fetch condition,
is inversely proportional to the square root of
their frequency, and is much larger than the phase
speed of linear water waves. For the individual
waves for each wind and fetch condition, there
exists statistically a conspicuous relationship of
the 3/2-power law [cf., Toba (1972, 1978a)] between
the normalized wave height and period. Consistently
529
with this and the phase speed relationships, the
steepness of the individual waves is statistically
constant. (4) Discussicn is presented as to the
possibility of approaching the above-mentioned
characteristics of the individual waves from the
similarity hypothesis and dimensional considerations.
Self-adjustment of the individual waves to the
local wind drift distribution is postulated to
explain the 3/2-power relationships, which may be
the basis of the possibility that the pure wind-wave
field is represented by a single dimensionless
parameter [Toba (1978a)]. (5) A new formulation
is presented for the roughness parameter or the
drag coefficient over the wind waves, incorporating
the single dimensionless parameter of the wind-wave
field. A physical interpretation of the form is
given from the internal flow pattern of individual
waves [Toba (1978b)].
1. INTRODUCTION
In a traditional model, the wind waves are treated
as phenomena, expansible to component free water
waves having weakly nonlinear interactions among
waves of different wave numbers. However, detailed
experimental studies on the actual conditions of
wind waves produced in wind-wave tunnels, have
shown that wind waves are much more strongly non-
linear phenomena, especially in their younger stages.
This report presents a review of recent studies
made in our laboratory, giving much emphasis to
the strong nonlinearities which are inherent in
wind waves.
2. INITIAL GENERATION OF WIND WAVES
The first topic starts with an approach from the
process of the initial generation. The wind waves
have long been assumed to be generated from a still
water surface by the effect of pressure fluctuations.
FIGURE 1. Flow visualization
of the initial stage of the
generation of wind waves by
use of hydrogen bubble lines
produced by the electrolysis
of water. The photographs were
taken from a viewpoint slightly
below the air-water interface,
so images reflected at the in-
terface are seen in the upper
1/4 of each picture. The hydro-
gen bubble lines are produced
near the left end as pulses of
0.002-s width at 0.04-s inter-
vals in a very slow, uniform
flow of water which was pro-
duced before the start of
wind. The wind was 6.2 m/s
blowing from left to right of
each picture. The filmed time
of each picture from the start
of the wind is shown in sec-
onds. The out-of-focus areas
were caused by some fluctuation
of the mean flow, for very shal-
low depth of the focus. In (e)
are seen the initial wavelets,
and in (f) is seen the onset of
turbulent mode. [Cited from
Okuda et al. (1976).]
| Ma
(a) 0.40 sec
4
(d) 3.29
A resonance mechanism for the initial generation
proposed by Phillips (1957) and an instability
mechanism for further growth proposed first by Miles
(1957) have been referred to on every occasion.
Valenzuela (1976) showed that the growth rate of
waves in the gravity-capillary range, observed by
Larson and Wright (1957) at the initial stage of
the generation, agrees with the expected growth
rate by the instability theory applied to a coupled
shear flow of the air and the water.
Kawai (1977 and 1978) of our laboratory has
arrived at the conclusion, by systematic experiments
together with theoretical analyses, that the mech-
anism of generation of the initial wavelets is the
instability in a two-layer shear flow of viscous
fluid of air and the water, as a selective amplifi-
cation of disturbances of the frequency at the
maximum growth rate.
The experiments were carried out mainly by use
of a wind-wave tunnel of 20 m length, 60 cm x 120 cm
cross-section, containing water of 70 cm depth.
After the sudden starting of wind on the still
surface of water, a shear flow first develops in
the uppermost thin layer of water, and several
seconds later, regular, long-crested initial wavelets
appears [Figure l(e)]. His theoretical analysis of
the shear flow instability of the two-layer viscous
fluids, using the actual profile of the shear flow
in water, shows that the system is unstable and
there exists a frequency at which the growth rate,
kCj, is maximum (Figure 2). The frequency of kCj-
maximum does not necessarily coincide with that of
Cr-minimum, or the minimum phase speed for the
gravity-capillary wave. Three properties of the
initial wavelets determined by the experiment, i-e.,
the frequency, the growth rate, and the phase speed
are all virtually coincident with those of the
theoretically predicted waves of the maximum growth
rate as shown in the following.
Figure 3 shows an evolution of the spectrum
calculated by the maximum entropy method, which may
be applicable to nonstationary processes. Each
spectrum represents an ensemble average of 8 runs.
- in Figure 7.
(b) 1.40 (c) 2.36
(f) 4.20
(e) 3.78
Wavelets of a constant frequency of about 15 Hz in
this case grow as shown in the figure with a smooth
spectrum. The peak then moves to a lower frequency
side showing the evolution to irregular wind waves
having the usual spectral form. In the stage of
constant frequency, Figure 4 shows the agreement of
the observed frequency of the initial wavelets with
the theoretical frequency for the kCj;-maximum, as
a function of the friction velocity of the air, u,,
but independent of the fetch. The frequency for the
Cy-minimum is around 14 to 13 Hz, and does not
coincide with the observed initial wavelets. Figure
5 shows the agreement in the phase speed, and Figure
6 the growth rate between the observed initial
wavelets and the theoretical initial wavelets for
kCj-maximum.
Thus, Kawai's conclusion is that the generation
of wind waves, whose initial stage is called initial
wavelets, is caused by the selective amplification
of small perturbations which inevitably occur in
the flow by the instability of the two-layer viscous
shear flow.
However, the duration of the exponential growth
of the initial wavelets was limited to from 1 to 8
seconds in the experiments. The transition from
the regular, long-crested initial wavelets to short-
crested, irregular wind waves takes place in a very
short time. The spectral peak, which has grown up
with an approximately constant frequency, starts
wandering at the transition, and then moves toward
the lower frequency side with the energy increased
in a general trend as seen in Figure 3, and also
The transition coincides with the
onset of turbulence at the water surface as revealed
in the next section.
3. INTERNAL FLOW PATTERN OF WIND WAVES — AN
EXPERIMENTAL SUBSTANTIATION OF THE STRONGLY
NONLINEAR PROCESSES
Irregularity is a character inherent in the wind
waves. This has been demonstrated by detailed
531
357 ay Lene UE 35 Ur Taal T ala =]
(a) (c)
% % wa
E
S 30+ 5 30h 4
y | ‘ eA r
r
Q2- 30 25F
kCi(8")
kC) (8)
3 4
k (crm)
studies of the internal flow pattern of wind waves
by use of flow visualization techniques [Toba
et al. (1975), Okuda et al. (1976, 1977) and
Okuda (1977) ].
Along the surface of individual undulations,
hereafter called individual waves, there is a strong
variation of the tangential stress exerted by the
wind. The stress value determined locally from the
distortion of hydrogen bubble lines, is several
times greater than the average wind stress value at
the windward face of the crest, and it is negligible
at the lee side of the crest as shown in Figure 8
as an example. The concentration of the shearing
stress results in the development of the local
surface wind drift forming a special region under
the crest where the strong vorticity is concentrated.
The vorticity concentration causes the forced con-
vection or turbulence, irrespective of whether or
not the air entrainment, or the breaking in a usual
sense occurs. As seen in Figure 9, small polystyrene
particles of 0.99 specific gravity placed just
beneath the water surface prior to the start of
the wind, begin to disperse into the interior by
the forced convection, coincidentally with the
t(Hz)
(tz)
FIGURE 2. Theoretically obtained correlation
of the amplification factor kC;, the phase
speed, Cy, and the frequency, f, to the wave
number, k, in the instability of coupled shear
flow of the air and the water, for four values
of the friction velocity of the air uy, of (a)
13.6, (b) 17.0, (c) 21.4, (da) 24.8 cm/s.
(Cited from Kawai (1977).]
transition of the initial wavelets to the irregular
wind waves. The main stage of the growth of wind
waves thus seems to proceed as a strongly nonlinear
processes.
4. COMPONENT WAVES AND INDIVIDUAL WAVES AS PHYSICAL
MODEL OF WIND WAVES
Despite the fact that the wind waves are thus a
strongly nonlinear phenomenon, they have been
assumed as expansible to component waves, having
phase speeds obeying the dispersion relation of
free water waves, and weak wave-wave interactions
have been considered.
Recently there have been some articles reporting
that the phase speeds of component waves do not
necessarily satisfy the dispersion relation, notably
by Ramamonjiarisoa (1974) for the one dimensional
case and Rikiishi (1978) for two-dimensional com-
ponent waves. Rikiishi developed an experimental
technique for the determination of the directional
structure of the phase speed of component waves
without pre-assuming the dispersion relation, and
=
ft)
N
E
2
i¥9)
10 F KK
L \
[ \
=)
rr
ic t(s) t(s) at(s)
intta) wavelets
[ —— 8.00~ 864 .005 128
Simei 0-64) -019:28) r=
239) EF= OC oo
r * 992-1056 -
— = 1056-1120 -
developing wind waves
10° —--- 1024~ 1536 01
= 1280-1792 -
— — 15.36 ~ 2048 =
— — 17.92 ~ 23.04
2048 ~ 2.60 -
statronary wand wares
—— 40.96 - 74.24 MQ 1664
=9
10 nn fLennnll
1 10 100
f(Hz)
FIGURE 3. A sequence of spectra for the initial stage
of the generation of wind waves, showing the growth of
initial wavelets at a constant frequency of about 15
Hz, and the transition to irregular wind waves. The
spectra were calculated by
the Maximum Entropy Method,
and each line represents an ensemble average of eight
cases. The fetch was 8 m and the nominal wind speed
was 5.1 m/s.
[Cited from Kawai (1977).]
Experiment
e F=3m, NS=1
1
1
8
0 10 20 30
ux (cm s")
FIGURE 4. Observed theoretically predicted frequency
of the initial wavelets, f., as a function of the
friction velocity of air, u,- Theoretical values are
for the condition of the maximum growth rate, where
U)/u, represents the dimensionless thickness of the
viscous boundary layer of the air. [Cited from Kawai
(Le) 01]
isa | a S|
Experiment
8- © F=3m, NS=1
° 6 17 Ex.! |
© 9 1
x 8 8 Ex.0 1
Tr x
6
O U,/ue = 5.0
A 8.0
E of i
/
/
= BAAN S
oo 4 noe 4
cas fe
Or 7 e
L Sf |
Oh
ore Ky
7 4
2r Oo” 6 ef 5
e Ad ye
VX of
hy ay 7
oe
ol sas |
0 10 20 30
u, (cm s*)
FIGURE 5. The growth rate of the initial wavelets,
8, as a function of u,- Theoretical values correspond
to those for waves of the maximum growth rate. [Cited
from Kawai (1977).]
{e)
{e)
30,- 2 oe 4
A S a A
%
£ 20+ |
&
o
10/- O Experiment Ex. 1 =|
O Theory Ui/ux =5.0
A ” " 8.0
O|L__ { silt
0 10 20 30
Ux (cm Ss")
FIGURE 6. The phase speed of the initial wavelets C
as a function of u,. Theoretical values correspond td
those for waves of the maximum growth rate. The theo-
retical values were calculated by use of the observed
velocity profiles in the water at the critical time of
the first appearance of the initial wavelets. The ob-
served values for higher three wind speeds were deter-
mined at the critical time, whereas that for the lowest
wind speed was determined about 3.5 s after the critical
time because of the experimental difficulty, and this
delay may presumably explain the observed higher value
533
ea,
WS ns
Sp(cm? s )
Mi
E
'
Q
¢
1
'
8
iH
~
10 20
f(Hz)
FIGURE 7. An example of minute inspection of the ob-
served time series of the spectral peak for the initial
stage of the generation of wind waves. The lapse of
time is indicated in alphabetical order, the interval
between successive points being 0.32 s. After the growth
of regular initial wavelets at a constant frequency of
about 15 Hz, the spectral peak shows an irregular mo-
tion corresponding to the transition to irregular wind
than the theoretical ones. [Cited from Kawai (1977).] waves. [Cited from Kawai (1977) .]
Seen a a a ae an ae ne ne
zi
lL 4
C U= 62 m/sec J
[ F2285m
zt
WIND —> |
15 2
Peat : : | : =
q 9 -180 0 180
$ Raton
§ ° oo
. | 5 aor “og 5
Per maasp: coca o | FIGURE 8. Observed values of the
Sa? of, io local shearing stress along the
oes) VPale) ae ] surface of representative wind
5 go 9 on 269 | waves. The abscissa is the phase
P99 9 5] relative to the peak point of the
cot a e cal crest and the ordinate is ex-
pressed as the square of the fric-
pti 88, tion velocity of the water. The
Oro 5
ops? aie re fe on'e ih wind speed was 6.2 m/s and the
Phase
so
fetch was 2.85 m. [Cited from
Okuda et al. (1977).]
20 150
FIGURE 9. Flow visualization of the
initial stage of the generation of wind
waves by use of polystyrene particles
which had a 2-mm diameter and the specific
gravity of about 0.99, and which were
placed just beneath the water surface
prior to the start of the wind. The wind
blows from the left to the right. The
wind speed in the tunnel section was
8.6 m/s, and the fetch was 2.85 m. The
time measured from the start of the wind
is shown in seconds. In 2.58 s, initial
wavelets may be recognized by streaks of
light in the water, and some particles
have already begun to disperse into the
water. In 4.78 s, waves are already ir-
regular wind waves and more particles are
dispersed. In 13.6 s, particles are dis-
persed down to more than 10 cm, corre-
sponding approximately to a half of the
representative wave length. [Cited from
Toba et al. (1975) .]
found that the phase speed was virtually independent
of the frequency, and had the same value as that
of the waves of the spectral maximum, at respective
fetches. These experimental results are interpreted
as indicating that the assumption of wind waves as
expansible to component free waves with weak non-
linearity is not necessarily appropriate for young
growing wind waves.
On the other hand, since individual waves as
instantaneous surface undulations have a specific
shearing stress distribution, and a specific interval
flow pattern, they may carry some factors as a phys-
ical element. We have examined, in a wind-wave
tunnel of 15-cm width, energy density distributions
for individual waves, as well as their phase speeds,
and compared them with those obtained by usual com-
ponent wave model for the same experimental data
{Tokuda and Toba (1978)].*
First, a normalized energy spectrum for individual
waves has been newly defined and calculated from the
statistical distribution of two kinds of the individ-
ual waves: zero-crossing, trough-to-trough and all
trough-to-trough on our wave records, as illustrated
in Figure 10. The definition of the normalized
individual-wave spectral density, O8y, is
8y (fy) = 6yAE/(AE/E,)E (1)
where vim
HABE debh 9)
are Seatee at ;
6;Af = > imt G peak: alae, Soon pant ae, db
and where m; is the number of individual waves of
the period class, T;, (frequency from f to f + Af),
Af = 1/(2nAt), where we used At = 0.02 Shen) 00),
and Af - 0.25 Hz, and also
*Tokuda, M., and Y. Toba (1978): Component waves
and individual waves as physical model of wind waves.
To be published.
is the frequency of the energy maximum.
The A-spectrum
and where f
Figure 10 shows the comparison.
is the normalized spectra by the traditional com-
ponent wave model in which the secondary peak is
seen at the normalized frequency of 2. The B-spectrum
is for individual waves of zero-crossing, trough-to-
trough, and the C-spectrum for all trough-to-trough
on our wave records. In the main frequency range
from 0.7 to 1.5, which is the value normalized by
the peak frequency, the spectra are virtually
equivalent with one another. The second peak at
frequency of 2 in the A-spectrum completely dis-
appears in the individual-wave spectra. The slope
of these straight lines is £-2 for the high frequency
side, and £2 for the low frequency, sides ethexe>
spectrum is considered to give a better represen-
tation of the high frequency side which is exactly
on the £79 line, and the B-spectrum represents the
low frequency side better, which is more similar to
the traditional A-spectrum. We may infer that much
energy of the higher frequency part of traditional
component waves, which is clearly shown as the
energy at higher harmonics of the spectra, is a
manifestation of the distorted shape of individual
waves of the main frequency range, as was already
suggested by Toba (1973).
Figure 11 shows the normalized phase speed of
individual waves determined by two adjacent wave
gauges. It is inversely proportional to the square
root of the frequency, in contrast to the phase
speed of linear waves which is inversely proportional
to the frequency. In addition, the phase speed of
the individual waves is much larger than that of
linear waves as shown later. In the case of the
phase speed of component waves of one-dimensional
10'
535
INDIVIDUAL WAVES
ZERO-CROSSING TROUGH-TO-TROUGH
ALL TROUGH-TO-TROUGH
10°
~~
z
=
~~
z
oe
{|
10
10°
fr
COMPONENT WAVES
a a ee
+
+
sls
FIGURE 10.
x
Tj
Comparison of three kinds of normalized energy spectra from the same wind-wave records in the wind
wave tunnel. A: Traditional energy spectra by the component-wave model. B: Energy spectra for individual waves
of zero-crossing trough-to-trough. C: Energy spectra for individual waves of all trough-to-trough. [Cited from
Tokuda and Toba (1978).]
spectra, which was obtained from the cross-spectra
of the records of two adjacent wave gauges (Figure
12), approximately the same phase speed is obtained
in the before-mentioned main frequency range, where
the coherence is close to unity. However, in the
higher frequency range, it is virtually constant
in agreement with Ramamonjiarisoa's 1974 measurement.
The original values are shown in Figure 13, in
which locations of the spectral peak are shown by
arrows for the shortest and the longest fetches,
respectively, and as the peak frequency moves to
the left, the phase speed of the component waves
becomes larger. In the figure, the full line shows
the phase speed of linear waves. Figure 12 is the
normalization of Figure 13, and Figure 14 shows an
example of the comparison of phase speeds of com-
ponent waves and individual waves. It should be
noted that, as the distance of two wave gauges
becomes wider, the range of high coherence becomes
narrower, and the phase speed of component waves
tends to be more uniform and obscure. However, it
is at least evident that phase speeds for both com-
ponent waves and individual waves have the same
value near the peak frequency, and are inversely
proportional to the square root of frequency, and
much higher than the values of linear waves. It
536
06 0.8 1-0 1.2 1.4 1.6
fn
FIGURE 11. Phase speed distribution of individual
waves (zero-crossing trough-to-trough), determined
by a photographic method, and normalized by values
for waves of maximum energy density. Dispersion re-
lation for water waves are also entered by the dotted
line. [Cited from Tokuda and Toba (1978).]
is caused by the effect of the wind drift, which is
concentrated near the crests.
Thus, by using appropriate normalization, we
may express the energy distribution of physically
substantial waves by the energy spectra of individual
waves for some local frequency ranges, excluding
false energy density. The above mentioned B-spectrum
and C-spectrum are two examples of these. Further,
we may reinterprete- the traditional energy spectrum
for the main frequency range as representing the
energy distribution of individual waves, rather
than the usual interpretation of a linear combination
of small amplitudes of freely travelling component
waves. In other words, the elementary physical
substance of wind waves is rather in the individual
waves, which have a specific distribution of local
wind stress and flow pattern, and an apparent phase
speed inversely proportional to the square root of
the frequency.
Further, Figure 15 shows that, for the individual
waves in the main frequency range for each wind and
fetch condition, there exists a conspicuous statis-
tical relation between normalized wave height and
period, for significant waves which Toba (1972)
proposed as the 3/2 power law:
H* = prx3/2 (2)
where H* = gH/u 2 and T* = T/u, represents the
g * g
COHERENCE
FIGURE 12. Phase speed distri-
bution of one-dimensional compo-
nent waves, obtained from the
cross-spectra of records of adja-
cent two wave gauges, and nor-
malized by values for waves of
maximum energy density. The
coherence of the cross-spectra
is shown in the upper part.
[Cited from Tokuda and Toba
(1978) .]
Fetch
m
(cm s-!)
Fo05,,9 5.87
cot ke
eee .e
° ©
sob 8 88a g000, eco cog lg
coe Senet scoce
eo
°
Oper)
0062250009
T °ee
00m
°°
wooo 2°
° 0%,
PHASE SPEED
0 2 4 6 8 10 12
f (Hz )
FIGURE 13. Original values of the phase speed distri-
bution, for eight fetches, before the normalization
shown in Figure 12. Peak frequencies for the shortest
and the longest fetches are indicated by arrows, other
cases being in between of these. The phase speed of
linear water waves is indicated by the full line.
[Cited from Tokuda and Toba (1978).]
dimensionless height and period, respectively,
normalized by use of the acceleration of gravity
g and the friction velocity of the air u,. The
figure shows the data for individual waves for
various fetches. Except for very short fetches up
to about 4m, the factor of proportionality B is
constant of about 0.045.
It should be noted that although the spectral
form of wind waves in wind-wave tunnels is different
from that in the sea as discussed, e.g., by Kawai
et al. (1977), nevertheless the above power law
holds for both cases, although the constant, B, is
slightly different [cf, also Toba (1978a)]. Figure
16 shows another representation of the same relation:
between the wave height and the frequency, normalized
for those waves of maximum energy. The slope of
the line is -3/2.
Consistently with this relation and the above-
mentioned apparent phase speed, the steepness of the
individual waves determined by a photographic method
is approximately constant, statistically. It is
FROM CROSS-SPECTRUM
FOR INDIVIDUAL WAVES
AND STANDARD DEVIATION
c=(9/k)'”?
(cm s-')
PHASE SPEED
1.0 p22? ocoo.
:
COHERENCE
OO 0 2QID BSG YU Ww WA
FREQUENCY (Hz)
FIGURE 14. An example of the comparison of one-
dimensional phase speeds of wind waves, determined
from cross-spectra of records of two wave gauges, and
determined for individual waves (zero-crossing trough-
to-trough) together with the standard deviation, and
the dispersion relation for water waves. At the bottom
is shown the coherence of the cross-spectra. The fy,
represents the frequency at the energy maximum. [Cited
from Tokuda and Toba (1978) .]
inferred that these facts strongly indicate the
existence of similarity in the individual waves or
in the field of wind waves, presumably as a result
of the strong nonlinearity.
Fetch 1-00m 1-70
10° 10!
537
5. APPROACH BY SIMILARITY HYPOTHESIS AND
DIMENSIONAL CONSIDERATION
In cases of strongly nonlinear processes, such as
turbulence, it is hard to approach problems from
the rigorous way of solving a closed system of
equations. In these cases, some assumptions based
on physical considerations are sometimes introduced
to supplement the system of equations, to arrive at
useful results. In the case of wind waves, it
seems that an approach by the traditional model of
component irrotational free waves with their weak
interactions is not necessarily realistic as has
been shown. There is another approach, in which
a kind of similarity structure in the field of wind
waves is assumed, and a regularity in gross structure
is sought by invoking dimensional considerations.
An example of this line of approach has been
attempted as partly described in a paper by me
[Toba (1978a)].
Since the local wind stress distribution along
the surface of individual waves is as shown in
Figure 7, the local wind drift is forced to be
stronger near the crest and weaker near the trough.
Water particles near the surface travel a longer
distance when they are near the crest than when
near the trough. On the other hand, water waves of
finite amplitude cause the wave current, resulting
from the difference between the foreward and the
backward movements of the water particles. Some
self-adjustment should occur for individual waves
in such a manner than the forward and the backward
movements by the waves are coincident with the
difference in the local wind drift as to the phase.
The wave current ug of the individual waves of
amplitude, a, and angular frequency, o, is now
approximated by that of the second order Stokes
wave:
ug = a*a3/g
Number Density
@ 02 -
e ONS = O12
© 010 - O15
* 005 - 0.10
0.005 - 0.05
Significant Wave
Standard Deviation
FIGURE 15. Examples showing that the main part of individual waves in the wind-wave tunnel (zero-crossing
trough-to-trough) satisfies the 3/2 power law between the normalized wave height H* and the period T*. The
u, was 68 cm/s. [Cited from Tokuda and Toba (1978) .]
538
1.5
NUMBER DENSITY
1.0 ji 0.005 ~ 0.05
a 005 - 0.10
08 0.10 015
015 0.20
0.20
06
z
a
04
02 4
05 1.0 2.0 3.0 4.0
fy
FIGURE 16. Another representation of the 3/2 power law
for individual waves (all trough-to-trough). H. and f
represent the wave height and the frequency, respectively,
normalized by values for waves of the maximum energy
density. [Cited from Tokuda and Toba (1978) .]
Since the difference in the local wind drift is
caused by the mean wind stress, the self-adjustment
is expressed by the condition that the wave current
is proportional to ux, namely,
a*o3/gu, = constant (3)
This is transformed immediately to
H* = B'o*73/2 (4)
which is equivalent to (2), where o* = u,d/g.
The condition of constant steepness may arise
from the similarity requirement. The combination
of the 3/2-power law relationship and the constant
steepness condition leads to the apparent phase
speed proportional to the square root of the
frequency. These three relationships, which have
been shown by the experiments to be satisfied by
the individual waves, are self-consistent with one
another, and may thus result from the strongly
nonlinear effects.
The 3/2-power law makes it possible that the
wind-wave field is represented by a single dimension-
less parameter of the frequency at the energy
maximum as discussed by Toba (1978a). One of the
consequences of the above paper is that the growth
of the wind wave field is expressed by the evolution
of the dimensionless single parameter in a form of
error function of the parameter itself, in which
the value of the parameter approaches a final value
as a simple stochastic process, irrespective of its
initial conditons, through a rapid self-adjustment
of the state.
6. WIND STRESS OVER WIND WAVES
The final topic of this paper concerns the expression
of wind stress over wind waves. It has been pointed
out on Many occasions that the roughness length, or
equivalently the drag coefficient of the water sur-
face, depends not only on the wind speed but also
on the state of the water surface. Various attempts
have been made to obtain a functional form of the
roughness length incorporating the state of wind
waves or the wave breaking. However, in view of
the complexity of the expressions, together with
the wide scattering of data points, a simple
dimensional formula by Charnock (1955) has been
cited most frequently, but with various values of
a constant of proportionality, although the formula
contains only a parameter representing the wind
field.
A dimensional consideration leads to an expression:
Zo* = zo*(ux*, On*) (5)
where z9* = z9/v is the dimensionless roughness
parameter, ux* = u,3/gv the dimensionless friction
velocity representing the overall wind effect, and
Om* = ux0p/g the single parameter representing the
wind-wave field as stated in the previous section,
where Om is the frequency at the energy maximum.
Charnock's formula
Zo = Bu,°/g (6)
zo* = Bu,* (7)
which is a form of (5) in which o,* is disregarded.
It is shown that another simple form for zo*, using
symbols, o and o*, instead of om and o,* hereafter:
FETCH=13.6m
TOBA (1972)
KAWAI et al. (1977)
KUNISHI (1963)
MITSUYASU et al. (1971)
20 = 0.035 ug/g
UpZo/V
10! 10? es 108
ue / gu
FIGURE 17. Data plots for the relationship (6) in
a dimensionless form. Data by Toba (1972) and
Kunishi (1963) are from wind-wave tunnel experiments,
and data by Kawai et al. (1977) and Mitsuyasu et al.
(1971) are from tower-station observations. [Cited
from Toba (1978b).]
FETCH#13.6m
TOBA (1972)
KAWAI et al. (1977)
KUNISHI (1963)
MITSUYASU et al. (1971)
20*0.025 un/o
u 2 3 4
10 10 Selo 10
FIGURE 18. Data plots for the relationship (8) ina
dimensionless form. The same data with Figure 17 is
used. [Cited from Toba (1978b).]
Zo* = au,*/o* = au,2/vo, a = 0.025 (8)
is a better representation [Toba (1978b)]*. In
Figures 17 and 18 are shown plots of some available
data in the forms of (6) and (8) including wind-wave
tunnel experiments and field observations. It
should be said that the new formula is better at
least. It is seen from Figure 19 that the breaking
of wind waves is also expressed as a function of
the parameter, u,2/vo, for data from the wind-wave
tunnel and the sea. The ordinate is the percentage
of the breaking crests among individual waves
travelling through a fixed point, and it was deter-
mined by the same procedure for both cases. The
breaking of wind waves occurs for the condition
ux? vo > 103.
Equation (8) corresponds to an elimination of g
from the form of (5). In view of the recent
recognition since Munk (1955) that the waves of
high frequency components play a major role in the
transfer of momentum from the wind to the sea, it
seems rather unreasonable that Eq. (7) contains
information only of energy containing waves as o
in the denominator. However, since ug2/V S A =
du/un represents the magnitude of the average wind
stress, and o-! « T is a measure of the integration
time associated with individual waves, u,?/va is
interpreted as a measure of the accumulation of
the shearing stress or the concentration of the
vorticity at each crest of the individual waves,
conveying the horizontal momentum transferred from
the air into the interior of the water through
forced convection, whether or not the waves are
breaking, as stated in Section 3. As this effect
*Toba, Y. (1978b). A formula of wind stress over
wind waves. To be published.
539
increases, the total momentum transfer, as well as
the probability of the occurrence of the breaking
increases.
The form of (8) may be transformed to
Zo = B'u,*/g, (SY = eh Aw, (9)
which may be interpreted as an extension of Charnock's
formula (6) to include information of wind waves
in the form of the wave age, c/u,, where c is the
phase speed of the dominant waves. Also, the drag
coefficient, Cp, may be expressed from (8) as
G. = k*/[ In (z190/au,)]~2 (10)
where k is the von Karman constant and z;9 the
reference height of 10 m. According to (10), Cy
is more sensitive to the wind waves than to the
wind speed.
7. SHORT SUMMARY
We may summarize the review paper as follows. First,
the initial wavelets are generated by an instability
of two-layer viscous shear flow of a type of insta-
bility that immediately transfers to three
dimensional turbulence. Second, the main phase of
the growth of wind waves is regarded as the conse-
quent, strongly nonlinear processes. Third, the
traditional component wave model is not necessarily
realistic, and the elementary physical substance
might better be treated by individual waves,
especially for younger stages as observed in wind-
wave tunnels. Fourth, the individual waves
represent a conspicuous and characteristic similarity
of structure, presumably as a result of the strong
nonlinearities, and this may be the basis for the
pure wind-wave field being represented by a single
dimensionless parameter. Finally, a new stress
formula over the wind-wave field is presented.
40
Oo FETCH: 13.6m
10.0 TOBA (1972)
6.9
TOBA et al. (1971)
oa
(eo)
i)
oO
PERCENTAGE OF BREAKING CRESTS
9°
Got tite) 10%
FIGURE 19. Percentage of breaking crests among indi-
vidual waves traveling through a fixed point, may be
expressed as a function of the same parameter with
Figure 18. Toba et al. (1971) data are from tower sta-
tion observations, which are common with data of Kawai
et al. (1977) used in Figure 18. [Cited from Toba
(1978b) .]
540
ACKNOWLEDGMENTS
The author expresses many thanks to Messrs. M.
Tokuda, K. Okuda, and Dr. S. Kawai of his laboratory
for continuous cooperation and discussion, and to
Professor H. Kunishi of Kyoto University, Professor
K. Kajiura of University of Tokyo, Professor H.
Mitsuyasu of Kyushu University, and Dr. N. Iwata of
National Research Center for Disaster Prevention
for valuable discussion and comments. He also
thanks Mrs. F. Ishii for her continuous assistance.
REFERENCES
Charnock, H. (1955). Wind-stress on a water surface.
Quart*. J. Roy). Met. Soc. 81, 639.
Kawai, S. (1977). Study on the generation of wind
waves. Ph. D. Dissertation at Tohoku University,
100 pp.
Kawai, S. (1978). Generation of initial wavelets
by instability of a coupled shear flow and their
evolution to wind waves. Submitted to J. Fluid
Mech.
Kawai, S., K. Okada, and Y. Toba (1977). Support
of the three-halves power law and the gu,o74
-spectral form for growing wind waves with field
observational data. J. Oceanogr. Soc. Japan 33,
WSi7/5 .
Larson, T. R., and J. W. Wright (1975). Wind-
generated gravity-capillary waves: Laboratory
measurements of temporal growth rates using
microwave backscatter. J. Fluid Mech. 70, 417.
Miles, J. W. (1957). On the generation of surface
waves by shear flows. J. Fluid Mech. 3, 185.
Munk, W. H. (1955). Wind stress on water: an
hypothesis. Quart. J. Roy. Met. Soc. 81, 320.
Okuda, J. (1977). Internal flow pattern of wind
waves, Proc. Nineth Symp. on Turbulence, Inst.
Space and Aeronautical Sci. Univ. Tokyo, June
1977, 54.
Okuda, K., S. Kawai, M. Tokuda, and Y. Toba (1976).
Detailed observation of the wind-exerted surface
flow by use of flow visualization methods. J.
Oceanogr. Soc. Japan 32, 51.
Okuda, J., S. Kawai, and Y. Toba (1977). Measurement
of skin friction distribution along the surface
of wind waves. J. Oceanogr. Soc. Japan 33, 190.
Phillips, O. M. (1957). On the generation of waves
by turbulent wind. J. Fluid Mech. 2, 417.
Ramamonjiarisoa, A. (1974). Contribution a4 1'étude
de la structure statistique et des mécanismes
de génération des vagues de vent, Thése a4
L'Université de Provence Le Grade de Docteur és
Sciences, 160 pp.
Rikiishi, K. (1978). A new method for measuring
the directional wave spectrum. Part II. Measure-
ment of the directional spectrum and phase
velocity of laboratory wind waves. J. Phys.
Oceanogr. 8, 518.
Toba, Y. (1972). Local balance in the air-sea
boundary processes, I. On the growth process
of wind waves. J. Oceanogr. Soc. Japan 28, 109.
Toba, Y. (1973). Local balance in the air-sea
boundary processes, III. On the spectrum of
wind waves. J. Oceanogr. Soc. Japan 29, 209.
Toba, Y. (1978a). Stochastic form of the growth
of wind waves in a single-parameter representation
with physical implications. J. Phys. Oceanogr.
8, 494.
Toba, Y., M. Tokuda, K. Okuda, and S. Kawai (1975).
Forced convection accompanying wind waves. J.
Oceanogr. Soc. Japan 31, 192.
Valenzuela, G. R. (1976). The growth of gravity-
capillary waves in a coupled shear flow. J.
Fluid Mech. 76, 229.
Wilson, B. W. (1965). Numerical prediction of
ocean waves in the North Atlantic for December,
1959. Deut. Hydrogr. Z. 18, 114.
An Interaction Mechanism between
Large and Small Scales for
Wind-Generated Water Waves
Marten Landahl,
Sheila Widnall,
and
Lennart Hultgren
Massachusetts Institute of Technology
Cambridge, Massachusetts
ABSTRACT
By aid of a non-linear two-scale analysis it is
shown that large-scale water waves can experience
growth due to spatial non-uniformities in the
growth rate of the small-scale waves in the non-
uniform wind field associated with the large-scale
waves. The growth rate is shown to be proportional
to the mean-square slope of the small-scale waves
and their growth rates, but inversely proportional
to the difference between the phase velocity of the
large-scale wave and the group velocity of the small-
scale waves. It is suggested that this mechanism
can transfer wind energy to short gravity waves at
a higher rate than the direct linear transfer
mechanism of Miles (1962). The analysis also
predicts that a large-scale wave moving against the
wind will be damped by the action of the small-scale
waves.
1. INTRODUCTION
The mechanism whereby wind generates water waves
has long proven a difficult and challenging problem
in theoretical fluid mechanics which has not yet
been satisfactorily resolved. The simple linear
mechanism of forcing by pressure fluctuations
[Phillips (1957)] and by instability induced by
the mean wind field [Miles (1957), 1962)] have
been found inadequate to account for the high values
of energy transfer from wind to waves observed for
longer waves, both in the laboratory and in the
open sea. For short waves in the capillary regime,
laboratory experiments [Larson and Wright (1975) ]
have given good agreement between observed growth
rates and Miles' instability theory, particularly
when the surface drift velocity in the water is
taken into account [Valenzuela (1976)]. For waves
in the short gravity range, however, recent experi-
ments by Plant and Wright (1977) give growth rates
much in excess of that predicted by the instability
541
theory with the discrepancy beginning at a wave
length of about 10 cm and increasing with wave
length. Open-sea measurements have also produced
energy transfer rates for gravity waves which are
much in excess of the values according to Miles.
[See, for example, the recent review of Barnett
and Kenyon (1975)].
In view of the failure of linear theory one is
forced to look for nonlinear mechanisms for energy
transfer. Nonlinear interaction between waves in
the gravity range [Phillips (1966)] is a compara-—
tively weak process (of third order in amplitude)
which causes redistribution of the energy from
waves of intermediate wave numbers to waves of lower
and higher wave numbers. This could be effective
for the eventual saturation of the spectrum but is
unlikely to be strong enough to make a large change
in the initial growth. A more tenable proposition
is that the modification of the turbulence in the
air by the wave induced velocity field could change
the phase shift between surface elevation and the
pressure so as to alter the energy transfer rate.
This effect has been investigated by many authors
[Manton (1972), Davies (1972), and Townsend (1972),
among others] employing different turbulence models.
These investigations point to the possibility that
the modulation of the turbulence by the wind could
have an important effect, but it is difficult to
assess the adequacy of the postulated turbulence
models employed.
An interesting possibility for transfer of energy
to gravity waves is through nonlinear interaction
with capillary waves which can draw energy from
wind at a much higher rate than the longer waves.
The interaction between short and long surface
waves has been subject to a great deal of discussion
in the literature. A train of short waves riding
on a long wave becomes modulated by the orbital
velocity field of the long wave so as to make their
wave length smaller - and hence their amplitude
greater - in the region near the crest of the long
wave. Longuet-Higgins (1969) argued that the
542
radiation stress then set up by the short-wave train
would act to transfer momentum to the long wave.
In particular, if the short wave were to reach an
amplitude at the crest of the long wave high enough
for breaking, it would give up all its momentum to
the long wave. This maser-like mechanism was
examined critically by Hasselman (1971) who showed
that the change in potential energy in the surface
layer due to Stokes' transport by the short waves
would give a contribution to the energy transfer
to the large waves which would exactly cancel that
arising from Longuet-Higgins' momentum transfer
term. Hasselman's analysis did not take into
account any transfer due to modulation of surface
wind stress or short wave growth rate, however.
[This effect has been analysed by Valenzuela and
Wright (1976)]. Also, his analysis concerned
primarily gravity waves, for which resonant inter-
action between wave number triads only occurs to
third order. For capillary-gravity waves, however,
the dispersion relation allows resonant interaction
at second order. Valenzuela and Laing (1972) have
developed a theory for this, and Plant and Wright
(1977) suggest that part of the measured excess
growth rate in the low gravity wave range could be
attributed to capillary-gravity resonant interaction.
Benny (1976) has also shown that under certain
conditions, a long gravity wave may grow in the
presence of small scale capillary waves; the wind
field was not included in his analysis.
The present paper reveals yet another possible
mechanism for the transfer of energy from capillary
to short gravity waves. The theory presented takes
into account the effect of shear flow modulation
on the local growth rate of the capillaries. It
is found that this variation gives rise to a modu-
lation of the Stokes' drift which is in phase with
the long-wave surface slope and therefore makes
possible an energy interchange with the long wave.
It is found that the energy transfer rate due to
this mechanism is positive for capillaries with a
group velocity higher than the phase velocity of
the long wave so that it can provide an increase
in the long-wave growth for waves in the short
gravity wave regime. For waves running against the
wind the transfer rate is found to ke negative, so
that the presence: of the capillaries would always
increase the decay rate of the long waves.
2. INTERACTION BETWEEN LONG AND SHORT WAVES
We shall consider the situation depicted in Figure
1 with two-dimensional surface wave of small wave
length, 4', riding on a large-scale wave of wave
length, A. An asymptotic analysis will be carried
out under the assumption that
Ee = A'/d << 1 (1)
(Prime refers to the short and tilde to the long
waves). The waves are excited by a wind field
blowing over the water surface. Only the normal
stress induced by the wind on the wavy surface is
considered in this process, the effect of shear
stresses being neglected. Of particular interest
is whether the presence of the small-scale waves
could change the growth rate of small-amplitude
long waves.
To arrive at the simplest possible analysis,
terms that are of higher order than linear in the
long-wave slope are neglected. For the short waves,
only quadratic and lower-order terms in the wave
slope are retained. Further, it will be assumed
that the flow in the water is irrotational, i.e.,
the effects of surface drift currents are neglected.
This allows the use of potential-flow theory leading
to the following boundary-value problem for the
velocity potential 6 in deep water:
V4e= ob +6 =0 (2)
xx ZZ
with boundary conditions
Oo eae Oe. (3)
at z=:
PW 1 3/2
Ty ee lee oN bag / EZ)
(4)
at z= -™: @=0 (5)
Here, © = C(x,t) is the surface deflection, P. the
surface pressure due to the wind, and T the surface
tension. Since cubic terms are neglected through-
out, the denominator in the last term of (4) will
be set equal to unity henceforth. We now separate
large and small scales by introducing into the
equations of motion
ES ie oie (6)
6=6 + 6! (7)
Be = Py + Pe (8)
For the boundary conditions it is useful first to
transfer them to the surface of the large-scale
motion, z = t, by a Taylor series expansion. Thus,
O(x,5) = 0 (x,0) + 5'0,,(x,5) +
= O,(x,o) + O' (x,o) ae 1G Y (a sn)
; 3
+ Or (er) stvemete (9)
etc. By neglecting terms involving triple and
higher products one finds from (4) and (5) the
following boundary conditions to be applied at
Zi iGes :
water
FIGURE 1. Long-wave short-wave interactions in a
shear flow.
' ' ' t) ' '
Oy OE S ty PER sb (ee ODS, Ge [ise & EA)
a (10)
ie ' = = ' & ' "(6 oO!
lly Pte) = ils 8 ee) = Ws ee Oey Oe he an ee)
1 b ry 2 6 ry 2
acy (ee NS se (Oe GANS
+ To + Oe) EO OD ((aLgb))
ibs) (Cabal) P, is the surface pressure in the absence of
the short waves and Pw the additional surface pres-
sure added due to the presence of the short waves.
In deriving (10), partial use has been made of (2),
which holds for ® and 6! separately. To arrive
at equations for the long wave, (10) and (11) are
averaged over the large scales. This is most
conveniently done by taking the ensemble average
of a large number of realizations differing only
as to the phase of the short waves, which is assumed
to be randomly distributed among the members of
the ensemble. This procedure yields
Do & Be tb Ole, ER te oc (12)
2 a
We = 2? ss 2 2
= GS > Os Sale = Oe) eae
= a= (E94) = AO = OOF) os GS)
ate
at Z= co where the tilde denotes the average over
the large scales and
aN
Siw a (14)
is the Stokes' drift due to the small-scale motion.
In deriving (13), use has been made of the linearized
boundary conditions for the small scales, for
example,
“—~ SN
O~j0 = Feuyey
5 oe Se tt ee 3
DR ee Pa ao Ce
ae (BIER) = Ber 063 Se y= oa
£
+ (15)
The long waves are to be determined as a solution
of Laplace's equation
v26 = 0 (16)
subject to the boundary conditions (12) and (13)
and the condition that disturbances vanish at large
depths, i.e.,
iBere 14 = = © (17)
The corresponding boundary conditions for the short
waves are obtained by subtracting those for the
long waves from the full equations.
543
'- 6! +77! - 6 6! - 6 oO!
tt) > ie x6 are a Zz
F pope (19)
both to be applied at z = t. The last bracketed
term of (18) and the last two of (19) will give
rise to higher harmonics. Their contribution to
the large-scale motion will be of higher order,
and they can hence be neglected. In deriving (19),
use was made of the linearized boundary conditions
for the long waves. Thus, for example, the term
eel in (19) arises from replacing Ore, in (11) by
Tet, which will give a negligible error to within
the approximation employed.
Since the major aim of the analysis is to deter-
mine the lowest-order effect of the short waves on
the growth rate of the long waves, it is sufficient
to retain only linear terms. However, all terms
linear in the large-scale motion which modulate
the small-scale wave train must be retained. The
long wave will be taken as a uniform, infinite
wave train of wave number k = 21/A. Its phase
velocity differs from the linearized value,
c= vg/k + kT (20)
by terms proportional to the square of the small-
scale wave slope, and by terms due to the wind,
which are proportional to the density ratio between
air and water both of which may be expected to be
small. The short waves driven by the wind may also
give rise to slow growth, or decay, of the large-
scale waves. For the subsequent analysis, it is
convenient to introduce the following nondimensional
"slow" variables:
k(x - ct) (21)
aa!
i
t=ket (22)
The solution for the long wave is sought in the
form (real part always implied)
Ze Bi@e” (23)
ete tkz (24)
The variation of the surface deflection and potential
with the "slow" time, t, allows for the effects of
wind, and the presence of the short waves, to have
a weak influence on the growth rate, and the phase
(and consequently also the phase velocity) of the
long waves. . Without the wind and the short waves
both f and ® would be constants.
For the short-waves, on the other hand, both
the phase velocity and wave number will vary slowly
along the long wave because of the modulation by
the latter. We therefore set
Bo = Lae aye Gr) (25)
et (x,t)
Oo = ou(&,zZ" 70) (26)
544
where 8 is the phase,
ie = Oe (27)
is the wave number,
TS Oe (28)
is the frequency (measured in a fixed coordinate
system), and
z= k"(z = C). (29)
The assumption of a slowly varying wave train allows
one to regard k' and w' as functions of the "slow"
variables, t and &.
An approximate asymptotic solution for the short
waves if sound by expansion in the small quantity
€ = k/k' (30)
(That € thus defined is a slowly varying quantity
causes no special difficulty). Substitution of
(25) - (29) into (18) and (19) and omission of all
terms of order lerle, Jerct |, and higher, as well
as of terms of order e“ and higher, gives the
following boundary conditions for the small-scale
motion to be satisfied at z' = 0:
ef = - itt (c'-a) + cust + ec(Zt - of) + iekt,s
Saricrs (31)
Pwo = - (g + k!2n)c' + ik'(c'-u)é"
p ~ a en a “ n
+ are ee: 5 g) + 2kuTey + kp Te")
Se KO 70 Wa (Gra oe) ] ‘Foo ¢ (32)
where
ce! = w/k' (33)
u and w are the perturbation velocities, 6. and
x
@, respectively, of the large-scale-flow evaluated
at z= C and ist is defined by
So 2 a 80 34)
a S pla
0 0 Py (
The terms neglected as being of higher order in ¢€
include the term Geeon in (19), which expressed in
the slow coordinates becomes
Paiva
ea ra
and is hence negligible compared to the term k'2Tc'.
For the long waves one finds similarly
@, = w= k[e(ct, - it) + 81] +... (35)
Sn SARA yee Rata a
0 15
Sere aa ene
= 2x1 [id + 6} |2 - 05/7] +... (36)
where
ete es
SS emai Gol a CLE) (37)
and the star denotes complex conjugate. The velocity
potential must satisfy Laplace's equation. Substi-
tution of (26)-(29) into (2) gives
se (@U ee al) ct iek'[ ko" + 2k'o}
0m Ss eee
ae ZA" ke Sek Yee oe
+ 0(e26') = 0 (38)
This equation may be solved approximately by series
expansion in €. One finds in a straight-forward
manner that
' 12k;
as ee at See elt ean ae EA 2
0) Go Atl = aie e et Itz, Cp) 1A iezA¢ + 0(e*A) }
(39)
where A = A(E,T) is to be determined by aid of the
kinematic boundary condition (31). By substitution
of (39) into (31) and expanding in powers of € one
finds
S(T iaty Se e(Shet)) = => let = 158) }
+ 0(e20") (40)
Combination of (40) and (32) yields
= =(g + kt? = k'(c'=u) 212" + ick" {fe(ey > ae
+ (et—u)u (ete) (c's 1a) eae
+ 2c(c'-u)o!
+ [(e'-a) (c'=2e + u) + 2k"T)eE } (41)
The induced surface pressure due to the wind may be
assumed to be related to the small-scale surface
deflection in a quasilinear manner that takes into
account the modulation of the wind field by the
long waves. The following expression is chosen:
Pe z Shes
— = k' (eu) (a 2B") (2 = ake)ice
(42)
p
where a' and §' are aerodynamic coefficients (having
the dimension of velocity) giving the in-phase and
out-of-phase components, respectively, of the induced
pressure. The modulation of the wind field due to
the presence of the long waves is accounted for by
the factor (1 - akt). For long waves running with
the wind and having a phase velocity less than the
wind-speed, the air flow at the crest will slow
down in the region below the matched layer where
U= c, and the small-scale growth rate will thus
be reduced in this region. Conversely, the air
speed will increase over the troughs leading to an
increased growth rate there. Hence, the coefficient,-
a, will be positive for such waves. For waves run-
ning against the wind, however, or for waves with
c greater than the wind speed, a will be negative.
To determine the numerical value of a, one must
carry out calculations based on the Orr-Sommerfeld
equation. First the wind field modulation due
to a long wave of small amplitude is calculated.
Then, the pressure on the short waves is computed
on the basis of quasilinear theory, whereafter the
effect of wind field modulations may be extracted
from the results. In Section 3, we derive the
governing equations for the local growth rate for
short waves in the modulated flow of the long waves.
Numerical results for a are presented in Section 4.
Consistency of the two-scale expansion requires
that the wind-induced growth rate is small, which
is indeed the case, since it is proportional to
the air-to-water density ratio. Accordingly, we
shall set, formally,
a’ + i8' = e(a+ iB) (43)
Substituting (42) and (43) into (41) and remembering
that all the quantities involved are real, we find
the following pair of relations:
g + k'?n - k'(c'-u)* + ek'G(c'-u) (1 -akZ) = 0 (44)
[e(c'-u), + (c'-c) (c'-u) + (c'-a) tg
o ial = B(c'-u) (1 - akt)]c
+ Zel(cueu)oe + [(e'=u) (c'=2e + u)
i 2k'T] oe =0 (45)
From (44) it thus follows that
c' =u +t vg/k' + k'T + O(c) = u + v(kK') - O(c)
(46)
Inspection of (35) and (36) reveals that the long
waves receive their growth both directly from wind
pressure and indirectly from interaction with the
short waves, the latter effect being proportional
to the mean-square slope of the short waves. Thus,
since the variation of long-wave parameters with
time is small, little error is incurred by taking
u in (46) to be a function of — alone. Furthermore,
the frequency of the short waves must then be
constant in a coordinate system travelling with the
long waves so that
ie (P= eS) Sw (47)
which, together with (46) determines how the wave
number for the short waves varies along the wave
train. Differentiation of (47) gives
k'u
Ceo
a ma c!-¢c 3)
g
where oe is the group velocity,
eg =k'y. tv tu (49)
and where v(k') is defined by (46). With the aid
of (46)-(49), (45) may thus be written
2(e'-a) [6b + (cg - E)EL] = {(c'-a)B(1 - ake)
- (c'-0) Ge + [(e"=e) vy" + T]
ae /A(Cee) (50)
This equation may be readily solved by integration
along the characteristic line
eS. 8
7 dé/dt, = (cg - c)/e (51)
545
Since only the terms which are linear in the large-
scale perturbation are to be retained, one may
ignore the variation of k' with & when carrying
out this integration.
- =i : E BCE 2
oo a @ exp {5 eo a -~—
c 2 (cg-c) 2(cg-C)
kha hes
+ Bw (o,=8)2 [(c “c)v,, ap abl Jy (52)
where C is a constant to be determined from the
initial value of ¢. By inserting this expression
into (37) one finds
u
S; = - gi2 any == tee ap GU Pit
g g
= = fie? Oey 2 aT 53
(es) k! ee)
g
where
= iL A
gi2 = lee |= (54)
is the mean-square slope of the short waves. (In
Appendix A an alternative derivation, based on
kinematic wave theory, is given.) In the second
bracketed term U may be expressed in terms of f
by the use of the linearized expression
W = kez (55)
Thus, Pe may be written
st= A'T ae BIZ
E (56)
where (ignoring terms which are nonlinear in C)
At = -s'2 as (57)
-12~
oS = {=c" = ec" + 20
Sy
k! ' 4 '
dP (ele) [(c -e) vy + T]} (58)
Gf
The boundary conditions for the long waves may now
be written. Substitution of the solution for the
short waves, and (24), into (35) and (36), gives
6 = eon = Ae) 4b Ne ee inde (59)
=- (g + k2r)z - KE (o_ - id) + O(e2) (60)
For the wave-induced pressure an expression similar
to (42) is used, namely
= ke(& + if)c (61)
Substitution of this and (59) into (60) and separa-
tion into real and imaginary parts yields
0 = [g + k2T - ke? - KE(B + G)IZ - kc aoa + REC
546
(@ - A')t = (26 - B')Z (63)
From (63) we find
18 ns
Spee B-a'
GC = S exp ii 5G=nUn at, (64)
By use of this, ee and Gone may be expressed in
terms of & and an eigenvalue relation obtained by
substitution into (62). This then gives
G = G/RIEEERITD be. (65)
with correction terms proportional to the mean
square slope of the short waves and to the air-to-
water density ratio, both of which are likely to
be small corrections of little importance. The
major result of the analysis is that given by (63),
(64) namely that
aB'c!'
= os LBB <1 2
(2nt) = =e 3 = oP 26 (e!-8) Ss (66)
ae
dt
i.e., the growth of the long-wave amplitude is
given by the sum of the growth due to direct action
of pressures in the manner of Miles (1957, 1962)
and the indirect growth due to the Stokes' transport
by the growing short waves. The second term may be
large compared to the first term, if ci is close
to ¢. However, the analysis presented does not
hold in the immediate neighborhood of cg = ¢ but a
separate (and nonlinear) analysis is then required.
For waves running against the wind, c', cl, anda
will be negative, so that the presence pf the short
waves will always increase the decay rate of the
long wave.
3. THE WIND-INDUCED GROWTH OF SHORT WAVES IN THE
PRESENCE OF LONG WAVES
The perturbation equation governing the modification
of short waves on the wind-water interface by the
long-wave field is derived from the momentum equation
by the procedure used to derive the Orr-Sommerfeld
equation. Additional effects arise because the
short waves see not only the mean wind field, U(z),
but long-wave fluctuations, ti and W. The large-
scale field is governed by a linear equation,
the small-scale field by an equation linear in u',
w' which also contains terms linear in U and w.
As in 2, we take the water to be inviscid and
the flow potential but we consider the air to be
viscous: with no surface current, and continuous
tangential velocity between air and water, this
corresponds to the limit un, > ©, vy > 0 with Us
and v, finite, justified by the large density ratio
between water and air. Both fluids are taken to
be incompressible. :
We begin with the Navier-Stokes equations for
two-dimensional flow in the air
du du du opie ily
Chee ge = ee
rye us + Woe a 6 + = V“u (67)
ow dw dw op 1 ik 9)
DEVEOR Moz = 5 os Ge REG (ee)
where velocities are scaled to free stream velocity
outside the boundary layer over the water, and
lengths scaled to boundary-layer thickness 6 ; R
is the Reynolds number based on 6. (In Section 2,
lengths were scaled to k', the short-scale wave
number) .
To derive the Orr-Sommerfeld equation, these
equations are cross-differentiated and subtracted
to eliminate the pressure. Some use is made of
the continuity equation and the result is
32u o2w
fu 1 y29u
dzot dxdt
- uV2w + wV2u - = = (en = 0
az R 3x
(69)
The flow in the air is taken to be a horizontal
shear flow plus two wave perturbations of disparate
scales: the fast scale, x and t; and the slow
scale, % = ex, and t = et where ec = k/k'. The
variation with z is set by the shear profile and
viscous effects and will be taken to be the same
order for both wave fields. The long wave field
is a function of X,z, and £ only; the short wave
field is a function of x,z, and t and in addition
will be influenced by the long scale waves so that
U(z) + a(X,z,t) + ul (x,z,t;%,t)
ll
u
Wie w(%,z,t) + w! (x,z,t;%,t) (70)
The surface deflection is taken as
BEG Cpe) > GY Cesare)
The major effect of long waves in a parallel
shear flow on the behavior of the short waves will
come from changes in the local growth rate and
convection velocities as well as an unsteady lifting
of the small scale as the large waves pass. There-
fore the small scales will be assumed to be of the
form
w' (x,Z,t:#,t) = wiz-c(z,t)ler °™ = wiz en”
Pb) | Meee ee oO = Gene aay
where z' = z-f and c = c(X,t). Changes in the wave
number, k, are O(cedU/3X); such terms will be ignored
in this local analysis. For this assumed form of
w' and u', the continuity equation becomes
ee + iku' - ¢€ — =) 10) (72)
a ae Ds Bo eee Re
a 1 dw ere Ww i dw wl dw
=-——+ — C~ —G =F = - ST 73
O5 cs 8 apt cs Se ae 3)
since el~ = -w/c. The presence of ia in the assumed
form for w' and u' introduces several terms into
the equation for the small scale. In addition,
the velocity perturbations, U and w, also appear.
The equation for the large scale is obtained by
a phase average of (69) written with the assumed
form (70) and (71). The non-linear coupling of the
small-scale motions will not be included although
the corresponding effects in the water are the
main subject of this paper and are worked out in
Section 2. We anticipate further work to complete
the study of non-linear coupling in both the air
and the water.
The large-scale motions are taken as
> 8 aed) 2 8 Ghee
SPs elk (x ct) em ae) (x-ct) (74)
soo, & A IRGHSE
min Bae ests)
To ease the process of working with products of
wave perturbations, two distinct complex variables
i and j are introduced.
Under these assumptions, the large-scale mo-
tions are governed by the linear homogeneous Orr-
Sommerfeld equation
w'" — 229" + kw - 3kR((U-G) (w" - k2w) - wu") = 0
(75a)
and a is related to a through the continuity
equation
a = j0'/k (75b)
where from now on primes will denote derivatives
with respect to z.
In the equation for the small scale, we will
keep all terms linear in the small-scale perturba-
tions including products of the small-scale and
large-scale perturbations.
When the assumed form for the perturbations (71)
is used in (69) together with the continuity equa-
tion (73) we obtain the following equation for the
small scale
w'" — 2k2W" + k4w - ikR[ (U-c) (w"-k2w)- wU"]
= Riwa-eSe] (w" - k*w') + ikR{(U'Z + di) (w"-k7w)
= Spey Oes GEA ew ORS "a!
ikR (= + U"'S)wt ERs I (2k (U-c) + U"]w
= U(w"! = k2w') fy U"'w ra U' (w" ne k2w')}
= ro (wW,w,z) (76)
where we have introduced the symbol xo (w,Ww,Z) for
the right-hand side of (76). The various terms in
(69) are worked out in Appendix C.
In deriving (76), terms of 0(k2t) have been ig-
nored, however terms such as 32a/a22 in air have been
kept since these can be large in a viscous flow. In
terms of 0(€00/dx) a viscous correction has also been
neglected since all other terms are proportional to R.
We are interested in the local equilibrium and
more specifically the local growth rate of short
waves in the modified wind-water field. Thus in
the assumed form of solution for the short waves,
for a given k the eigenvalue, c, will be a slowly
varying function of space
@ = Gy v Gp (eS) (77)
where Co is the eigenvalue of the short wave field
in the presence of the wind shear field only; c, (x)
will be at most O(z), the amplitude of the long
wave.
Thus the governing equation for w is the Orr-
Sommerfeld equation with additional terms arising
from the long-wave perturbations. Some of these
terms could be obtained directly by replacing U by
U + u in the Orr-Sommerfeld equation; additional
547
terms come from the unsteady lifting and distortion
of the small scale flow by the long waves.
The boundary conditions that are satisfied at
the free water surface, z =~ + c', will now be
derived for both the large and small scale motions.
The first boundary condition is that the tangential
velocity is continuous at the interface, z' = Z',
ac C14
Win Un SM mee > Ww O50
Expanding the velocities from (70 and 71) ina
Taylor series about z' = 0, and keeping terms linear
in the large scale and small scales we obtain for
the large scale
Wie Sh, = wh, at z = 0 (78)
for the small scale [to 0 (kt) ]
Ui at ea ul =a ae oO at z=0 (79)
The term 0U,,/dz = Uy has been ignored in deriving
(79) since it is 0(k*Z) and a*u/dz2 (0) has been
taken to be zero.
Conditions (78 and 79) can be expressed in the
vertical velocity, w' (and w), through use of the
kinematic boundary condition; that the substantial
derivative of the surface displacement function,
S(x,z,t), is zero for both the air and water flow
at the interface, S = 0. That is, if S(x,z,t) =
Zou sthenwDS/DtE = Omatnou——OM(zi rat)
D C) a a
where a we U(Z) ae + w(T) Bp
Expanding the velocity field for both air and
water about z' = 0, and again keeping terms linear
in the large and small scales, we obtain in the
long-wave limit for the large scale
a, WE =O
for the air aE 5 w= 0
for the water O8 L. w=0
at Fi
and for the small scale
for the air
iket - [U't + UjJikz - efu'c' + u'] ae w= 0
at z' =0 (80)
and for the water
ret CU ae oars A Og, 6 :
OS, > Wheall<ie = eu, re Fe = Oat zi a=a0)
From (78) to (80) we see that
ae -
i at z' =0
and W = Wy
From (39) and (40), the velocities in the water at
z' = 0 are related to the displacement & by the
expressions
z : > 1 DE
Wy = ~ik(e -— u) (1 + tesg)o"
Ae ee Ba) Ge aoe Set 81
uy = (c — U,) ( tens (81)
548
“ w u .W
-& ee es
and t ke (@l sp = iz)
where € oe i Z
ox c
thus Uy = iw. as in fixed coordinates.
For the large scale motions,
and 5 (82)
Ae
With (75b) and (82), condition (78) for the large
scale motion becomes
Uwteéw' -wké=0 at z=0 (83)
and with (73) and (81) condition (79) for the small
scale motion [to 0(kt)] becomes
a R = » Bly thy ~
U'w + cw' - wke = - ou -w aul [— + ity
z Zi é
Cn WX ES (84)
= a = 0
iw a The (w) at z
where we have introduced the symbol 19 (w) for the
right hand side of (84).
The remaining boundary condition to be satisfied
at z = t and then transferred to z' = 0, is the
balance of pressure or more precisely of normal
stress with surface tension
as (85)
Ps ~ [nn = Pw Sebel: 9x2
The viscous normal stress at z = t is given by
3
2 Zn
)
where hn = > Wee Gls eS ae aes Spe
Because uwU' (0) HaUd (0) inl the) Damitt uy 2
there is some cancellation in the stress condition
and the final result for the large scale is Spy =
2 w'/R.
For the small scale, all terms involving the
large scale perturbations are negligible for k<<k'
so that Caan = 2 w'/R.
The pressure in the air at the surface, z=,
is obtained by expanding the pressure about z' O.
p(t) = plo) +¢ & (0) (86)
az
where p(o) and dp/dz(o) are available from the
momentum equation (67).
After considerable manipulation we obtain the
following formula for pg - Onn
ikIB, - Opp] = p,[- ¢ G' - wu" + — (w"' - 3k2w')
SSIS nn’ “a Rk
du = 0 ~ du ww! Ow
Cae a ea uae t ie See MaKe OZ
w a, GE k2w w"
= (23 — - 22 - ice 87
z (2ikcw + a 2 = icy )] (87)
We obtain p_ - k2T~ at z' = ¢' [to 0(kZt)] directly
from (41)
By - k2Te = pylk(e - &)? - (g + k*n)]E" (88)
Using (81) to rewrite (88) in terms of w, introduc-
ing s = Pa/p,, and using (84) to rewrite the group,
cw' - U'w, we obtain the final form of the pressure
boundary condition (85) to be satisfied at z' = 0.
SiGe & (Gam = = gx - a wl!
u
= wlke? - (g+k*T)] ( = + iw/é) - w 2ci,
ow uy ay
a 2K D5 piel = 0 (ae iw
+ s(Gw'e + iw'c aa ke COU ian ( z + 5
ae: = = Aa Ze
we Sr Wy a ML “eats Aw k*w) — ~
+ awe =) s z (2ikew + ae 2 A ) = Yoo (w)
We will introduce the variables p,Q,b, and y and
rewrite (89) as Pw + Qw + bw"' =
Yoo (w) where
P = [-k(1-s)c? + (gtk2T)]
Q= ASK b = -isc/Rk
and Y59 (w) is the right hand side of (89).
The corresponding boundary condition for the
large scale is homogeneous and of the form
Bw + Qw' + bw"' = 0 (90)
where
P = [-k(1-s)¢* + (g+k2T)]
Q
jc3sk/R; b= —js/Rk
To summarize, the long waves satisfy the ordinary
Orr-Sommerfeld equation (75) with the appropriate
linear boundary conditions for a free water (83)
and (90) plus w (©) = w' (2) =o. The resulting
homogeneous eigenvalue problem is solved numerically
to determine 4, w, and ¢ for a given long wave
amplitude, a wave number, k, and R.
The short waves satisfy a modified Orr-Sommerfeld
equation, (75) with the effects of the long-wave
perturbations appearing also in the boundary con-
ditions (84) and (89).
To solve this short-wave local-equilibrium
problem we resort to techniques that by now have
become standard in stability theory for perturbed
eigenvalue problems.
We assume that the short-wave solutions can be
expanded about the perturbed solution (no long waves
present) in the form
aw=uU, +24,
w=w, +o w,
BIS ey ee Sp (91)
“where z is the large wave amplitude. The eigenvalue,
c, is also expanded
c=c) +c, (91b)
For & = 0 the problem of short-wave dynamics
reduces to the ordinary Orr-Sommerfeld equation
with free-surface boundary conditions.
Wo" - 2k2we + kw - GkRL (U-c) (w" —k?w9)-woU"] = 0
cows + (= - key) Wo = 0
at 2)" = 0
Pw) + Q wi +b wi! =) (0)
and Wo (*) = wi (~) = 0 (92)
The eigenvalue, cj, which determines the growth
rate of the short waves in the parallel shear flow,
is determined by a numerical solution of these
equations for a given k and R.
The equations governing the modification to the
flow due to the long-wave perturbations are derived
by using (91) in (76) and equating terms of 0(¢).
In this operation the as-yet-unknown correction
to the eigenvalue, c cy, will appear multiplying the
lowest order solution, Wo- The resulting problem
for Wy is written
wy -.2k2wt + kYw, - GkR[(U-cg) (WY -k2W,) —w 0")
= c,r,(w,z) + x, (W,w,z) (93)
where r, (w,z) and rp (w,W,Z) are known functions of
the long-wave perturbation, w(z), and the lowest-
order short-wave perturbation, wy (2). From (84)
and (91)
x, (W,Z) = - ikR(w" - k?wy) (94)
and r»(w,w,z) is defined by (76) with the long
wave perturbations normalized by ¢.
The boundary conditions for W, have homogeneous
operations that are identical to those for w, but
the equations are non-homogeneous with terms that
depend on w_, the long-wave perturbations, w, and
the unknown correction to eigenvalue, c)-
i du A ate
' — = =
cow, + (a kc) Wy Yq c, + Thi (w,w )
at z=0
Bw) iO) wi oe by wh = Ven Ci Va (W,Wo) (95)
and wy (~) = wy (7) = 0
where y (w) and Yoo (w) are defined in equations
(86) and (90) with the long wave perturbations
normalized by f
phe eee
and Yay Wo kw)
= 2c )k(1-s)wy = ik3s/Rw) + is/kRw)
Very,
This problem is similar to that considered by
Stuart (1960) and many others in later studies.
In Stuart (1960) we have the problem
L(w,) = r(z)
with w, (0) = Ww! (0) =w,(@) =w! (=) =0 (96)
where L is the Orr-Sommerfeld operator.
The solvability condition [Ince (1926) pg. 214]
for this problem is
549
where v is the solution to the adjoint problem
L(v) = 0
with v (0) = y"'(~) =0 (98)
i
<
S
Mn
<
ay
Condition (97) is then used to determine the modifi-
cations of the flow due to non-linearities.
The present problem differs from that in (96) -
(98) in that the boundary conditions of (95) involve
linear combinations of the derivatives of Wy at z
= O and are non-homogeneous.
In Appendix B, we show that the adjoint boundary
conditions that replace (98) in the determination
of v are
v" (0) =u" (@) =7 (e) =(0 (99)
and [B-ikRcU' + o(U'-ke)]v + [U'-kc]v" + cv"' = 09
where ey ae
G = - (2k? + ikR(u-e)]ip Bi = =e - 2(u"-ke)
and the extended solvability condition for non-
homogeneous boundary conditions is
co
J xvae = { [c) ry (z)) + Ly (z) ]vaz
0 0
ll
z
a
+
\
dq
S
+ (ye + His On (100)
The solvability condition (100) is then used to
determine ¢, and thus the correction to the local
growth rate due to the presence of the long wave
perturbations.
vi {ood Fa NOEs Grae zs
i eee OE eg EN, ca Js (z) vdz
Ca 2
Vv oO : Q a Vv
eg Oar Beg) 7 (0) I+, Y ¥ (0) - Wes (z) vaz
(101)
After c has been determined from (101), the
normal stress on the small-scale waves in the water
due to the air flow may be determined. The sim-
plest approach is to use (88) and infer Pa = Crm
directly from Poy, using the momentum equation in the
water (or Bernoulli's equation).
Retaining the terms linear in the large-scale
quantities, we have
Day = Cres = 0h kz (c =H > = (gi TkA)cl (102)
where c is given by (91b) with cy from (101). The
correction to the growth rate, Cy, is doubly complex
in that it has both real and imaginary parts (cee
and c,) that are in phase and out of phase with @.
550
To complete the calculation of Section 2 for
long-wave growth rate due to the non-uniformity of
short-wave growth rate, we require the part of c.
that is in phase with ee For the analysis of short
waves, Section 2 uses an expression equivalent to
Duesonry =fosku (cla aa) (user BU) (le ake) cn (42)
where all quantities are real. This assumes that
the real and imaginary part of c are modulated by
the large scale in exactly the same proportions.
Thus by this assumption,
<5 (Ze, - uw)
aky = -2 Gera (103)
since
co = V g/k+kT + ACen eee c' = Cy +U
Should these assumptions not be exactly correct, a
would have a small imaginary part, which will be
ignored.
4. NUMERICAL RESULTS
We have carried out the calculations described in
3 using the Orr-Sommerfeld solver developed by
Gustavsson (1977). This is an implicit method
which uses an Adam's -integration technique. One
particularly attractive feature of the program is
its variable step size. Thus it is possible with
a reasonable number of points to have a fine mesh
in the "wall" layer and other regions of high gra-
dients and to coarsen the mesh as one moves out
into the boundary layer. The programs and results
will be more fully described in a subsequent publi-
cation. Only one set of calculations will be
reported here.
The shear flow profile and its derivatives are
modelled with continuous functions that approximate
the mean profile of a turbulent boundary layer.
Calculations were done at a friction velocity, uT,
of 30 cm/sec; conditions were chosen so that the
ratio, ut/Uw, was .05, a typical value for wind-
tunnel experiments. Interaction between long waves
of 100, 75, 50, 36, 20, and 16.5 cm with short
waves of 2, 1, 0.75, and 0.6 cm were investigated.
Although many interesting features of the flow can
be investigated using this approach (such as the
distortion of the mean profile as the large wave
passes, and the variation of the wave speed, local
‘growth rate, and amplitude of the short waves along
the large waves), the only systematic investigation
we have yet performed concerns the energy input to
the large waves due to the modulation of the short-
wave Stokes drift.
The linear temporal growth rate of wind-driven
waves 2; = ke; is of course a direct output of the
calculations. Figure 2 shows Ma = sec-l as a
function of wave number, k ~ cm-l, for we = 30 cm/
sec. The growth rates we obtained are slightly
higher than Miles's viscous calculations [Miles
(1962)] but when we used his shear-flow profile, we
obtain close agreement. For Wee = 30 cm/sec, all
the waves we investigated were viscously dominated,
that is, their critical layers were sufficiently
close to the free surface to be essentially merged
with the surface viscous layer. Thus, little in-
sight to the behavior of these flows can be obtained
10. 100
= 1S)
1 ®
1S) 2)
® Ss
=
2 rs)
Gg LO 10 2
x =
" o
SG oO
Oui 1 ess ate 1.0
0.I | 10
k~cm!
FIGURE 2. Linear temporal growth rate @ u_ = 30 cm/sec;
1—-—-—-present calculations; —W— present calcu-
lations with miles profile U, = 5uU*.
from an inviscid model of the behavior of shear
flows. The real part of the wave speed, c,, also
shown in Figure 2, differs very little from the
free wave speed of gravity-capillary waves, co.
The energy input to the large waves from the
small waves is given by (66). With t = kct, and
s'2 from (54), the dimensional temporal growth rate
of the large waves can be written
ae
— 1272
ae i S61] (104)
= = aB' ~ c!
= fo [2 1 Tas =
—C}
g
where 2; is the linear growth rate Bk/2.
the coupling coefficient C as
-aB'k
c= 7 (105)
We define
where the minus sign is introduced because, contrary
to our expectations, a turned out to be negative
for the cases we investigated. Thus the growth of
the large-wave amplitude is given by
dé
— ican 2
ae Jkt" | 2] (106)
= Gy lia
Thus for C positive, an energy input to long waves
comes from short waves whose group velocity, c!,
is slightly less than the long wave phase velocity,
c¢. The theory also predicts that long waves will
decay if c} > ¢. Since waves satisfying this con-
dition will be shorter capillary waves which will
be more strongly damped by viscosity, we expect a
net energy input to the large waves. Of course
the theory does not hold at ¢ = c, where non-linear
“interactions must be considered.
Numerical values of the coupling coefficient, C,
are shown in Figure 3 as a function of A for various
A'. C is certainly 0(1) having a maximum value of
3 at 4' = 1 cm. It is also a slowly varying function
of A. It has its maximum value about \’' = 1 cm
which corresponds to the maximum in the linear
growth rate for short waves for these conditions.
It drops off more rapidly with decreasing wave
length, A', than does the linear growth rate, Qa
(A'). The long-wave linear growth rate, Qin is
also shown for comparison; it is much smaller. Of
course the interaction growth rate also involves
(k'c') 2 of the short waves which would be typically
0.01 but the division by cmc would somewhat offset
the effect of small slope. One calculation for an
upstream travelling long wave verified that a was
negative and energy was removed from the long wave
by interaction with the short wave. We have not
carried the calculations further to date.
Some idea of the wavelengths, involved in any
practical application of these ideas can be seen
from Figure 4 which shows the group velocity and
phase velocities for gravity-capillary waves. The
requirement for strong coupling is ¢ x ci. We
further note that waves shorter than say 0.3 cm
are unlikely to be important in a viscous fluid.
Thus short waves in the range 0.3 cm could interact
with a 20 cm long wave in the manner we have dis-
cussed but waves longer than 20 cm would be unlikely
to be affected.
Although the effects of surface drift are not
yet included in our calculations, the range of
affected long waves can be somewhat broadened by
considering surface drift. Drift velocities are
typically 5% of the wind velocity; this is the same
order as the friction velocity which we have taken
as ut = 0.05 U.. If we assume that a surface layer
will advect the short waves [Valuenzuela (1976) ]
but leave the phase velocity of the long waves
unaffected (Valenzuela's calculations did not extend
to long waves), we can consider a broader range of
interaction possibilities, as sketched in Figure 4.
For a group velocity augmented by a surface current
of 30 cm/sec, interactions between a long wave of
about 50 cm and waves longer than 0.3 cm become
possible and a 20 cm wave may interact with waves
of order 1.4 cm.
Experimental data in the range of wave lengths
and friction velocities of interest for the inter-
actions we have investigated here was presented by
Plant and Wright (1977). Some of their results are
reproduced in Figure 5, showing the temporal growth
rate vs. wave number for several values of friction
velocity. Of particular interest is that while the
short-wave growth rate is accurately predicted by
linear theory, there is a departure of theory and
FIGURE 3. Coupling coefficients for long-wave and
short-wave interaction; linear temporal growth rate
Qs. aes 30 cm/sec.
100
\~cm
L4cm
Cg+ 30cm /sec
A Se
LLL)
“10 100
C ~ cm/sec
FIGURE 4. Group velocity and phase velocity for
gravity-capillary waves.
(0) sea enn
O.I 1.0 10
k~cm!
FIGURE 5. Measured temporal growth rates for various
uu. cm/sec; from Plant and Wright (1977).
55a
552
experiment for waves longer than about 10 cm. This
is close to the first possible long wave that can
strongly interact with a short wave whose group
velocity is equal to the long wave phase velocity.
Thus the results we have obtained to date indi-
cate that the long waves can receive energy due to
their interaction with wind driven short waves.
The interaction mechanism we have investigated
requires the presence of the wind and the variation
of the short wave growth rate along the surface of
the long wave due to changes in the local wind
field caused by the passage of the long wave. Of
course further work remains to be done to explore
the full implications of these results, to complete
the calculations and to make fuller comparison with
experiment.
5. ACKNOWLEDGMENTS
This work was supported by the National Science
Foundation under Grant ENG7617265. We acknowledge
many stimulating discussions with our colleague,
Professor E. Mollo-Christensen.
APPENDIX A.
DERIVATION OF STOKES' DRIFT MODULATION FROM
KINEMATIC WAVE THEORY
Kinematic wave theory, modified to allow for small
dissipation or growth due to energy interchange
with the wind, gives the following conservation
equation for the wave action density, A", of the
train of short waves:
dA" 3 -
a= tS UA = 20S A A.l
at ax (a ) ac ( )
where &' is the temporal growth rate. The wave
action density for waves on a current is given by
[Bretherton and Garrett (1968) ]
A' = E'/Q' 2 (A. 2)
where E' is the energy density and 2' = k'(c' - 0)
the frequency relative to the fluid at rest. By
introduction of
Be = ki (eu a) (a |2 = (c' — ws" (A.3)
(A.1) may be cast as a conservation equation for
the Stokes' drift
as'
e) tar =
gee Ucegue
1 dk u ' ok" ' '
gice | Six ) + 2011s
Il
7
fob] Keb)
x lee
ap QV) Se (A.4)
L
With Qe = k'B" (1-akZ) /2 and expressed in the vari-
ables T and & this takes the form
=U 6 2 ey OL Rp ie vi
[c carat ee c) dE ]2n S' = B(1-akz)
u
aus ' eo oe ee eee
Ste oe +c 20 (re) O) Ven te LN)
(A.5)
By neglecting the variation of the left-hand side
with t one finds from this
s! set oe
s (cg-c) (ee -ti)
k!
= Cra) [(c'-e)v,, + T)} (A.6)
which, with S' = (c=) s"2/k", is found to agree
with (53).
APPENDIX B.
THE EXTENDED SOLVABILITY CONDITION
We first determine the adjoint to the homogeneous
problem for a shear flow over a water surface.
This problem is written
L(w) = 0
Wi ew (U" - kc) w' =0O
Wo = Pw + Qw’ + bw"' =
a ow . o at z=0
w(e) = w'(~) = 0 (B.1)
where L is the Orr-Sommerfeld operator.
The adjoint to the Orr-Sommerfeld equation is
[Sturart (1960) ]
L(v) = v"" + ov" - 2ikRU'v' + [k* + ik3R(U-c)]v = 0
o = -2k2-ikR(U-c) (B.2)
From the Lagrange identity [Ince (1926) pp. 210,
214)
J tenes) - wL(v) }dz = P(w,v)
0 0
where P(w,v) is the bilinear concomitant. The
boundary conditions on v that will complete the
statement of the adjoint problem are found by the
requirement that P(w,v) be zero at both end points.
Since w and its derivatives are zero at Zz = o, this
leaves the conditions on v to be found for z = 0.
P(w,v) is written in bilinear form as
(hepa) UA NAL UA | attastail 1
(oa Ww
-o 0-10 w'
@Q al © © w"
= v-[U] -w il 0) 0) © w"' (B.3)
The free surface boundary conditions for w may be
written
v'-ke c (0) 0 w
P Q 0 b w'
w"' = 0
ww" (B.4)
(B.4) is an underdetermined set of equations
that will yield two solution vectors with arbitrary
coefficients. They are not unique and any linear
combination will also be a solution. Two such
solution vectors are
Ww, = {0,0,1,0} and Wy = (-c,U'-kc,0,8) (B.5)
where
B = - [Q(U'-kc)-cP]/b
We now enforce the requirement that P(w,v) be
zero. This requires that certain linear combinations
of v, v', v", v'"' be zero and these are of course
the required adjoint boundary conditions.
Consider the solution vector Wy: For P(w,v) to
be zero
P(w,v) =v °[U] - th 2 wo 0 |
(B.6)
This requires that
0 = © (B.7)
Consider the solution vector We: For P(w,v) to
be zero,
aloIsaoj) ap @(WUae)) ap (8
P(w,v) =v °[U] We = Wo =o
(Y= Tee
Cc
so that (B.8)
[-icRU'k+o(U'-kc) + B]v - cov' + (U'-kce)v"
re Ww = ©
Since v' = 0, this term may be eliminated from this
relationship. Thus given
WwW, (7) 2 ey 4 (! = ina = 0 |
at z=0
W, (w) = Pw + Qw' + bw" = of (B.9)
for P(w,v) to be zero requires
(v) = [-ikcRU' + o(U" - ke) + B]v + (U' - ke)v"
tev = 0) at z= 0 (B.10)
It can be shown that if (B.9) and (B.10) are used
to construct P(w,v), the result is identically zero.
Thus (B.10) are the boundary conditions for the
adjoint problem.
The solvability condition for a problem of the
form [Ince (1926) ]
553
L(w) =r
W, (w) = Va, aly = hyo) (B. 11)
is that
[ vrdz = i Von + 15 Wo yaeh + O0000 (B.12)
where the Von'S are determined such that P(w,v)
= Wy Wyn Wo Worm +... and v is a solution of
the adjoint system.
V, () ="0) i =1,n (B.13)
For the present problem, only Y, and Yo are
non-zero. By standard techniques, we have deter-
mined the additional linear combination of w and
Vv,
W3 (w) = - A
Wy (w) = - w" - ow
V3 (v) = v/b
Wh, (CA) = We a (Ce = OAsxe))a7 (B.14)
such that the bilinear concomitant (B.3) may be
written in the form
P(w,v) = W Vy + W> V3 ta Ws Vo + Wy Vv
1 1
where W, and W2 are the boundary conditions from
(B.1) and V, and V, are the adjoint boundary con-
ditions from (B.10).
Thus the solvability condition for non-homogeneous
problem with non-homogeneous free water boundary
conditions is
i rvdz = Y, Vv, 1 Foy V3 (B.15)
0
where
Wn = We uz (fe = Ore)
at z=0
V3 = v/b
and v is a solution of the adjoint system
L(v) = 0
V, (v(o)) = 0
Vy (v(0)) = 0
v(~) = v'(~) = 0 (B.16)
APPENDIX C.
In this section we give the expressions for the
various terms in (69) for the assumed form of the
small scale (71). The continuity equation (73)
has been used to express u' in terms of w'.
The results are as follows
sin. Oe Oa a Oi OS. dle Dts OS
dtez dtoz d22 k 023 dt k 9023 0x
(Gea)
o2w i Aw Poe eeOWsO at ow!
RO 7 © REDE Gy BEG ae HEE BE ae Meca
Vw = daw Ant alent Be Es + Vw (C.3)
az2 3 ox i
She0 > ya Pose
pvesies Cy Bh, Be oe Otw
IES ea ae ve om fen vo ae Bee
+V2R + U" + UNZ (Cc. 4)
89 _ i atw 32w A, © 3t aw A 33w cs
ie a eee dz2 «k2 O& Oz5 923 Ox
(Va) Un! (C.5)
az
Die ae open) oy cary SR Be
ox > dz Ox dz (Ox
Oo
+ eaE Vw (C.6)
The equation for the small scale will contain
coefficients involving the mean flow expressed as
a function of z'.
U(z")) = U(z) + UN(zZ)) Z
and u"(z") = U"(z) + U"' (z) & (G7)
so that, for example, the term uV2w, with only
linear terms retained, becomes :
2
uV2w = UV2w + [U + U'E + a] a k2w"]
Oz
nee & eon OS, See
+ u'VCw €2ikU aE Oz (C.8)
and
2 ~ i 33w! dw'
v2 = ao at ' mo4 mt aay RCE e ‘Wo
wV*u wi w'[{U (0-7) whe aa3 iky ]
coe (c.9)
of these, v2w and 32a/ax2 will be ignored.
The viscous term is manipulated as follows
1,42 dw 2 du 1 2 ow 2 ou
eye {co Se a oc
RL ax v az! R Ny ox az 1
1 32w! 2 i otw! 32w!
+ - ") - = i
plik G72 Ewe) k dzt 7 fe
tee Oe. 6 Oke | Mh Beh
+ RI3k 5 a2 = Dee ] (C.10)
The fifth derivative is obtained from the Orr-
Sommerfeld equation. Some cancellation occurs
among these terms to yield the final result (76).
REFERENCES
Benny, D. J. (1976). Significant interactions
between small and large scale surface waves.
Studies in App. Math. 55, 93.
Bretherton, F. P., and C.J.R. Garrett (1968).
Wavetrains in inhomogeneous moving media. Proc.
ROU ESOC TAy mS ODO 2a.
Davis, R. E. (1972). On prediction of the turbulent
flow over a wavy boundary. J. Fluid Mech. 52,
287.
Gustavsson, L. H. (1977). Private communication.
Hasselman, K. (1971). On the mass and momentum
transfer between short gravity waves and larger
scale motions. J. Fluid Mech. 50, 189.
Hinze, J. O. (1975). Turbulence, McGraw-Hill,
New York.
Ince, E. L. (1926). Ordinary Differential Equations,
Dover Publications, New York.
Larson, T. R., and J. W. Wright (1975). Wind-
generated gravity-capillary waves: laboratory
measurements of temporal growth rates using
microwave backscatter. J. Fluid Mech. 70, 417.
Longuet-Higgins, M. S. (1969). A nonlinear mech-
anism for the generation of sea waves. Proc.
ROUEE SOC -wrAt ae StL maith
Manton, M. J. (1972). On the generation of sea
waves by a turbulent wind. Boundary Layer
Meterology 2, 348.
Miles, J. W. (1957). On the generation of waves
by shear flow. J. Fluid Mech. 3, 185.
Miles, J. W. (1962). On the generation of surface
waves by shear flow, Part 4. J. Fluid Mech. 13,
433.
Phillips, O. M. (1957). On the generation of waves
by turbulent wind. J. Fluid Mech. 2, 417.
Phillips, O. M. (1960). On the dynamics of unsteady
gravity waves of finite amplitude, Part l. J.
Fluid Mech. 9, 193.
Phillips, O:. M. (1966). The Dynamics of the Upper
Ocean, Cambridge University Press, 79-87.
Plant, W. J. and J. W. Wright (1977). Growth and
equilibrium of short gravity waves in a wind-
wave tank. J. Fluid Mech. 82, 767.
Stuart, J. T. (1960). On the non-linear mechanics
of wave disturbances in stable and unstable
parallel flows, Part 1. J. Fluid Mech. 9, 353.
Townsend, A. A. (1972). Flow in a deep turbulent
boundary layer over a surface distorted by water
waves. J. Fluid Mech. 55, 719.
Valenzuela, G. R. (1976). The growth of gravity-
capillary waves in’a coupled shear flow. J.
Fluid Mech. 76, 229.
Valenzuela, G. R. and M. B. Laing (1972). Nonlinear
energy transfer in gravity-capillary wave spectra,
with applications. J. Fluid Mech. 54, 507.
Valenzuela, G. R. and J. W. Wright (1976). Growth
of waves by modulated wind stress. J. Geophysical
Research 81, 5795.
Preliminary Results of Some
Stereophotographic Sorties Flown Over
the Sea Surface
L. H. HOLTHUIJSEN
Delft University of Technology
The Netherlands
SYNOPSIS
Preliminary results are presented of a study which
is concerned with the directional characteristics
of wind generated waves. The basic approach adopted
was to measure the actual sea surface elevation as a
function of horizontal coordinates by means of stereo-
photogrammetric techniques. The surface representa-
tions thus obtained were Fourier transformed to
estimate two-dimensional wave number spectra.
Basic considerations concerning the photogram-
metrical process, the transformation rules and the
statistical significance of the results are described.
The required stereo photographs were obtained during
photographic missions carried out in 1973 and 1976
off the island of Sylt (Germany) and off the coast
of Holland. So far three two-dimensional spectra,
each from a different flight, have been calculated.
The sea and weather conditions during these flights
are briefly stated. The wind direction in these
flights was off-shore.
Frequency spectra computed from the observed wave
number spectra are compared with an assumed frequency
spectrum and an observed frequency spectrum. The
agreement is reasonable but some discrepancy needs
to be resolved. For two of the three observations
the directional distribution of the wave energy is
strongly asymmetrical around the wind direction.
This asymmetry seems to correspond to asymmetry in
the up-wind coast line.
From the observed spectra a directional spreading
parameter has been computed as a function of wave
number. The results in normalized form agree well
with published data. The absolute values of the
spreading parameter for two spectra are within 30%
of the anticipated values. For the third spectrum
the values were almost five times too large but a
comparison in this case may not be proper. In one
of the spectra some indications of bi-modality around
the wind direction have been observed in the direc-
tional distribution function near the peak of the
spectrum.
555
1. INTRODUCTION
Observations of the two-dimensional spectrum of wind
generated waves are relatively few and are mostly
based on methods with rather poor directional resolu-
tion. The techniques which are used for the observa-
tions may be based on such systems as a sparse wave
gauge array [e.g., Panicker and Borgman (1970)] ora
buoy capable of detecting directional characteristics
of the sea surface [e.g., Longuet-Higgins et al.
(1963)]. The few detailed observations which have
been published were based on other techniques such
as high-frequency radio-wave backscatter [e.g.,
Tyler et al. (1974)], analysis of the sea surface
brightness [e.g., Stilwell (1969), Sugimori (1975) ]
or stereophotography [e.g., Cote et al. (1960)].
These provided information with a high directional
resolution but the analysis of the results in terms
of wave characteristics has not been very extensive.
The Delft University of Technology and the Min-
istry of Public Works in the Netherlands have devel-
oped a system based on stereophotography which
monitors the instantaneous sea surface elevation as
a function of horizontal coordinates. It has been
used in this and other studies and it is anticipated
that it will also be used in future studies of wave
phenomena such as wave transformation in the surf
zone or wave patterns around marine structures. The
present study, which is a joint effort of the Uni-
versity and the Ministry, is aimed at observing and
interpreting two-dimensional spectra of wind gener-
ated waves in a variety of atmospheric conditions.
The study is primarily directed towards the evalu-
ation of the shape characteristics of the directional
energy distribution of the waves.
For this study a few hundred stereo pictures have
been taken since 1973 and the analysis has just be-
gun. The results reported here are preliminary in
that the number of analyzed pictures is only a frac-
tion of the total and in that the interpretation of
these pictures has not as yet been completed. The
spectra which are presented here were calculated
556
from three sets of pictures, each containing ten Cote et al. (1960)] and the present system is es-
stereo pairs. These sets were chosen on two bases. sentially a revised version of the system used in
One is the photographic quality which was judged by SWOP.
photogrammetric experts, the other is the scientific It will suffice here to comment only briefly on
interest. In this stage of the study it was felt the operational system. Actually two independent
that wave fields generated by off-shore winds would systems were built. One is based on Hasselblad
be of most interest because the boundary conditions cameras and has been described in detail elsewhere
are well defined. Also, results of past investiga- [Holthuijsen et al. (1974)]. The other is an almost
tions of wave generation [Hasselmann et al. (1973), exact copy of that system except that the Hasselblad
Hasselmann et al. (1976)] suggests that observations cameras were replaced by UMK cameras of Jenoptik
in these conditions may be extrapolated to more com- which are superior in optical and metrical aspects.
plex conditions. The Hasselblad system was used for observations in
The first set of pictures which was analyzed was the area off Sylt and the UMK system was used in
taken in September 1973 during almost "ideal" off- the area off the coast of Holland. Synchronization
shore wind conditions in the area just west of the of the cameras was achieved by using a radio signal
German island of Sylt. These observations were that triggered a command pulse which was manipulated
carried out in the framework of an international electronically in such a way that it complied with
oceanographic project known as the Joint North Sea the timing characteristics of the receiving camera.
Wave Project (JONSWAP) which is concerned with the The synchronization error for the Hasselblad system
study of wave generation and prediction. A variety was less than 1 ms for all of the analyzed stereo
of articles directly related to JONSWAP has been pairs and for the UMK system the synchronization
published and more are being prepared for publica- error was less than 5 ms. To position the cameras
tion. Some references are: Hasselmann et al. (1973), two Alouette III helicopters were used. These heli-
Spiess (1975), Hasse et al. (1977), and Hiihnerfuss copters had a drop-door over which the cameras could
et al (1978). The two other sets of pictures were be mounted. The distance between the helicopters
taken in March and November 1976 in the area west of was estimated during the flight through a range
Holland near the town of Noordwijk, also in off- finder which was imposed on the viewer of a third
shore wind conditions. Wave observations at sea camera which looked from one helicopter to the other.
level during the first and last flights are avail- It took a picture of the other helicopter every time
able and these have been used for comparison with the downward looking cameras were activiated. From
the stereophotogrammetric results. these photographs the distance between the helicop-
ters could be computed and the scale of photography
could be determined.
2. STEREOPHOTOGRAMMETRY OF THE SEA SURFACE The specification for the helicopter formation
during a photographic sortie were largely based on
When an object is photographed from two slightly photogrammetric requirements. Only the altitude
different positions, the imagery in the two pictures was based on the anticipated sea state since the
will also be slightly different. The differences noise and resolution in the spectrum are directly
depend upon the geometry of the object. By measur- related to the altitude of photography. The upper
ing the differences, the elevation of the surface limit of the altitude was based on noise considera-—
relative to an arbitrary plane of reference can be tions. The standard deviation of the measurement
determined. The conventional technique of analysis error is estimated to be 0.03% of the altitude
requires human interpretation of the pictures and [Holthuijsen et al. (1973)]. Taking a noise to
complicated stereoscopic viewing devices. More ad- signal variance ratio of 1:10 as an acceptable upper
vanced procedures, which have only recently been limit, it can be shown that the altitude should be
developed, use a computer to carry out a correlation less than 1,000 times the standard deviation of the
between the images to arrive at the same results instantaneous sea surface elevation (or 250 times
[e.g., Crawley (1975)]. the significant wave height). The lower limit of
In the conventional geodetic aerial survey the the altitude is directly related to the resolution.
pictures are taken vertically in sequence from an If a resolution in the spectrum is required equiva-
airplane and the interval is chosen such that the lent to % of the peak wave number or better, it
pictures overlap in the area directly under the : appears that for the Hasselblad system the altitude
line of flight. An obvious condition is that the should be higher than 6.7 times the reciprocal of
object does not change between exposures. In land the peak wave number. For the UMK system the fac-
survey this poses no problem since the ground sur- tor is 4.0. For most "young" sea states these upper
face does not move. The sea surface, however, and lower limits are not in conflict. The final
changes very rapidly. To limit the distortions be- choice of the altitude was confined to multiples of
tweeen two successive pictures to an acceptable 250 ft for the pilot's convenience.
level, they should be taken within an interval of The size of the sea surface covered in stereo in
1-5 ms. The airplane cannot possibly fly from one stereo pair is usually too small to produce suf-
one required point of photography to the other within ficient data for a reliable estimate of the two-
this time lapse. The consequence is that not one but . dimensional spectrum. To increase the amount of
two cameras are needed which take the pictures "simul- data more pictures were taken in sequence with a
taneously," that is, within an interval of 1 - 5 ms space interval sufficiently large to ensure photog-
and that two aircraft are needed to position the two raphy of non-overlapping sea areas. The correspond-
cameras. Apart from these technical differences in ing time interval between the exposures would be
obtaining the stereo pairs, the methods and pro- typically between 4 s and 20 s (depending on camera
cedures used in this study are standard in geodetic type, ground speed, and altitude). The photographic
survey and they have been used in the past by various operation to obtain this sequence is called a sortie.
Oceanographic investigators. A well publicized ef- In principle, the pictures can be analyzed with
fort is the Stereo Wave Observation Project [SWOP, recently developed, fully automated processes. The
facilities, however, were not available for the pres-
ent study and the conventional technique was used.
In the three-dimensional space which is reproduced
in the stereoscopic viewing devices a right-handed
system of coordinates was defined with the y-axis
in the direction of flight and the z-axis upward.
During the analysis the sea surface was read at a
square grid with spacing Ax = Ay, which was chosen
such that aliasing in the spectrum would be limited
to only a fraction of the total wave variance. For
each stereo pair the analysis was carried out ina
square field as large as possible and the elevations
were determined relative to an arbitrary plane of
reference. In the subsequent numerical analysis the
linear trend was removed through a least-squares
analysis. The fields obtained from a series of
stereo pairs were initially arbitrary in shape but
fairly close to a rectangle. Later they were clipped
or extended to a square of one common size of Ly.*Ly
as required in the spectral analysis. Sections where
no stereo information was available (mainly in the
areas of extension) were filled with zeros.
3. TRANSFORMATION AND STATISTICAL SIGNIFICANCE
The sea surface data from the stereophotogrammetric
analysis were Fourier transformed to estimate the
two-dimensional wave number spectrum (k-spectrum) .
To inspect the directional characteristics as a
function of wave number, the K-spectrum was trans—
formed to the wave-number, direction space to pro-
duce the k,9-spectrum. The k-spectrum was also
transformed to the frequency domain.
The k-Spectrum
The definition adopted here for the two-dimensional
wavenumber spectrum E(k) is given by Eqs. 1, 2, and
30
re
E(k) = lim < ee > (1)
Aoo
where
Ba > -i2tk:x 12
H(k) = |SS h(x) e ax (2)
>
R
A = Sf dx (3)
>
R
and <> denotes ensemble averaging. Observations of
E is estimate of E(k)
5
a
k-plane
grid in k-plane
E. linearly interpolated between EB and E
Ee linearly interpolated between E
Eo linearly interpolated between E
557,
h(x) were available from the stereo analysis ina
number of square fields and these fields were con-
sidered to be realizations of the ensemble. They
were Fourier transformed with a multi-dimensional,
multi-radix FFT procedure [Singleton (1969)] and
the final estimates were obtained by averaging the
results over the available realizations. The sea
surface data were not tapered and the spectral
estimates were not convolved; consequently the
spectral estimates are "raw" estimates. In analogy
with time series analysis [e.g., Bendat and Piersol
(1971)] the reliability is represented by a y2=
distribution with 2n degrees of freedom, where n
is the number of fields. The resolution denoted
by Ak, Aky is) on the order of (Ly ° Ly) ae
The k,9-Spectrum
ze
The transformation of the k-spectrum to the k,
6-spectrum is formally given by Eq. 4.
nm
E(k,6) = E(k) |J;| (4)
where k = magnitude of ik, 8 = orientation of k and
where the Jacobian Jj = k. Computing the values of
E(k,8) at a regular grid in the k,8-plane requires
the estimation of E(K) at corresponding values of
k. This was done by bi-linear interpolation of
E(K) at the proper values of K (see Figure 1).
The directional resolution can be estimated by
considering the angular distance between two
neighbouring, independent estimates of E(K) ona
circle in the ¥-plane centred in k = 0. On this
circle with arbitrary radius, k, approximately
27Tk/Ak independent estimates of E(K) are available
and the directional increment between these estimates
in radians is Ak/k. This would be a fair approxima-
tion of the directional resolution if all pictures
were oriented in the same direction. But actually
the orientation is a random variable due to the heli-
copter motion during the sortie. The directional
bandwidth to be added will be on the order of twice
the standard deviation (dg) of the helicopter yaw.
The final expression for the directional resolution
(A8) is given in Eq. 5.
A® = Ak/k + 20, (5)
The resolution in k will be on the order of the
increment between estimates of E(k) in the k-plane
which is LZ! = Lo
The reliability of the estimates of E(k,6) can
again be expressed in terms of a y2-distribution but
the number of degrees of freedom is not uniformly dis-
tributed over the k,@-plane. It constitutes an un-
2
3 and E,
5 and EG
+ gridpoint in k,@ plane transformed to
FIGURE 1. Bi-linear interpola-
tion in the k-plane.
558
FIGURE 2. Sites of the field operations. The areas in
the boxes are shown enlarged in Figures 3 and 4.
dulating function due to the fact that the estimated
value of E(k,@) is based on four values of E(k) which
are usually not equally weighted in the given in-
terpolation technique. They are equally weighted
only when a transformed gridpoint in the k,8-plane
coincides with the centre of a mesh in the ¥-plane.
In that case the number of degrees of freedom for
E(k,6) is four times the number of degrees of free-
dom for each individual estimate of E(K). This is
the upper extreme of the undulating function. The
lower extreme occurs when a transformed k,® grid-
point coincides with a gridpoint in the K-plane.
Then the number of degrees of freedom of the esti-
mate of E(k,8) is equal to the number of.degrees of
freedom of an individual estimate of E(k). The
values of the two extremes are 8n and 2n respectively.
The £-Spectrum
The f-spectrum is determined by integrating the f,
8-spectrum over the range (0,7) and multiplying the
result by two. The operation is given by Eq. 6.
T
E(f) = 2 f E(£,6)d0 (6)
0
The £,8-spectrum has been computed from the k-
spectrum. The relationship to transform from wave
number vector to frequency is based on the linear
dispersion relation for deep water corrected for
currents. This expression and the transformation
are given in Eqs. 7, 8, and 9.
L. >>
f = (gk/2m) * + k.V (7)
E(£,0) = E(k)|g.| (8)
By
Jo = Ds (g/2m) 3/2 + vk-! cos (0, - e)17? (9)
a
V is the current vector and V and 8. are its magni-
tude and orientation. To determine the values of
E(k) the same procedure as described above was used.
The resulting spectrum is the frequency spectrum as
Norway
\
Germany |
Great Britain |
Belgium
observed in a point stationary with respect to the
sea bottom. This was done so as to be able to com-—
pare the results with measurements carried out with
anchored buoys. Expressions for the approximate
resolution (Af) and number of degrees of freedom
(N) are given by Eqs. 10 and ll.
> Use el
KE ak Ak = on eis (10)
Phe)
ee ues ab
N= 8 G AK (11)
4. DESCRIPTION OF THE SITES AND THE WEATHER
CONDITIONS
Maps of the areas off Sylt and off Noordwijk and
two bottom profiles are given in Figures 2, 3, 4
and 5. It may be noted that both areas are similar
in general appearance but an important difference
seems to be that the coast near Sylt recedes sharply
North and South of the island and is strongly asym-
metric with respect to the off-shore direction,
25km
+ observation tower
FIGURE 3. The area of observation off Noordwijk. Lo-
cations of observations indicated by dots.
» Denmark
554
7e \ o
= EOE |
FIGURE 4. The area of observation off Sylt. Active
wave monitoring stations and station of observation
indicated by dots. Wind direction indicated by arrow.
whereas the coast near Noordwijk is more continuous
and symmetric. For both sites the water is effec-
tively deep for waves generated by an off-shore wind.
The sortie in the area west of Sylt was carried
out during the field operations of JONSWAP in 1973,
on September 18th, at 17:30 hr (local). Britimmer et
al. (1974) describe the large scale weather features
during the JONSWAP operations of 1973 and also give
results of meteorological observations from ships,
buoys, and balloons in the area. According to this
information the windspeed and direction prior to the
flight had been fairly constant for one day. Since
the wind was almost perfectly off-shore the situa-
tion was classified as an "ideal" generation case.
In the two hours prior to the flight the windspeed
and direction at station 8 (see Figure 4), at 10 m
elevation was approximately 13 m/s and 110° respec-
tively. The direction is only a few degrees off the
"ideal" off-shore direction of 107°.
The weather during this flight was poor for photo-
graphic operations and all pictures which were taken
were under-exposed, in spite of the best possible
photographic measures. Pictures were taken over
six stations of JONSWAP, including active wave mon-
itoring stations 5, 7 and 9 (see Figure 4). The
frequency spectra observed at these stations are
given in Figure 6 and they may be used for a direct
comparison with the results of stereo observations
over these stations. But in selecting the pictures
for preliminary investigation preference was given
to photographic quality rather than availability of
ground-true information and it appeared that the best
pictures were taken over station 10, which was other-
wise inactive during the flight.
TTT, ON
———— Noordwijk
FIGURE 5. Bottom profiles off Sylt (direction 287°)
and off Noordwijk (direction 300°).
559
The frequency spectrum at station 10 was estimated
with a "hindcast" procedure based on the JONSWAP pa-
rameter relationships [Hasselmann et al. (1973)].
The "hindcast" was attempted for stations 5, 7, and
9 with the observed windspeed of 13 m/s but the re-
sults (Figure 6) were rather poor, although they
seemed consistent with the statistical variation in
the observations of JONSWAP. The agreement improved
when a windspeed of 15 m/s was used (Figure 7). This
was the windspeed estimated just prior to the flight.
Since this fictitious windspeed produced more real-
istic results, in particular for station 9 which was
the nearest to station 10, it was used for the "hind-
cast" at station 10. The resulting spectrum is given
in Figure 8, the comparison with the stereophoto-
graphic results will be discussed in Section 5.
The second and third set of pictures to be ana-
lyzed were chosen from the pictures obtained in the
area off Noordwijk. The main reason for selecting
these pictures rather than the pictures taken off
Sylt was that the results of the sortie just de-
scribed indicated that the data were influenced by
the asymmetry of the coastline of Sylt. The coast
near Noordwijk is more symmetric for off-shore wind
directions. The information on the atmospheric con-
ditions during these flights was based on standard
synoptical observations which were received through
the office of the Royal Netherlands Meteorological
Institute. In addition a cup-anemometer and a wind-
cone were available at an observation tower located
9.5 km off-shore from Noordwijk (see Figure 3).
The second sortie (the sequence refers to the
sequence of analysis, not the time sequence of the
flights) was flown in off-shore wind conditions on
November 12, 1976, at 13:05 hr (local). From the
synoptical observations it was found that the wind
was rather weak over the entire North Sea and the
wind in the area of observation was mainly caused
Sylt 730918
ast JONSWAP spectrum _—
observed spectrum
17:51 start of record (duration 25 min.)
N stat.9
30+ 17:25hr
257
energy density (m*/Hz)
0.30 0.35 0.40
010 015 0.20
FIGURE 6. Observed frequency spectra at stations 5, 7,
and 9 and corresponding JONSWAP spectra for U = 13 m/s.
560
Sylt 730918
35}
stat.9 JONSWAP spectrum
30 | 17.25 hr
r | observed spectrum
| \ 17:51 start of record (duration 25 min.)
20
wn
i=)
spectral density ( m?/Hz)
os
Qo ——,
0.10 a15 020 025 030 035 040
frequency (Hz)
FIGURE 7. Observed frequency spectra at stations 5, 7,
and 9 and corresponding JONSWAP spectra for U = 15 m/s.
by a weak and fairly large low pressure area over
central France. Synoptical observations in the
coastal region 25 km North and 8 km South of Noord-
wijk indicated windspeeds of 4.5 m/s and 4.0 m/s
respectively and the wind directions of 100° and
160° respectively. The wind observation at the
platform was carried out at 23 m above mean sea
level. Averaged over the duration of the photo-
graphic operations (about 40 min.), the observed
windspeed was 6.4 m/s and the directions just prior
and just after the flight were approximately 140°.
The “ideal" off-shore direction would have been 120°.
To estimate the windspeed at 10 m elevation, the
observed value was corrected. The correction for
the bulk of the tower, is known from wind-tunnel
5.0
Sylt 730918
JONSWAP spectrum ------
Observed spectrum
40
(from stereo dota)
w
o
spectral density (m?/Hz)
Nn
i=)
is)
01 015 0.20 0.25 030 a35 f (Hz)
FIGURE 8. Spectrum inferred from stereo data and cor-
responding JONSWAP spectrum for U = 15 m/s.
tests, and the windspeed was extrapolated using a
logarithmic wind-profile with a drag coefficient,
cj9 = 1-5 x 10-3. The resulting windspeed is 6.0
m/s. The corrections for the wind direction are
marginal and well within the error of observation.
During this flight pictures were taken over the
observation tower and at locations 30 km and 50 km
from the coast (see Figure 3). The pictures taken
30 km off-shore seemed to contain sufficient stereo
information to obtain a relatively high directional
resolution and these were chosen for preliminary
investigation.
Wave observations at sea level were available
from a wave gauge at the observation tower and from
an accelerometer buoy at the location 30 km off-
shore. The spectrum of the buoy is given in Figure
9. It will be used for comparison with the stereo-
photographic results. During the flight some swell
coming from south-westerly directions was observed
from the helicopters.
The third sortie was flown off Noordwijk on 23
March 1976 at 12:20 hr (local). The wind was rather
weak over the entire North Sea and the direction
varied from ENE off the Dutch coast to SSW off the
Norwegian coast. This windfield was caused mainly
by a fairly weak high pressure ridge over the North
Sea and a low pressure area over central France.
Synoptical observations at the same coastal stations
as mentioned above indicated windspeeds of 11.0 m/s
and 8.0 m/s respectively and wind directions of 80°
and 70° respectively. The corrected wind speed and
direction at the observation tower (averaged over
20 min.) were 8.3 m/s and 70°. Since the "ideal"
off-shore wind direction would have been 120° the
wind is slanting across the coast line at an angle
of approximately 50°. Obviously this implies a
strong asymmetry of the coast line with respect to
the wind direction. Pictures were taken over the
observation tower and at locations 17 km and 30 km
off-shore. Since the pictures taken 17 km off-shore
seemed to be the best, they were analyzed. Unfor-
tunately no simultaneous wave observations in the
area were available.
0.50 Noordwijk 761112
buoy spectrum = = ———-—-
observed spectrum
0.40 (from stereo data)
030-
spectral density(m*/Hz)
010 0.20 030 0.40 0.50
frequency (Hz)
FIGURE 9. Spectrum inferred from stereo data and
spectrum from buoy measurement.
561
TABLE 1 Photogrammetric Sylt Noordwijk | Noordwijk
parameters Sept. 1973 | Nov. 1976 March 1976
altitude of photography 1500 ft
orientation of helicopters
elative to true North
percentage of zeros added
in stereo areas
umber of pictures
accepted for stereo analysis
2
stereo area per picture 220x220 m™ 156x156 ma 170x170 an”
es ey 2 2 2.
grid in x-plane 5), 33 5) in 3} big Sh am Dose~aos) iu
5. RESULTS resolution, Og was estimated at 0.06 [c-.f.,
Holthiujsen et al. (1974) ].
The values of a number of parameters relevant to the On closer inspection of the contour-line plot of
photogrammetric process are given in Table 1. In the spectrum of Sylt two wave fields can be identi-
view of the preceding paragraphs this table is fied: one coming from approximately 110° and one
largely self-explanatory but a few parameters will from approximately 155°. This is rather surprising
be discussed briefly. because neither the wind conditions nor the ground-
The altitudes of photography are based on antic-— true information gave such indication. The swell
ipated significant wave heights and peak wave num- in the second spectrum (off Noordwijk) coming from
bers. These were estimated by substituting the south-westerly directions was observed during the
windspeed and fetch in the JONSWAP parameter rela- flight. It is well separated from the locally
tionships [Hasselmann et al. (1973)]. For the sortie generated wind sea and it will be largely ignored
off Sylt the wind information was fairly good as it in the following discussion. The peak of the third
was based on ship observations in the area but for spectrum is, surprisingly, coming from Northerly
the sorties off Noordwijk this information was poorer, directions rather than from Easterly directions, as
partly because no observations prior to the flights may be antitipated from the wind direction.
were available. The helicopters were flying directly Instead of the k, -spectra, the normalized direc-—
into the wind during the sortie off Sylt. During tional distribution functions have been plotted in
the second and third sortie they were flying with Figures 13, 14 and 15. The definition of these
the wind in the left respectively right rear quarter functions is given by Eqs. 12 and 13.
with 11° drift. Ten stereo pairs were taken in each
sortie but one pair was rejected from the set taken E(k,8)
off Sylt because it covered too small an area. Us— DO BES) ° FOO ies (te)
ing the sea surface information from the stereophoto- Halen eek
grammetric analysis the three K&-spectra were computed
according to the procedures described in Section 3.
The results are presented in the form of countour- D(8;k) = 0 mone Wy SO) Ss Ay (13)
line plots in Figures 10, 11, and 12. Some isolated
regions in the k-plane have been indicated where the This seemed to be more illustrative than a contour-
spectra are thought to be seriously affected by line plot of the k,§-spectra, the normalized direc-—
noise. This noise is dealt with in the Appendix. primarily for the directional characteristics. An
Values of relevant spectral parameters are given in evaluation of these functions will be given in the
Table 2. For the determination of the directional next paragraph.
TABLE 2 Spectral
parameters
Sylt
Sept. 1973
Noordwijk | Noordwijk
Nov. 1976 | March, 1976
]
resolution in k-plane imal (220x220) 1 (156x156) (170x170)!
number of degrees of 20
freedom
1
] 0.0641 &*)
peak wave number (Gc) [m
directional resolution
atke=s ok
k = 2k"
k = 3k”
m
as
10
562
FIGURE 10. Contour-line plot of K-spectrum off Sylt,
Sept. 18th, 1973. Contour-line interval equivalent
to factor 2. Minor variations are dashed, shaded
areas seriously affected by noise. Orientation of
positive ky-axis 110°, k_-axis 200° from true North.
Wind direction 110°.
FIGURE 11. Contour-line plot of K-spectrum off
Noordwijk, Nov. 12th, 1976. Contour-line interval
equivalent to factor 2. Minor variations are dashed,
shaded areas seriously affected by noise. Orientation
of ky-axis 275°, negative k -axis 185° from true
North. Wind direction 140°.
FIGURE 12. Contour-line plot of k-spectrum off
Noordwijk, March 23rd, 1976. Contour-line interval
equivalent to factor 2. Minor variations are dashed,
shaded area seriously affected by noise. Orientation
of ky-axis 310°, k,-axis 40° from true North. Wind
direction 70°.
mt
is
Ber
B)e
els Sylt 730918
5 ae
(J resolution area Aky=Ak,4220m) 3 5
0.5 spectral density(m4) 0
. 1 10
ee 4
—_—_,
Q03 Q04 Q05 Q06 007 Q08 008 0.10
001 002
wavenumber component ky[m7"]
North {i
Wind
al Noordwijk 761112
een es
ol
v4
A ot
C] resolution area AK, =AKy=(156m) ©
0S spectral density (m4) F a
3|E a
3/8 = G
‘F015 =
Li
002 006 006 g08 O10 O12 014 016
wavenumber component k[m"]
North Wind
Ss Noordwijk 760323
heh Ot
re
oe
O resolution area AKx=AKy=(170m)~! e|s
05 spectral density (m4) i 8
Oren:
FD) g BS
01 015 02
wavenumber component - ky [m? ]
North
1.0
SCALE
20° 110° 200° k =1.90 km
k =0.71km k=2.14km
k =0.95km k =238km
k=119 km k=2.62km
k =1.43 km k =2.86 Km
k =157km k=3.10 km
KeTetSSixt Om min,
The f-spectrum has been computed from the ie
spectrum according to the procedures described in
Section 3. The result for the spectrum off Sylt
is given in Figure 8 along with the corresponding
JONSWAP spectrum. The resolution is about 0.02 Hz
near the peakfrequency, which is 0.165 Hz, and 0.01
Hz at twice the peak frequency. The number of de-
grees of freedom for frequencies greater than 0.13
Hz is 250 or more. Considering the scatter in the
original data set of JONSWAP and taking into account
the resolution, it is concluded that the agreement
between the two spectra is fair.
The frequency spectrum computed from the observed
k-spectrum of the second sortie is plotted in Fig-
ure 9 along with the frequency spectrum of the buoy.
The resolution of the spectrum based on the stereo
data is on the order of 0.02 Hz near the peak of
the swell and 0.015 Hz near the peak of the locally
generated wind sea. The number of degrees of free-
dom is 125 or more for frequencies greater than
0.10 Hz. For the spectrum of the buoy the resolu-
tion is about 0.02 Hz and the number of degrees of
freedom is about 48.
The spectrum based on the stereo data seems to
be shifted in energy density. This may have been
caused by noise and to appreciate this influence
the R-spectrum was corrected. The noise was assumed
563
Sylt 730918
k =3.33 Km
k =3.57km
k =3.80 km
k=4.28km
FIGURE 13. Normalized directional
distribution functions of the k-
spectrum off Sylt, Sept. 18th,
1973. Directions are relative to
true North.
to be uniformly distributed over the K-plane and the
variance was estimated at 0.002 m2 (based on the
anticipated measurement error of 0.03% of the alti-
tude of photography, see Section 2). Accordingly
a uniform noise level of 0.018 m* was subtracted
from the ¥-spectrum and the transformation was
carried out again. The differences were marginal
compared with the earlier results and the shift
cannot be explained with the anticipated noise uni-
formly distributed over the K-plane. Further in-
vestigation is needed to resolve the remaining
discrepancy.
The frequency spectrum of the third sortie is
given in Figure 16 but no attempt has been made
to compare this spectrum with a "hindcasted" spec—
trum because the relatively simple relationships
for off-shore wind situations cannot be applied.
6. DISCUSSION OF THE RESULTS
In the area off Sylt, where the wind was almost
perfectly off-shore and fairly homogeneous and
stationary, one would expect to find a frequency
spectrum with a shape similar to the shape found
earlier in JONSWAP. Finding a JONSWAP-type spectrum
in the conditions off Noordwijk seems to be less
564
k=0.2 km
k=0.3km
k=0.4 km
k=0.7 kp,
FIGURE 14. Normalized directional distribution
functions of the k-spectrum off Noordwijk,
Nov. 12th, 1976. Directions are relative to
true North. The peak wave number kn is related
to the locally generated wind sea.
likely because the differences between the wind ob-
servations at the coast and at the tower are fairly
large and the wind may have varied between the point
of observation and the coast. In particular for the
slanting wind conditions it is obvious that a JONSWAP-—
type spectrum would not be found, due to the asym-
metry in the coastline around the wind direction. On
the other hand, non-linear interactions in the spec—
trum may produce a JONSWAP-type spectrum, in spite
of the asymmetry and the variations in the windfield
[Hasselmann, et al. (1976)]. From an inspection of
Figure 8 it can be concluded that the frequency
spectrum in the sortie off Sylt is indeed JONSWAP-
like. The correspondence of the frequency spectra
off Noordwijk with a JONSWAP-type spectrum has not
yet been investigated.
For the k-spectra of the first two sorties one
would expect to find directional distribution func-
tions having some kind of standard shape, symmetrical
about the mean direction although some skewness may
be expected in the observation off Noordwijk because
the wind direction was not perfectly off-shore. For
SCALE
Noordwijk 761112
oo
=
3
=
W
is)
=
3
=
i
nN
=
3
k=26 km
km=6.41x 10-2 m_~!
k=18k rp
the third spectrum strong skewness may be anticipated
due to the slanting position of the coastline.
These expectations seem to be far from reality
in the k-spectrum off Sylt. The directional distri-
bution near the peak of the spectrum (see Figure 13)
is distinctly asymmetric with respect to the wind
direction with the highest peak at + 45° off the
wind direction (155° from true North). It is highly
improbable that the wave generation mechanism would
build a directional distribution as strongly asym-
metrical as this. An explanation for this unexpected
observation can perhaps be found through a detailed
study of the wind and wave fields, possibly using
"hindcasting" procedures. But in the context of this
paper one can only speculate on some possible causes.
The source function is symmetrical, as is the radia-
tive energy transfer, since bottom and current re-
fraction is virtually non-existent. It seems then
that the asymmetry stems from asymmetry in the wind
field or in the boundary conditions. As for the
wind field, a cursory inspection of the large scale
weather maps revealed no asymmetry. As for the
boundary conditions, the coast of Sylt, rather than
the main-land coast was deemed to be relevant as up-
wind boundary. This was based on the expectation
that the wave energy is propagating in a narrow
angular sector around the wind direction [e.g.,
Hasselman et al, (1973)] and since the coast of
Sylt is rather symmetric it should not cause asym—
metry in the wave field. But the coast to the North
and South of Sylt is strongly asymmetric. In fact,
the distance to shore in the direction of 155° (the
direction of the highest peak) is almost 2.5 times
the distance to shore in the direction of 65° (the
"symmetrical" direction, see Figure 4). If this
asymmetry in the windward boundary is indeed the
cause, then it seems that the "ideal" generation
cases of JONSWAP may be contaminated to some degree
by asymmetric boundary conditions. Still, relating
this conclusion to the observed K-spectrum is largely
speculative as long as it is not substantiated with
more data. In particular the shapes of the k-
1.07
SCALE / ;
———————er
310° 40° 130° k=2.00km
eee ey
k= 022 km k=2.22km
a
k=0.44 km k=2.44Km
| ele
ed (Be
k=0.67k 4 k=2.67 km
k=1.56km k=355 km
ale Bea
k= 1.78 km k=3.78 kp
FIGURE 15.
565
spectra at locations closer to shore may give some
clues.
The expectations regarding the directional dis-
tributions for the locally generated wind sea off
Noordwijk in the second sortie seem to be more
realistic, at least in an overall sense (Figure 14,
for k > k,). Any skewness is hard to identify
through visual inspection of the plots due to the
small scale variations in the functions. These
probably stem from the statistical variability of
the estimates. The swell peak (k = 0.3 con
0.6 k,) is unimodal and covers a narrow angular
sector with a half power width of about 35°.
The directional distribution functions of the
spectrum in the third sortie seem to be strongly
skewed for the lower wave numbers (Figure 15,
mS 2 kp, Say) but for higher wave numbers skewness
is hard to identify visually. As for the main di-
rection of the energy distribution, it varies almost
monotonously from approximately 80° at higher wave
Noordwijk 760323
=s00K =a ean eae k=600 Km
I ll
AN aie PIS AEN
eos = Nihewrsy
k=4.44 km k=6.44km
ee weir
(he
k=489 km
k=5.11 km k=711 km
ab
[jira ee eee
k=533m k=733 km
k=7.55km
k= 5:88 x1073m_!
k=5.78 km
as
Normalized directional distribution functions of the k-spectrum off Noordwijk,
March 23rd, 1976. Directions are relative to true North.
566
numbers to about O° for the lowest wave numbers (see
also Figure 17). The energy of the higher wave num-
bers travels more or less in the wind direction but
the main direction of the peak of the spectrum ap-
pears to be about 10° relative to true North; that
is about 60° from the wind direction and almost
parallel to the coast. This seems to be the most
remarkable feature of this spectrum as one would
expect to find a uniform main direction of 70°, con-
sidering the wind direction and the effects of non-
linear interactions [Hasselmann et al. (1976)].
Again, as with the spectrum off Sylt, it is felt
that the observed phenomenon is due to the asym-
metry of the coastline around the wind direction.
To substantiate this preliminary conclusion
qualitatively, a simplified "hindcasting" model was
implemented for homogeneous, stationary wind fields,
arbitrary coastlines, and deep water. In this model,
which is basically the same as suggested by Seymour
(1977), the wave components from different direc-—
tions are decoupled. In this version the parameter
relationships from JONSWAP [Hasselmann et al. (1973) ]
were taken and the suggestions of Mitsuyasu et al.
(1975) were used for the directional distribution
function. When applied to the situation of the
first and third sortie it did produce two-dimensional
£,6-spectra which at least qualitatively agreed with
the so far unexpected main directions in the observed
k-spectra.
This seems to be.in contradiction with the con-
clusions of Hasselmann et al. (1976) that the shape
of the spectrum is fairly insensitive to variations
in the wind field due to the non-linear interactions
in the spectrum. It should be noted however that
the distance to the coast, in terms of wave lengths,
seems to be rather short for the lower wave numbers
in the two spectra so that non-linear interactions
May not have been sufficiently effective to over-
0.40
Noordwijk 760323
frequency spectrum
| from stereo data
0.30
spectral density (m*/Hz)
0.10 0.20 0.30 0.40 050
frequency (Hz)
FIGURE 16. Spectrum inferred from stereo data of ob-
servation off Noordwijk, March 23rd, 1976.
+ Noordwijk, 761112, Ak=1/156m~"
1
8
270 © Sylt, 730918, Ak = 1/220 m7!
rm
26a 8 Noordwijk, 760323, Ak=1/170 m7
210 F net
180
0 - —e — Ll Uhre ts =}
LS 6 NT 8 IO ee 2618 ez O Nn 22 226 26 SONS AES
k=mAk
39
a
FIGURE 17. The mean direction of the waves relative
to true North, as function of wave number.
come the influence of the geometry of the coastline.
For the higher wave numbers the distance to shore
is relatively long and the non-linear interactions
may have produced the observed directional distribu-
tion functions which indeed seem to be hardly af-
fected by the asymmetry of the coastline. The ob-
servations therefore may still be consistent with
the theory of non-linear interactions and the con-
clusions of Hasselmann et al. (1976) if the relevant
space and time scales are considered.
In an "ideal" generation case the directional
distribution of the wave energy is often approximated
with a simple unimodal function. The observed situa-
tions are distinctly multi-modal, but one such func-—
tion, given in Eq. 14, has been fitted to the data.
This was done mainly to compare the results with the
published data.
lL WG = A) a= 8
Di(e) = Se We ay cos 2s (Come) (14)
In this expression s is the spreading parameter and
8m is the mean direction, both of which may vary
with k. The values of 8,, and s have been computed
using a least-squares technique. The results for
8m as a function of wave number are given in Figure
15. Noise in the spectra (see Appendix) did influ-
ence these results and outliers had to be identified.
As a criterion for acceptation, the rate of change
of 6, along the wave number axis has been chosen.
An accepted value of Om should be within 30° of its
neighboring values on the wave number axis.* This
is equivalent to a rate of change of approximately
0.0024 m for the first sortie, 0.0033 m for the
second sortie, and 0.0031 m for the third sortie.
This allows for slow but significant variations in
6m which is required, for instance, in the spectrum
of the third sortie. The resulting set of accepted
values of 6,, is also indicated in Figure 17. The
values of s at the corresponding values of the wave
number have been plotted in Figure 18 in a format
*
The value of 30° was chosen arbitrarily.
suitable for a comparison with data published by
Mitsuyasu et al. (1975).
Mitsuyasu et al. (1975) presented results of a
number of measurements (five) which were carried
out with a cloverleaf buoy at several locations
around the Japanese islands. The observed wave
fields were generated by various types of wind
fields, including on-shore and off-shore winds. It
appears from the ratio of the wind speed to the phase
speed of the peak frequency of these observations,
that the state of development of the wave fields was
rather advanced (the ratios ranging from 0.75 to
1.25). Based on the observed values of s, relation-
ships in the frequency domain were suggested. The
relevant expressions have been transformed here to
the wave number domain to produce Eqs. 15 and 16.
s = k7}-25 for k 2
S| = k259 for k <1 (15)
= 268)
s 1.5 (u/c) (16)
where § = S/S and k = k/ky, a is the maximum value
of s, ky is the peak wave number, cm is the phase
speed of the peak wave number, and U is the wind
speed. The data of Mitsuyasu et al. (1975) are
probably obtained in situations where tidal currents
were negligible and in the above transformation the
deep water linear relationship between frequency and
wave number was used.
Equations 15 and 16 are also plotted in Figure 18
and the agreement is fair, the scatter being on the
same order of magnitude as the scatter in the data
of Mitsuyasu et al. (1975). The values of sp com-
puted from the stereo data are 6.0 for the spectrum
off Sylt, 5.0 for the first spectrum off Noordwijk.
These are also in fair agreement with the values
suggested by Mitsuyasu et al. (1975) which are 4.6
and 6.1 respectively. However, for the second spec-—
trum off Noordwijk the observed value of s is 27.4
whereas the value following from expression 16 is
5.9. This is a very large discrepancy which is
possibly due to the rather extreme asymmetry of the
coastline around the wind direction where the sug-
© Sylt 730518
A Noordwijk 760323
+ Noordwijk 761112
Sf=15)/Sim
R= K/Kym
(74
OIF
FIGURE 18. The normalized spreading parameter as a
function of the normalized wave number.
567
gestions of Mitsuyasu et al (1975) may not be ap-
plicable.
The above discussion concerned rather overall-
characteristics of the directional distributions.
It is planned to investigate these functions more
in detail. For instance, in the &-spectrum off
Sylt one aspect which will require closer study is
the shape of the directional distribution near the
peak of the spectrum in a sector around the wind
direction. Two peaks at + and - 15° relative to
the wind direction can be identified and this phe-
nomenon seems to be "real" in the sense that the
directional resolution seems sufficiently high (20°)
to resolve these peaks in terms of statistical sig-
nificance. The resonance theory of Phillips (1957)
predicts a bimodal distribution for frequencies in
the initial stage of development, but the components
around the peak have passed that stage and there is
no relation with this theory. More relevant seem
to be the theory and calculations of Hasselmann
(1963), Longuet-Higgins (1976), and Fox (1976) which
produce a non-linear energy transfer in wave number
space with two lobes towards the lower wave numbers
and two lobes towards the higher wave numbers. Fox
(1976) noted that this function resembles a "butter-
fly." Also the results of Tyler et al. (1976), who
observed directional distributions of wind generated
waves with high-frequency radio-wave backscatter,
may be of interest since some of the distributions
have a bimodal character around the mean direction.
7. CONCLUSIONS
Three, two-dimensional, wave number spectra have
been computed from stereophotographic data obtained
in off-shore wind conditions. The agreement with
ground-true information is reasonable but some dis-
crepancy needs to be resolved.
The directional distribution of the wave energy
near the peak of the first spectrum is strongly
asymmetric. In the third spectrum the main direc-—
tion of the waves differs appreciably from the wind
direction. It is speculated that these phenomena
are due to asymmetry in the up-wind coastline. The
directional distribution functions of the second
spectrum are more symmetric and unimodal, at least
in an overall sense.
A bimodality in a sector around the wind direction
is observed near the peak of the first spectrum.
This bimodality may be related to a multi-modal non-
linear interaction in the spectrum.
The observed normalized directional spreading
parameter as function of a normalized wave number
is in fair agreement with published data. The ab-
solute values are about 30% larger for the first
spectrum and about 20% lower for the second spectrum.
The values for the third spectrum are almost five
times too large. This may be due to the rather
extreme asymmetry of the coastline where a compari-
son with the published data may not be proper.
The results reported herein are preliminary.
Additional analysis of available data is being
carried out.
ACKNOWLEDGMENTS
The helicopters were provided by the Royal Nether-
lands Air Force and they were flown by the Search
and Rescue team of Soesterberg airbase (the Nether—
568
lands). This is gratefully acknowledged. Consider-
able support in terms of logistics, groundtruth
data, meteorological observations, etc. was received
from colleagues in the framework of JONSWAP and this
is greatly appreciated.
NOTATION
A area of spatial integration
Cm phase speed of component fp
Cg group velocity
D(8) standard directional distribution function
E(k) spectral density in k-space
spectral density in k,8-space
spectral density in f,8-space
E(£) spectral density in f-space
£ frequency
g acceleration due to gravity
ES instantaneous surface elevation
H (k) Fourier transform of surface elevation
J Jacobian 3)
k wavenumber vector k = (kx, Ky)
k wavenumber, modulus of wavenumber vector
km wavenumber at peak of wavenumber vector
spectrum of locally generated wind sea
dimension of area of analysis in x-direction
dimension of area of analysis in y-direction
number of degrees of freedom
number of transformations
boundary of spatial integration
directional spreading parameter
maximum value of s
dimensionless spreading parameters s/sp
windspeed at 10 m elevation
tidal current vector
magnitude of V
place vector x= (x,y)
ASP EB spatial coordinates
increment
direction, orientation of wavenumber vector
orientation of tidal current
mean direction
standard deviation of helicopter yaw
i
*
2 Q YS et J
TOEPXK ¥I am
QD
=]
REFERENCES
Bendat, J. S., and A. G. Piersol (1971). Random
Data: Analysis and Measurement Procedures,
Wiley-Interscience, New York.
Brummer, B., D. Heinrich, L. Kriigermeyer, and D.
Prim (1974). The Large-Scale Weather Features
over the North Sea during the JONSWAP II Experi-
ment. Berichte des Instituts fur Radiometeor-
ologie und Maritime Meteorologie, Universitat
Hamburg, Institut der Frauenhofer Gesellschaft,
24.
Cote, L. J., J. O. Davis, W. Marks, R. J. McCough,
E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephen-
son, and R. Vetter (1960). The Directional
Spectrum of a Wind-generated Sea as determined
from Data obtained by the Stereo Wave Observation
Project. Meteorological Papers, 2, No. 6, New
York University.
Crawley, G. B. (1975). Automatic contouring on the
Gestalt photomapper, testing and evaluation.
American Society of Photogrammetry, Workshop IIT,
San Antonio, Texas, U.S.A.
Fox, M. J. H. (1976). On the non-linear transfer of
energy in the peak of a gravity-wave spectrum.
II, Proceedings Royal Society of London, A. 348,
467.
Hasse, L., M. Griinewald, and D. E. Hasselmann (1977).
Field observations of flow above the waves. Pre-
print from the Proceedings of the NATO-Symposium
on "Turbulent Fluxes through the Sea Surface,
Wave Dynamics and Prediction," Bendol, to be
published by Plenum Press (New York, London) .
Hasselmann, K. (1963). On the non-linear energy
transfer in a gravity-wave spectrum. Part 3.
Evaluation of the energy flux and swell-sea in-
teraction for a Neumann spectrum. Journal of
Fluid Mechanics, 15, 385.
Hasselmann, K., R. P. Barnett, E. Bouws, H. Carlson,
D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp,
D. E. Hasselmann, P. Krusemann, A. Meerburg, P.
Muller, D. J. Olbers, K. Richter, W. Sell, and
H. Walden (1973). Measurements of Wind-Wave
Growth and Swell Decay during the Joint North
Sea Wave Project (JONSWAP). Ergd&nzungsheft zur
Deutschen Hydrographischen Zeitschrift, Reihe
A (EO), Wis 1A
Hasselmann, K., D. B. Ross, P. Muller, and W. Sell
(1976). A parametric Wave Prediction Model.
Journal of Physical Oceanography, 6, 2; 200.
Holthuijsen, L. H., M. Tienstra, and G. J. v.d.
Vliet (1974). Stereophotography of the Sea Sur-
face, an Experiment, Proceedings of the Inter-
national Symposium on Ocean Wave Measurement and
Analysis, American Society of Civil Engineers,
alsyako
Huhnerfuss, J., W. Alpers, and L. Jones (1978).
Measurements at 13.9 GHz of the radar backscat-
tering cross section of the North Sea covered
with an artifical surface film to be published
in Radio Science.
Longuet-Higgins, M. S., D. E. Cartwright, and N. D.
Smith (1963). Observations of the directional
spectrum of sea waves using the motions of a
floating buoy. Ocean Wave Spectra, 111-132,
Prentice Hall, Inc., New Jersey.
Longuet-Higgins, M. S. (1976). On the non-linear
transfer of energy in the peak of a gravity-wave
spectrum: a simplified model. Proceedings
Royal Society of London, A. 347, 311.
Mitsuyasu, H., F. Tasai, T. Suhara, S. Mizuno, M.
Ohkusu, T. Honda, and K. Rikiishi (1975). Ob-
servations of the Directional Spectrum of Ocean
Waves Using a Cloverleaf Buoy. Journal of
Physical Oceanography, 5, 750.
Panicker, N. N., and L. E. Borgman (1970). Direc—
tional Spectra from Wave Gage Arrays. Proceed-
ings of the 12th International Conference on
Coastal Engineering, Washington, D.C., p. 117.
Phillips, O. M. (1957). On the generation of waves
by turbulent wind. Journal of Fluid Mechanics,
2 aie
Singleton, R. C. (1960). An algorithm for computing
the Mixed Radix Fast Fourier Transform, IEEE
Transactions on Audio and Electroacoustics, AU-17,
An \S)sh<
Spiess, F. N. (1975). Joint North Sea Wave Project
(JONSWAP) progress - an observer's report. Re-
port ONRL-C-8-75, Office of Naval Research,
London.
Seymour, R. J. (1977).
on restricted fetches.
ican Society of Civil Engineers.
Estimating wave generation
Proceedings of the Amer-
Journal of the
Waterway, Port, Coastal and Ocean Division, WW2,
paper 12924, p. 251.
Stilwell, D. (1969). Directional Energy Spectra of
the Sea from Photographs. Journal of Geophysical
Research, 74, 8; 1974.
Sugimori, Y. (1975). A study of the application of
the holographic method to the determination of
APPENDIX
NOISE
Inspection of contour-line maps of the sea surface
obtained from the observation off Sylt revealed a
dome-shaped distortion. This distortion is probably
caused by the fact that the pictures could not be
positioned in the stereoscopic viewing devices with
the accuracy normally obtained with high grade pic-
tures. When this positioning is not optimal, a
dome-shaped distortion is to be expected. Unfor-
tunately the exact distortion cannot be determined,
but in the k-plane it seems to be well separated
from the wave information (area No. 1 in Figure 19)
and the data in this area was removed in the sub-
sequent analysis.
The other noise-affected areas are related to a
phenomenon introduced by the manner of scanning
the pictures during the photogrammetrical process:
the sea surface elevation at even-numbered lines
wavenumber [m~]
t ky
© componen
o, L
ao
569
the directional spectrum of ocean waves.
Sea Research, 22, 339.
Dyllew,, GCG. Lie, Cu (Ce. Teague, Ro H. Stewart), A. Ms
Peterson, W. H. Munk, and J. W. Joy (1974). Wave
directional spectra from synthetic aperture ob-
servations of radio scatter. Deep-Sea Research,
21, 989.
Deep-
(scanned in positive y-direction) is systematically
slightly too low, while the elevation at odd-
numbered lines (scanned in negative y-direction)
are systematically slightly too high. This effect
has been observed earlier in the analysis of stereo
photos of regular waves generated in a hydraulic
laboratory. The principal wave length and direction
of this distortion correspond with the location of
area No. 2 in Figure 19, which is the location of
the Nyquist wavenumber in x-direction. This spectral
information was removed from the spectra in the sub-
sequent analysis. The noise in areas No. 3, 4, and
5 was labeled as such mainly because of the delta-
type behavior of the directional distribution func-
tions in these regions. It is probably due to
variations in the error introduced by the scanning
and possibly also by "leakage" from area No. 2. In
the k-spectrum off Sylt this noise was not removed.
In the k-spectrum off Noordwijk the noise in the
indicated region in Figure 11 has been removed.
4
areaS (ee
—1____1
ay | GS | 0.01 002 003 004 005 006 007 008 ao9 010
wavenumber component k,[m"]
IGURE 19. Location of noise in
F
>
k-plane.
Gerstner Edge Waves in a Stratified Fluid
Rotating about a Vertical Axis
Erik Molo-Christensen
Massachusetts Institute of Technology
Cambridge, Massachusetts
ABSTRACT
An exact solution is obtained for edge waves along
one inclined planar boundary in a fluid rotating
about a vertical axis. The solution is based on a
modification of Gerstner's rotational waves, and
includes the effect of mean drift. The solution re-
duces to Yih's edge wave solution for zero rotation
and to Pollard's rotational deep water Gerstner waves
in rotating flow. Satellite observations of sea sur-
face are shown which reveal patterns similar to those
which would be generated by Gerstner edge waves.
1. INTRODUCTION
The early, exact solution by Gerstner (1802, see
1932, p. 419) was rediscovered by Rankine (1863),
discussed by Lamb (1932), found to be valid for free
surface waves in an arbitrarily stratified flow by
Dubreil-Jacotin (1932), further modified to describe
edge waves by Yih (1966), and free surface waves in
a rotating flow by Pollard (1970). However, there
has been a tendency to dismiss Gerstner waves as of
limited applicability to phenomena in nature. As
Lamb (1932) has pointed out, the generation of
Gerstner, free surface waves by the application of
surface stresses requires a certain mean vorticity
distribution to exist in the fluid. It can be argued
that in a nonrotating fluid of uniform density it is
difficult to conceive how the required vorticity dis-
tribution can be established. However, in a strati-
fied and rotating fluid, there are mechanisms capable
of generating vorticity without viscous diffusion.
In a stratified fluid, the baroclinic term, express-
ing the action of a pressure gradient normal to a
density gradient in generating vorticity will be
capable of establishing a horizontal vorticity field.
In a rotating fluid, the effects of vortex stretch-
ing and compression can establish distributed vertical
vorticity.
There, in a rotating stratified flow, waves simi-
570
lar to Gerstner waves are more likely to be encoun-
tered. In fact, the uniform flow, usually assumed
as the mean flow on which small perturbation waves
may ride, would be less likely to occur in a rotating
stratified fluid. But the small perturbation solu-
tions for waves, as well as exact, finite amplitude
solutions, are all useful as approximate descriptions
of real phenomena and actual observations.
If such solutions do not fit the exact circum—
stances, they can possibly serve as starting points
for perturbation expansions. Furthermore, we may
learn about some of the special features of finite
amplitude exact wave solutions; there is a tendency
to forget some of these facts when preoccupied with
linear wave solutions.
In the following, I shall present a Lagrangian
description of an edge wave field, point out where
it differs from previous solutions, and develop the
dispersion relation for the waves.
2. COORDINATE SYSTEMS AND DISPLACEMENT FIELD
Coordinate System
The waves propagate in the x - direction, normal to
the plane of Figure 1. In the planes normal to the
x - direction we define the oyZ- coordinates, with
o0Z vertical and the oyz-coordinates, with oy in the
plane of the inclined boundary, inclined at an angle
a with the vertical. The particle motion will be in
planes parallel to xy.
While Yih (1966) could let the amplitude of
particle motion decay with negative y-distance, and
Pollard (1970), for deep water waves away from a
side boundary, made the obvious and correct choice
of letting the particle motion decay with decreasing
vertical position; here I have to make a different
choice. The amplitude of particle motion will decay
along a direction - or, shown in Figure 1 as another
coordinate system, ors.
—> n> DO]
FIGURE 1. Coordinate system, looking along the direc-
tion of wave propagation, ox, and along the labeling
coordinate direction, oq.
Displacement Field
Using labeling variables, q, r, s, to identify fluid
particles, define the field of particle positions in
terms of 1, r, s and time, t, as follows:
x = q + Ut - a (exp mr) sin (kq - ot) (1)
y =x cos 8 - s sin B
+ a (exp mr) cos (kq - ot) (2)
BS 1 Sali 5} a S Cos § (3)
for m SIRs O
U is a constant mean particle velocity in the x-
direction, a is an oscillation amplitude parameter,
m is an inverse decay distance measure, K is wave-
number and o is the frequency of particle motion.
First consider the kinematics of wave motion,
next find the condition for incompressibility before
proceeding to apply dynamics to give the dispersion
relation. A surface defined by letting r be a func-
tion of s will have waves that proceed in the x-
direction. For example, a string (line) of particles
defined by fixed values of r and s will have maxima
in y-displacement at
kq - ot = 2ntT (4)
From Eq. 1, substituting for q from Eq. 4 gives the
x-positions of crests to be at
Ke aee = [2nt + (o + Uk)t]/k (5)
The crests move at a speed of
c = (o + Uk)k = w/k (6)
571
w is the wave encounter frequency, and differs from
the particle oscillation frequency by the Doppler
SHEsEe Uke
Mass Conservation
The displacement field defined by Eqs. 1, 2, and 3
can be made to satisfy the requirement that the
density of a fluid particle is independent of time
by requiring that the Jacobian:
d(x,y,2)/9(q,xr,S)
= 1 - a@km (exp2mr) cos 8
+ (m cos 8 - k) a(exp mr) cos (kq - ot) (7)
is independent of time. This requires
k =m cos £ (8)
Now proceed to apply the momentum equations to cal-
culate the pressure, which in turn will be set con-
stant at the free surface.
3. PRESSURE FLUCTUATIONS
The momentum equation in Lagrangian variables gives,
for the derivative of pressure with respect to the
labeling variable q:
“Pg/P = (% + z £ sina - y f cos Oe
+ (¥ + x £ cos ON, + (2 - x f sin Ne
(9)
N>
ar Cj g
The equations for the r and s-derivatives are
similar. f = 22 is the angular velocity of rotation
of the coordinate system, the angular velocity being
vertical as mentioned before. Substituting for x,
Vin and ez etromeEqs..15,, 27) sand 3) into) Eq= 197) onesob=
tains:
-Pg/P [o2 - £ cos a(o + Uk)
-gk sin a] a (exp mr) sin 6 (10)
-p,/p = - [0% - f 0 cos aja? exp2mr
+ [-o2 cos 8 + £ o cos(a + 8)
+ fUm cos a+ gm sin aJa(exp mr) cos 6
+ fU cos (a + 8) + g sin (a + 8) (11)
-ps/0 = [o2 sin B - £ o sin (a + 8)]a(exp mr) cos 6
= £Ul san) (0) +18) +g) cosh (a7 298) (12)
where § = kg - ot is the phase of particle oscilla-
tion.
At the free surface, which consists of particles
with a specified relation between r and s, and with
values of labeling variable, q, from - ~ to + ~, the
pressure must be independent of q and t. This is
satisfied, as can be seen from Eqs. 10, 11, and 12,
572
if the pressure is independent of phase 6, and Pq
Py, and Pp, are independent of 6.
From Eq. 12, pz is independent of 6 when
cot 6 = an = (elon (3 (13)
f sina
Since a is given by the slope of the boundary, Eq.
13 gives 8 for a given o and a. Equation 10 shows
Pq to be independent of 8 when
o* - £ cos a(o + Uk) - g k sina = 0 (14)
For a given value of o, Eq. 14 yields k, and m is
then found from Eqs. 8 and 13.
This leaves Eq. 11 unused, but it can be shown
that the requirement that Py be independent of 6 is
not independent of Eqs. 13 and 14. Equation 11 also
shows that there will be a mean pressure gradient
across the wave propagation direction, proportional
to a*. This is a nonlinear effect of the presence
of waves.
4. DISCUSSION
The equivalent to a linear dispersion relation con-
sists of Eqs. 8, 13, and 14, relating particle fre-
quency, 0, decay direction angle, 8, horizontal
wavenumber, k, and decay parameter, m, with f, a,
and U as parameters.
Note that the introduction of a mean drift veloc-
ity, U, has a now-trivial effect on dispersion, as
can be seen from Eq. 14, where the effect is not a
simple Doppler shift in frequency. The equations
of rotating fluids are not invariant to Galilean
transformations. Also note that the dispersion is
independent of the amplitude parameter, a; this is
an unexpected result for non-linear waves. But the
amplitude of particle motion parallel to oy is really
a exp[2mR(s)], where R is the value of r at the sur-
face. Since m is found from the equations involved
in determining dispersion, one cannot really claim
that dispersion is independent of amplitude.
With the dependence on phase, 9, eliminated in
Eqs. 10, 11, and 12 by satisfying the dispersion
relations, one can see that the mean surface slope
across the wave propagation direction will vary with
wave amplitude and with y- position.
As pointed out by Dubreil-Jacotin (1932), and
later by Yih (1966) the results are valid for a
fluid of arbitrary stable density stratification.
The solutions given here can be further extended
to replace the free surface by an interface between
the given flow field and a homogeneous wave trapped
fluid, giving the gravitational billows described
elsewhere [Mollo-Christensen (1978)]. This will re-
place the acceleration of gravity, g, by g' =
g(Ap/p), where Apis the density difference between
the two fluids and p the density of the lower fluid
at the interface.
Similarly, the flow field at the off-shore or
inside end may be bounded by a field of geostrophic
billows or a combination of gravitational and geo-
strophic billows [see Mollo-Christensen (1978)].
FIGURE 2. High-passed and contrast enhanced satellite
infrared images from January 27, 1975, at 1600, 1700,
and 1800 hrs., GMT. Florida on the right side, Gulf
Coast on top.
5. SOME EXAMPLES OF OBSERVATIONS OF FINITE AMPLITUDE
WAVES ALONG A SLOPING BOUNDARY
By processing satellite data on sea surface infrared
emission one can see moving patterns of sea surface
temperature in the Gulf of Mexico between the con-
tinental shelf edge and the coast.
A sequence of processed satellite images taken
one hour apart is shown in Figure 2. Because the
mean current, U, at the time of observation is not
known, one cannot say whether these waves satisfy
the dispersion relations for the kind of edge waves
discussed here. All one can say at this point is
that it appears possible to satisfy the dispersion
relations given with wavelengths, bottom slopes,
and currents of reasonable orders of magnitude, but
one needs to refine the observations further before
one can reach any definite conclusions.
6. CONCLUSIONS
Nonlinear edge waves of finite amplitude can have
dispersion relations defined by a set of equations
relating particle oscillation frequency, encounter
frequency, wave number, and other parameters in a
way that can be solved systematically if one starts
by specifying a suitable wave variable, in the pres-
ent case, frequency.
The observations which inspired the present anal-
ysis show Gerstner edge waves or possibly waves of
a different kind; one cannot tell with the evidence
now at hand.
573
ACKNOWLEDGMENT
The research reported here was supported by the
Office of Naval Research under Contract No. NOO014-
76-C-0413. The observations cited were made with
support from the Office of Naval Research under
Contract No. NO0014-75-C-0291. The satellite images
were processed using the facilities of Air Force
Geophysics Laboratory, Lincoln, Mass.
REFERENCES
Dubreil-Jacotin (1932). Sur les oudes de type
permanente dans les liquides heterogenes. Atti.
Accad. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., 6,
15; 814-819.
Gerstner, F. (1802). Theorie der Wellen. Abh. d.
Koénigl. Boéhmische Ges. d. Wissenschaften zu Prag
fur das Jahr 1802.
Lamb, H. (1932). Hydrodynamics, 738 pp.
New York, 1945.
Mollo-Christensen, E. (1978). Gravitational and geo-
strophic billows, some exact solutions. J. Atmos.
Sci. To be published.
Pollard, R. T. (1970). Surface waves with rotation:
an exact solution. J. Geophys. Res., 75, 5895-
5898.
Rankine, W. J. M. (1863). On the exact form of
waves near the surface of deep water. Phil.
Trans., 127-138.
Yann Co S35 (SG).
fied fluid.
Dover,
Note on edge waves in a strati-
J. Fluid Mech., 24, 765-767.
The Origin of the
Oceanic Microstructure
Gia abo
Barenblatt and A. S. Monin
P. P. Shirshov Institute of Oceanology
Moscow, USSR
ABSTRACT
Microstructure of hydrodynamical fields, a well-
known phenomenon in the ocean, is attributed to the
formation and development of turbulent spots gener-
ated due to the loss of stability or breaking of
internal waves. Under some general assumptions the
relations are obtained governing the development of
turbulent spots at various 'stages of their evolution
It is shown that the longest and slowest stage of the
extension of a turbulent spot is the final, viscous
one. Simple self-similar laws of the extension of
turbulent spots are obtained for this stage and com-
pared with experiment. Long-standing turbulent
layers of the "blini" shape, sharply bound by am-
bient non-turbulent stratified fluid, are identified
with turbulent spots of the above-mentioned origin
which are in the final viscous stage of their evolu-
tion. The relations are also obtained governing
viscous intrusion of the bottom seawater into the
body of the ocean.
1. INTRODUCTION
Under strongly stable stratification, turbulent mix-
ing is inhibited due to large losses of the turbulent
energy for the work against the buoyancy forces.
der natural conditions, therefore, turbulence cannot
be present in the whole body of the fluid during
rather long periods of time [Woods (1968), Monin et
al. (1977), Federov (1976)]. In fact, it is concen-
trated only in separate turbulent layers having the
shape of "blini," vertically quasi-homogeneous due
to mixing, and separated by thin streaks with micro-
jumps of temperature, electrical conductivity, sound
velocity, salinity, density, refraction index, and
other thermodynamic parameters of sea water some-
times accompanied by microjumps of flow velocity.
Such thin-layered vertical structure, which is ap-
parent from inhomogeneities ("steps") on the verti-
cal profiles of density and other thermodynamic
Un-
574
parameters (see schematic drawing in Figure 1) or
even more sharply from multiple peaks on the pro-
files of vertical gradients of these parameters, is
called microstructure or fine structure of hydro-
dynamical fields. Numerous measurements performed
using the method of continuous vertical sounding in
the cruises of the research vessels of the Institute
of Oceanology, USSR Academy of Sciences, and re-
search vessels of other countries showed that the
microstructure exists always and everywhere in the
World Ocean (the lack of microstructure may be ex-
pected only for the regions of macroconvection which
occur rather seldom in the ocean, at least in the
low and temperate latitudes).
Smoothing over the microstructural "steps" on
the profile of a thermodynamic parameter, e.g.,
density or temperature, we obtain a smooth curve
characterizing large-scale stratification of the
ocean (gross-stratification). We have to emphasize
that from the point of view of the Richardson cri-
terion gross-stratification is nearly always stable
- the Richardson number computed for it, Ri(z), as
a rule, is essentially larger than its critical
value, 1/4. How can the turbulence be generated
under such conditions? Graphs of Ri(z), taking
into account the "steps" of microstructure, show
values of Ri < 1/4 in several layers of the micro-
structure - apprently in these very layers, at the
momeht of sounding, the generation of small-scale
turbulence took place (in other layers where Ri >
1/4 turbulence decayed with time). The appropriate
conditions for local generation of turbulence at
stable gross-stratification may be created by in-
ternal waves. Indeed, in the field of internal
waves in the regions near their crests and hollows
the local values of the Richardson number can be
reduced lower than the critical value, 1/4, and the
turbulence spots would then be formed there. The
internal waves can also break. For the turbulent
spots formed after the breaking of internal waves,
the formation is characteristic of continuous spec-
trum, i.e., of developed turbulence immediately
FIGURE 1.
chronous vertical distribution of
density and shear in the ocean. The
dashed line shows the shear distri-
bution for intrusions.
Schematic form of syn-
after the breaking [Belyaev et al. (1975)].
The evolution of a newly-formed turbulent spot
appears to be the following. The turbulent mixing
makes the spot vertically quasi-homogeneous, there-
fore, within the spot the density of the water be-
comes uniform. For stable stratification, when the
density grows with depth, the density in the upper
half of the mixed spot is higher and in the lower
half of the spot lower than at the same levels in
ambient fluid. Therefore, under the action of the
buoyancy forces, the upper half of the spot should
go down and the lower half of the spot should rise
to its middle level. Therefore, the spot should
"collapse," simultaneously spreading and transform-
ing itself into a thin "blin." The intrusion of
such a "blin" into the body of surrounding strati-
fied fluid creates in it a new layer of microstruc-—
ture.
If the initial internal wave has a long period
and wave length (e.g., internal waves with tide
periods may be generated by tide forming forces
and tides themselves) turbulent spots formed by
this wave are large and corresponding turbulent
layers are very thick. Internal waves of smaller
periods and lengths may develop on these layers
forming turbulent spots of smaller sizes and layers
of microstructure of smaller thicknesses, etc.;
internal waves of minimum periods and lengths,
turbulent spots of minimum sizes and layers of
microstructure of minimum thicknesses. Thus, the
answer to the question "which came first, the chicken
or the egg?" consists for this case in the indica-
tion of a cascade process "internal waves > turbulent
spots > layers of microstructure +> internal waves
etc." This cascade process may lead to the forma-
tion of a quasi-steady spectrum of internal waves,
intermittent turbulence, and layers of microstructure
(although in real nature the action of some other
processes influencing real spectra is possible, in-
cluding storms and quasi-steady horizontal inhomo-
geneities of geographic and dynamic origin). The
turbulent spots also take part in a rising cascade
generated by local instabilities of available shear
flows, breaking of surface waves, sinking of cooled
575
fluid from the turbulized surface layer, etc. As
distinct from the classical Kolmogorov cascade in
non-stratified fluid, here, in passing from a larger
scale to a smaller one, the energy is not preserved,
being left in turbulent spots in the final stage of
their evolution where internal waves do not gener-
ate. Thus, in stratified fluid turbulent spots of
various scales are continuously generated and the
process of their evolution is of considerable
interest.
The first stages of the evolution of turbulent
spots* where the radiation of internal waves takes
place are rather short: by estimates of J. Wu
(1969) and T. W. Kao (1976) they come to an end in
a time interval of the order of several tens of
n7! (N is the Brunt-Vaisdla frequency) after the
beginning of the process. The final stage of the
evolution of turbulent spots is much longer. This
stage is much less known: in the paper of J. Wu
(1969) concerning this stage it is mentioned only
that viscosity is of significance at this stage and
it is noted that the profile of the spot is pre-
served during this stage. The analysis presented
here shows that the velocity of the extension of
turbulent spots at the viscous stage is essentially
lower than at the initial stages. It is our opin-
ion that the "blini"-shaped turbulent structures
are the intrusions of the turbulent spots of various
scales into surrounding stratified fluid which are
mainly at the final stage of their evolution.
Thus, let a turbulent spot (Figure 1) be formed in
a stable continuously density-stratified (linearly
for definiteness) fluid due to some reason (breaking
of internal waves, local loss of stability of shear
flow, penetration of denser fluid from the turbulent
surface layer, etc.). The density of fluid within
the turbulent spot due to mixing is uniform in con-
trast to an ambient continuously stratified fluid
being in a state of rest or laminar motion. Certain
potential energy is stored due to mixing in the tur-
bulent spot, so the state of the mixed fluid-
stratified environment system ceases to be in
equilibrium. Mixed turbulent fluid starts to strike
(Figure 2) into stratified non-turbulent fluid by
tongues - "intrusions" which are formed at the
level, z = 2), (z is the vertical coordinate) where
the density of stratified fluid is equal to the
density of mixed fluid.
Potential energy, stored by the fluid at initial
turbulization and mixing in the spot, dissipates
during the intrusion of mixed fluid into stratified
non-turbulent fluid. It is natural to consider
three stages of the evolution of the spot:
(1) Initial stage of free intrusion. The motive
force of the intrusion at this stage exceeds greatly
the drag forces. The turbulent spot extends slightly
but the internal waves are intensively formed by the
spot.
(2) Intermediate steady state. The motive force
at this stage is balanced mainly by form drag and
wave drag due to radiation of internal waves by an
extending turbulent spot. The acceleration of the
tongue is negligible.
*the classification of stages of the evolution of the spot of
mixed fluid in the continuously density-stratified fluid goes
back to the fundamental work of J. Wu (1969) where the ex-
perimental investigation of the initial stages of this process
was performed for the wake of circular initial cross-section.
T. W. Kao (1976) performed semi-empirical theoretical investi-
gation for the initial stages of the evolution of such wakes.
576
FIGURE 2. The intrusion of a turbulent spot into con-
tinuously stratified fluid.
(3) Final viscous stage. The motive force is
balanced at this stage mainly by viscous drag.
Of course, between the first and second and the
second and third stages there exist intermediate
transitional periods. When the third stage comes
to the end the spot is mixed due to diffusion with
ambient fluid and disappears.
The turbulent motion inside of the intrusion
tongue is supported by general shear stress together
with eddy motions inside of the intrusion due to the
difference of the velocities of the tongue and en-
vironmental non-turbulent fluid. The boundary of
turbulent and non-turbulent fluid is sharp and if
the thickness of the intrusion is not too small,
the shear required for supporting the turbulence
within the intrusion is not large.
Indeed, let us consider the equation of the
balance of turbulent energy in a shear flow of
stratified fluid neglecting, as usually, the viscous
transfer term [Monin and Yaglom (1971)-]
3,5 + 3 {w'E' + p'w'}
= - pw'g - pe - p u'w' au (1)
Here t is the time, E the turbulent energy of
unit mass, € the dissipation rate per unit mass,
u the longitudinal and w the vertical velocity
components, p the pressure. The flow is considered,
for the estimates we need, as horizontally homogen-
eous and the Boussinesque approximation is accepted,
i.e., the density variation is taken into account
only if it is multiplied by very large factor -
gravity acceleration g.
Let us accept for the terms of the equation of
balance of turbulent energy, the Kolmogorov approxi-
mations [Monin and Yaglom (1971) ]
wwii} ap joy eS pvp. 38
u'w' = - 2vB aa, é = y'*p3/272 (2)
Here 8 = E/p is the mean turbulent energy per
unit mass, 2 the external turbulent scale. Thus,
the equation of balance of turbulent energy takes
the form
3,8 = a LB 3,8 - pw'g/p
+ 278 (2 a)? - y'tB3/272 (3)
The mathematical nature of sharp interface between
the turbulent and the non-turbulent regions becomes
completely transparent from this equation. In fact,
Eq. (3) is a non-linear equation of heat conductivity
type with heat inflow where the coefficient of trans-
fer of turbulent energy equal to ave tends to zero
with turbulent energy itself. For such equations
under zero initial conditions the disturbed region,
in contrast to the linear heat conductivity equation,
is always finite; this explains (cf. below) mathe-
matically the existence of a sharp interface between
the turbulent and the non-turbulent regions.
It is important that, due to mixing following the
generation of a turbulent spot, the losses of turbu-
lent energy for the work of suspending a stratified
fluid [the second term of the right-hand side of the
Eq. (3)] disappear because the density within the
spot becomes uniform. Furthermore, the first term
of the right-hand side of (3) governs the diffusional
transfer of turbulent energy within the mixed region
and does not influence the averaged, through the
spot, value of turbulent energy. Therefore, the
decay of turbulent energy within the spot is governed
by the balance of the two last terms of the right-
hand side of the equation (3) representing genera-
tion and dissipation of turbulent energy,
respectively.
It seems natural to accept that the external scale
of turbulence 2, within a factor of the order of
unity, coincides with the transverse size of the
tongue of intrusion h; the constant y by estimates
has a value of about 0.5. Thus, the shear d,u ~
VB/h is sufficient to support the turbulence within
the spot at a steady level together with the state
of mixing within the spot. If h has the value of
tens of centimeters - one meter or more, then for
the value VB ~ 1 cm/sec, characteristic of oceanic
turbulence, the shear required for supporting steady
turbulence is small. In thin layers it is large;
therefore, the turbulence in thin layers decays
rather quickly and the spot of mixed fluid exists
during the time interval required only for the dif-
fusional mixing of the spot with the ambient strati-
fied fluid.
Furthermore, available experimental data show
(J. Wu (1969)] that turbulent entrainment and the
erosion of a turbulent spot may be neglected, start-
ing from a very early stage of the evolution till
rather late stages of this process. Therefore, we
shall take the volume of turbulent spot constant at
all stages of its collapse to be described.
For simplicity we shall further suppose that the
initial form of a turbulent spot is symmetric in
respect to the equilibrium plane where the densities
of stratified fluid and mixed fluid coincide.
2. INITIAL STAGES OF THE EVOLUTION OF THE SPOT OF
MIXED FLUID
At the first stage, free fall (lifting from below)
of the particles of mixed fluid to the equilibrium
plane takes place, followed by the spreading of
fluid particles along this plane. Therefore, the
rate of change of the area of horizontal projection,
S, of a turbulent spot is proportional at this stage
to the product of the actual area by the rate of
fluid influx to the equilibrium plane. The latter
quantity is equal to the product of the acceleration
of free fall proportional to N2 and time t. Thus,
we obtain for the initial stage
dS/dt ~ sN*t (4)
For small Nt we obtain by integration
= = ee
(S S,)/S, Nft (5)
(Sg is the initial area of horizontal projection of
the spot). Thus, at the first stage the character-
istic size of the plan form of the turbulent spot,
L, changes proportionally to the square of time
(L-L)/L. ~ n2t2 aL/dt ~ L N2t (6)
fo) fo} {o)
[for the wake, S ~ L, and the relation (6) follows
from (5) in an elementary way; for the spot of the
circular plan form, S ~ L2, but at (L = Lo) SS) op
we = Tae ~ 2 (L - Lo)Lo and (6) follows again from
(5) Ic
The relations of the type of (6) were obtained
by J. Wu (1969) from the experimental investigation
for a spot having the form of a cylinder with a
horizontal axis; they were confirmed by some nu-
merical investigations [see Kao (1976)]. Actually
they were confirmed to be valid to Nt ~ 2.5.
At the intermediate stage the motive force of
the intrusion is balanced by form drag and wave
drag, thus, the velocity of the propagation of the
intrusion tongue is governed by the parameter of
stratification - Brunt-Vaisala frequency N - to-
gether with the actual height of the tongue, h,
whence by dimensional considerations we obtain
aL/dt ~ Nh (7)
We see that at this stage the dependence of the
velocity of the extension of the intrusion tongue
is different for various geometries of the problem.
In fact, the volume of the turbulent spot V is
constant; for the cylindrical spot h ~ V/LH (H is
the longitudinal size of the spot) and h ~ V/L2 for
a spot of the circular plane form. Therefore, we
obtain for the cylindrical spot
dL2/at ~ NV/H , L ~ VYNV(t —- to) (8)
(to is a conditional time moment of the beginning
of the second stage), whereas for the spot of the
circular plane form
3
aL3/dt ~ NV , L~ YNV(t — to) (9)
The relations of the type (8) were obtained by
J. Wu (1969) from the experimental data for collapse
of a turbulent wake of initial circular cross-
FIGURE 3. Elementary particle of the diffusion tongue.
577
section. They were confirmed to be valid for
3S ie S BS,
3. FINAL, VISCOUS STAGE OF THE INTRUSION
Under accepted assumptions the equation of mass con-
servation for a mixed fluid takes the following form
in hydraulic approximation.
a,h + div (hy) = 0 (10)
Here h(x,y,t) is the height of the intrusion
tongue; x,y are the spatial horizontal coordinates,
t is the time, v is the velocity of fluid displace-
ment averaged through the height of the tongue.
For the determination of the velocity, v, let us
consider the system of forces acting on the cylin-
drical particle of the intrusion tongue leaning upon
the area 56S (Figure 3). The motive force of this
particle is caused by the action of the gradient of
redundant pressure, P, and spatial variation of the
height of the tongue of intrusion
Fm = - grad(ph) 6s (11)
Furthermore, the drag force per unit area of a
particle surface due to the viscous character of
the drag at the final stage of the intrusion under
consideration is governed by the velocity, v, of
the particle relative to ambient fluid, viscosity
of the fluid, u, and particle height, h. The di-
mensional considerations give the viscous drag
force per unit area of particle surface proportional
to uv/h. Therefore, the viscous drag force acting
on the particle leaning upon the area, 6S, is equal
to
Fr = CuvéS/h (12)
where C is a constant, under given assumptions - a
universal one. For estimating the constant, C, the
well-known solution of the problem of viscous flow
between flat plates may be used. This solution
gives for the viscous drag the value 12uvéS/h, whence
C = 12. Equaling drag force to motive force (the
inertia force, as at the second stage, is supposed
to be a negligible one) we find
v = - hgrad(ph) /Cu (13)
To complete the statement of the problem we have
to find the redundant pressure in the mixed fluid.
In stratified fluids the density varies linearly with
height. The intrusion tongue propagates symmetri-
cally, thus, the equilibrium plane divides the
height of the tongue in half. Let us denote by pj
and ~,, correspondingly, the pressure and the density
in stratified fluid at the level, 2 = zy. ‘Then,
evidently, the pressure in the stratified fluid
varies with depth following the relation
Pp = pi ~— 919(2 - 2)
+ p{N2(z - 2)) 2/2 (14)
Here, as before, N is the Brunt-Vdisdla frequency
N2 = ag, g is the gravity acceleration, a = (dp/dz)p .
Thus the pressure at the upper and the lower points of
a vertical section of the tongue z = z; + h/2 are
equal, respectively, to
578
Ur = 2,2
P = Pp] — Pigh/2 + p\N°h*/8
Pp = pj + pigh/2 + p\N2h2/8 (15)
because at the upper and the lower points the pres-
sure in the tongue coincides with the pressure in
ambient stratified fluid. Hence, the pressure
within the tongue is distributed according to the
hydrostatic law
P = Pp) — p1g(z - 21) + p1N*h?/8 (16)
The pressure averaged over the section of the tongue
is equal to
Ki 2y2
Pog 7 Pl + ei h“/8 (17)
The pressure averaged in the same way in the strati-
fied fluid due to (14) is equal to
22
= ot 4 18
Be Pl p1Nch*/2 (18)
Thus, the redundant pressure entering the expres-
sion of motive force of the intrusion tongue at a
given vertical line is
= = = 2n2/12
2) See ee = ho/, (19)
The relations (13) and (19) give
p,NA aN
1 1
= 3) eo
M 12Cu lore telte) 4cyu
hegrad(h) (20)
Putting this expression into the equation of
mass conservation of mixed fluid (10) we obtain
for h a non-linear equation of the heat’ conductivity
type
a,h - nAh®> = 0 , n = p,N2/20Cy = N2/20Cv = (211)
Here A is the Laplace operator, v the kinematic
viscosity of the fluid. In particular, for one-
dimensional motions Eq. (21) takes the form
d.h - nd*2 h°2 = 0 (22)
iS xx
= 5 =
dh n(1/r) 0x0 h 0) (23)
for the plane and the axisymmetrical cases, respec-
tively. Here x is the horizontal Cartesian co-
ordinate, r the horizontal polar radius.
4. SELF-SIMILAR ASYMPTOTIC LAWS OF TURBULENT SPOT
EXTENSION AT THE VISCOUS STAGE
We neglected turbulent entrainment and the erosion
of a turbulent region; therefore, the volume of the
turbulent mixed region is considered to be constant
and equal to the initial volume of the turbulent
spot. It stands to reason that this assumption at
the viscous stage is valid for sufficiently high
stratification only. If the characteristic dimen-
sions of the plane form of a turbulent spot are
nearly equal, it is natural to expect that the ex-
tension of the intrusion starts already to be axi-
symmetric at the end of the intermediate stage and
deliberately is axisymmetric at the viscous stage.
Hence, Eq. (23) may be applied for its description.
Thus, the condition of conservation of the volume
of a turbulent spot takes the form
co
27 ff rh(r,t)dr = V = Const (24)
fo)
The asymptotic stage of the spreading of the spot
is of primary interest when the plane size of the
intrusion exceeds the corresponding initial size
of the turbulent spot. At this stage the details
of the initial distribution h(r,0) cease to be
essential and for an asymptotic description or the
viscous stage of the intrusion the initial distribu-
tion may be represented in the form of an instantan-
eous point source
h(r,t;) = 0 (r #0), 2m f rh(x,t))dr = V (25)
(0)
Here, t; is the conditional time moment of the be-
ginning of the viscous stage.
The solutions of such type for non-linear heat
conductivity equations with the power-type non-
linearity to which Eqs. (22, 23) belong were con-
sidered in the papers of Ya. B. Zel'dovich, A. S.
Kompaneets, and one of the present authors [see
Barenblatt et al. (1972)]. In our case the solu-
tion depends on the quantities t - tj, n, V, r.
The dimensional considerations show that it is a
self similar one:
@ 1/5
a Anais > 1851) £(o)
-1/10
t= r[vin(t - t))/l6n*] (26)
Putting (26) into Eq. (23) and integrating the
ordinary differential equation obtained for the
function, f(t), we find
(aol ye A r2 1/4
6 ( ae ) gOSES So
0,520.5 103/572 = 2 (27)
(GG) =
Thus, at each moment of time the intrusion tongue
stretches for a finite distance: this is (cf. Sec-
tion 1) the peculiar feature of non-linearity dis-
tinguishing the equation of intrusion from the
linear equation of heat conductivity. The edge of
the intrusion propagates following the law
ro(t) = 2(vin(t = ey fen 2/2 (28)
The form of the intrusion tongue represented by
the curve 1 in Figure 4 also is peculiar: the
thickness of the tongue changes slowly to the very
edge where it comes abruptly to naught. The maxi-
mum spot thickness, ho (t) = h(o,t), also changes
very slowly with time
0 0.5 70
FIGURE 4. The distribution of thickness along an
intrusion.
anne (2 1/4 ( Vv ) 1/5 (29)
fe) 6 Cra (te = te)
Equation (28) seems very simple and accessible
for experimental confirmation: confirmation of
this equation will give some confidence in the
validity of the model proposed here. The experi-
mental checking of Eq. (28) was performed by A. G.
Zatsepin, K. N. Federov, S. I. Voropaev, and A. M.
Pavlov. They used the following scheme for the ex-
periment (Figure 5). An open plexiglass tank having
the form of a rectangular parallelepiped contained a
stable, temperature-stratified fluid. A hollow
cylindrical tube was introduced from above under
the surface of the fluid. The fluid in the tube was
mixed and then the tube was raised, leaving in its
place a spot of mixed fluid which immediately started
penetrating the ambient stratified fluid. The ob-
servations, photo- and movie camera, were performed
using a shadow device. The experiment allowed one
to observe clearly the two last stages of spot evolu-
tion; the spot extension at the viscous stage is
represented in Figure 6. The mixed fluid volume in
the spot was fixed for all experiments, as well as
the kinematic viscosity of the fluid and the diameter
of the tube. Therefore, if Eq. (28) is correct, the
experimental data in the coordinates Lg[2ro(t)/D],
&gIN (t - t))] had to fall on a single straight line
with the slope 0.1. This is confirmed by the graph
of Figure 6 where the slope of the solid straight
line is 0.1 and t; = -10 sec. Thus, the law of one
tenth Eq. (28) for the viscous extension of a spot
was confirmed by the experiments of A. G. Zatsepin,
K. N. Federov, S. I. Voropaev, and A. M. Pavlov with
a Satisfactory accuracy.
Analogously, in the case when the form of the
turbulent spot is close to the cylinder with a
horizontal axis Eq. (22) for the height of the in-
trusion tongue will hold, where x is the horizontal
coordinate normal to the axis of the spot. The con-
dition of conservation of the volume of the spot of
mixed fluid takes, for this case, the form
H ff h(x,t)dx = V = Const (30)
where H is the longitudinal size of the cylindrical
spot. The initial conditions corresponding to the
579
asymptotic solution of the instantaneous point
source type may be written in the form
Die = ONG 70) { h(x,t))dx = Vv (31)
a)
and the asymptotic solution itself due to the same
reasons, as before, may be represented in the form
v2 1/6
h 4n(t - t,)H2 =i)
1/6
5 = x[Vin(t - t,)/l6Ht]
MG. = Be Aye ae Se
(0) 10)
2/3
1/6
2 =U 0, 62%. = Us)” om 2 3).(6
Ae (5, 27s) 4? 2 0.97 (32)
so that the leading edge of the intrusion, x = X(t),
propagates according to the law
ee?
(2) = co lvin(t - t,)/16H* (33)
o}
while the maximum thickness of the intrusion, ho(t)
= h(o,t), decays with time according to
/6
hot(t) = 0.97(v2/4H2n(t - ty))? (34)
Thus, in both cases a strong deceleration of the
extension of intrusion was characteristic for a
turbulent spot in the transition to the viscous
stage. Indeed, at the free intrusion stage the ex-
tension of a turbulent spot is proportional to the
The scheme of the experimental checking of
FIGURE 5.
the law of viscous extension of a spot of mixed fluid.
1) The tank, 2) Point light source with collimator,
3) Lens, 4) Vertical elevator with electromotor, 5)
Mixer, 6) Tube, 7) Screen, 8) Movie camera.
eee |
Hi. 1
Bes 6 6 20
FIGURE 6. The one tenth law as confirmed by laboratory
experiments of A. G. Zatsepin, K. N. Fedorov, S. I.
Voropaev, and A. M. Pavlov.
square of time; at the intermediate stage it is
proportional to the square root of time for a cylin-
drical spot and to the cube root of time for an axi-
symmetric spot. At the viscous stage the extension
is proportional to time; to one sixth in the case of
a cylindrical spot and to one tenth in the case of
an axisymmetric spot. Thus the extension of the
spot is sharply decelerated at the viscous stage in
comparison with the initial stages.
It seems plausible to us that the "blini" shaped
regions of constant density and temperature observed
in the ocean are turbulent spots of various scales
generated by the loss of stability or breaking of
internal waves, local instability of shear flows,
penetration of cooled turbulized fluid from the
curbulized surface layer, etc. which are mainly in
the last, viscous stage of their evolution. Note
that along with the states in which turbulence is
preserved within the spot, the states are possible
and apparently rather frequent, especially for spots
of small scales, in which turbulence within the spot
has disappeared but the fluid remains mixed and homog-
eneous. This assumption is supported qualitatively
by some data of simultaneous measurements of vertical
distributions of density and velocity gradient
[Federov (1976)]. These distributions have the form
presented by solid lines in Figure 1. Indeed, if
the regions of constant density are intrusions, then
the shear should increase near their boundaries com-
pared to ambient fluid (cf., Figure 2). However, in
this case the shear should be reduced near the cen-
tral line of intrusion (dashed line in Figure l).
It is plausible that the resolution in these mea-
surements was not sufficient to observe this shear
reduction.
5. THE INTRUSION OF BOTTOM SEA WATER INTO THE
BODY OF THE OCEAN
The intrusion of mixed fluid into a continuously
stratified medium is widely distributed in nature;
IO) 100
it is of interest from the point of view of the
evolution of turbulent spots in stratified fluid.
A characteristic example - the intrusion of the
bottom Mediterranean water into the body of the
Atlantic (Figure 7). The bottom water descends
through the Straits of Gibraltar down the contin-
ental slope and enters the body of the ocean in an
intermediate layer where the density of the ocean
water is equal to its own density. The intrusion
of the bottom water of the Red Sea into the body of
the Indian Ocean is completely analogous. The in-
trusion of bottom water is a slow process and we
may assume that for its description, Eq. (22),
corresponding to a pure viscous mechanism of the
intrusion drag, is valid.
The intrusion of bottom sea water into the body
of the ocean goes by separate portions [Federov
(1976) ] and it is possible to assume that, at the
beginning of the intrusion of a new portion, the
bottom fluid that intruded earlier is carried suf-
ficiently far away so that the initial condition
holds
(<= (0) (35)
Here h, as before, is the height of the intrusion
tongue, x the horizontal coordinate in the direction
of intrusion from its origin. Let us suppose that
FIGURE 7. The intrusion of sea bottom water into the
body of the ocean.
the height of the bottom water layer at the origin
of intrusion does not depend on time:
IM(Opi) = hy = (Const (36)
The solution of Eq. (22) under conditions (35)
and (36) is also self-similar and has the form
= = Yoh +
h hot, (a) + t x/ nh t (37)
where the function f9(t) which satisfies the equa-
tion
a2£5° df>
A a SER = @ (38)
ac? dt
under the conditions
10) (39)
(0) Si, sy)
is continuous and has a continuous derivative
df°/at (the last requirement follows from the
continuity of the flow of bottom fluid). The solu-
tion, f(t), is represented in Figure 4 (curve 2).
It is also different from zero only in a finite
interval 0 = G = To 6 «61.66, so that the leading
edge of the intrusion x,(t) propagates as
REFERENCES
Woods, J. D. (1968). Wave-induced shear instability
in the summer thermocline. J. Fluid Mech. 32,
791.
Monin, A. S., V. M. Kamenkovich, and V. G. Kort
(1977). Variability of the ocean. J. Wiley.
Federov, K. N. (1976). Fine thermohaline structure
of oceanic waters. Gidrometeoizdat, Leningrad.
Belyaev, V. S., I. D. Lozovatsky, and R. V. Ozmidov
(1975). On the relation between the small-scale
turbulence parameters and the local stratifica-
tion conditions in the ocean. JIzv. AN SSR, Ser.
Physics of Atmosphere and Ocean II, 718.
Wu Jin (1969). Mixed region collapse with internal
wave generation in a density stratiied medium.
J. Fluid Mech. 35, 531.
Kao, T. W. (1976). Principal stage of wake collapse
in a stratified fluid: two-dimensional theory.
Physics of Fluids 19, 1071.
Monin, A. S., A. M. Yaglom (1971). Statistical
hydromechanics. Part I. The MIT Press.
Barenblatt, G. I., V. M. Entov, and V. M. Ryzhik
(1972). Theory of non-steady filtration of liquid
and gaS. Nedra, Moscow.
v 4
che
a
en
may a
ar 2
i i ex AS
I
u 1
i = Ve :
ca i i
ey I Ai
= i ; —
; i
ry
1 L 1 ‘
he { zs
;
‘ . b
|
~
1
“1%
eld OP, Raw fees tated marta Me™
ne
Z
eee: Ey pain’ 2
Session VITT
GEOPHYSICAL FLUID DYNAMICS
LOUIS N. HOWARD
Session Chairman
Massachusetts Institute of Technology
Cambridge, Massachusetts
The Rise of a Strong Inversion
Caused by Heating at the Ground
Robert R. Long and Lakshmi H. Kantha
The Johns Hopkins University
Baltimore, Maryland
ABSTRACT
A theory is offered for the rise of a strong inver-
sion in the atmosphere caused by heating at the
ground. The heating, specified by the buoyancy
flux, q,;, near the ground, causes turbulence in a
growing layer of depth, D, above the ground with an
inversion or interfacial layer of thickness, h,
separating the mixed layer from the non-turbulent
air above. There is a buoyancy jump, Ab, across
the interfacial layer and the air above the inver-
sion has a buoyancy gradient, No:
The lower surface of the inversion layer rises
(at a speed, Us = dD/dt) because of two processes.
One is related to the mean temperature rise of the
mixed layer which, in the present model, leaves h +
D unaffected but which causes the interfacial thick-
ness, h, to decrease and therefore D to increase at
a rate proportional to eee where Ri = DAb/w% is
the Richardson number and wx = (q,D) ? is the con-
vective velocity typical of the rms velocities in
the main portion of the mixed layer. The second
process, increasing both h and D, is the erosion of
the stable fluid by the turbulence in the mixed
layer and the intermittent turbulence in the inter-
facial layer. This causes D to increase at a rate
proportional to Rea a The total effect is con-
tained in the equation
Ye
— = aRi7! + cRi-7/*
Wy
where a and c are universal constants. Other re-
sults are presented, notably the ratio, lqo/ay|, where
qo is the (negative) buoyancy flux near the level
Z=D. This ratio decreases with increase of sta-
bility as observed in experiments of Willis and Dear-
dorff. |qo/q,| ~ Ri-3/7*.
1. INTRODUCTION
When the sun rises and begins to heat the ground,
the atmosphere is normally in a stable state (po-
585
tential temperature increases with height). If we
neglect the effect of mean wind for the moment, the
heating creates instability and turbulence near the
ground and a mixed layer of depth, D, appears, capped
by an inversion. This phenomenon is called penetra-
tive convection. The potential temperature of the
mixed layer is nearly constant with height except
very close to the ground, where a superadiabatic
lapse rate exists in a thin layer, and just below
the inversion base where there is weak stability.
The inversion base rises because of two processes.
The first is heating alone which tends to decrease
the thickness, h, of the inversion layer, (IL), and
so increase D. The second is the entrainment effect
of the turbulent eddies just below the inversion
base. We do not have a detailed understanding of
this erosion process but laboratory experiments with
mechanical stirring [Moore and Long (1971), Linden
(1973) ] suggest that the eddies in the mixed layer
deflect the IL upward storing potential energy. When
this is released by downward motion, a portion of
the lighter fluid in the IL is ejected into the
homogeneous layer where it is carried away by the
turbulent eddies, leaving the lower surface of the
IL sharp again.
If there is no mean wind, the energy for the tur-
bulence comes from the energy flux divergence term
and from the buoyancy flux term in the energy equa-
tion, where q = -w'b" is the buoyancy flux*. When
there is a mean wind, as is usual in the atmosphere,
the shear yields another energy source. This serves
to increase the turbulence energy and thus to in-
crease the entrainment effect through greater agita-
tion of the IL. In addition, the shear may cause
Kelvin-Helmholtz instability and consequent wave
breaking at the interface and thereby enhance ero-
sion.
On the other hand, the effect of shear should be
“SHOVES in an incompressible fluid is defined as b =
g(p - P9)/p9 where g is gravity, p is density and pg is a
representative density. In the atmosphere, p and pg are
potential densities. We may also write b = g(@ -89)/8o
where 6 is a potential temperature.
586
negligible if the mixed layer depth is much greater
than the Monin-Obukhov length, L = -u3/q), [Monin
and Yaglom (1971, p. 427)] where ux is the friction
velocity. Thus (-L/D) 173 is proportional to the
ratio, ux/wy, of the turbulent velocity in the mixed
layer associated with shear to the turbulent veloc-
ity associated with convection, wx = (q,D) 173. The
shear effect becomes less important as this ratio
decreases. Lenschow (1970, 1974) presents aircraft
measurements, which appear to confirm the unimpor-
tance of energy production by the shear for the
turbulence near the inversion if |L/D| is small
enough.
The purpose of this paper is to construct a
theory for the rise of an inversion in the atmo-
sphere neglecting the effect of shear. The analysis
is similar in some respects to that in a recent paper
by the first author, [Long (1977b), hereinafter
referred to as MISF] in which a theory is developed
for turbulence in a stably stratified liquid, as
for example in the experiments of Rouse and Dodu
(1955), Turner (1968), Wolanski (1972), Linden (1973),
Crapper and Linden (1974), Linden (1975), Thompson and
Turner (1975), Wolanski and Brush (1975), and Hop-
finger and Toly (1976). In these experiments a
stably stratified fluid is agitated by a grid oscil-
lating up and down near the bottom of the vessel
(Figure 1). A growing mixed layer of depth, D,
appears in the lower portion of the fluid separated
from the non-turbulent fluid above, in which the
buoyancy gradient is given, by an IL of thickness,
h. Observations indicate that the lower mixed layer
has a very weak mean buoyancy gradient. The buoyancy
difference across the IL is relatively large and is
denoted by Ab.
As indicated by the experiments of Thompson and
Turner and Hopfinger and Toly, and derived by the
first author in a recent paper [Long (1977a)], the
turbulence generated by the grid in a homogeneous
fluid is nearly isotropic, and if u is the rms veloc-
ity and 2 is the integral length scale, the quantity,
uZ(proportional to eddy viscosity), is constant with
height. When there is stratification, the mixed
layer is nearly homogeneous and us = K is again con-
stant near the grid [Hopfinger and Toly (1976)].
Since 2 is proportional to the depth, D, the veloc—
ity, u, = K/D, is characteristic of the turbulent
velocities in the mixed layer. The quantity, K,
may be taken to be characteristic of the "action"
of the energy source (grid).
On the basis of observations, experimenters have
FIGURE 1.
the grid.)
Oscillating grid experiment. (S = stroke of
proposed that the entrainment velocity u, = dD/dt
is expressible in the form
Ue «3/2 _* DAb
fs Ri , Rl = Feg2 (1)
where Ri* is the overall Richardson number, f is the
frequency, and S is the stroke of the grid. The
measurements correspond to large values of Ri* so
that attention is confined to the usual situation
in nature in which the Richardson number is large.
In terms of the "action" K of the grid, another
Richardson number is
Mee | msi (2)
ea
This is very similar to the number Ri = 2Ab/u2
proposed by Turner (1973), where 2 and u are the
integral length scale and rms velocity measured at
the level z = D in a homogeneous fluid agitated by
the same grid at the same grid frequency and stroke.
In MISF and in the present paper, the role of the
IL separating the mixed layer from the non-turbulent
fluid above is essential. This contrasts with ear-
lier theories in which h is neglected despite ex-
perimental evidence [Linden (1975)] that h is
proportional to D and is not particularly small
(h/D = 1/4). Observations [for example, Wolanski
and Brush (1975)] indicate that the IL with its
large density gradient is typified by wave motion.
Wolanski and Brush found that the frequency of dis-
turbances in this layer was proportional to the
Brunt-Vaisdla frequency (Ab/h)? although numerically
one order of magnitude smaller. Certainly turbulence
of some kind exists in the IL and since the density
gradient there is strong rather than weak as in the
mixed layer, it is reasonable to assume that the
turbulence in the IL is intermittent and that this
intermittent, weak turbuience transfers the buoyancy
in the layer. In MISF the intermittency factor de-
creases with increase of stability so that for the
large Richardson numbers of the asymptotic theory
the layer is, for the most part, in laminar wave
motion with occasional breaking waves in the interior
and at the lower surface of the interface.
Similar ideas may be applied to the present prob-
lem in: which the turbulence in the mixed layer is
caused by heating at the lower surface. The princi-
pal differences are the effect of heating in causing
h to decrease and D to increase, and the differences
in the sources of turbulence kinetic energy. The
energy equation is
' Dae 12 12 ey
+
Open w(2 +% a ) Vu & (3)
C4 Po 2
where the first term is the energy flux divergence;
u', v', w' are the instantaneous velocities, p' is
the pressure, Po is a reference density, q = -w'b"
-is the buoyancy flux, and € is the energy dissipa-
tion. In the present problem the buoyancy flux
term, -w'b', is of basic importance and corresponds
to the conversion of potential energy to kinetic
energy. This effect is missing of course, in the
case of mechanical stirring in a homogeneous fluid.
Equation (3) omits the local time rate-of-change
of kinetic energy although, in fact, the inversion
is rising and conditions are therefore unsteady.
With respect to the mixed layer, the kinetic energy
Heated Surface
FIGURE 2. Model of entrainment at an interface by
heating from below. The curve on the left is the mean
buoyancy, b, with an assumed linear profile above the
interfacial layer. The curve for buoyancy flux, q, is
on the right. The superadiabatic layer near z = 0 is
not shown.
is proportional to the square of the convective
velocity, (q,D) 273, so that the ratio of the time
rate-of-change term to the other terms in Eq. (3) is
Ue/Wx. This ratio is of order one if the convective
motions are spreading upward at a speed, ug, in
initial conditions of neutral stability. Evena
fairly weak inversion will cause a great slowdown
and ue/wy will be small. Similar remarks apply to
the IL and we are assured that the time dependence
is negligible in the stable conditions of the paper,
although it has received some attention in considera-
tions of the real atmosphere [Zilitinkevich (1975)].
We may conclude this introduction with reference
to work on penetrative convection in the atmosphere
and oceans including atmospheric observations:
Lettau and Davidson (1957), Ball (1960), Veronis
(1963), Izumi (1964), Summers (1965), Deardorff
(1967), Kraus and Turner (1967), Lilly (1968), Dear-
domki (972) Betts) (197/3)",, Carson) (197/s)),) Stull
(1973), Tennekes (1973a,b, and 1975), Adrian (1975),
Farmer (1975), Zilitinkevich (1975), Kuo & Sun (1976),
Stull (1976a,b,c), and Zeman and Tennekes (1977).
Related experiments have been run by Deardorff,
Willis, and Lilly (1969), Willis and Deardorff (1974),
and Hedit (1977). A second-order closure model has
been given by Zeman and Lumley (1977). More recent
field observations have been made by Kaimal, et al.
(1976). Mixed layer deepening in the upper layers
of the ocean, which is almost always associated with
wind stirring has been discussed by Niiler and Kraus
(ID 77) 6
2. RELATION OF FLUXES TO THE BUOYANCY JUMP AND TO
MIXING LAYER AND INTERFACIAL LAYER THICKNESSES
In the theory of the paper we ignore rotation, radia-
tive heating, water vapor, and horizontal variations
of mean quantities. The model is shown in Figure 2
which contains curves for the mean buoyancy and buoy-
587
ancy flux. The mean buoyancy curve above the IL is
assumed to be linear with buoyancy gradient N2. In
one case we assume that N* = 0 so that the inversion
rises and weakens, eventually disappearing. When
n2 # O we assume that the air was at rest with uni-
form buoyancy gradient when heating began. Then the
inversion strength increases with time. Since the
theory of this paper is concerned with very stable
conditions, the solutions hold for large values of
the Richardson number.
The buoyancy flux curve is derived below from the
assumed buoyancy distribution. The latter is assumed
to be linear in the IL (region R3). This is an ex-
cellent approximation* in certain circumstances at
least, for example in the mechanical stirring ex-
periments of Wolanski and Brush (1975). Observa-
tions in the mixed layer [Willis and Deardorff
(1974) ] indicate that there is very little mean
buoyancy variation in this layer except for some
indication of a stable mean gradient near the heated
plate. If we ignore these gradients for the moment,
the equation
db aq
at Oz i)
indicates that q is a linear function of z. In fact,
experiments show that q is nearly linear [Willis and
Deardorff (1974)] so that the neglect of mean buoy-
ancy variations in the mixed layer in the model of
Figure 2 seems reasonable. The lower surface is
heated and the buoyancy flux q = -w'b' (proportional
to the heat flux) is held constant at the lower sur-
face where it is denoted by q,. The mean buoyancy
in the mixed layer is
b =b.. = N2(D + h) + Ab (5)
m 00
where Ab is the buoyancy jump across the interfacial
layer and bog is constant equal to the buoyancy at
the surface 1f the linear gradient above is extra-
polated down to the surface. Integrating (4) over
the mixed layer, we get the flux, qo, just below the
IL. IRS Als)
Se ky GD
De Ch
N2D S= (D + h) (6)
On physical grounds qo must be negative (Figure 2)
and this is confirmed by laboratory measurements
(Willis and Deardorff (1974)]. In the IL, the mean
buoyancy is
- N2(D = h) (7)
= Ab
Ney) Ne) ee (ES 1D) Ie 1a
Integrating (4), we get the flux at a given level
in the interfacial layer
2 2
dAb t 2 dh z ap)
Bou? G+ a) @ ESS Ee
aD . dh
peter sem (lola tet 8
wee (2+ B) (3)
where T= z- D. At z=D+h, the buoyancy flux
is zero so that
*
Even when the approximation is only fair, the error in as-
suming a linear profile is small. We discuss this in Section
6.
Using (6) we get
ei = = x [(D + sh) Ab - 4N?(D + h) 2] (10)
The integral of (10) is
(D + sh) Ab - 4N2(D + h)? = Vo - q,t (11)
where
We = (oe ‘sh ) Abg - 4N?(D, + ny)? (12)
and the zero subscript denotes values at t = 0.
Tennekes (1973b) obtained (11) and (12) with h and
ho missing. As we have indicated, the interfacial
layer thickness h plays an important role in the
theory of this paper. The time to = vp /a1 is the
time for an initial buoyancy difference to disappear
when the upper air has a uniform potential tempera-
ture [Tennekes (1973b) ].-
3. THE INTERFACIAL LAYER (REGION R3)
According to the discussion in Section 1, the IL
in our model is turbulent with intermittency factor,
I3, defined here as the ratio of the volume in tur-
bulent motion to the whole volume*. Much of the
layer is in wave motion in which all of the compo-
nents of the fluid velocity are of the same order,
i.e., the ratios w3/u3, w3/v3 are independent of
the Richardson number. The intermittent turbulence
is caused by the intermittent breaking of these
waves. Since the wave amplitude is of the order of
the wave length when the wave breaks, we should have
u3 ~ V3 ~ w3 initially in the breaking waves as well
and we assume this. Of course the "homogeneous"
fluid in the breaking patch will tend to flatten
out and the vertical velocities in the patch will
decrease relatively as time goes on. In our model
we ignore the patch after a time of order (h/Ab)%
and consider that the local heat transfer has al-
ready been accomplished. In actual fact this trans-
fer is accomplished by the spreading of the patch
over a larger time interval and the ultimate trans-
fer by molecular processes. Since buoyancy flux
occurs only in the turbulent portions of this layer,
we get, at any level in the IL,
a3 = ~ Byu3b313 (13)
where b3 is the rms buoyancy fluctuation in the
interfacial layer. B , is a universal constant? but
“the introduction of intermittency may result in confusion
if one inadvertently thinks of the IL as a’surface or even
as a layer with thickness of the order of the amplitude of
the wave disturbances. The latter is not excluded as a
possibility in this section but, in fact, as we see in
Eq. (26) the wave amplitude is much smaller than the thick-
ness of the IL so that I is not the ratio of the times that
a fixed point is in the upper (non-turbulent) and lower (tur-
bulent) fluid.
twe use symbols B), Byj,... to denote universal constants.
Later, "constants" arise which, at first glance at least,
may be functions of s = N2/(Ab/h) , i.e., the ratio of the
stabilities of the upper "quiescent" layer and the inter-
facial layer. We denote these "constants" by Aj,Ap,..--
b3, u3, and I3 may vary with height. The turbulence
is certainly strongly influenced by buoyancy in this
layer so that kinetic and available potential ener-
gies [Long (1977d)] are of the same order not only
in the waves but in the turbulent patches, i-.e.,
eS BS, S B05 a (14)
where 63 is the order of the size of the disturbances
and because of the tendency for conservation of buoy-
ancy, we assume b3 is proportional to 63(Ab/h). Us-
ing (14), Eq. (13) becomes
B2B 5
coils erase a ab)
43 aa (4 1 (15)
Let us now find the dissipation. This occurs only
in the turbulent patches and we assume that the
local dissipation Gia = £(u3,63, b3). Since us ~
b353, we get Ep ~ 3/63 and
Bu? ls Ab 45
a
€3 = 13 4 = B,Bju3 (2) T3 (16)
63 h
Equations (15) and (16) show that €3 ~ q3. Since
these are both dissipative, it follows that they are
of the order of the energy flux divergence. At the
upper boundary of the IL, the kinetic energy of the
waves has been so reduced by losses to potential
energy and dissipation, that there can no longer be
wave breaking and turbulence. Thus h is the depth
of penetration of the turbulence. At the height z
= D+h, the energy flux is too weak to support tur-
bulence so that it has apparently decreased to a
value well below that at the bottom of the IL.
Therefore, the increment in energy flux over the IL
is proportional to the value at the bottom of the
IL. Integrating Eq. (3) between levels in the layer
near the upper and lower surface, we find that q3h
is of the order of the energy flux just below the
inversion where q3 is the average buoyancy flux in
R3-. Since the interface is being distorted by the
vertical motions (inducing pressure fluctuations) ,
the energy flux should be proportional to W5P5/00
~ we in Ro. We may write
= 3
q3h SO Aowo (17)
Equation (17) has a form superficially similar to
that proposed by others in a number of papers [for
example Long (1975), Zeman and Tennekes (1977)] on
the basis of assumptions about the size of terms in
the mixed layer. In present notation, these authors
propose qoD ~ we and this leads rather directly to
the Ri-! law for the entrainment. Equation (17) is
really quite different. If the upper fluid is ho-
mogeneous, Ay should be a universal constant. How-
“ever, when the upper layer is stratified, losses of
energy may occur by wave radiation and Ap may then
be a function of s = N2/(Ab/h).
Using (6), (8), (14), (15), (17), we get
3
Agw? dip nh dAb) Ab dh aD
iy See nO EO Be Sn, Se
h ae) cles 9G Ge DS Ge
2 1 d
+ q; - N*“(D + %h) ae (D + h) (18)
1,
=e ma \ 2
= eff,
So = B3 (2) wo (19)
BB 1s
od Ab dAb d
yaa) = ceo Dl a = anes
where the subscript "2" denotes values at a level
just above z = D. Equation (19), which follows from
Eq. (18), is consistent with the assumption that the
pressure fluctuations in eddies in region Rp of fre-
quency wo/59 of order of the natural frequency
(Ab/h)2 are generating the breaking waves by reso-
nance.
4. TURBULENCE IN THE MIXED LAYER
According to (17) the vertical turbulence velocity
in Ro is related to the average buoyancy flux in
the interfacial layer. The latter is related to
the entrainment velocity so that it is essential to
relate w 2 to turbulence in the main portion of the
mixing layer, or to w, = (qb) 173 This is often
called the convective velocity. A great deal of
confusion has arisen regarding this problem because
of two explicit or implicit assumptions often made:
(1) that the turbulence near the interface is quasi-
isotropic, i.e., ug ~ vo ~ wo, and (2) that wo ~ wy.
We will try to show that both of these assumptions
are incorrect*.
In laboratory experiments with mechanical mixing,
measurements indicate that the mean buoyancy gradi-
ent in the mixed layer is very weak and, in fact,
approaches zero as the Richardson number increases
(Wolanski (1972)]. Instantaneously, the lower sur-
face of the interfacial layer is very sharp (perhaps
a discontinuity for infinite Reynolds numbers!) .
This surface is agitated by the disturbances of the
mixed layer so that the mean buoyancy curve varies
continuously, although rapidly in the region, R».
It seems quite safe, however, to neglect effects of
buoyancy on the turbulence of the instantaneous mixed
layer. Let us do this tentatively although we will
return to this point later. Since, for the highly
stable conditions of this paper, the interface dis-
turbances will be very small, the inversion will act
like a 'rigid lid" with slipt and the turbulence will
be similar to turbulence between a rigid heated plate
at z = 0 and a rigid plate az=D. The first ques-
tion to face, then, is the nature of the turbulence
at some level € = D - z near the upper "plate." To
do this, we first consider the findings in two recent
papers by Hunt (1977) and Hunt and Graham (1977) re-
garding the distorting effect of a rigid plane on
homogeneous turbulence. The corresponding labora-
tory experiment is produced by passing air through a
grid in a wind tunnel. The rigid plane is a moving
belt along one wall of the wind tunnel with speed
equal to the mean wind. This serves to eliminate
the shear near the wall and the corresponding energy
source. The wall causes two boundary layers (Fig-
ure 3). One is a very thin viscous layer of thick-
ness dy near the wall in which all three components
of velocity go to zero, and the other, called a
source layer of thickness 6,, extends from the vis-
*
We mean by A ~ B that A/B is finite and non-zero in the limit
as ixul $7 dp
"This is the opinion also of Zeman and Temnekes (1977).
589
FIGURE 3. Turbulence near a wall.
cous layer to a level at which the disturbing ef-
fects of the wall are negligible. The vertical
velocity must decrease throughout the source layer
because it is very small at the top of the viscous
layer, but there is no obvious reason for a decrease
of the horizontal velocity components in the source
layer. This is confirmed by experiment and by the
mathematical analysis by Hunt and Graham who derive
the following results of interest in the present
problem: The rms vertical velocity in the lower
portions of the source layer is wo = B(et) 1/3, where
B is a universal constant and € is the dissipation
function far from the wall, and the rms horizontal
velocities are of the same order as those far from
the wall although somewhat larger. It is useful to
obtain these and other results more intuitively.
In a recent paper, the first author [Long (1977c) ]
has shown that turbulence at high Reynolds number in
a wind tunnel far from a wall is determined com-
pletely by two quantities, K and u/x, where K is a
quantity of dimensions L?T-! characteristic of the
grid and proportional to ul. u is the mean velocity
and x is distance downstream from the grid (or more
accurately from a virtual energy source replacing
the grid). For example, the dissipation function
far from the wall is e€ ~ Ku2/x*, the rms velocity
is u ~ (Ku/x)%, and the integral length scale is
2 ~ (Kx/u)*s.
Obviously the source layer thickness is 6, ~ 2
[Hunt (1977)] and the dissipation in the source
layer is
u
e, = ef a (21)
K ey 2
Just outside of the viscous layer, Es is Eso or
rt
bya?
ein 68 \| Soae (22)
K°x?
INS Ws Op Sy > 0O, and, since ¢€ must be independent
of viscosity for high Reynolds number turbulence,
aq @ Co
At small ¢, eddies of length much less than f
will not feel the distorting effect of the surface
and will be isotropic. Eddies of length much greater
than ¢ will feel the surface very strongly and will
be strongly flattened. Eddies of length of order
& << & will feel the surface but will remain quasi-
isotropic. From the equation of continuity the
large flattened eddies of horizontal dimensions D
yield vertical velocities of order ujZ/D ~ KE/D2.
The quasi-isotropic eddies are much smaller and for
590
high Reynolds numbers will lie in the inertial sub-
range. They will have a spectrum function
2 5
Sg ie ae wd Spall (23)
E
1 (k) oe k Aids (6
where k is the wave number so that ,the contribution
to the vertical velocity is et6°cl/3, This is much
larger than the contribution from the flattened
eddies so that w, ~ elf3zi/3 or we ~el/3 ti/3, as
derived rigorously by Hunt and Graham (1977) .
In the mixing experiments the surface at z = D
is not rigid but is agitated by disturbances of
amplitude 65. Assuming that eddies of this size
are in the inertial subrange, we get vertical veloc-
ities of order BMSINiee and again these, rather than
the eddies of size D, contribute most to the rms.
Then wo ~ el/35,1/3, Since € ~ K3/p4, we get, as
in MISF,
=B (24)
The problem of the present paper is somewhat more
complicated but the distorting effect of the inter-
face should be the same since the buoyancy varia-
tions in the mixed layer are very small. The air
in the main portion of the mixed layer has velocities
of order (q)D)!/3 rather than K/D and in Ro the
buoyancy flux is similar to that in the case of
mechanical stirring. Equation (24) takes the form
3
WwW
Ee BCH (25)
85 :
This result, together with (19), implies wo ~
W*Ri-4(h/D)% , where Ri = DAb/w2, and differs
fundamentally from that of Tennekes (1973b) who
assumed wo ~ w, by arbitrarily equating the buoy-
ancy flux and the energy flux divergence. Tennekes
has acknowledged [Zeman and Tennekes (1977) ] the
inadequacy of this assumption.
The drop-off of w as the interface is approached
is revealed in the data of Willis and Deardorff
(1974). As shown by Hunt and Graham (1977), the
total kinetic energy is the same near the distorting
surface as it is far away so that the horizontal com-
ponents of rms velocity should increase toward the
interface. There is an indication of this also in
the data of Willis and Deardorff.
It is also interesting that we may predict the
same type of behavior near the lower heated surface.
In fact, earlier data of Deardorff and Willis (1967)
as well as the more recent data of Willis and Dear-
dorff (1974) show that the vertical velocity near
the heated plate increases with height, roughly in
accordance with similarity theory [Prandtl (1932)],
but that the horizontal velocity decreases with
height. Thus, it is possible to apply similarity
theory to obtain the vertical component, w, but not
to obtain the horizontal components, u and v. The
dimensional analysis for the horizontal components
at large Rayleigh number must include D as well as
q, and z no matter how small the ratio, z/D! There
are experimental indications that the classical
arguments of "localness" are also incorrect in prob-
lems of turbulent shear flow [Tritton (1977, p.
Using (25), the relations in (18)-(20) and the
expression for wy are
283)].
oat oD 3
Sy a iain is
LONER PS
=p 2a. 8 A) 27
pe techy cel (2 bean)
mies ie oy an = @= hb) = (28)
Bp 119162 = at at a
3
-a,q)2 h\dAb Ab dh 1 aD
2th
=> — —_— + —_- — — —
Spe gus ea ae ( Sig ) cae | 2S ae
ht (Ab) 4
Se = hy ESL ae ey) (29)
BPD sa aie) as
Seanes
3
q
where Q5 = A,B) 1/33 5
5. DIFFERENTIAL EQUATIONS
Equation (29) is a single differential equation in
three unknowns, D, h, Ab. Let us now seek additional
information. The quantity, 03/53, is the dissipa-
tion in the turbulent patches in the interfacial
layer. We have seen that it is independent of Ri
in the lower portions of the layer. Obviously it
will vary continuously with < (now defined as z - D)
in the layer and, to the first order, will remain
independent of Ri although it may vary with the
quantity s = N2h/Ab when the upper fluid has a
linear buoyancy field. We may therefore write
U3 t
oe = Ci 4? (é, s) (30)
or using (14)
4 nye
4 h ic
u3 = B3 Ae) ¥7 (, s) (31)
We may obtain another expression for us by in-
tegrating the energy equation over the interfacial
layer. We have already seen that Je3|~|a3| and
assuming that the energy flux is proportional to
u3 in this layer*, we have from the energy equation
Ww
3
3 = Brigg (32)
Using (8) and integrating, we get
dab (72 3
ud = wi + Bio [ ans + MB(E- =)
(DE) (33)
*
We have seen that the energy flux at the bottom of the layer
1s proporticnal to ugae To the first order it should be pro-
portional to us in the rest of the layer, i.e., independent of
Ri.
Comparing (31) and (33) and using (27), we get
3
ga aes 2 ne a tr eC Ee
Yy (Es) =e ge Tr usp + 25 (34)
where? A3, Ay, and As may depend on s. Equating
coefficients in (34) we get (29) again and the
following
ied
2 4
dAb a Gl ie aor
Ghy WPMD) are" 7 NED) a (D + h) = -a3 3 (35)
D(Ab) #
33
2 rane
Dddb , db aD _ N*D a 7 1
5 Ge Do as a ae (O° 2) = ch 3 (36)
D(Ab) *
32
24
D2 aAb | D2Ab dh q, h
- = + 5 = a5 (37)
6h dat 6h2 at 3
D(Ab) *
my BYAR na ;
where Oj] = A;/B)2B3 (i = 3,4,5). Equations (35)-
(37), (29), and (11) are five equations in the three
unknowns. They determine the solution to the first
order for large Ri, although we must make sure that
all equations are satisfied to that order. In this
regard, if we use (35)-(37), (29), and the deriva-
tive of (11), i.e., (10), we may consider these as
five homogeneous linear, algebraic cquat tous in five
EET SA FA dAb/dt, dD/dt, dh/dt, q,, and qi? 24374 7p
(Ab) 3 +. The determinant of these eguations vanishes
and we satisfy compatibility.
6. HOMOGENEOUS CASE (N = 0)
If N = O, the upper fluid is homogeneous and (11)
becomes
1 2
(D + 5h) Ab = Vp - ayt (38)
2 9 Aad
where Vp is a constant related to initial conditions.
We use (35) and (38) to eliminate Ab in (29), (36),
and (37). We get
3 3 3
9.4 4 4
d a, °h (D sr 2 )
at (Ab) = - ea a3 52 ViEgaeene i! (39)
(V5 - q,t)
dh qih (D+ oh)
dt D
(vi = Cite)
Si o
q, 2h" ines tn)
+ Cy = 6a5 D = 7 = (0) (40)
ON = qyt)*
+ 5 be PD: A
.For arbitrary Ri, the quantities A,; may depend on Ri. As
Ri > ©, however, A, will approach "constants" which may, of
course, be zero.
591
1
dap h (D+ gh)
SG
at 1 >) 4
(Vo - q,t)
a7 As
aon (D + 5h) 4
= (a3 + 204) =0 (41)
D2 2 Z
(Vo - a, t)
1
Bc e anya penap ny aie
dt dt 2) De
(Vo - q,t)
307 7
ae
D2 D q,°h! (D + zh)"
sue 6a9 oe 603 a 203 = (0) (1))
h 2 f
Two effects occur in (40) and (41). We may separate
them by adding the two equations. We get
3 7
24 1 4
él h\ 41 h }D) Sr oh
—_ + = + _ _
dt (D h) (2u, 6a5 ae 2 (43)
Vo ~— qt
The term on the right of (43) expresses the upward
motion of the boundary between (intermittently)
turbulent and non-turbulent fluid due to turbulence
in the interfacial layer causing entrainment of the
upper, non-turbulent fluid. On the other hand, the
second terms in (40) and (41) express the upward
motion of the boundary between fully turbulent and
intermittently turbulent fluid (and the consequent
decrease of h) due to heating alone. This contribu-
tion to the entrainment velocity is proportional to
the interfacial thickness, h, and disappears when
the common approximation is made that h = 0.
Let us find an approximate solution to (40)-(42).
If we let Dp and ho be the values of D and h at t =
0, we make the following definitions:
ho h D
S'S =) = h = = jp
Do a , Do 1 , Do i
2 2 1
3 3 a J.
q
a8 pet (44)
Vo Do3
Then equations (40)-(42) may be written
1
+ 5 hj)
dh i By, Uo
dt Dy (1 - 6t)
L, ilies
4 ii yoy
2, ye SOE Ze =0 (45)
+ a3 > 645 Dy 2 Z
: Dy (AL > Ox)
1
aD, hy (D, ar zhy) ;
dt Dy (lL = Or)
ih 1 Ho
GD, & Bin, ) B08
2 (Gg > 2p) 7 0 (46)
Do (2 - 6t)*
592
D Dj hi
(602 ne 6a3 hy + 604 + 605 7
72 Le
q 1 rar
ot SS ST ee, (47)
2 £
Dy (1 - at)4
Solutions are of the form
h i q,t
—_ = _ + = —_
mh aL (al 74) ve
3 7 sa
= < 2n2
4 Pace fay Das
= A(Cley = Seley) Gi: (al + $a) at
v2
0
Qe
mate
+5 a(2t+ayo 1+... (48)
Vo
Gast
Zen aG so) =
Do 2 V6
atl
fr il Dee Dye
ae (ie), ae Aton) ery (al ae 58) aetricn
Vo2
Gace
- 5 a2(2 + a)? ae (49)
Vo
Pee 2 Se
vy 2+a ve
0
Sel
iL 1 c=
7 oF Pip ne
O13 ) a a y qy Do
G 204 3asa}a{l+ 5 z
Vo
2 D2
a(2 + a) qjt
ee aC SOO (50)
4
4 Vo
where hg and Dp are related by the equation
6a9 - 6a3a + 6ay4a2 a 6as5a2 = 0 (51)
The entrainment velocity, ug = dD/dt, may be ex-
pressed in terms of the Richardson number, Ri =
DAb/w? , by using
6(1 + Sa) = fal a 25)
Y 2
np Ce (Ox 4 2a4)a* (52)
The first term is of the same form as the non-
dimensional entrainment velocity of Tennekes (1973)
but, as already pointed out, the derivation and
physical mechanism are very different. It is easy
to trace the error in (52) arising from the simpli-
fication of Section 2 that the IL has a linear
buoyancy field. The error is proportional to
(u,/w,) ab./Ab where b, is the maximum difference
between the actual buoyancy in the IL and the as-
sumed buoyancy. Since a is 1/6 or so and be/Ab is
fairly small, this error is negligible. Notice also
that the theory concerns strongly stable conditions
so that (52) does not apply in the limit as Ru, == Op
As Ri tends to order one ue becomes of order wy as
one would expect.
The ratio go/q, is of interest. Using (6), we get
a2 D_ dAb
— si ¢ = S— (53)
qi qi dat
Using (50) and (52), we get
Z
3 £
a2 a TR = a
|| = yRi * , y = — (3+ 2ay + asa) (54)
ql (ea)
2
The expressions (26)-(28) are
1
Td 5 3
Cy aguas Op BHae: Se
—_—= at atri # = 8 ua Rea '
We im iB 3.3
B3! rears
B3 alt
3
I SeE Rae (55)
2 ak ae ee
BiB) *
These relations are identical to those in MISF. The
result that the disturbances in the IL are small
compared to the thickness of the IL is contrary to
speculation [Stull (1973) and Zeman and Tennekes
(1977)] that h is the depth of penetration of the
eddies into the stable region.
7. LINEAR BUOYANCY FIELD IN THE UPPER LAYER (N # 0).
We consider initial conditions in which the fluid is
at rest initially with a linear buoyancy field, so
that D, h, Ab are zero at t = 0. Equation (11) be-
comes
n2
(D + sh)ab = (D+ ny? = - gt (56)
Equations (56), (29), and (35)-(37) determine the
problem. The approximate solutions* are
*
The solutions, as throughout the paper, are for strong
stability, which implies here that Nt is large. Thens +1,
and %9,043,4,45 are independent of s.
i L
2
(2q)t) (2q)) “by
= ,
N Ne
1
l 2
= (2q,)“b
2a 1 2
h = (2q)t) z + 3
n2
at ee
Ab = (2q,t)2aN + (2q)) 2b3N? (57)
where
1
2: b a
a=-1+(Q+2 Pyles ME Se) \ See L
Oy ay i ay 3
22
1
ao 2 ide} = lok) =
—=a*(1 +a) , ———= 22a (58)
(omn 3) Oy
Using the relationship
3 3 3
42 uf b b aS
Ri 3a 1 sl Ban oh
Nt = 3 1 eee Ri (59)
2a? 22
we obtain for the entrainment velocity
u x Tek math
— = ari7! + 22b,a'Ri * (60)
We
The ratio of fluxes is
3 ees
|*| = 22p aki 4 (61)
qi
Notice that Ab/h > N2 as t > © so that the IL be-
comes indistinguishable from the upper layer as the
turbulence in it weakens (becomes more intermittent).
This contrasts with MISF in which the stability in
the IL is several times larger than the stability
in the upper fluid. Notice also that s + 1 implies
09...d5 are universal constants. More accurately,
Nie
[Nd 2.
ae = 1+ ay (<2) (62)
We see from (32) that a, > 0 so that the buoyancy
gradient in the IL is more stable than in the air
above. These results suggest that an interfacial
layer will be difficult to identify when there is
a stable buoyancy gradient aloft. This is certainly
the case in the experiments of Deardorff, Willis,
and Lilly (1969) and Willis and Deardorff (1974).
8. DISCUSSION
We have already contrasted the theory of this paper
with that of Tennekes (1973). He obtains
593
Ue OY
Beanie (63)
*
which has the same form as the first term in (52)
or (60). The present theory should not, however,
be regarded as an extension or modification of the
Tennekes' theory because, as we have noted in sev-
eral places, the two theories differ fundamentally.
This is also evident in the difference in the nature
of the two constants of proportionality for the
Ri71 term in the two theories. The a, in (63) may
be identified physically as the ratio |qo/q | which
is a universal constant in the Tennekes' theory.
The constant, a, in (52) or (60), however, is a
universal constant equal to the asymptotic value
of the ratio of the inversion layer thickness to
the thickness of the mixed layer. Tennekes assumed
a value of 0.2 or so for a; and it is a coincidence
that this is also a reasonable choice for a.
We may attempt to estimate the constants in the
expressions
7 3
u a air qo a) oe
e ates a. 4 0
— = aRi7! + cRi * f Asal 4 (64)
Wy ql
using the data of Willis and Deardorff (1974)*.
Approximate estimates for the two cases:
Sily iD) = Se) Guy, in = S) on, Ae S 157%,
0.39 cm/sec, Qo = 0.18°C cm/sec,
WwW 1.3 cm/sec, Ri = U5 5,5 4 = Ostler
¢ = 1.09, y = 1.61
B48) te) S FSeidp iy SS 55) vem, (aw = Be,
A
He the
0.69 cm/sec?, Qo = 0.22°C cm/sec,
1.4 cm/sec, Ri = 20, a = 0.15,
@ = 1,05, 7 = 1.05
Ee top
te ak
We may also attempt to compare with atmospheric data.
For example, using the 1200 observation on Day 33
for the Wangara data, [Zeman and Tennekes (1977)],
we obtain
2 S10" Guy AO S 2°C;,
20°C cm/sec,
3, 4 = O58 C= 22
D = 1.1 10°cm, h =
13 cm/sec’, Qo
194 cm/sec, Ri
Ile
Cc
o
Idk
These computations indicate that the two terms in
the expression for u, in Eq. (64) are roughly
similar in magnitude for atmospheric and laboratory
conditions.
It is interesting to compare the theory of the
erosion of a linear buoyancy field with a numerical
experiment of Zeman and Lumley (1977) using a
second-order closure model. The numerical calcula-
tion began from an initial instant, ty, at which
eshte) IY S Dip Wa = Wag) = (qaDo)it7 2. The present
theory sat timelt) =) tS tp ais
D T 9)
— = te SP ooo Sn) = WDE AW
Do i So 7 SH) Uf
where we have assumed that (tgN)’s is large. The
numerical’ curves [Figure 1 of the paper of Zeman
and Lumley] are nearly linear after tT exceeds 2 or
so although, as (57) would indicate, D/Dop increases
somewhat more slowly after considerable time. The
i .
Supplemented by information in a personal communication from
Dr. Willis.
594
FIGURE 4. Comparison of present theory and numerical
experiment of Zeman and Tennekes (1977). The curves
correspond to values of S, in (65).
most important comparison, however, is that the
curves of Zeman and Lumley for various Sg collapse
rather well when plotted against t/Sg instead of Tt
as in Figure 1 of Zeman and Lumley. Conversely, we
may reproduce Figure 1 of Zeman and Lumley together
with plots of D/Do in (65) for the same values of
Sg chosen by Zeman and Lumley. This is shown in
Figure 4. The agreement is good, especially at
large stabilities where the approximation in (65)
should be best. This indicates that the two models
have some similar features.
ACKNOWLEDGMENT
This research was supported by the National Science
Foundation under Grant Nos. ATM/6-22284, ATM/6-04050,
and OCE76-18887, and by the Office of Naval Research,
Fluid Dynamics, under Contract No. NO0014-75-C-0805.
REFERENCES.
Adrian, R. J. (1975). Turbulent convection in water
over ice. J. Fluid Mech. 69, 753.
Ball, F. K. (1960). Control of inversion height by
surface heating. Quart. J. Roy. Met. Soc. 86,
483.
Betts, A. K. (1973). Non-precipitating cumulus con-
vection and its parameterization. Quart. J. Roy.
Met. Soc. 99, 178.
Carson, D. J. (1973). The development of a dry,
inversion-capped convectively unstable boundary
layer. Quart. J. Roy. Met. Soc. 99, 450.
Crapper, P. F., and P. F. Linden (1974). The struc-
ture of turbulent density interfaces. J. Fluid
Mech. 65, 45.
Deardorff, J. W. (1967). Empirical dependence of
the eddy coefficient for heat upon stability
above the lowest 50 m. J. Appl. Meteor. 6, 631.
Deardorff, J. W. (1972). Parameterization of the
planetary boundary layer for use in general cir-
culation models. Mon. Wea. Rev. 100, 73.
Deardorff, J. W., G. E. Willis, and D. K. Lilly
(1969). Laboratory investigation of non-steady
penetrative convection. J. Fluid Mech. 35, 7.
Farmer, D. M. (1975). Penetrative convection in
the absence of mean shear. Quart. J. Roy. Met.
Soc... LOD, 869)
Heidt, F. -D. (1977). Comparison of laboratory ex-
periments on penetrative convection with measure-
ments in nature. Heat Transfer and Turbulent
Buoyant Convection, 1. D. Spalding and N. Afgan,
ede), 199-2010;
Hopfinger, E. J. and J. A. Toly (1976). Spatially
decaying turbulence and its relation to mixing
across density interfaces. J. Fluid Mech. 78,
I1G)5}<
Hunt, J. C. R. (1977). A review of the theory of
rapidly distorted turbulent flows and its appli-
cations. XII Biennial Fluid Dynamics Symposium,
Poland.
Hunt, J. C. R., and J. M. R. Graham (1977). Free
stream turbulence near plane boundaries. J. Fluid
Mech. (In press).
Izumi, T. (1964). Evolution of temperature and
velocity profiles during a breakdown of a noctur-
nal inversion and a low level jet. JG. Appl.
Meteor. 3, 70.
Kaimal, J. C., J. C. Wyngaard, D. A. Haugen, O. R.
Cote, Y. Izumi, S. J. Caughey, and C. J. Readings
(1976). Turbulence structure in the convective
boundary layer. J. Atmos. Sci. 33, 2152.
Kraus, E. B., and J. S. Turner (1967). A one-
dimensional model of the seasonal thermocline.
Part II. The general theory and its consequences.
Tellus) 19), 98.
Kuo, H. ie, and Ww. Y. Sun (1976). Convection win
the lower atmosphere and its effects. J. Atmos.
Seals Bp Zils
Lenschow, D. H. (1970). Airplane measurements of
planetary boundary layer structure. J. Appl.
Meteor. 9, 874.
Lenschow, D. H. (1974). Model of the height varia-
tion of the turbulence kinetic energy budget in
the unstable planetary boundary layer. J. Atmos.
Sexo sel, AXS5\5
Lettau, H. H., and B. Davidson (1957). Exploring
the Atmosphere's First Mile., Vols. 1 and 2.
Pergamon Press, New York.
Lilly, D. K. (1968). Models of cloud-topped mixed
layers under a strong inversion. Quart. J. Roy.
Meteor. Soc. 94, 292.
Linden, BP. 'F.' (1973). The interaction o£ jay vortex
ring with a sharp density interface: a model
for turbulent entrainment. J. Fluid Mech. 60,
467.
Linden, P. F. (1975). The deepening of a mixed
layer in a stratified fluid. J. Fluid Mech. 71,
385.
Long, R. R. (1975). The influence of shear on mix-
ing across density interfaces. J. Fluid Mech. 70,
305.
Long, R. R. (1977a). A theory of turbulence in a
homogeneous fluid induced by an oscillating grid.
Submitted to Physics of Fluids.
Long, R. R. (1977b). A theory of mixing in a stably
; stratified fluid. Submitted to Physics of Fluids.
Long, R. R. (1977c). The decay of turbulence. Sub-
mitted to Physics of Fluids.
Long, R. R. (1977d). Some aspects of turbulence in
geophysical systems. Advances in Applied Me-
eChaniiicsil/a melts
Monin, A. S. and A. M. Yaglom (1971). Statistical
Fluid Mechanics: Mechanics of Turbulence 1. The
MIT press., Mass.
Moore, M. J., and R. R. Long (1971). An experimental
investigation of turbulent stratified shearing
flow. J. Fluid Mech. 49, 635.
Nawdiern, P. Pi. .and E., Bi. Kraus) (9/77). \One=
dimensional models of the upper ocean. Modelling
and Prediction of the Upper Layers of the Ocean,
E. B. Kraus, ed., Pergamon Press, Oxford, England,
pp. 143-172.
Prandtl, L. (1932). Meteorlogische Andwendungen der
Strémungslehre. Beitr. Phys. fr. Atmos. 19, 188.
Rouse, H. and J. Dodu (1955). Turbulent diffusion
across a density discontinuity. Houille Blanche
10, 405.
Stull, R. B.(1973). Inversion rise model based on
penetrative convection. J. Atmos. Sci. 30, 1092.
Stull, R. B. (1976a). The energetics of entrainment
across a density interface. J. Atmos. Sci. 33,
1260.
Stull, R. B. (1976b). Mixed-layer depth model based
on turbulent energetics. J. Atmos. Sci. 33, 1268.
Stull, R. B. (1976c). Internal gravity waves gen-
erated by penetrative convection. J. Atmos. Sci.
33, IAT).
Summers, P. W. (1965). An urban heat island model:
Its role in air pollution problems with special
application to Montreal. Proc. First Canadian
Conf. Micrometeorology, Toronto, Atmos. Environ.
Service.
Tennekes, H. (1973a). Similarity laws and scale
relations in planetary boundary layers. Workshop
on Micrometeorology, D. A. Haugen, ed., AMS,
Boston.
Tennekes, H. (1973b). A model for the dynamics of
inversion above a convective boundary layer. J.
AEMOS a SCL SO) DD Sl.
595
Tennekes, H. (1975). Reply (to Zilitinkevich, S.
Soo Mo Aewess Metis st, hq
Thompson, S. M., and J. S. Turner (1975). Mixing
across an interface due to turbulence generated
by an oscillating grid. J. Fluid Mech. 67, 349.
Tritton, D. J. (1977). Physical Fluid Dynamics,
Van Nostrand Reinhold, New York.
Turner, J. S. (1968). The influence of molecular
diffusivity on turbulent entrainment across a
density interface. J. Fluid Mech. 33, 639.
Turner, J. S. (1973). Buoyancy Effects in Fluids,
Chapter 9, Cambridge Univ. Press.
Veronis, G. (1963). Penetrative convection. Astro-
phys. J. 137, 641.
Willis, G. F., and J. W. Deardorff (1974). A labora-
tory model of the unstable planetary boundary
layer. J. Atmos. Sci. 31, 1297.
Wolanski, E. J. (1972). Turbulent entrainment across
stable density-stratified liquids and suspensions.
Ph.D. Thesis, The Johns Hopkins Univ., Baltimore,
Md.
Wolanski, E. J., and L. M. Brush (1975). Turbulent
entrainment across stable density step structures.
RELI) Ap PBa)c
Zeman, O. and J. L. Lumley (1977). A second-order
model for buoyancy driven mixed layers. Heat
Transfer and Turbulent Buoyant Convection, Vol. l,
D. Spalding and N. Afgan, ed., pp. 65-76.
Zeman, O., and H. Tennekes (1977). Parameterization
of the turbulent energy budget at the top of the
daytime atmospheric boundary layer. J. Atmos.
seit, S44, dali,
Zilitinkevich, S. S. (1975). Comments on 'A Model
for the Dynamics of the Inversion above a Con-
vective Boundary Layer.' J. Atmos Sci. 32, 991.
Laboratory Models of Double-Diffusive
Processes in the Ocean
J.
Stewart Turner
Australian National University
Canberra,
ABSTRACT
There is now good observational evidence to support
the ideas that double-diffusive processes, i.e.,
those for which the differential diffusion of heat
and salt are important, can affect the rates of
vertical transport of these properties in the ocean,
and are responsible for the formation of certain
types of microstructure. Much of our detailed
understanding of these effects has come from related
laboratory experiments, but new phenomena are still
being discovered which are as yet untested by direct
measurements in the ocean. It is the purpose of
this paper first to review the background to this
Australia
subject, and then to describe the more recent experi-
ments which suggest further double-diffusive effects
likely to be significant in various oceanographic
contexts.
A convenient laboratory technique has been to
use two solutes (commonly salt and sugar) to model
the T-S variations; some of these experiments with
closer diffusivities are in fact directly relevant
to the ocean. When more than two diffusing compo-
nents are present it has been shown that even small
differences in molecular diffusivity can signifi-
cantly affect the relative rates of transfer of
solutes through an interface, and this should be
considered more carefully in geochemical studies.
Strong double-diffusive layering is often associated
with large horizontal gradients of T and S, and
related effects have been studied in our laboratory
in three different geometries: the circulation
produced by a block of ice in a salinity gradient;
a line source of one fluid intruding at its own
density level into a gradient with different prop-
erties; and the spreading across a frontal surface
separating two fluids having the same vertical
density but different T-S structures.
596
1. INTRODUCTION
It is past the stage when the relevance of double-
diffusive effects has to be justified ab initio to
an audience of oceanographers. Over the last few
years, there have been many observations of fine-
structure and microstructure in the deep ocean
which can only be explained in these terms. Wherever
there is a systematic association between T and S
variations, with both properties increasing or
decreasing together (so that their effect on the
density is in opposite senses), then it is clear
that the difference in molecular diffusivities for
heat and salt can affect the vertical structure
and the transports of the two properties. It is
not then sufficient to base predictions of mixing
on the net density distribution alone.
Our understanding of these processes has been
greatly influenced by related laboratory experiments
[see Turner (1973, 1974)]. Much of the detailed
work has concentrated on the properties of sharp in-
terfaces separating relatively well-mixed layers:
it has been shown that when there are compensating
T-S gradients, a smoothly stratified water column
typically breaks up into a series of steps, and
molecular processes must be more important across
such interfaces. Once layers have formed there
remains little doubt that the coupled transports
can be estimated using the laboratory results. It
is much less certain, however, that the processes
of formation of layers have always been adequately
modelled in the laboratory, where most of the experi-
ments have been one-dimensional in form.
More recent experiments [Turner and Chen (1974),
Huppert and Turner (1978), Turner (1978)] have begun
to explore a variety of two-dimensional effects, and
it is these which will be given most attention in
the verbal presentation of this paper. It should be
admitted right at the beginning that these experi-
ments are still largely qualitative, and that much
more remains to be done, but already they suggest
new explanations of some existing observations in
the ocean, and allow us to predict what might be
measurable in future work.
2. ONE-DIMENSIONAL PROCESSES
Formation of Layers from a Gradient
For completeness, the fundamental physics of the
double-diffusive convection will be outlined briefly
by referring to the simpler early experiments. The
review of one-dimensional experiments will then be
brought up to date and specific oceanographic
examples of these processes will also be described.
The necessary conditions for double-diffusive
convection to occur in a fluid are firstly that
there should be two or more components having
different molecular diffusivities, and secondly
that these components should make compensating
contributions to the density. It is remarkable
that under these conditions strong convective
motions can arise even when the net density distri-
bution increases downwards. The overall density is
"statically stable' in this sense in all the cases
described here. Motions are nevertheless generated
since the action of molecular diffusion, at different
rates for the two components, makes it possible to
release the potential energy in the component which
is heavy at the top. This can drive convection in
relatively well-mixed layers, while the second
(stably distributed) component preserves the density
difference across the interfaces separating them.
There are two cases to be considered, depending
on the relation between the diffusivities and the
density gradients, i.e., on whether the driving
energy comes from the component having the higher
FIGURE 1.
Layering produced from an initially
smooth salinity gradient by heating from below.
Three well-mixed layers are marked by fluorescein
dye, lit from the top. (Tank diameter, 300mm. )
597
or lower diffusivity. The simplest example of
the former is a linear stable salinity gradient,
heated from below. An unstratified tank would over-
turn from top to bottom, but because of the stabi-
lizing salinity gradient only a thin temperature
boundary layer is formed at first, which breaks
down through an overstable oscillation [Shirtcliffe
(1967)] to form a shallow convecting layer. This
layer grows by incorporating fluid from the gradient
above it, in such a way that the steps of S and T
are nearly compensating, and there is no disconti-
nuity of density, only of density gradient.
When the thermal boundary layer ahead of the
convecting region reaches a critical Rayleigh number,
it too becomes unstable. A second layer then forms
above, and eventually many other layers form in
succession (See Figure 1). The vertical scale of
these layers increases as the heating rate is
increased, and decreases with larger salinity gra-
dients. Turner (1968) has shown that the first
layer stops growing when
Je =D Nee (1)
Here d, is the critical depth, D is a dimensional
constant which depends on the critical Rayleigh
number and the molecular properties, B = -gaFm/pC
is the imposed buoyancy flux corresponding to a
heat flux Fp (a being the coefficient of expansion
and C the specific heat), and Ng = [(g/p) (dp/dz]%
is the initial buoyancy frequency of the stabilizing
salinity distribution. The criterion for the for-
mation of further layers is currently being studied
by Huppert and Linden (personal communication).
A device which has proved very helpful in elim-
inating uncontrolled sidewall heat losses (as well
as providing results directly relevant to the ocean)
is to carry out experiments with two solutes, say
sucrose and sodium chloride solutions, instead of
salt and heat. Essentially the same phenomena can
be observed, although the diffusivities are much
more nearly equal (the ratio T = Ksg/Kz, where Km
denotes the larger and ks the smaller diffusivity
in each case, is about 1/3 for sugar and salt,
compared with = 1072 for salt and heat).
Linden (1976) has in this way extended the
"heated gradient" experiments to study the case
where there is a destabilizing salt (T) gradient
partially compensating the stabilizing sugar (S)
gradient in the interior. He has shown, both
theoretically and experimentally, that during the
formation of layers the relative contributions of
the energy provided by the boundary flux, and that
released in the interior, change systematically
with the ratio of the vertical T and S gradients.
In the limit where these gradients become equal,
all the energy comes from the destabilizing compo-
nent in the interior, and the ultimate layer depth
is finite and proportional to N,-% (where N, is the
buoyancy frequency corresponding to the stabilizing
component) .
Once layers and interfaces have formed, it is
important to understand what governs the fluxes of
S and T across them. For this purpose two or more
layers can be set up directly, and the interfaces
examined using a variety of optical techniques.
For example, Figure 2 is a shadowgraph picture of
a very sharp interface formed between a layer of
salt solution above a layer of sugar solution, which
is equivalent to colder fresh water above hot salty
water. Note that salt is here the analogue of heat,
FIGURE 2. Shadowgraph picture of a sharp "diffusive"
interface, formed between a layer of salt solution above
a denser sugar solution. Note the convective plumes each
side of the interface, evidence of strong interfacial
transports. (Scale: the tank is 150mm. wide.)
and sugar the analogue of salt, since in each case
the convection is maintained, and the interface
kept sharp, by the more rapid vertical transfer of
the faster diffusing component. Such interfaces
have been called "diffusive interfaces", for reasons
which will become clearer in the following section.
Fluxes through Diffusive Interfaces
Quantitative laboratory measurements have been made
of the S and T fluxes across the interface between
a hot salty layer below a cold fresh layer, and
they have been interpreted in terms of an extension
of well-known results for simple thermal convection
at high Rayleigh number. Explicitly, Turner (1965),
Crapper (1975), and Marmorino and Caldwell (1976)
have shown that the heat flux oF», (in density units)
is described by
473
OF = AL (aAT) (2)
where Aj has the dimensions of velocity. Fora
specified pair of diffusing substances, A, is a
function of the density ratio Ro=BAS/aAT, where 8
is the corresponding "coefficient of expansion"
relating salinity to density differences. The
deviation of A, from the constant A obtained using
solid boundaries, with a heat flux but no salt flux,
is a measure of the effect of AS on Fm. When Rp
is less than about 2, A, > A due to the increased
mobility of the interface, and when Rp > 2, Ay
falls progressively below A as R, increases and
more energy is used to transport salt across the
interface. The empirical form
A,/A = 3.8 (BAS/aAT) 7 (3)
{Huppert (1971)] provides a good fit to the obser-
vations over the whole of the measured range 1.3<
Rp <7.
The salt flux also depends systematically on
Rp, and has the same dependence on AT as does the
heat flux. Thus the ratio of salt to heat fluxes
(both expressed in density units) should be a
function of Rp alone for given diffusing substances:
BF./oF, = £, (8AS/aAT) (4)
The results reproduced in Figure 3 support this
relation, and they also reveal the striking feature
that the flux ratio is substantially constant (=0-.15)
for 2<R, <7. [The more recent experiments of
Marmorino and Caldwell (1976) suggest that the flux
ratio can be as high as 0.4 with much smaller heat
fluxes, but the reason for this discrepancy is not
yet resolved]. Experiments by Shirtcliffe (1973),
using a layer of salt solution above sugar solution,
have shown a much stronger dependence of Fm on Rp
than (3), but again a constant flux ratio, the
measured value (for NaCl and sucrose) being
BF</OF p x 0.60. Note that the flux ratio must
always be <l, for energetic reasons: the increase
in potential energy of the driven component must
always be less than that released by the component
providing the energy. This implies that the density
difference between two layers must always increase
as a result of a double-diffusive transport between
them.
Direct measurements through the interface in
Shirtcliffe's experiment suggest that this has a
diffusive core, in which the transport is entirely
molecular, and which is bounded above and below by
unstable boundary layers. The "thermal burst"
model of Howard (1964) has recently been extended
to this two-component case by Linden and Shirtcliffe
(1978), to predict both the fluxes and flux ratios.
The constant range of flux ratio can be explained
in the following way. Boundary layers of both T
and S grow by diffusion to thicknesses proportional
to Ken and Keer and then both break away intermit-—
tently. If only the statically unstable part at
the edge of the double boundary layer is removed
(such that aAT=8AS), then the fluxes will be in
the ratio tz, in reasonable agreement with the
laboratory results for the two values of T used.
Linden (1974a) has given a mechanistic argument to
explain the increase of flux ratio at lower values
of Ry, which he attributes to the direct entrainment
of both properties across the interface.
It is worth noting in passing that Huppert (1971)
10
0-8
FIGURE 3. The ratio of the fluxes of salt and heat (in
density units) across an interface between a layer of
hot, salty water below colder, fresh water, plotted as
a function of the density ratio R.. [From Turner
(1965) .] "
has shown theoretically that an intermediate layer,
or series of layers, is stable if the overall S
and T differences lie in the range where the flux
ratio is constant, and unstable if the flux ratio
varies with Rj. Observations of stable layers in
the ocean seem to be consistent with this criterion.
The merging of layers by this and other mechanisms
has been studied experimentally by Linden (1976).
Some measurements have also been made in the
case where several solutes with different diffusivi-
ties, Ky, are driven across an interface by heating
from below. Turner, Shirtcliffe, and Brewer (1970)
showed that the individual eddy-transport coeffi-
cients can be different, and suggested that they
are proportional to yee More recent work by
Griffiths (personal communication) predicts theo-
retically that the ratios of transports of pairs
of solutes should be proportional to T2 at low
solute-heat density ratios, and to Tt at higher
ratios. His much more accurate and extensive
experiments show an even larger variation, for
reasons which are still unexplained. These results
are potentially of great importance for the inter-
pretation of geochemical data, as will be discussed
further below.
Observations of Diffusive Interfaces
There are now many observations of layering in the
ocean which can unambiguously be associated with
"diffusive" interfaces, and where a one-dimensional
interpretation seems appropriate. The regularity
of the steps and the systematic increase of both
S and T with depth serves to distinguish these from
layers produced in other ways (by internal wave
breaking, for example). Neal et al. (1969) and
Neshyba et al. (1971) have observed layers about
5 m thick, underneath a drifting ice island in the
Arctic where cold fresh melt water overlies warm
salty water. A common observation in Norwegian
fjords is that cold fresh water, formed by melting
snow, can often form a thin layer on top of warmer
seawater, with an interface which remains extremely
sharp, and thickens much less rapidly than expected.
This is due to double-diffusive convection driven
by the heat flux from below, which will stir the
layers on each side of the interface (independently
of any wind stirring at the surface) and thus keep
the interface sharpened.
There are also fresh-water lakes in various parts
of the world which have become stratified in the
past by the intrusion of sea water. Some of these
are heated at the bottom by solar radiation, and
convectively mixed layers separated by diffusive in-
terfaces are formed. A particularly well-documented
example is Lake Vanda in the Antarctic [Hoare (1968),
Shirtcliffe and Calhaem (1968)]. Since these lakes
are not complicated by horizontal advection pro-
cesses, Huppert and Turner (1972) were able to
use the Lake Vanda data to show that the one-
dimensional laboratory result (3) can be applied
quantitatively to comparable large-scale motions.
Other striking examples are the multiple steps
observed in a lake in the East African Rift zone,
which is heated geothermally by the injection of
hot saline water at the bottom [Newman (1976)],
and the layers of hot salty water found at the
bottom of various Deeps in the Red Sea [Degens and
Ross (1969)]. These layers are nearly saturated
with salts of geothermal origin, including a high
599
proportion of heavy metals, and are of special
interest because of the potential commerical value
of the associated thick sediments. [Another related
application, to the genesis of ore deposits on the
sea floor, has recently been proposed by Turner and
Gustafson (1978)].
The existence of many components in these layers
raises another question which should be explored
more systematically in the oceanic context.
Griffiths' laboratory measurements mentioned above
indicate that different solutes are transferred
across diffusive interfaces at different rates,
depending on their molecular diffusivities. The
"Mixing rate" for a tracer is thus not necessarily
a good indicator of the transport of a major com-
ponent if interfaces are important. In the absence
of definite knowledge of the mixing mechanisms
which have operated between the sources and the
sampling point, the assumption that all components
are mixed simultaneously (i.e., that a single "eddy
diffusivity" should be used) seems likely to lead
to large errors, and even to gross misinterpretations
of geochemical data.
Double-diffusive processes can also be important
in other systems besides aqueous solutions. A
situation of oceanographic interest arises if liquid
natural gas (LNG) or some other liquid gas spills
(following a tanker accident for instance) onto
the sea surface [Fay and MacKenzie (1972)]. The
liquid quickly evaporates to form a layer of cold
gas, predominantly methane, which would be lighter
than the air above it except that it is much colder.
Since methane, and also water vapour picked up from
the sea surface, have larger diffusivities than heat
in air, double-diffusive effects can again be
important in this gaseous system. The driving
energy comes from the distribution of methane and
water vapour, so the interface is "diffusive".
The limited observations available suggest that
the top of such a layer is very sharp, and its rate
of spread vertically small, which is consistent
with a self-stabilizing double-diffusive transport
across the interface. Another application, to
explain the phenomenon of "rollover" in LNG storage
tanks, will not be described in detail here, but it
too depends on double-diffusive effects, this time
in the liquified gas [see Sarsten (1972)].
Salt Fingers and Related Phenomena
We now turn to the second type of double-diffusive
convection, that for which the driving energy is
derived from the component having the lower molecular
diffusivity. Though this is associated with the
very different phenomenon of "salt fingers", there
are many similarities between it and the "diffusive"
case already presented, and these will be emphasized
in the following discussion.
When a small amount of hot salty water is poured
on top of cooler fresh water, long narrow convection
cells or "salt fingers" rapidly form. These motions
were first predicted by Stern (1960) [and see Stern
(1975) for a more up to date account of the theoret-—
ical work]. They are sustained by the slower
horizontal diffusion of salt relative to heat, which
permits the release of the potential energy in the
salt field. Again, fingers may be produced using
two solutes with much closer diffusivities, and
when there are strong contrasts of properties, the
fingers are confined to an interface. Figure 4
600
FIGURE 4.
Shadowgraph of a thickened "finger" inter-
face, formed between a layer of sugar solution on top
of salt solution. (Scale: the tank is 150mm. wide.)
shows a shadowgraph picture of such an interface
between a layer of sugar solution (S) above heavier
salt solution (T). This is bounded by sharp edges,
where the fingers break down and feed an unstable
buoyancy flux into the convecting layers on either
side.
Finger interfaces between two such layers have
been shown to thicken linearly in time [Stern and
Turner (1969), Linden (1973)]. They have also been
observed in plan by Shirtcliffe and Turner (1970)
who showed that the convection cells have a square
cross section, with upward and downward motions
alternating in a close-packed array. The initial
stability of an interface has been examined quanti-
tatively by Huppert and Manins (1973). Whena
layer of S is placed on a layer of T, the sharp
boundary thickens by diffusion; the condition for
formation of fingers depends on the magnitude of
the gradients and the ratio of diffusivities, T,
and is related to the overall differences by
BAS/aAT > tv¥t . (5)
These results can be extended to three components,
as can the earlier linear stability theories
[Griffiths (1978)]. For heat and salt, (5) shows
that fingers should form with very small destabiliz-
ing salinity differences, and suggests that they
will be ubiquitous phenomena in the ocean.
Our confidence in applying these results ona
geophysical scale has recently been increased greatly
by the direct observations of fingers (using an
optical method) by Williams (1974, 1975) under
conditions close to those predicted by Linden (1973)
on the basis of laboratory results. Magnell (1976)
has also measured horizontal conductivity variations
with the right scale (=2cm.) to support this inter-
pretation.
As mentioned above, there is not as big a differ-
ence between the "diffusive" and "finger" cases as
there appears to be when we simply compare the
interfaces illustrated in Figures 2 and 4. Layers
can be produced from a smooth gradient in the latter
case too, by supplying a flux of S at the edge of
a gradient of T; this was first demonstrated, using
a sugar flux above a salt gradient, by Stern and
Turner (1969). When viewed on the scale of the
convecting layers, there is in fact a close corres-
pondence between the two systems. The inequality
of diffusivities results in an unstable buoyancy
flux across statically stable interfaces in both.
cases, and this maintains convection above and below.
Only the mechanism of interfacial transport differs,
and it is here that the detailed structure of the
interface enters. Across a finger interface the
buoyancy flux is dominated by the destabilizing
component, S, and salt is transported faster than
heat, whereas the opposite holds a diffusive inter-
face.
Corresponding laboratory measurements of the
two coupled fluxes have been made for finger inter-
faces. Again there is a strong dependence on the
density ratio across the interface, and the ratio
of heat to salt fluxes is constant over a consider-
able range. Turner (1967) has shown in the heat
salt case that the salt flux is about 50 times as
large at R)* = aAT/BAS + 1 as it would be if the
same salinity difference were maintained at solid
boundaries, and falls slowly as R,* increases.
He also obtained a value for the flux ratio aFp/8F,
= 0.56 over the range 2 < Ro* < 10. Linden (1973)
has made direct observations of the structure of
salt fingers and the velocity in them that support
these estimates of the salt flux. His estimate of
the flux ratio was much lower, but recent experiments
in our laboratory have supported the earlier value.
These new experiments have concentrated on achieving
as small a value of Rp* as possible, but measurements
in the "variable" range of flux ratio are still
elusive. This range could, however, be of great
importance in the ocean, where Ro* is often close
to unity.
It is also of interest to mention the experiments
of Linden (1974b) who applied a shear across a
salt-finger interface. He showed that a steady
shear has little effect on the fluxes, though it
changes the fingers into two-dimensional sheets |
aligned down shear. Unsteady shears (i-.e., stirring
on both sides of the interface) can, on the other
hand, rapidly disrupt the interface, and actually
decrease the salt flux.
There are now many examples of layering in the
ocean which are consistent with the "fingering"
process. These are observed in situations where
both the mean salinity and the temperature decrease
with increasing depth, and often occur under warm
salt intrusions of one water mass into another.
The first observations were made by Tait and Howe
(1968, 1971) under the Mediterranean outflow, and
a good summary of other measurements is to be found
in Fedorov (1976). For reasons which will be dis-
cussed more fully in later sections, it is difficult
to find cases where one can be sure that the forma-
tion of layers bounded by finger interfaces has
been the result of one-dimensional processes,
strictly analogous to those studied in the labora-
tory. Once layers have formed, however, the effects
‘of the fluxes through the finger interfaces between
them can properly be discussed in these terms, and
two practical examples will be given.
The first arises in the context of sewage disposal
in the sea. Fischer (1971) has discussed the case
where effluent, which can be regarded for this
purpose as nearly fresh (though polluted) water, is
ejected from a pipe laid along the bottom, and
rises as a line plume into sea water which is strat-—
ified in temperature. Careful design of the outfall
ensures that the effluent, diluted with many times
its volume of cold sea water, will spread out ina
layer below the thermocline. But this layer will
remain colder and fresher than the water above it,
so the salt finger mechanism can cause it to thicken
vertically, and even extend to the surface. A
related case, in which the environmental effects
could be even more serious, arises in the disposal
of effluent from a desalination plant. Suppose
that the brine from which water has been evaporated,
and the heated water from the cooling plant, are
mixed together to be disposed of as a single effluent.
This hot, salty water will have about the same
density as the original sea water - according to
the precise design conditions, it can be slightly
heavier or slightly lighter. If it is made heavier,
and forms a layer along the bottom, a diffusive
interface will be formed, and the coupled transports
will tend to increase the density difference and
thus keep the layer distinct. If it is put in at
the surface, or at an intermediate level ina
gradient, fingers will form, and there will be more
rapid vertical mixing. One thing is certain: the
rate of mixing cannot be determined using only the
net density distribution and leaving out of account
the double-diffusive effects.
3. TWO-DIMENSIONAL EFFECTS
Side-wall Heating and Related Processes
It became clear in early laboratory experiments on
double-diffusive convection that layers will readily
form from a salt gradient in another way, if it is
heated from the side. This effect was studied
systematically by Thorpe, Hutt, and Soulsby (1969)
and by Chen, Briggs, and Wirtz (1971), and their
results can be summarized as follows. The thermal
boundary layer at a heated vertical wall grows by
conduction and begins to rise. Salt is lifted to
a level where the net density is close to that in
the interior; then fluid flows out away from the
wall, producing a series of layers that form
simultaneously at all levels and grow inwards from
the boundary. The layer thickness is close to the
length-scale
aAT
= Bas/da &)
which is the height to which a fluid element with
temperature difference AT would rise in the initial
salinity gradient.
The stability problem corresponding to sidewall
heating of a wide container has not been solved,
though Stern (1967) has shown theoretically how
lateral gradients could lead to the generation of
layers. Thorpe, Hutt, and Soulsby (1969) have
analyzed the simpler case of a fluid containing
compensating linear horizontal gradients of S and
T, contained in a narrow vertical slot and Hart
(1971) improved their analysis; both theories
predict slightly inclined cells extending right
across the gap, with a spacing in fair agreement
with the measurements.
Similar layers are formed when the salinity as
well as the temperature of the vertical boundary
does not match that in the interior, for example
when a block of ice is inserted into a salinity
gradient and allowed to melt. A qualitative experi-
ment of this kind was reported by Turner (1975),
601
FIGURE 5. Showing the tilted layers formed by insert-
ing a block of ice into salt-stratified water at room
temperature. Fluorescein was frozen into the ice, and
was illuminated from the side, so that the spread of
the dye indicates the distribution of the melt water.
(Negative print.)
but interest in the process has increased recently,
because of the application to melting icebergs.
Huppert and Turner (1978) have carried out a more
extensive set of experiments with this problem in
mind.
An understanding of the melting of icebergs
could be important in various contexts. Several
groups are currently examining the feasibility of
towing icebergs to their coasts and melting them
to provide fresh water, but there are many unsolved
scientific and engineering problems [see, for
example, Bader (1977)]. It has been proposed that
fresh water could be obtained by building a shallow
pen round a grounded iceberg, allowing the melt
water to collect in this, and siphoning it off the
surface. On the other hand Neshyba (1977) has
suggested that the melt water produced by icebergs
would mix with the surrounding sea water, and could
thus be effective in lifting water and nutrients
from deeper layers to the surface, where it would
increase biological production.
Huppert and Turner's (1978) experiments have
shown, however, that neither idea is likely to be
valid, because of the neglect of the stable salinity
gradient which exists in the upper layers of the
oceans where icebergs are found. As demonstrated
in Figure 5, the presence of horizontal S and T
differences then produces a regular series of tilted
convecting layers, which feed most of the meltwater
into the interior; very little rises to the surface.
A more detailed analysis of the experiments is
continuing. At present it appears that for a
cooled sidewall the layer depths are similar whether
melting is occurring or not, and that they are not
described simply by (6) but depend more weakly on
the initial salinity gradient. Another phenomenon
which deserves more careful study is the series of
grooves and ridges produced by non-uniform melting
associated with the circulation in the layers (see
Figure 6).
Sloping Boundaries
Phenomena analogous to those described above can
be observed in systems containing smooth gradients
of more slowly diffusing solutes. The essential
physical feature of the heated sidewall process is
FIGURE 6.
Shadowgraph photograph of a melting ice-
block in a salinity gradient. Note the regularly spaced
scallops and ridges, caused by uneven melting asso-
ciated with the convection in layers.
that the boundary conditions (on temperature or
salinity or both) do not match the conditions in
the interior. In tanks containing opposing gradients
of two components, with say a maximum salt concen-
tration at the top falling linearly to zero at the
bottom, and a maximum (slightly larger) sugar
concentration at the bottom falling to zero at the
top, the same kind of instability can be produced
in another way. With vertical side walls, the
surfaces of constant concentration are normal to
the boundaries, and the no-flux boundary condition
is automatically satisfied. But when an inclined
boundary is inserted, diffusion will distort the
surfaces of constant concentration away from the
horizontal, so that they become normal to the
boundary. Density anomalies are produced which
tend to drive flows along the wall; these cannot
remain steady, but instead turn out into the interior
and produce a series of layers.
This process was first investigated experimentally
by Turner and Chen (1974), with the initial strati-
fication in the "diffusive" sense. A prominent
feature of the intruding layers is the local reversal
of gradients in the extending "noses", where fingers
are prominent. In the later stages of that experiment,
the advancing noses have become independent of the
mechanism which produced them, and this suggested
the systematic study of double-diffusive sources
in various environments which is pursued below.
Linden and Weber (1977) have investigated layer
formation in the "finger" case; they have also
discussed the instability of the boundary layer
at the sloping wall, and the criteria determining
the layer depths. In the limit where the opposing
gradients are nearly equal, the characteristic
vertical lengthscale depends mostly on the initial
vertical distributions of S and T, and little on
the mechanism triggering the instability.
A different effect of sloping boundaries should
be mentioned here. In a two-layer system, in which
the layer depths vary because one wall of the
containing vessel is inclined, large-scale quasi-
horizontal motions can be set up even when the
buoyancy flux across the horizontal interface is
uniform. This effect is a purely geometrical
consequence of the sloping boundary. The net result
of the double-diffusive transports across the
interface is to provide an unstable buoyancy flux
which makes the bottom layer heavier. A given flux
produces more rapid density changes in shallower
regions where there is less dilution, and this sets
up a circulation in the sense which includes a flow
down the slope. Gill and Turner (1969) have shown
that this flow can reverse the relative gradients
of the two components, for example, giving rise to
salt fingers at the bottom of a tank originally
stratified in the diffusive sense. They have also
suggested an application to the formation of bottom
water near the Antarctic continent. Similar effects
have been observed Ly Turner and Chen (1974) when
a sloping interface, rather than a solid sloping
boundary, produces the non-uniformity of depth,
and this too can have implications for the formation
of bottom water in deeper water.
Double-diffusive Intrusions
The experiments described in the two preceding
sections have recognized the importance of horizontal
gradients, but they still have not dealt with the
common situation where fluid with one set of T-S
properties intrudes into another having different
properties. This question has recently been
addressed by Turner (1978), using sources of sugar
and salt solutions released into gradients of
various kinds.
The basic intrusion process with which other
phenomena can be compared is the two-dimensional
flow of a uniform fluid at its own density level
into a linear gradient set up using the same property.
Figure 7 shows the behaviour of a (dyed) source of
salt solution released into a salinity gradient.
This is what we might intuitively expect: the
intruding fluid just displaces its surroundings
upwards and downwards, and is kept confined to a
horizontal layer by the denisty gradient. Detailed
studies of this process have been reported by
Maxworthy (1972), Manins (1976), and Imberger,
Thompson, and Fandry (1976). Note praticularly the
"upstream wake" effect, leading to a considerable
disturbance of the environment ahead of the advancing
nose.
When the source of salt is replaced by sugar
solution (S), while the same salinity gradient (T)
is retained in the environment, the behaviour is
very different. (It is worth keeping in mind
throughout the following, the analogous situation
with temperature and salinity: this corresponds
to the intrusions of a layer of warmer, saltier
water into a stable temperature gradient). As
shown in Figure 8, there is strong vertical convec-
tion near the source: this is produced by a
mechanism which also occurs with a uniform ambient
fluid close to the same density as that injected.
The more rapid diffusion of T relative to S across
the plume boundary causes it to become heavier,
and its immediate surroundings lighter, than the
fluid at the level of the source. The vertical
spread is limited by the stratification, and "noses"
begin to spread out at levels above and below the
source. The process of vertical convection continues,
and further layers appear as the layers first formed
extend away from the source. The total volume of
fluid affected by mixing is many times that of the
input, showing that the intrusions are overtaking
and incorporating the environment, rather than
just displacing it as in the experiment of Figure
7. The implication for the ocean is, of course,
that large scale intrusions will tend to break up
into thinner noses and layers, as is indeed observed.
Each individual nose as it spreads contains an
excess of S relative to its environment, so that
conditions are favourable for the formation of a
diffusive interface above and fingers below, as can
be seen in Figure 8. This also implies that there
will be a local decrease with depth or an inversion
of T through each layer, and that the density
gradient above such an intrusion will be greater
than that below. These features have been demon-
strated in oceanic data by Howe and Tait (1972),
Gregg (1975), and Gargett (1976).
Note too the slight upward tilt of each layer
as it extends, which can be interpreted as follows.
Above and below an intrusion, the net density
differences are small and the double-diffusive
fluxes therefore large. The one-dimensional labo-
ratory observations indicate that the transports
across a finger interface (both in the sugar-salt
FIGURE 8. The flow produced by releasing
Sugar solution at its own density level into
a salinity gradient. Strong vertical convec-
tion occurs, followed by intrusion at several
levels. The density gradient and flow-rate,
and the scale of the photograph, are approxi-
mately the same as for Figure 7.
603
FIGURE 7. The intrusion of dyed salt solu-
tion into a salinity gradient at its own
density level. The distorted dye streaks
show that the fluid in the environment be-
gins to flow well ahead of the advancing
fluid. (The region shown is about 400mm.
wide.)
and salt-heat case) are larger than those across
a comparable diffusive interface. Thus the flux
of positive buoyance through the fingers from below
can exceed the negative flux from above, so a layer
becomes lighter and rises across isopycnals as
it advances away from the source. There is also
a systematic shear flow associated with the inclined
layers, and both these features would seem worth
looking for when observations are made of oceanic
finestructure in the future.
The interpretation of the layer slope in terms
of the differences in fluxes across the two inter-
faces is supported by experiments carried out in
the inverse sense. With a source of salt solution
(T) flowing at its own density level into a gradient
of sugar solution (S), the behaviour is as shown
in Figure 9. Vertical convection near the source
is again followed by the spread of noses at various
levels, but now with diffusive interfaces below
and fingers above, corresponding to the excess of
T in the noses relative to their S environment.
There is a systematic downward tilt as the noses
advance, due again to the dominance of the buoyancy
flux at the finger interfaces, which now causes
the layers to become heavier as they extend. The
sense of the internal shear is also consistent
with this picture: the motion is inclined slightly
down and away from the source at the bottom of the
604
FIGURE 9. The flow produced by releasing salt
solution into a gradient of sugar solution,
using conditions comparable with, but the in-
verse of those shown in Figure 8.
fingers and above the diffusive interfaces, indica-
ting again that there is an increase in density due
to the continuing flux in the fingers.
Two other features of the laboratory observations
which have important implications for the ocean
should also be mentioned. The most rapid formation
of layers in the series of experiments reported by
Turner (1978) occurred when the tank was stratified
in the "finger" sense, and the fingers were allowed
to run down towards a marginally stable state.
When source fluid was introduced, layers formed
more rapidly and regularly than before, because of
the potential energy already available in the
ambient fluid. This implies that "reactivation"
of layers in a region where they have previously
formed will proceed more quickly than the original
layering process. It suggests that the patches of
strong layers, under the Mediterranean outflow
into the Atlantic for example, are associated with
the arrival of a fresh pulse of intruding fluid.
The second related observation is that the further
stage of overturning to produce nearly uniformly-
mixed layers, bounded above and below by finger
interfaces is also more likely to be reached near
the source of the intruding water. The relationship
between the two types of layering has been demon-
strated directly in the measurements of Gregg (1975),
which show that inversions of intrusive origin can
in the course of time break down to form well-mixed
layers.
Layer Formation at Fronts
An important geometry which merits separate study
is a discontinuity of T-S properties over a vertical
or inclined surface, i.e., a front. The motions
produced when an inclined boundary is inserted into
a fluid stratified with opposing gradients (Section
3) have some of the required features, but the
presence of the solid wall is clearly undesirable.
Fronts can be set up in the laboratory in several
ways. Large horizontal T and S gradients can, for
example, be produced just by pouring fluid with
contrasting properties into one end of a stratified
tank at several levels, or by stirring it in
throughout the depth. A somewhat more controllable
method is to insert a vertical barrier in a previ-
ously stratified tank, to introduce the extra fluid
on one side of it, and allow the disturbances to
This tech-
die away before removing the barrier.
nique has been used in the experiment shown in
Figure 10. It is difficult to get the two vertical
gradients exactly matched, and so when the barrier
is removed internal waves are set up, which soon
die away, leaving the isopycnals horizontal but
the front distorted. The initial state illustrated
in Figure 10a is completely determined by the
readjustment of the density field, but note that
diffusive interfaces have already developed in the
sense to be expected with an excess of S on the
left. At a later stage (Figure 10b) the frontal
surface is spread out horizontally by the inter-
leaving of inclined layers, the scale of which is
unrelated to that of the initial adjustment, and
which are driven entirely by the local density
anomalies produced by double-diffusive transports.
A more sophisticated version of this experiment
is currently being studied by Ruddick (personal
communication). He has set up identical vertical
density distributions on two sides of a barrier,
using sugar (S) in one half and salt (T) in the
other. When the barrier is withdrawn, there is
some small scale mixing, but virtually no larger
scale distortion. A series of regular, interleaving
layers then develops, with a spacing and speed of
advance which are systematically related to the
horizontal property differences.
There are now many. measurements which support
the view that the prominence and strength of
layering in the ocean are related to the magnitude
of the horizontal gradients of properties. To
cite just two examples: profiles across the Antarc-
tic polar front [Gordon et al. (1977)] reveal
inversions which decrease in strength with increasing
distance away from the front. Coastal fronts
between colder fresh water on a continental shelf
and warmer salty water offshore also exhibit strong
interleaving [Voorhis, Webb, and Millard (1976)].
A general conclusion which can already be drawn
from the laboratory experiments described in this
section is that the formation and propagation of
interleaving double-diffusive layers is a self-
driven process, sustained by local density anomalies
due to double-diffusive transports. Once a series
of noses and layers has formed, the changes of T
and S within them can be described in terms of the
one-dimensional (vertical) transport processes
previously studied. It should eventually be possible
[Joyce (1977)] to parameterize the effective
increase in the horizontal diffusion of T and S,
produced by interleaving, in terms of the horizontal
gradients and these quasi-vertical fluxes.
ACKNOWLEDGMENT
I am grateful to R. Wylde-Browne for his assistance
with many of the recent experiments described here,
and particularly with the photography. Discussions
with R. W. Griffiths and B. R. Ruddick about their
current work have been very stimulating.
REFERENCES
Bader, H. (1977). A critical look at the iceberg
project. First International Conference on
Iceberg Utilization, Iowa State University.
Chen CoE DAmGHEBralggsmand Re Al Wastzan(lhOv/as)ir
Stability of thermal convection in a salinity
gradient due to lateral heating. Int. J. Heat
and Mass Trans. 14, 57-65.
Crapper, P. F. (1975). Measurements across a
diffusive interface. Deep-Sea Res. 22, 537-545.
Degens, E. T, and D. A. Ross (eds.) (1969). Hot
brines and recent heavy metal deposits in the
Red Sea. Berlin: Springer-Verlag.
605
FIGURE 10. Showing the spread of a front,
formed by withdrawing a barrier from the center
of an experimental tank. The initial state cor-
responded to "run down" fingers, with extra
(dyed) sugar solution mixed in at the left.
Upper: 2-1/2 mins. after barrier was removed,
showing intrusions formed by the den-
sity adjustment.
Lower: 65 mins. after barrier removed, showing
the smaller scale motions driven by
double-diffusive processes across the
frontal region.
Fay, J. A., and J. J. MacKenzie (1972). Cold cargo.
Environment 14, 21-29.
Fedorov, K. N. (1976). The thermohaline finestruc-—
ture of the ocean. Gidrometeoizdat, Leningrad.
Fischer, H. B. (1971). The dilution of an undersea
sewage cloud by salt fingers. Water Research
57 DOP—Gi5.
Gargett, A. E. (1976). An investigation of the
occurrence of oceanic turbulence with respect to
finestructure. J. Phys. Oceanog. 6, 139-156.
Gilly A Heeand. Jie Se Turner (969) Some mnew
ideas about the formation of Antarctic bottom
water. Nature 224, 1287-1288.
Gregg, M. C. (1975). Microstructure and intrusions
in the California current. J. Phys. Oceanog. 5,
253=2 15
Griffiths, R. W. (1978). The influence of a third
diffusing component upon the onset of convection.
J. Fluid Mech. (in the press)
Gordon, A. L., Ds i. Georgi, and! H.) We Taylor (977)
Antarctic polar front zone in the western Scotia
Sea - summer 1975. J. Phys. Oceanog. 7, 309-328.
Hart, J. E. (1971). On sideways diffusive instabil-
ity. J. Fluid Mech. 49, 279-288.
Hoare, R. A. (1968). Thermohaline convection in
Lake Vanda, Antarctica. J. Geophys. Res. 73,
607-612.
Howard, L. N. (1964). Convection at high Rayleigh
number. Proc. eleventh Int. Congress Applied
606
Mechanics, Munich (ed. H. Gértler), 1109-1115.
Berlin: Springer-Verlag.
Howe, M. R., and R. I. Tait (1972). The role of
temperature inversions in the mixing processes
of the deep ocean. Deep-Sea Res. 19, 781-791.
Huppert, H. E. (1971). On the stability of a
series of double-diffusive layers. Deep-Sea
Res. 18, 1005-1021.
Huppert, H. E., and P. C. Manins (1973). Limiting
conditions for salt-fingering at an interface.
Deep-Sea Res. 20, 315-323.
Huppert, H. E., and J. S. Turner (1972). Double-
diffusive convection and its implications for
the temperature and salinity structure of the
ocean and Lake Vanda. J. Phys. Oceanog. 2,
456-461.
Huppert, H. E., and J. S. Turner (1978).
icebergs. Nature 271, 46-48.
Imberger, J., R. Thompson, and C. Fandry (1976).
Selective withdrawal from a finite rectangular
tank. J. Fluid Mech. 78, 489-512.
Joyce, T. M. (1977). A note on the lateral mixing
of water masses. J. Phys. Oceanog. 7, 626-629.
Linden, P. F. (1973). On the structure of salt
fingers. Deep-Sea Res. 20, 325-340.
Linden, P. F. (1974a). A note on the transport
across a diffusive interface. Deep-Sea Res. 21,
283-287.
Linden, P. F.
On melting
(1974b). Salt fingers in a steady
shear flow. Geophys. Fluid Dynam. 6, 1-27.
Linden, P. F. (1976). The formation and destruction
of fine-structure by double-diffusive processes.
Deep-Sea Res. 23, 895-908.
Linden, P. F., and T. G. L. Shirtcliffe (1978). The
diffusive interface in double-diffusive convection.
J. Fluid Mech. (in the press)
Linden, P. F., and J. E. Weber (1977). ‘The formation
of layers in a double-diffusive system with a
sloping boundary. J. Fluid Mech. 81, 757-774.
Magnell, B. (1976). Salt fingers observed in the
Mediterranean outflow region (34°N, 11°W) using
a towed sensor. J. Phys. Oceanog. 6, 511-523.
Manins, P. C. (1976). Intrusion into a stratified
fluid. J. Fluid Mech. 74, 547-560.
Marmorino, G. O., and D. R. Caldwell (1976).
librium heat and salt transport through a
diffusive thermohaline interface. Deep-Sea Res.
23, 59-68.
Neal, V. T., S. Neshyba, and W. Denner (1969).
Thermal stratification in the Arctic Ocean.
Science 166, 373-374.
Neshyba, S. (1977). Upwelling by icebergs.
267, 507-508.
Neshyba, S., V. T. Neal, and W. Denner (1971).
Temperature and conductivity measurements under
Ice Island T-3. J. Geophys. Res. 76, 8107-8120.
Newman, F. C. (1976). Temperature steps in Lake
Kivu: a bottom heated saline lake. J. Phys.
Oceanog. 6, 157-163.
Sarsten, J. A. (1972).
Equi-
Nature
LNG stratification and
rollover. Pipeline and Gas Journal. Sept.
1972, 37-39. :
Shirtcliffe, T. G. L. (1967). Thermosolutal convec-
tion: observation of an overstable mode. Nature,
213, 489-490.
Shirtcliffe, T. G. L. (1973). Transport and profile
measurements of the diffusive interface in
double-diffusive convection with similar diffusi-
vities. J. Fluid Mech. 57, 27-43.
Shirtcliffe, T. G. L., and I. M. Calhaem (1968).
Measurements of temperature and electrical
conductivity in Lake Vanda, Victoria Land,
Antarctica. WN. Z. J. Geol. Geophys. 11, 976-981.
Shirtcliffe, T. G. L., and J. S. Turner (1970).
Observations of the cell structure of salt
fingers. J. Fluid Mech. 41, 707-719.
Stern, M. E. (1960). The ‘salt fountain' and
thermohaline convection. Tellus 12, 172-175.
Stern, M. E. (1967). Lateral mixing of water
Masses. Deep-Sea Res. 14, 747-753.
Stern, M. E. (1969). Collective instability of
salt fingers. J. Fluid Mech. 35, 209-218.
Stern, M. E. (1975). Ocean circulation physics.
Academic Press, pp. 246.
Stern, M. E., and J. S. Turner (1969). Salt fingers
and convecting layers. Deep-Sea Res. 16,
497-511.
Tait, R. I., and M. R. Howe (1968). Some observa-
tions of thermohaline stratification in the deep
ocean. Deep-Sea Res. 15, 275-280.
Tait, R. I., and M. R. Howe (1971).
staircase. Nature, 231, 178-179.
Thorpe, S. A., P. K. Hutt, and R. Soulsby (1969).
The effect of horizontal gradients on thermohaline
convection. J. Fluid Mech. 38, 375-400.
Turner, J. S. (1965). The coupled turbulent trans-
ports of salt and heat across a sharp density
Thermohaline
interace. Int. J. Heat and Mass Trans. 8,
759-767.
Turner, J. S. (1967). Salt fingers across a density
interface. Deep-Sea Res. 14, 599-611.
The behaviour of a stable
J. Fluid
Turney ist rol LOOS is
salinity gradient heated from below.
Mech. 33, 183-200.
Turner, J. S. (1973). Buoyancy effects in fluids.
Cambridge University Press, 367 pp.
Turner, J. S. (1974). Double-diffusive phenomena.
Ann. Rev. Fluid Mech. 6, 37-56.
Turner, J. S. (1975). Laboratory experiments on
double-diffusive instabilities. Adv. Chem.
Phys. 32, 135-149.
Turner, J. S. (1978).
into a density gradient.
the press)
Turner, J. S., and C. F. Chen (1974). ‘Two-=
dimensional effects in double-diffusive convec-—
tion. J. Fluid Mech. 63, 577-592.
Turner, J. S., and L. B. Gustafson (1978). The
flow of hot, saline solutions from vents in the
sea floor - some implications for exhalative
sulfide deposits. Econ. Geol. (in the press)
Turners, Ji. Si, Le Ga Le. Shirtclitre\andmebesGe
Brewer (1970). Elemental variations of trans-
port coefficients across density interfaces in
multiple-diffusive systems. Nature 228, 1083-
1084,
Voorhis, A. D., D. C. Webb, and R. C. Millard (1976).
Current structure and mixing in the shelf-slope
water front south of New England. J. Geophys.
Res. 81, 3695-3708.
Williams, A. J. (1974).
the Mediterranean out-flow.
Double-diffusive intrusions
J. Geophys. Res. (in
Salt fingers observed in
Science 185,
941-943.
Williams, A. J. (1975). Images of ocean micro-
structure. Deep-Sea Res. 22, 811-829.
Buoyant Plumes in a Transverse Wind
Chia-Shun Yih
The University of Michigan
Ann Arbor, Michigan
ABSTRACT
With the rise in energy needs and the consequent
proliferation of cooling towers (not to mention
smoke stacks) on the one hand, and society's
enchanced concern with the environment on the other,
the study of buoyant plumes caused by heat sources
in a transverse wind has become important. Buoyant
plumes may also occur in the ocean, such as when
a deeply submerged heat source moves horizontally
in it. The fluid mechanics involved in buoyant
plumes is very nearly the same, be they atmospheric
or submarine.
In this paper a similarity solution for turbulent
buoyant plumes due to a point heat source in a
transverse wind is presented. By a set of trans-
formations the mathematical dimension of the
problem is reduced from 3 to 2. Analytical solutions
for the first and second approximations are obtained
for the temperature and velocity fields. The
solution exhibits the often observed pair of longi-
tudinal counter-rotating vortices. As a result of
buoyancy, the point of highest temperature and the
"eyes" of the vortices at any section normal to
the wind direction continuously rise as the longi-
tudinal distance from the heat source increases.
1. INTRODUCTION
As industry expands and energy needs rise, the
buoyant plumes caused by ever-increasing cooling
towers and smoke stacks have become an important
concern for societies anxious to protect their
environment. Much effort has been expanded on the
so-called numerical modeling of the phenomenon
of plumes both in the United States and in Europe.
In most of the numerical studies, the eddy viscosity
is assumed constant, and its value is chosen to
make the results agree with whatever gross observa-
tions are available. The power of modern computers
has made it possible to obtain numerical solutions
607
for partial differential equations with very
irregular data, such as wind and temperature profiles
in the atmosphere. On the other hand, one can
only carry out a number of these special solutions,
and while the power of the computer makes computa-
tion possible it also makes the intermediate steps
so opaque that one can only have faith in the
accuracy of the results and the correctness of the
programing; and one can attempt to interpret the
results and understand the phenomenon only at the
very end, when numerical results are available.
One can hardly see, for example, the effects of
changing one single parameter of the problem, without
giving that parameter several values and going to
the computer again and again. It is in view of
this condition that even people most concerned with
the immediate applicability of calculated results
desire a certain measure of transparency in the
analysis of the phenomenon.
At the same time systematic and detailed experi-
ments on buoyant plumes in transverse winds, with
temperature and velocity measurements, are lacking.
This being so, it seems that an analytical solution
of the problem is most desirable and timely, even
if it must of necessity be constructed by assuming
certain quantities (such as the turbulence level
in the plume) on the basis of whatever related
experimental results are available. The assumed
quantities (or quantity) will appear in the analysis
as unspecified coefficients (or coefficient, as in
this analysis), to be determined by experiments
later. In the present work only one coefficient
related to the turbulence level is left unspecified,
to be determined by future experiments. But the
probable range in which it lies is given.
The solution is based on a set of transformations
that reduces the mathematical dimension of the
phenomenon from 3 to 2 is thus characterized by the
striking feature of similarity between cross sections
normal to the wind direction. The laws of decay
of the temperature and velocity fields are given
in simple, explicit terms. Thus, apart from the
608
quantitative predictions that this analysis is
intended to furnish, I hope that the general features
of the solution will be found especially useful.
2. THE DIFFERENTIAL EQUATIONS
The two basic assumptions underlying the analysis
are that the longitudinal velocity component in
the direction of the wind is constant and that an
eddy viscosity, €, is constant in any cross section
normal to the wind direction. It can be shown that
the first assumption ceases to be true only at
stages of approximations later than those arrived
at in the present analysis, and its violation is
therefore not very important. The second assumption
mentioned above has been made in all analytic
solutions for turbulent jets and plumes, according
to Prandtl's simplified theory. These solutions
are well known. See, for example, the paper by
Yih (1977) on turbulent plumes for the latest
application of that theory. One feels reassured
that for a calculation of the mean temperature and
velocity fields, this theory can again be used.
We shall take the direction of the wind to be
the direction of increasing x, and the z direction
to be vertically upward. The y direction will then
be a horizontal direction transverse to the x direc-
tion. In general ¢ depends on x, y, and z. But it
has been repeatedly shown before in other studies
of jets and plumes that in their core, € can be
taken as constant at a constant value of x, and
that only at their outer edges does the nonuniformity
in the y-z plane introduce some errors in the
calculated mean quantities. (Very far away from
the jets and plumes the value of € is immaterial
for the determination of the temperature and velocity
distributions). Accepting these outer-edge errors,
which are fairly small, we shall take € to be a
function of x only, apart from the parameters of
the problem to be defined later. We note that if
an eddy viscosity is used to determine the velocity
distribution in turbulent flow in a circular pipe,
Laufer's (1953) measurements show that in the core,
that is, away from the narrow region near the pipe
wall, € is nearly constant.
The equations of motion are then, with subscripts
denoting partial differentiations,
Wii ar WAI SP ni = O
al
x y Zz p Py + E(Vyy a7 View (1)
), (2)
Uw, + vw, + wwo = - e = 60) ap eX te
m8 y A 0 Pz GI e Wy Woz
in which U is the wind velocity, assumed constant,
v and w are the velocity components in the directions
of increasing y and z, respectively, p is the
density, p is the pressure, and g is the gravita-
tional acceleration. The variable 8 is defined by
Ap
a=",
- (3)
where Ap is the variation of the density from the
ambient density p, assumed constant. Thus the
Boussinesq approximation has been used in Eqs. (1)
and (2). Since § is small and the pressure vari-
ation in the plume, though important for determina-
tion of the flow field, is unimportant in the deter-
mination of Ap from the temperature variation by
the equation of state, 8 can also be written, by
virtue of the equation of state of ideal gases.
oa
T
where AT is the temperature variation and T the
ambient temperature. For a liquid, the relationship
between Ap and AT is still linear if 8 is small,
and the constant. of proportionality is determined
by the property of the liquid.
We shall assume the eddy viscoity for heat
diffusion to be the same as that for momentum
diffusion. This may not be strictly true, for the
turbulent Prandtl number may be slightly different
from 1. The effect of this difference, if any, is
not of great importance in our attempt to determine
the mean temperature and velocity fields. The
equation for heat diffusion can then be written in
the form
+ wO, = EN + 6 (4)
x y y zz)-
Longitudinal diffusion of heat or of momentum is
ignored in Eq. (1), (2), and (4). This is justified
in the same way as in other works that use the
boundary-layer theory.
The equation of continuity is, since the longi-
tudinal velocity component is assumed constant,
Yoo WS Oe (5)
The heat source, located at the origin, is
measured by the quantity
UOdydz. (6)
Note that solid boundaries are assumed to be far
away from the source, so that their effects are
negligible. Equations (1), (2), (4), (5), and (6),
with appropriate boundary conditions, govern the
phenomenon under investigation.
The equation of continuity (5) allows the use
of a stream function | in terms of which v and w
can be expressed:
V =z, w = -Wy. (7)
By cross-differentiation of Eqs.
obtain the vorticity equation
(1) and (2), we
Wiese or vey + wi, = € (Evy AP So) = 980 (8)
in which é is the x component of the vorticity and
is given by
E=w
ne te Wy S Vado (9)
3. THE FORM OF THE EDDY VISCOSITY
We assume the terms in Eqs. (1) or (2) or (4) to
be of the same order of magnitude. In particular,
this means that the diffusive and the convective
terms are of the same order of magnitude in any of
these equations. It also means that in Eq. (2) the
buoyancy term is of the same order of magnitude as
the convective and diffusive terms for w. This
assumption underlies all existing analytical studies
of jets and plumes and can be regarded as amply
justified.
Comparing the first and last terms in Eq. (2),
then, we have
(10)
in which 2, and 2, are the length scales for the
x and z directions. Comparing the first term in
Eq. (2) with the term g@, we have
Uw
§@ ~ ——, (11)
gh,
where 9 and w stand for the magnitudes of 9 and w,
rather than 8 and w rigorously, as they do also
in the following proportionalities. Equation (6)
gives, further,
8 Nea w 8, (12)
2
UL,
if we take 2, and 2, to be equal. From proportion-
alities (11) and (12) we have, after some rearrange-
ment,
gGr
Wy = (13)
UL,
But surely
Se ~ Woo (14)
Hence
gGe
x
Coy : (15)
UL,
From proportionalities (10) and (15) we have
G G
Oi ta ar (16)
U U
since the doer the scale of x, is just x. Thus (12),
(13), (14), and (16) give
Le a x2/3, G0 xl/3, w ~ alae 8 ~ ee
These results are unaffected when other comparisons
are made between terms in either Eq. (1), (2), (4),
@xe ((5)) o
From porportionalities (15) and (16) we have
a= Se? : (17)
where a is a dimensionless constant to be determined
experimentally or estimated from known values of
€ in similar phenomena. We shall leave it free
throughout our analysis. Equation (17) gives the
form of € to be used in this paper.
It seems strange at first sight that € should
vary inversely as U. I believe that the interpre-
tation of € ~ U7l is that € increases with the
time that is required for the wind to travel a unit
distance in the x direction, because turbulence
needs time to develop.
609
4. THE TRANSFORMATIONS AND THE DIFFERENTIAL
SYSTEM TO BE SOLVED
The transformations to be used to obtain similarity
solutions are already suggested by (12), (13), and
(16) and are
U Wis)
Oia alae (5) Sin Un 7iB)) 7 (18)
g
1/3
(vyw) = = (3) (V,W), (19)
S/S}
U
(n,) = Gayi7z (69x) (y,z)- (20)
Then the equation of continuity (5) becomes
and Eqs. (7) become
Va Yrs Wil (21)
in which ¥ is the dimensionless stream function
related to yp by
me all
DS eae (Cae) 7 ne) o (22)
Equation (9) now takes the form
= = = ap bd
E we Ve Ce aaa (23)
where € is the dimensionless vorticity component in
the x direction.
With the transformations (18), (19), and (20),
Eq. (4) becomes
Lh = A(Vh_ + Wh_), (24)
n 6
where L is the linear operator defined by
92 92 3 3
= =——— —— — 5
an2 + 02 + 2n55 + 2057 + 4, (25)
and
VS ee (26)
Equation (8) now has the form
(> DE S S05 A(VE, + WE). (27)
Equations (23), (24), and (27) are the final equa-
tions governing the dynamics of the plume in a
transverse wind. They are to be solved with the
boundary conditions
(ij) by =O, G20, YaO, aU = 0 ae i = O-
(atat))) 394 0 at n = to or G = to.
lI
[e)
wy
i
jo)
E
tl
Boundary conditions (i) correspond to symmetry with
respect to the ¢ axis, and conditions (ii) ensure
that there is no temperature variation and no
610
velocity components v and w at infinity. The
integral relation (6) now takes the form
hdndg = 1, (28)
The mathematical problem is now completely specified.
5. THE METHOD OF SOLUTION
The mathematical problem just formulated can be
solved numerically once \ is known. But consider-
able effort is required for this solution, since
there are three second-order partial differential
equations to be solved, two of which are nonlinear.
It is true that computers can deal with nonlinear-
ities, but the domain is infinite, and some estimate
has to be made of how far to go in the numerical
computation. Furthermore the integral condition
(28) can only be imposed after the computations
are done for h, and this makes the computation very
cumbersome.
For arbitrarily large values of \ an analytical
solution is extremely difficult because the non-
linearities present formidable difficulties. We
shall attempt a power-series solution of the form
h = hy + Ah, + Near ie Daetiel hk
= Z2
BS Ba PAS 3 Be ae ool Gg (29)
a 2
Des, OMG AW ole eo
The success or failure of this approach depends
not only on the value of A, but also on the magni-
tudes of h,/ho, ho/hy, etc. Thus we need to make
an estimate of the range of X, and we have to find
out how fast hy, &,, and Y, decrease as n increases.
Furthermore, even the estimate of A cannot be made
without knowing the magnitudes of Yg. It turns
out that a reasonable estimate of i is
SOME OF
Using Eq. (29), we shall show in the following
sections that h,/ho, E1/eor and ¥1/¥ are all of
the order of 1072. Thus, if i = 30, stopping at
the second approximation, that is, at the terms
with the first power in X, would introduce an error
of about 10%, if we assume, as we evidently can,
that the ratio 1072 would apply to (An+1)/hpy etc.
for n equal and greater than 1. If X = 50 this
error would be about 25 to 30 percent, and it would
be necessary to go to at least the 042 terms to
reduce the error to less than 15%.
We shall delay the presentation of the estimate
of A until later and shall proceed with the solution
according to the approach in Eq. (29). In awaiting
the experimental determination of i, we shall carry
out the solution to the second approximation.
6. THE FIRST APPROXIMATION
The first approximation is governed by the equations
Lh, = 0, (30)
(L - 1) &9= -ho,, (31)
WY ar Md = SS, 1 (32)
0 Orr 0
with the boundary conditions (i) and (ii) stated
before, which we need not repeat here.
The solution of Eq. (30) is
2 2
- +
Mis ee oo)
and application of the integral condition (28) on
ho gives the value 1/m for C, so that
2
al ie
ho = =e F (33)
where
me! Ge 4b 62.
Then the solution of Eq. (31) is
2 2
2 75 2 =e
Eo =a a, 1e = ee Gels 8 re 0
where
Oe= tern & 4
n
Given Eo. Eq. (32) can be easily integrated by
separation of the variables r and 8. The result is
=
Y= ee (So he (34)
The isotherms given by Eq. (32) are just concen-
tric circles. But the streamlines given by Eq.
(34) are already interesting. They are shown in
Figure 1, which shows two very prominent vortices,
with the vorticity pointing in the x direction.
Thus the first approximation already shows the
prominent features of the flow pattern in any plane
normal to the x axis. Note that both the flow
pattern and the temperature field are symmetric
Cic
FIGURE 1. Flow pattern from the first approximation.
The horizontal axis is the n axis, the vertical axis
the © axis, and the arrow indicates the direction of
the gravitational acceleration. The value of 679 is
zero on the t axis. It increases toward the left and
decreases toward the right. The increments (or decre-
ments) are all 0.1.
with respect to the € axis, and that both ho and
Yo vanish at infinity, as desired.
The maximum vorticity is 0.09 and is at the
point
i) = WD, 6 = Op
at which both V and W are zero. The maximum vertical
velocity is 1/6m and is at the origin. The maximum
absolute value of Yo is 0.63817/61, which occurs
a © 2 © 2 wy te = alos.
7. THE SECOND APPROXIMATION
The equation for h, is
Lh, = Vghon + Woh (35)
1 Omi Ole”
where Vj) and Wo are the velocity components from
the first approximation. The right-hand side of
Eq. (35) can be written in polar coordinates as
KIFR
Cle) #9 dr
Hence Eq. (35) can be written as
A ae apd
iy, = = Sea P (oe? Yep
3r@r
where L, in its polar-coordinate form, is
32 i 1 92 3
b= Sl “= te’ at eZ te’
Writing
in 6
bh sets mm ©), (36)
1 2 1
31
we have
Ve 5
e Aig
L,H, = - (l-e Yip (37)
if we write L, for L with the operator 32/902 in
L replaced by -n2.
To solve Eq. (37), we let
sl SS ie ep (38)
so that Eq. (37) becomes
2 2
= (Loe Jo (a)
cel) see (2 = 3) £" + 2£ = =e
Then we approximate the right-hand side of this
equation by
2 2
2.-r x x ie x
r“e ( 5 B ) (40)
The greatest error occurs at r = 1.8, but it is
less than 6.5% of the maximum value of the quantity
approximated. Up to r = 1.2 the approximation is
excellent. It is expected that the local errors
around r = 1.8 will be diffused out when Eq. (39)
611
is integrated and will introduce negligible errors
in the result. After (40) is substituted into
Eq. (39), the latter is solved by repeated use of
the following formula for various values of n:
=r ZIn(n = 2) - 2(n - 1)r2]Jer¥
The result for f is put into Eq. (38), and we have
1S | Gi ya
H, = == - —— —— rt - ——
ie 3 G2 1G » Iya) Bona
r®) eons (41)
The function Hy is tabulated in Table 1. A look
at h, given by Eq. (36) then reveals that the
temperature is increased in the upper half of the
n-G plane and decreased in the lower-half plane,
making the isotherms more widely spaced in the
upper-half plane and more crowded in the lower-half
plane.
The tabulated values of H, show a very smooth
variation of H, with r, verifying the expectation
that the local irregular variation of (40) is
diffused away when Eq. (39) is solved with (40)
replacing its right-hand side.
The next step is to solve
Gb = ais, 2 Sa + VoEon + Woboc- (42)
1n
A simplification is possible before we attempt to
solve Eq. (42). Differentiating Eq. (35), we have
(L + 2) = Vohonn + WoPonz + VonPon
+ WonPoc: (43)
Let
hy
f= aa + q. (44)
Then Eq. (42) becomes
BT il
(L- lq + (L + 2) S~ = FZ Wohgnn + Wohonz) » (45)
since
1
S09 = 3 Aon
By virtue of (43), Eq. (45) becomes
aL
(ieee) qa 3 Yonon ap WonPoc) - (46)
But
Von = (¥o)on, Wor eS -(¥9)nn,
so that ¥ is a stream function for the fictitious
velocity field (Vont Won)» and we can write Eq. (46)
as
612
TABLE 1 Values of H,, S, and F; for r > 4, -100F = 5.04 r72
r 0.1 0.2 0.3 0.4 OS 0.6 0.7 0.8 0.9 1.0
100H), 3.03 5.85 8.28 10.19 11.48 12.14 12.20 11.75 10.89 9.76
-100S 0.25 0.97 2.06 39 - 80 12 UoP3 8.00 8.41 8.43
-100F 0.03 0.11 0.24 0.41 0.60 0.80 1.00 iL AL 7/ igs 1.44
r itoal eo? tgs} 1.4 eo) 1.6 ed 1.8 ike) 2.0
100H, 8.47 PoU3} 5.84 4.65 3.62 2.74 2.04 1.48 1.06 0.75
-100S 8.11 7.51 6.70 5 7/8) 4.83 Soe)al 3.07 2.34 Leys} 1675)
-100F 1D 1.56 1G S)7/ 1.55 iLook 1.45 1.38 1.30 eer 212} iligals}
r PAL Deed, od) 2.4 Bod) 2.6 Bet 2.8 oe) 3.0
100H, 0.53 0.38 0.27 0.19 0.14 0.11 0.08 0.06 0.05 0.03
-100S 0.88 0.61 0.42 0.29 0.20 0.14 0.11 0.08 0.06 0.05
-100F 1.05 0.98 0.90 0.84 0.78 On75 0.68 0.63 O59 0.56
r Shoal Sie SSS) 3.4 355) 3.6 a7 3.8 3.9 4.0
100H, 0.02 0.02 0.01 0.01 0.01 0.00
-100S 0.04 . 0.03 0.02 0.02 0.01 0.01 0.01 0.00
-100F OF 2 0.49 0.46 0.44 0.41 OF39 0.37 0.35 0233 0.32
(L-Da=- = Z%y - ut Aa eer (2? és ) -2 Ses = art (elt Eee)
Remembering that for
p) sin 6 9 L GG oo one E) = en (22 a z)
aa aor e cos 8 We 2 8 24 8
aie
and with ¥o and ho given by Eqs. (34) and (32), we oy ‘
have, finally, and the last member of (49) can be approximated by
one eighth of (40). By repeated use of the formula
re al
(@ = 1)q =—— sin 26 |e Q f =) Pare Bans) a8
on BD inGe~ee, )) = & [n(n - 4) - (2n - 3)r?Je
7 2 for various values of n, we can then find the
- — enn (47) solution for (48), and the final result for q is
r2
2 f
To solve this, let q = —z sin 20 + Q, (50)
on
) :
qa= — epi 2G) 0 ae i, wanes
on 2 2) 2
al pee, Cre a al 116 5
= = - -=+
2 Bie fc Wa ( 8° 9945 *
Then Eq. (47) becomes e r
— Sh 6
A 3 =P 95472 42432
Te = ki (2x - =) IRD Sik BS © [e (eS) = a),
(48) Lgl sia) (51)
32640
where L is the linear operator defined by (48).
It is advantageous to write the right-hand side of With hy given by (36) and (41) and therefore with
(48) as hy, known, (44), (50), and (51) give
f= sin 26 + S(r), (52)
oT
where
2 2,
19 1 = aL 1OS283 a2
Ss = —— = += =
Se 2 fe DB) oT SAD
4r
114713 i, SSALz/ ¥
1670760 247520
x 181 ome aL id
68544 4320
==)
(53)
The values of S(r) are tabulated in Table 1, from
which it can be seen that the maximum absolute
value of S occurs at about r = 0.95 and is about
0.847. Since S is negative throughout, inspection
of Eq. (52) shows that the maximum value of &) is
at
= 095), 6 =
The effect of S is to reduce the strenths of the
vorticity for the lower-half plane, but to augment
them in the upper-half plane, thus to raise the
eyes of the vortices.
Finally, ¥; is to be found from
9 i, 6 iy Oe
Ty Yee, = A eee 2 Te) Yn = Ene
or 1G 08
Let
vy, z sin 2 F (r) (54)
9
Then
po Bop os Sop S este),
re 2
Two integrations by the method of variation of
parameters (since a complimentary solution of F is
simply rv?) gives, with due regard for the boundary
conditions,
a2 i2
=5
F = -r? Yr r3Sdr dr
0 a
Yr 12
a)
al -2 r3sdar - r2 nG l car 2 (55)
= a @
6
613
which is given in Table 1 also. The calculations
for the second approximation have now been accom-
plished.
8. ESTIMATE OF i
The terms involving € in Eq.
in the Reynolds stress terms
(2) have their origin
o) Ure Q 12
By (v'w') and De (KE) 5
where the primes indicate turbulent quantities.
The terms were originally on the left-hand side of
Eq. (2). The nonlinear terms on the left-hand
side of Eq. (2) can be written as
oh OE @
ay (vw) + ae (wo).
Thus the ratio of
2. G2 wo). 12
ae (w*) and aS (w'<)
is the ratio of
3 2
a5 (w-) and “SW 0
and this ratio has the magnitude of
-\W2/W .
/ c
The magnitude of Wy is 1/67, and the magnitude of
Wo, is 0.267/31, which is the maximum value of Wor
along the n axis. Thus, approximately,
Ar AL
0.267(12m) 5 me
where s is the square of w'/w. The convection in
the bent plume is like the convection in a two-
dimensional plume, since the plume is bent by the
wind to a nearly horizontal position. The measure-
ments of Kotsovinos (1977) for the plane plume
give the value 0.2 to s. This is considered by
some people to be too high. But for the problem
under investigation s may be even higher, because
any swaying or deformation of the vortices would
contribute a good deal to turbulence. Thus using
0.2 for s in Eq. (56) would overestimate A. Using
0.2 for s, we obtain from Eq. (2)
dX = 48.5.
This is probably too high. My estimate of i is
that it is somewhere in the range
50) S AS AIO),
The value 30 for A} corresponds to a value of 0.34
LOSI:
Let us now see what errors would be committed
for h, ¥, and & by stopping at the second approxi-
Mation. For A = 30, the errors (in ratio of the
614
FIGURE 2. Flow pattern from the second approximation.
The n axis is horizontal and the f axis vertical. The
arrow indicates the direction of the gravitational ac-
celeration. The value of 67¥ is, starting from the ¢
axis and going to the curves on the right, respectively,
Op Obi, “Os4p “Ossi “Wot, =—Oo5p —OaGg Einel —MoGEin Gus
values of 67¥ on the curves to the left of the f axis
have corresponding absolute values but are positive.
estimated* maximum value of the terms neglected to
the maximum value of the computed quantity) are,
respectively, less than 15%, 3%, and 10%. For A
= 40 these percentage errors are, respectively,
25%, 5%, and 18%. The most interesting thing to
note is that ¥ is the most accurately calculated
quantity. Figure 2 shows the flow pattern ina
plane normal to the x axis, and Figure 3 shows the
isotherms therein, all for } = 30. The flow pattern
in Figure 2 can be regarded as sufficiently accurate
to be representative of the actual flow pattern in
a plane normal to the x axis. As expected, the
hottest point and the "eyes" of the vortices occur
at positive values of f. That is to say, the plume
rises according to the x°/3 law. After the present
work was done, I found that this law had recently
been verified experimentally by Wright (1977),
although he did not measure the detailed velocity
and temperature distributions in the plume.
If later measurements show i is larger than 30,
higher approximations would be necessary.
9. DISCUSSION
It is perhaps surprising that the analysis shows
that the results in dimensionless terms are indepen-
dent of the parameter Gg2/u°. The explanation is
that the velocity (v,w) far downwind from the heat
source becomes vanishingly small, and whatever the
value of U, the transverse wind is asymptotically
always strong.
Near the heat source the flow indeed depends
very much on the magnitude of U. The plume may
*On the basis that hj/hp and (hn+)) /Ay are of the same order
of magnitude and that the same is true for ¥ and &.
FIGURE 3. Isotherms from the second approximation.
The n axis is horizontal and the ¢ axis vertical. The
arrow indicates the direction of the gravitational ac-
celeration. The value of th is 1.1 on the smallest
closed curve and 0.3 on the outermost curve. The incre-
ments are 0.1.
rise high in a weak wind before being bent suffici-
ently for the present theory to apply. In using
the present theory it is always necessary to
determine a virtual position for the heat source,
which for small value of U can be considerably
higher than its actual position.
ACKNOWLEDGMENT
This work was partially supported by the Office of
Naval Research. The subject of this work was
suggested to me by my friend Dr. Michel Hug, Director
of the Department of Equipment, Electricity of
France, through Mr. F. Boulot of the National
Hydraulics Laboratory at Chatou, France, during my
brief sojourn there in the summer of 1977. Their
interest in this work, as well as the interest of
Dr. A. Daubert, director of that laboratory, is
very much appreciated. The work, begun at Chatou,
was substantially improved and finished during the
tenure of my Humboldt Award, at the University of
Karlsruhe. To the Humboldt Foundation and my
Karlsruhe hosts I should like to express my sincere
appreciation.
REFERENCES
Kotsovinos, N. E. (1977). Plane turbulent buoyant
jets. Part 2. Turbulence structure. J. Fluid
Mech. 81, 45-62 (see P. 52, Figure 7).
Laufer, J. (1953). The structure of turbulence in
fully developed pipe flow. NACA Tech. Note
2954.
Wright, S. J. (1977). Report KH-R-36, Keck Labor-
atory, California Institute of Technology.
Yih, C.-S. (1977). Turbulent buoyant plumes.
Phys. Fluids 20, 1234-1237.
APPENDIX:
THE EFFECT OF NEGLECTING THE PRESSURE
GRADIENT IN CALCULATIONS FOR THE CONVECTION
PLUME IN A TRANSVERSE WIND
J. P. Benqué
Electricité de France
Chatou, France
In many previous studies on jets and plumes, the
pressure distribution in the jets or plumes is
assumed hydrostatic, so that if the body-force term
in the equation of motion is written in the form
-gAp, where Ap is the difference between the local
density and the ambient density, the pressure gradi-
ent can be neglected in the equations of motion.
If, further, the flow is two dimensional or axisym-
metric, only the equation of motion for the vertical
velocity component is then needed. After that
velocity component is determined, the equation of
continuity can be used to determine the other veloc-
ity component.
In the preceding paper by Yih, the assumption
that the x component of the velocity is constant
leaves only two other velocity components to be
determined, and it is tempting to adopt the usual
procedure of neglecting the pressure gradient. Yih
has resisted that temptation. But it is useful to
see what effects such a neglect would have on the
flow and to determine whether in the problem treated
by Yih such a neglect is allowable. This Appendix
is devoted to this question.
If the pressure distribution is assumed hydro-
static and the usual procedure is followed, one will
drop Eq. (1) and retain Eq. (2), with the first
term on its right-hand side dropped. [Equation
numbers in Yih's paper are retained.] Equations
(3) to (7) will remain but (8) and (9) will not be
needed.
Following Yih's development and using his nota-
tion, then, we have, as the dimensionless equations
to solve, (24) and
(L - 3)W = -h + A(VW, + WW). (A.1)
Using the A-series (29), we have again (33) for the
solution of hy. The equation for Wor obtained from
(A.1), however, is now
(L - 3)Wy = hy: (A. 2)
The solution of this equation, satisfying all the
boundary conditions for W stated in Yih's paper, is
1 =y2 1 -n2-72
0 3 - aC s (A. 3)
Although it can be readily verified that Eq.
(A.3) satisfies Eq. (A.2), it is not obvious that
Eq. (A.3) is the unique solution. We shall show
in the following that it indeed is the unique
solution. The complementary solution Wo¢ of Eq.
(A.2) satisfies
(L - 3)Woc = O (A.4)
and must be even in both n and ft. Let
Woo = E(Mg(Z),
615
where the f is in no manner the same as the f in
Eq. (38) of Yih's paper, we have
£" 4) 2n£! Gat =" 0}, (A.5)
g' + 20g' + bg = 0, (A.6)
where
a+b = 1. (A.7)
Now let
-n2/
fi) =e 28 (h) - (A.8)
Then Eq. (A.5) becomes
BM = (n2 + b)R = O- (A.9)
Similarly, if we let
=P
g(t) =e b /2(c) .
Then
y" - (t2 + a)y = 0. (Al10)
Because of Eq. (A.7), a or b must be positive. Let
b be positive. (The argument is strictly similar
if a is positive.) Because of the symmetry with
respect to the f axis,
B' (O)| = 0.
Then Eq. (A.9) shows that 8 will approach infinity
as n° approaches infinity, if 8(0) is not zero.
[If 8(0) = 0 then 8B = O throughout.] To see how
£(n) behaves at infinity, it is necessary to see
how 8(n) behaves asymptotically. A simple calcula-
tion shows that the two solutions of Eq. (A.9)
behave, for large values of n2, like
1
exp |- (n2 - a - 2)%dn| and exp (n2 - a) 2an|.
As we have seen, 8 must contain the second solution
since 8 approaches infinity as n2 + ©, Using the
second solution as the dominant term (a constant
multiplier being understood), and recalling that
Df en di ai part io
(n a) n mn + 0 ( 2 ,
we see from Eq. (A.8) that for large 2
aqy.< lal 2, (A.11)
which can be seen to satisfy Eq. (A.5) asymptotically.
If a is negative, (A.11) shows that f(n), and there-
fore Wo, cannot satisfy the condition on Wp at
infinity. If a is positive, it must be less than
1, because of (A.7) and because b is positive.
Then if Wo contains Woor
616
ol which shows that at In| = ©, Vj does not vanish.
We must then, if we adopt the procedure of neglecting
the pressure gradient, not demand that Vo vanish
at infinity, but instead demand
SS 0
This boundary condition for V, must, for consistency,
be demanded of V, i.e., of Vj, V2, etc., as we
proceed to higher and higher approximations.
In this connection we can also see that it is
not possible to add a multiple of Woo to the Wy
given by Eq. (A.3) to make V, vanish at infinity.
For, in order to make Vo vanish at infinity, the
only possibility is to add to the Wo given by Eq.
(A.3) a multiple of
FIGURE A.1. Flow pattern for (V,W)). Aém¥, = 0.2.
2
Worse) a(n) (A.15)
co co
where f satisfies
I= W )dndz = oe, (A.12) £" + 2nf' - f = 0.
That means
a=-l,
But this cannot be true, because integration of
(A.2), by parts if necessary, gives and Eq. (A.11) gives
1
c} i) £(n) ~ [In| 2,
=spl SS = hydnds —tiLy, which makes Woar and therefore Wo if it contains
Woc, infinite at |n| = ©, Any other dependence of
Wo on © than exp (-67) would, of course, not make
rae rc a4 Vo vanish at infinity, for the part of Vo that
so that arises from Wo, would not be able to cancel out Eq.
cael (AQ13) at |=.
3) Hence Wy and V) are uniquely given by Eqs. (A.3)
and (A.13). Using them in
Hence W cannot contain a multiple of Woar and Eq.
(A.3) is the unique solution. Lh, = Voho, + Wohors (A.16)
Then the equation of continuity gives
and
: 2 2
2 — + - — -
V2 f te (n G Fane (A.13) (L - 3)W, hy + VoWo, + WoWors (A.17)
g we find that
> hy
0,10 W) = 5 (A.18)
I have computed h, numerically from Eq. (A.16)
and the boundary conditions, and therefore W}).
The velocity component, V,, is then found from the
by equation of continuity. The flow pattern corres-
ponding to (V,,W)) is given in Figure A.1, where
250
¢ the streamlines are shown, with ¥; = 0 on the ¢
: axis and Aém¥, = 0.2.
Then the flow field for
oF V=vV
ee o + AV, and W = Wy + AW)
is shown in Figure A.2, with X = 30, where the
AetY = 0.2.
It is clear that the "streamlines" do not close
to form closed eddies, as in the figures of Yih's
paper. Thus the effect of the pressure gradient
Flow pattern for (Vj + AV, Wo + AW))- cannot be neglected in the problem studied by Yih.
In past studies of jets and plumes, where the
FIGURE
A6rY =
Oop
OS)
pressure gradient has been successfully neglected,
the velocity component other than the one retained
is one order of magnitude smaller than the one
retained. Thus the equation of motion for it can
be neglected together with the gradient of (the
dynamic part of) the pressure, and the flow pattern
617
can be determined from the equation of continuity
once the principal component of the velocity is
determined. Such is not the case in the problem
under discussion here, and therefore for this prob-
lem it is necessary to retain the pressure gradient,
as Yih has done.
Internal Waves
OF Me SPheslaaip's
Johns Hopkins University
Baltimore, Maryland
ABSTRACT
It has become evident in the past few years that
the wave-number, frequency spectrum of deep ocean
oscillations has a remarkably consistent form close
to that which would be expected for statistical
equilibrium among the modes under wave-wave resonant
interactions. The energy sources that maintain deep
oceanic internal waves are, however, not well under-
stood. -
In the vicinity of the thermocline, the energy
density (per unit mass) of internal wave activity
is generally much greater than in the ocean depths.
Relatively high frequency internal waves, generated
in a variety of ways, are to a first approximation,
trapped in this region. Disturbances whose fre-
quencies are less than Ng, the deep stability fre-
quency, do however radiate downwards effectively.
Also, groups of high frequency, low mode waves
generate second order mean perturbations to the
thermocline structure, and if the group frequency
is less than Ng, again energy radiates down. The
flux of energy into the deep ocean is illustrated
first in a simple model in which a sharp pycnocline
lies over uniformly weakly stratified water. The
more general problem involving an arbitrary strati-
fication is formulated and some preliminary asymp-
totic solutions are presented.
1. INTRODUCTION
During the last 10 years or so, a variety of new
and ingeneous oceanographic observations has been
made on the structure of internal waves fluctua-
tions in the ocean. Twelve years ago, in the first
edition of The Dynamics of the Upper Ocean, I was
forced to write that in view of the difficulty and
expense involved in the systematic study of oceanic
internal waves, "those (measurements) that do exist
are correspondingly rare and valuable." The present
situation is gratifyingly different. Deep oceanic
618
observations of internal waves are no longer rare,
but they remain valuable; Cairns (1975), Katz (1975),
Gould, Simmons, and Wunsch (1974), and a number of
others have provided different kinds of observations
from which a consistent pattern is emerging. It ap-
pears that the deep oceanic internal wave spectrum
has a remarkably universal form close to that speci-
fied by the Garrett-Munk (1975) spectrum, though why
this is so cannot yet, I think, be asserted with con-
fidence. McComas' (1975) calculations on resonant
wave-wave interactions indicate that the Garrett-
Munk spectrum is close to what one would expect in
a state of statistical equilibrium under the balance
of these interactions. On the other hand, there are
indications, such as the occurrence of sporadic,
isolated patches of turbulence in the stably strati-
fied regions of the ocean which suggest that local
instabilities may be limiting the wave spectral
density.
Soviet investigations, such as those of
Brekhovskikh et al. (1975) have concentrated on the
low mode structure in the thermocline region whose
energy density (per unit mass) exceeds, usually by
an order of magnitude, that of the deep oceanic in-
ternal waves. The characteristic frequencies are
also about an order of magnitude higher. The cal-
culations of Watson, West, and Cohen (1975) among
others indicate that the lowest modes are generated
quite rapidly by interactions among surface wave
components; a number of studies along these lines
are described in the useful review by Thorpe (1975)
and by the present author (1977). The upper ocean
is certainly the site of considerable dynamical
activity, but how much of it is radiated downwards
-to provide a source for those motions encountered
in the deeper, less strongly stratified region be-
low? According to the usual linear analysis, the
low mode, relatively high frequency waves are trapped
to the strongly stratified thermocline region; only
the low frequency high modes have structure that can
penetrate great depth.
Yet the description of deep oceanic motions as a
linear superposition of high modes may make little
sense. A linear mode can itself be considered the
superposition of two disturbance trains, one propa-
gating downwards and the other upwards with reflec-
tions either at the bottom or at a region where the
buoyancy (or stability) frequency N drops below the
wave frequency. McComas! calculations indicate that
the non-linear interaction time of such components
at the spectral densities found in the deep ocean,
is remarkably short, only a few wave periods in
many wave cases. Accordingly, a train of waves
generated, say, near the thermocline will in actu-
ality have little opportunity to travel to the
bottom, reflect upwards, and combine with a down-
wards travelling wave to produce a 'mode' as usually
conceived. More realistic would be the view of dis-
turbances generated in the more active thermocline
region, radiated downwards but being 'scrambled' by
wave-wave interactions into a more diffuse spectral
background.
This contribution is concerned with some aspects
of the energy flux downwards from high frequency,
low mode internal waves at the thermocline. If the
internal wave frequency is greater than the stability
frequency Ng below the thermocline, the waves are
of course trapped to the thermocline region. How-
ever, as their frequency decreases below Ng, they
become 'leaky' and their energy radiates rapidly
downwards as the simple analysis of the next section
will demonstrate. Yet, if Brekhofskikh et al. (1975)
measurements are at all typical, most of the energy
of the low mode internal waves in the thermocline
region is at frequencies considerably above Nagi
indeed, in view of the efficiency with which such
low frequency energy is propagated downwards, we
would not expect to find much energy at these fre-
quencies in the main thermocline. However, one
possible link is suggested by the work of McIntyre
(1973) who showed that groups of internal waves in
a fluid of constant frequency N, confined between
horizontal boundaries, produce second order 'mean'
motions, modulated as are the wave groups. There
is no reason to believe that these second order dis-
turbances are confined only to the particularly
simple case that he considered, and indeed in Sec-
tion 3 it is shown that they are not.
internal waves, occurring in groups and trapped
within the main thermocline, produce second order
low frequency disturbances; if the group frequency
is less than Ng, their energy is radiated downwards
at the group frequency.
The results presented here are preliminary but
intended to provoke consideration of this mechanism
as a source of oceanic internal waves. The simplest
case of a sharp thermocline overlying a deep, uni-
formly stratified region is described in some detail.
The more realistic (and complicated) case with a
general distribution of N(z) can be considered by
asymptotic methods and these results will be de-
scribed elsewhere.
2. RADIATION DOWNWARDS--A "LEAKY MODE"
Consider the following experiment: a laboratory
tank (Figure 1) is stratified with a layer of uni-
form density lying over a density jump 6p below
which the fluid is uniformly stratified, with N? =
(-p7lg d0/dz) = constant. A wave-maker at the end
of the tank generates a periodic disturbance with
(real) frequency n. What are the characteristics
of the motion induced?
High frequency
619
It is, I think intuitively evident that if n > N
an interfacial wave mode will propagate. The struc-
ture of the mode below the pycnocline will be in-
fluenced by the stratification but at these high
frequencies, no internal waves can be supported in
the lower layer and the interfacial wave will propa-
gate without loss. If, however, n < N, internal
waves induced in the lower region by the interfacial
disturbance can carry energy downwards so that the
interfacial wave will attenuate. The question is:
how rapidly does this occur?
A linear analyses suffices. Suppose the pycno-
cline displacement is represented by the real part
of © = a exp i(kx - nt), where n is real and k may
be real or complex. Above the pycnocline at z = 0,
the motion is irrotational with u = Vd and V2 = 0.
In the uniformly stratified region below, the vert-
ical velocity component, w, obeys the internal wave
equation
a2 2 Dep We
cen VA Row 2 © , (1)
where Vine is the horizontal Laplacian operator,
32 /ax2 in this two-dimensional problem. At the
upper free surface at z = d, w = O to sufficient
accuracy; at the pycnocline the vertical displace-
ment and the pressure must both be continuous and
as z>- ©”, the disturbance must either die away
or represent internal waves with an energy flux
downwards.
In the upper region, the solution for 9 is
readily found to be
_ ina cosh k(z - dq) F 2
> = Te sinh ka CxPi (kx Me) p (2)
while in the lower layer, if
w = - ina exp [kz + i(kx - nt)] , (3)
(which satisfies the condition of continuity of w
at z = 0), then substitution into (1) requires that
(Pje silo Gye c (4)
Note that since n is real, «/k is either purely real
(Gin 2 N) or purely imaginary (if n < N).
The dispersional relation is obtained from the
condition that the pressure be continuous at z = f.
In the upper region of density p,
Pp, = - pg = 22 |
- palg + (n*/k) coth kd] exp i(kx -nt) , (5)
Wave
Absorbers
FIGURE 1. Tank stratified with a layer of uniform
density over a density jump below which the fluid is
uniformly stratified.
620
to the first order in the wave amplitude. In the
lower region, where the density is p + 6p - oN2z/g,
the horizontal pressure gradient
DS 2 ou (ond Woy Se
to the lowest order,
ax ot
i(p + 6p)an2(k/k) exp i(kx - nt)
at z = 0 from (3), so that
Po = (9 + 6p) an2(k/k2) exp i(kx - nt)
At z = T, below the pycnocline,
Py =- (p + dp)a(g - n2K/k?) exp i(kx - nt) (6)
From (5) and (6) it follows that
2 _ __(6p/p) gk Bie ek ae (7)
coth kd + (k/k) coth kd + (K/k) ’
to the Boussinesq approximation, when é6p/p << 1, and
where b is the contrast in buoyancy across the pycno-
cline.
For high frequency oscillations, when n 2 N,
equation (4) shows that k/k is real and less than
unity; from (7) k is real and the waves propagate
without attenuation. The additional restoring
forces provided by the stratification below do how-
ever increase the wave frequency for given k and b
above the value for an unstratified lower layer by
the ratio
[coth kd + 1] / [coth kd + (1 - N2/n2)*%]
The case when n < N is algebraically simplest when
|ka| +e, In view of the upper boundary conditions,
the real part of k > 0, while from (4)
2\}5
K N -
mm = 2 i(a - a) =e ta stan 6), (8)
where n = N cos @. From (7)
Pe = (Gai) (il Hh ee Ch) G (9)
Since the interfacial waves attenuate in the posi-
tive x-direction as energy leaks downwards, the
positive sign in (9) is relevant and the vertical
wave-number
ray
i]
iktan 6,
(n2/b) ( - tan26 + i tan 6) . (10)
The motion of the pycnocline is therefore repre-
sented by ;
2 2 \
= _ nx fetes hotel
C=a exp ( Spe tan 8) exp 1 (a ne) ,
n?x
= a(x) exp i SS nt p (11)
where
2
a(x) = a exp (- n* tan e) :
The ratio of the spatial attenuation rate to the
wave-number is simply tan 6 = (N2/n2 = 1)”; when n
is significantly less than N the attenuation dis-—
tance is short as the energy leaks downwards very
effectively.
Expressions for the motion in the upper and
lower regions can be written down simply. In the
lower layer energy flows along the characteristics
—& = x cos @ + z sin 8 = const., and the distribu-
tion of vertical velocity is
wes fm e@ (- nieint . (25 aoe t)
*P bcos26 ©XP + \beosé u
2
mete aff rte!
=)/=no!7)(&))) exp i( ST oe ) fF (12)
Zz
g
6 x
S
IS q
Ne SS
N ~
SS SS
S
where a)(&) is the amplitude of the interfacial
wave at the point where the characteristic inter-
sects the pycnocline. The horizontal component
of the velocity field in the lower layer is u =
- w tan 6, since the motion here consists of alter-
nate layers sliding relative to one another along
the characteristic surface inclined at an angle 98
to the vertical. The pressure fluctuation can be
found most simply from the horizontal momentum
equation:
n2E
p = - a1(&)bsin6(sin® + icos§)exp i eaeeat =
The vertical energy flux is therefore
2
ES = -4noa ](&)b sin@cosé ,
and the total energy flux, directed downwards along
the characteristics — = const is
E = mnaj(—) bsine . (14)
In the upper region, the fluctuations in pressure
are found from (2):
pe DD 2 es
DRE) seca neces, [-kz + i (kx - nt)] ,
when kd >> 1, whose real part, in view of (9) and
(12) reduces to
IS. = & ei(Ge op zcot6) b(cos*8cosy + cos@sin@sinyx) , (15)
ae
where
n2
xX = pb (* - ztané@) - nt P
The real part of the horizontal velocity field is
likewise
UL. = (0$/3x) = - na(x + zcot@) cosy ;
so that the horizontal energy flux in the upper
layer
the horizontal divergence of which
ab = - kna?(x)b sin@cosé (17)
provides for the radiative flux in the lower layer.
This simple example illustrates the way that
energy can be radiated downwards by the low fre-
quency perturbations produced by groups of high
frequency waves, but they have a deeper theoretical
interest. Gaster (1977) has pointed out that if
the dispersion relation for waves involves complex
wave-numbers or frequencies, the usual kinematic
definition of group velocity may not be correct,
and a simple calculation shows that the solution
is an example of this failure. Here the wave-
numbers are complex as the energy leaks into the
lower layer, but the energy flux is not at the rate
represented by the local energy density, n2a}2/2
cos*6, times the ordinary group velocity Vw =
c tan 6 = (b/n) sin @. The correct interpretation
of these situations will be considered elsewhere.
3. ENERGY RADIATION DOWNWARDS FROM GROUPS OF
INTERFACTAL WAVES
To illustrate the way in which groups of internal
waves produce 'mean,' second order disturbances locked
to the wave group, let us consider the same basic
stratification as in the previous section, with
fluid of depth d and constant density lying over
a buoyancy jump b below which the stability fre-
quency N is constant. Suppose that interfacial
waves with frequency n > N are maintained by high
frequency forcing £ from the upper layer, perhaps
by the surface wave-wave interactions described by
Watson, West, and Cohen (1976). If the internal
wave amplitude is characterised by a and the wave-
number by k, then, to order E2 = (ak)*, the condition
or continuity of pressure across the interface can
be expressed as
du pY4 ( 2u )
S)\)-bp = =- tu: V ; 8
(2) b Aen +f I Wie azct u u (18)
ae B= O, wae M( )} = Cn = ( Yag the difference
across the density jumps. Since
aie 2 - pO harness a GS
C= we uy VG =wo +f a2 I, we ae 0
= = oe (u ) (19)
0 3x 0S ,
to order E2, where the suffixes, tT and 0, represent
quantities measured at z = t,0, then the condition
that tf be continuous across the interface assumes
the form
621
(sie)
Aw = A ox
at z=0O . (20)
Finally, in the lower layer,
nae Wo a Ne ae SO oy (21)
93
2 axdzot go wey
- Cee * Vb + oa + Vw) ) (22)
ox = dt .
Variations in energy density of the primary waves
will propagate with the group velocity, c,; let us
therefore average these equations at aeslnres fixed
with respect to the wave groups but over random
phases of the waves themselves, a process repre-
sented by brackets [ ]. The averaged interfacial
conditions are then, to order E% = (ak)2,
a, | oe S95 22d Nel eee gon gyi (23)
at ox azot ~ U
a
AS ev} = Nee (x6) ' (24)
ox |
both at z = 0, and
(2) = fe) = & Teel (25)
ox 4
also at z= 0. The averaged field equation for the
lower layer follows similarly from (21) and (22).
The linear fluctuating internal wave motion is
as given in the previous section when n > N; through
the non-linear terms on the right of (23)-(25), this
forces a second order mean disturbance [zc], [wl],
etc., that moves with the velocity of the wave groups.
The pycnocline disturbance can be represented as
t = ka{cos(k'x - n't) + cos(k"x - n"t)} .
The form of the forcing functions is simplest when
the pycnocline depth is such that kd >> 1, and it
is found that (23) reduces to
ou eC)
ae | Bes [c]
2
= % a2 uipts
4a es {2 («) |
(53S i ‘8)
g
ips
(c_ + 4e)Sin k
g g
{1 + O(ak, n? fn?) 3 0
i}
c
- k a2k NZ (<2 + 1) Salina Ik (5 © C38) 5 (26)
g c 2 g g
Wont Be = et ca jel Ng = n' - n", and cy, represent
the wave-number, frequency, and velocity of the
groups. Similarly, from (24)
622
1 2 f K Q
Mw = q an ae + «) sin xX) 7 (27)
where X = kg(x - cgt) and from (25)
A al :
(Es) [wil g ank, Same’
i]
an
|
aly igs ;
[w]_ 3 2 is e/a) sini! 7 (28)
where [ ], and [ ]J_ represent averages taken just
above and below the discontinuity in density.
These matching conditions to be applied as z = 0
involve the non-linear forcing provided by the wave
groups. The field equations are, however, linear
to this order.
we have Laplace's equation for the averaged velocity
lols O. | (29)
while in the averaged internal wave equation (21)
for z < 0, the non-linear terms are smaller by at
least (ng/n) 2 << 1 than those in the matching con-
ditions, since they involve two horizontal deriva-
tives (or one x and one t derivative) of averaged
second order quantities. Accordingly, to sufficient
accuracy,
Oe ea 2 22 =
DED Vi[w] + N aes Kal SO 5 1a <6 © «6 (30)
Since the length of the wave groups is large
compared with the wavelength of the interfacial
waves, k. << k and it is consistent to assume that
kgd << 1, even though kd >> 1. Furthermore n/N <<
1 while NgN = O(1). Under these conditions the
solutions for the mean pycnocline displacement and
the low frequency internal waves radiated downwards
are found to be
a2nc
aed
[co] =- ba COs ka (x - ea) g (31)
[w] 2» - 5 arnk, (2 + a cos OSes ar Sa = ye) nS)
where
1
2 cy)
Baral paga are (33)
g
is the vertical wave-number of the radiated field.
The horizontal velocity component in the internal
wave motion below the pycnocline
{u] = [w] tanyp ,
where cos j = ng/N and the energy density (twice
the kinetic energy density) is
E =» (Tae + Twi),
patkg2n2n2 eN2
~ 126n2 | ( + £) G
4 2n2
an. 2pa n“N
ea ; (34)
g
Above the pycnocline, when d > z > O,
since n/N >> 1 and kK/k = 1. The vertical component
of the group velocity of the radiated waves is fg
cos ~ sin ~ where c, is the group of the inter-
facial waves, so that the vertical energy flux is
m7
to
n2 *
= (9/128) a'tn2Nk ( = el ) : (35)
g N2
Although this representation of the density dis-
tribution by a discontinuity at the pycnocline,
followed by a uniform stratification below, is a
gross simplification of typical oceanic conditions,
it is of interest to examine the order of magnitude
of the vertical energy flux that might be generated
in this way. If the interfacial wave amplitude is
10 m at a frequency of 5 c.p.h., having groups 1 km
in length and if N = 2 c.p.h., the downwards energy
flux is about 2 erg/cm* sec., which is of the same
order as the 5 erg/cm* sec. estimated by Garrett
and Munk (1972) for the rate of energy dissipation
from internal waves by sheer instability. This
correspondence is sufficiently close to encourage
a more detailed study with N(z) arbitrary, the re-
sults of which will be presented elsewhere.
ACKNOWLEDGMENT
This work was supported by the Fluid Dynamics
Branch of the Office of Naval Research under con-
tract NR 062-245.
REFERENCES
Brekhovskikhk, L. M., K. V. Konjaev, K. D. Sabinin,
and A. N. Serikov (1975). Short period internal
waves in the sea. J. Geophys. Res., 80, 856-64.
Cairns, J. L. (1975). Internal wave measurements
from a midwater float. J. Geophys. Res., 80,
299-306.
Garrett, C., and W. H. Munk (1972). Space-time
scales of internal waves. J. Geophys. Fluid
Dyn., 3), 225-64.
Garrett, C., and W. H. Munk (1975). Space-time
scales of internal waves: a progress report.
J. Geophys. Res., 80, 291-7.
Gaster, M. (1977). On the application of ray math-
ematics to nonconservative systems. Geofluid-
dynamical wave mathematics, Appl. Math. Gp.,
U. Washington, 61-6.
Gould, W. J., W. J. Schmitz, and C. Wunsch (1974).
Preliminary field results of a mid-ocean dynamics
experiment (MODE-0). Deep-sea Res., 21, 911-32.
Katz, E. J. (1975). Tow spectra from MODE. J.
Geophys. Res., 80, 1163-7.
McComas, C. H., and F. P. Bretherton (1977). Reso-
nant interaction of oceanic internal waves. J.
Geophys. Res., 82, 1397-1412.
McIntyre, M. E. (1973). Mean motions and impulse
of a guided internal wave packet. J. Fluid
Mech., 60, 801-11.
Phillips, O. M. (1977). Dynamics of the Upper Ocean
2nd ed., Cambridge University Press.
Thorpe, S. A. (1975). The excitation, dissipation
and interaction of internal waves in the deep
ocean. J. Geophys. Res., 80, 328-38.
Watson, K. M., B. West, and B. I. Cohen (1976).
Coupling of surface and internal gravity waves:
a Hamiltonian model. J. Fluid Mech., 77, 185-
208.
Breaking Internal Waves in Shear Flow
Si iS whorpe
Institute of Oceanographic Sciences,
Wormley, United Kingdom
ABSTRACT
During and following periods of strong winds, the
Richardson number (the square of the ratio of the
Brunt-Vdisdla frequency to the shear) in the
thermocline is of order unity, and the shear becomes
an important factor in determining the properties
of internal gravity waves. These properties are
discussed and the shape and breaking of waves ina
shear flow is investigated in laboratory experiments.
These experiments show that the waves may break at
their crests or their troughs depending on the sign
of a certain vector scalar product. An analogy
between surface waves and interfacial waves is
invoked to account for this behaviour. Breaking
is observed to occur by particles of fluid moving
forward more rapidly than the wave crest advances,
leading to gravitational instability. The effect
of breaking in the ocean will not only enhance
diffusion rates, but it will modify the directional
spectrum of the internal waves.
Although many acoustic backscatter observations
from ships reveal clearly the presence of internal
waves in the ocean seasonal thermocline, very few
have been published which appear to show signs of
their breaking. This is surprising in view of the
clear and not infrequent evidence of 'breaking
events' in the equivalent acoustic or Doppler radar
Measurements in the atmosphere. Our knowledge of
internal wave breaking in the ocean still rests
almost entirely on the direct observations by divers
using dye in the Mediterranean thermocline [Woods
(1968)]. The present towed, moored, or dropped
instruments give inadequate information on the
nature or structure of the intermittent mixing events
in the ocean to be certain of their cause, or even
of the scales of motion which contribute most to
diffusion across density surfaces in spite of its
great importance to the prediction of the thermo-
cline structure of the upper ocean.
It is against this background of poorly known
dynamical structures that this paper is presented.
623
One aim is to describe the patterns which accompany
wave breaking, for without a knowledge of such
patterns it is difficult to design the appropriate
experiment to detect wave breaking or, conversely,
to correctly identify the processes involved once
observations are available.
It would be naive to ignore the effect of wind
in a description of breaking waves on the surface
of the sea in deep water [see, for example, Phillips
and Banner (1974)]. (Wave breaking on a beach is
a different matter). It is similarly inappropriate
to ignore the effect of mean shear on internal waves
in the seasonal thermocline, since the Richardson
number there is low, especially during, and follow-
ing, storms [Halpern (1974)]. Internal gravity
waves can exist and propagate in a shear flow just
as they can when a mean flow is absent. These waves
belong to a group which Banks, Drazin, and Zaturska
(1976) have classified as 'modified' (-by shear)
"internal gravity waves'. They may sometimes coexist
with a set of wavelike disturbances which grow in
amplitude (the 'unstable wave solutions' of the
Taylor-Goldstein equation) and which may eventually
lead to turbulence (Figure 1). It is known however
that (for steady mean flows) the latter solution cor-
responding to Kelvin-Helmholtz instability (K-H.I)
only exists if the Richardson number, Ri, in the flow
is somewhere less than a quarter [Miles (1961),
Howard (1961)] and even then in some flows an un-
stable solution may not exist. One way in which
internal gravity waves may break is by themselves
causing or augmenting a mean shear to induce regions
of such low Ri that small-scale disturbances may
grow as K-H.I and generate turbulence. It appears
that Woods' (1968) billows were generated in this
way, and similar structures in Loch Ness [Thorpe,
Hall, Taylor, and Allen (1976)] may have a like cause.
It is however known that internal waves may break in
quite a different way, by what has been termed 'con-
vective instability' [Orlanski and Bryan (1969) ].
This form of instability becomes much more likely in
the presence of a mean shear.
FIGURE 1. The development of Kelvin Helmholtz In-
stability (K-H.I) in a stratified shear flow [from
Thorpe (1971)].
Shear affects internal gravity waves in several
ways. Perhaps the most important concern the wave
speed. Bell (1974) has shown that for any wave
mode, the phase speed, c, is a decreasing function
of wavenumber, k, which, for waves moving faster
than the mean flow at any level, tends to k71Nnax
+ U, as k increases indefinitely, where Nna
max x
is of the Brunt-Vaisdld frequency, N, and Uneee che
maximum mean flow. (A similar result holds for
waves travelling more slowly than the mean flow.)
This result reduces to the well-known property,
OS Rep Cie internal waves in the absence of shear
[Groen (1948)] where o = ck is the wave frequency
relative to the mean flow. It implies that even
in a shear flow the wave frequency is less than
Nmax Provided the waves are viewed in frame of
reference which moves forward at the speed, Upax-
Banks et al. showed further that, at least for
simple mean flow profiles, the speed of waves of
a given mode and wavenumber tends to Where (from
above) as Ri decreases. We see a consequence of
this result later.
The vertical structure of internal waves is also
changed by shear. Figure 2 shows how the distri-
bution of the amplitude of a small amplitude wave
of given k varies with z as the shear increases
for (a) plane Couette flow of a fluid with constant
N and (b) hyperbolic tangent profiles of mean speed
and density. The profiles are distorted as Ri
decreases with the largest amplitudes displaced
towards the level at which the mean speed in the
direction of wave propagation is greatest. We shall
find it convenient to distinguish these cases by
the sign of x = c.g X 2 where 2 is the mean flow
vorticity and c the phase speed of the waves in a
frame of reference in which the depth averaged mean
flow is zero. Positive U,) in Figure 2 corresponds
to x > 0, and conversely.
The shape of waves in a fluid with density and
velocity distributed as tanh z (corresponding to
Figure 2b) is shown in Figure 3 for (a) backward
relative motion in the upper layer, x < 0, (b) no
shear, (c) forward motion in the upper layer, x > 0.
The waves in (b) and (c) have narrower crests than
troughs, whilst the waves in (a) have wide crests
and narrow troughs.
This second-order effect is not unexpected. It
may easily be shown [Thorpe (1974, Appendix C)]
that interfacial waves (see Figure 4) which move
forward with the speed of the upper layer (the
limit, as we have seen, towards which the phase
speed of the internal waves tends as Ri decreases)
have exactly the same shape as have surface waves
on a fluid of depth equal to the lower layer. Con-
versely those moving at the speed of the lower layer
have the shape of surface waves on a fluid of depth
equal to the upper layer, but inverted. This is
just the trend shown in Figure 3. The limiting
form of the surface wave is one with a sharp apex
of 120°. Such an angle can exist in a two-layer
flow only in the cases we have considered where
the wave speed is the same as the flow in one of
the two layers. Otherwise there is a relative flow
around the apex in the upper (or lower) fluid
leading to a singularity of infinite flow in the
irrotational fluid. In general, some other limiting
profile must appear, although it is likely to tend
in a continuous way towards the limiting sharp apex
profile. Recent work on breaking surface waves
[Cokelet (1977)] cannot be applied even in the
special case for the analogy is valid only for
steady waves.
Experiments, however, [Thorpe (1968)] demonstrate
how internal waves break in a shear flow. Figure
5 shows wave breaking for x > 0. A jet of fluid
moves forward (that is faster than the waves advance)
from the wave crest above the level of the mean
interface where we saw in Figure 2 that the dis-
placement was concentrated, and, in Figure 3, where
the curvature was greatest. The fluid particles
move forward (at speed C,) more rapidly than the
wave advances and this leads to a layered structure
with a region of slightly denser fluid overlying
less dense fluid with the potential consequence of
gravitational instability. Similar 'forward'
breaking occurs at the wave troughs when X < 0.
The experiments demonstrate clearly the difference
between K-H.I of the mean flow (seen in Figure 5}j)
and the convective instability of the waves. In
the former the wave-like disturbances grow, extract-
ing energy from the mean flow, whilst in the latter
the waves do not grow in amplitude and lose energy
as a consequence of instability.
The condition for convective instability to
occur (C, = c) has been used in a calculation to
produce the stability diagrams of Figure 6. These
are appropriate only to a particular wavelength
and show the wave slope at which instability will
occur for a given Ri. The Couette flow (Figure 6a)
is stable in the absence of waves for all Ri > 0,
but the hyperbolic tangent profile (Figure 6b) is
unstable at Ri = 0.25 and the dashed lines show
the value Ri = 0.25 at the interface marking the
boundary at which K-H.I will occur in a quasi steady
flow. These diagrams demonstrate how shear greatly
reduces the wave slope at which convective instabil-
ity sets in, a partial consequence of the trend of
the phase speed toward Umax and hence a reduction
of the wave particle speed necessary to promote net
speeds, Cpr which exceed the phase speed. The non-
linear terms are also very important however, the
finite amplitude change in the phase speed being
as important as other non-linear effects.
625
FIGURE 2. The amplitude of the displace-
ment of lines of constant density in
internal waves of the first mode with wave
number k = m/H calculated from linear theory
(i.e., from the Taylor-Goldstein equation)
at various Richardson numbers (as labelled)
in
(a) Couette flow, U = Ug(2z/H - 1), with
constant density gradient. Ug is posi-
tive for the left hand set of curves
and negative for the right hand set.
(b) Hyperbolic tangent profiles, U =
Uptanh y and density p = pg(1 -
Atanh y) where y = 20z/H - 15.
Up is positive for the first three
curves at the left, zero for Ri = =
and negative for the three curves on
the right. The value of Ri marked on
these curves is the minimum mean flow
value at z = 3H/4.
We may press the analogy between interfacial
internal waves in a shear flow and surface waves
further. The shape of surface gravity waves
(narrower crests than troughs) and their habit of
breaking forwards at the crests seems universal,
in that it is independent of water depth, being
observed and (where theory is available) predicted
for both shallow and deep water waves. The internal
waves observed in the experiments have similar prop-
erties, accepting that the profile is inverted with
respect to the surface waves if x < 0, even though
they are not strictly interfacial waves or moving
at the speed of one of the layers. This suggests
that the shape and breaking, by convective overturn,
of long first mode internal waves on a relatively
narrow interface between two uniform layers follow
the pattern observed in the experiments, independent
of the depths of the layers, provided that the
Richardson number of the mean flow in the interfacial
region is small.
Figure 6b is not symmetrical, a consequence of
the asymmetry introduced by having unequal layer
thicknesses above and below the interface. Trans-—
lated to a situation in which wind is driving a
flow above a shallow thermocline, the diagram
implies that internal waves travelling with the
wind (x > 0) will break at a greater amplitude (or
later if the shear flow is increasing) then waves
of the same length travelling against the wind.
This result also follows from our analogy with
surface waves since, for a given wavelength, surface
waves of limiting (120° apex) amplitude in deep
water (corresponding to the forward moving, x > 0,
internal gravity waves) are higher than waves in
shallow water (which correspond to the backward
moving waves). Waves moving across the flow will
TAAL AAAT NNN RAT IAAAL IA T
FIGURE 3. Internal waves in a shear flow with profiles of U and p similar to those of Figure
2(b), except that the interface is at z = H/4 and the mean, depth averaged, flow is zero. The
waves propagate to the left and in (a) the mean flow in the upper layer is to the right, lower
to the left (y<0), in (b) there is no mean flow, whilst in (c) the mean flow in the upper layer
is to the right and in the lower layer to the left (y > 0).
h FIGURE 4. Interfacial waves in a two-layer fluid. In
(a) the phase speed of the waves, c, is equal to the
speed of the lower layer, Up. The wave shape is identi-
cal to that of surface waves on a layer of depth hj,
h, but inverted. (This corresponds to xy < 0). In (b),
c = U;, and the wave shape is identical to that of
surface waves on a layer of depth ho.
——=
——
627
FIGURE 5. The onset of wave breaking for xy > 0. The waves are moving to the left. The mean Richardson number at the in-
terface in the accelerating flow is approximately (a) 2.5 (b)
(c) 0.36 (d) 0.25 (e) 0.18 (f) 0.14 (g) 0.11 (h) 0.09
(i) 0.07 (j) 0.06 [from Thorpe (1968)]. Convective overturn is seen to begin at (c) and K-H.I at (i). The instability is
not seen at the critical value of Ri because of the time needed for growth in the accelerating flow.
not be unaffected by it. This process may be
important in producing asymmetric directional wave
spectra in the seasonal thermocline.
In practice of course unidirectional flows and
long trains of internal waves do not occur in the
ocean. The component of the mean flow velocity
normal to the direction of wave propagation appears
to play no part in the breaking or dynamics of the
waves, and the results should be valid for long
crested waves even in (Ekman) spiral flows. A
periodic shear flow applied to a wave, as when one
internal wave moves through another, may produce
locally the conditions for convective overturn of
the kind we have described. The final stages of
the experiments of Keulegan and Carpenter (1961)
or Davis and Acrivos (1967) illustrate this process.
In these experiments a short second mode wave is
driven by resonant interaction from a long first
mode wave, itself generated by a wavemaker. The
shorter wave eventually breaks in the shear field
of the longer first mode wave.
Flow acceleration accompanies both the periodic
flows in a wave field and the motion of the upper
layers of the ocean during periods of wind forcing.
In the experiments shown here breaking was induced
by allowing the flow to accelerate uniformly. It
was discovered that the energy of the fluctuating
wave components was reduced very rapidly as a
result of this acceleration. The consequent Rey—
nolds stress working on the mean velocity gradient
transferred energy to the mean flow. This inter-
action may have important consequences on the
development of the seasonal thermocline during
1:0
Wave
slope
0-5
UNSTABLE
0:5 1 5 (a)
Rig
0-4
Wave 1
slope
UNSTABLE UNSTABLE
Limit for
convective overturn
o
nN
nN
Ww
ail at
XEON Rie XCHIONR NS
FIGURE 6. Stability diagrams corresponding
to the waves described in Figure 2, based
on a calculation extended to third order
(Thorpe, 1968). (a) Couette flow (b) Hyper-
bolic tangent profiles.
628
periods of wind forcing and the acceleration of the
mixing layer, but they are beyond the scope of this
paper.
It seems likely that in the seasonal thermocline
short internal waves may break predominantly by
convective overturn whilst the longer are more
prone to K-H.I, but the balance of effects is not
known. The importance of non-linearities in
determining the condition of convective overturn
and the unknown structure of the density and veloc-—
ity fields make the problem difficult to resolve
theoretically, and some effort is being directed
towards an observational, and hence empirical,
solution using small arrays of thermistors with
rapid response times, and sensitive CTDs.
REFERENCES
Banks, W. H. H., P. G. Drazin, and M. B. Zaturska
(1976). On the normal modes of parallel flow
of inviscid stratified fluid. J. Fluid Mech.
75, 149.
Bell, T. H. (1974). Effects of shear on the prop-
erties of internal gravity wave modes. Dt.
jeitlehgooies 45 Bip Silo
Cokelet, E. D. (1977). Breaking waves. Nature
2677, 169%
Davis, R. E, and A. Acrivos (1967). The stability
of oscillating intérnal waves. J. Fluid Mech.
30, 723.
Groen, P. (1948). Two fundamental theorems on
gravity waves in inhomogeneous incompressible
fluids. Physica 14, 294.
Halpern, D. (1974). Observations of the deepening
of the wind-mixed layer in the Northeast Pacific
Ocean. J. Phys. Oceanog. 4, 454.
Howard, L. N. (1961). Note on a paper by John W.
Miles. J. Fluid Mech. 10, 509.
Keulegan, G. H., and L. H. Carpenter (1961). An
experimental study of internal progressive
oscillatory waves. Wat. Bur. Stand. Rep. No.
7319.
Miles, J. W. (1961). On the stability of hetero-
geneous shear flows. J. Fluid Mech. 10, 496.
Orlanski, I., and K. Bryan (1969). Formation of
the thermocline step structure by large amplitude
internal gravity waves. J. Geophys. Res. 74,
6975Se
Phillips, O. M., and M. L. Banner (1974). Wave
breaking in the presence of wind drift and swell.
J. Fluid Mech. 66, 625.
Thorpe, S. A. (1971). Experiments on the instabil-
ity of stratified shear flows: miscible fluids.
J. Fluid Mech. 46, 299.
Thorpe, S. A. (1974). Near-resonant forcing in a
shallow two-layer fluid: a model for the
internal surge in Loch Ness? J. Fluid Mech.
O37, SOL)-
Thorpe, S. As, A. J. Hall; es Taylor, and ai Alden
(1977). Billows in Loch Ness. Deep-Sea Res.
Zaye Sills
Thorpe, S. A. (1978). On the shape and breaking
of finite amplitude internal gravity waves in
a shear flow. J. Fluid Mech. 85, 7.
Woods, J. D. (1968). Wave-induced shear instability
in the summer thermocline. J. Fluid Mech. 32,
Tbe
List of Participants
Allan J. Acosta, California Institute of Technology,
Pasadena, USA
Bruce H. Adee, University of Washington, Seattle,
USA
Jose A. Alaez, Canal de Experiencias Hidrodinamicas,
Madrid, Spain
Klaus Albrecht, Institut fur Hydroakustik, Ottobrunn,
Federal Republic of Germany
Vladimir K. Ankudinov, Hydronautics, Inc., Laurel,
USA
Robert E. Apfel, Yale University, New Haven, USA
Roger E. A. Arndt, University of Minnesota,
Minneapolis, USA
Glenn M. Ashe, U. S. Coast Guard, Washington, USA
Daniel G. Bagnell, U. S. Coast Guard, Washington,
USA
Kwang-June Bai, David Taylor Naval Ship R & D
Center, Bethesda, USA
Ignacio Baquerizo Briones, Spanish Society of
Naval Architects, Madrid, Spain
Goran B. R. Bark, Swedish State Shipbuilding
Experimental Tank, Goteborg, Sweden
Steven J. Barker, University of California,
Los Angeles, USA
Franco C. Bau, Cantieri Navali Riuniti, Genoa,
Italy
Robert F. Beck, University of Michigan, Ann Arbor,
USA
Michael L. Billet, Pennsylvania State University,
State College, USA
William K. Blake, David Taylor Naval Ship R & D
Center, Bethesda, USA
Christian Bratu, Institut Francais Du Petrole,
Rueil-Malmaison, France
John P. Breslin, Stevens Institute of Technology,
Hoboken, USA
Neal G. Brower, Johns Hopkins University, Baltimore,
USA
Samuel H. Brown, David Taylor Naval Ship R & D
Center, Bethesda, USA
Donald R. Burklew, Operations Research, Inc.,
Silver Spring, USA
629
Otto Bussemaker, Schottel-Nederland B. V., The
Hague, Netherlands
Ben J. Cagle, Office of Naval Research, Pasadena,
USA
Nicholas Caracostas, Advanced Marine Enterprises,
Inc., Washington, USA
George F. Carrier, Harvard University, Cambridge,
USA
F. Sherman Cauldwell, Naval Ship Engineering
Center, Washington, USA
Tuncer Cebeci, Douglas Aircraft Company, Long
Beach, USA
Georges L. Chahine, Ecole Nationale Superieure de
Techniques Avancees, Paris, France
Ming-Shun Chang, David Taylor Naval Ship R & D
Center, Bethesda, USA
Richard B. Chapman, Science Applications, Inc.,
San Diego, USA
Howard A: Chatterton, U. S. Coast Guard, Washington,
USA
Michael A. Chaszeyka, Office of Naval Research,
Chicago, USA
Henry M. Cheng, Office of the Chief of Naval
Operations, Washington, USA
Teresa Chereskin, Massachusetts Institute of
Technology, Cambridge, USA
George H. Christoph, Sun Shipbuilding & Dry Dock
Company, Chester, USA
Allen T. Chwang, California Institute of Technology,
Pasadena, USA
David W. Coder, David Taylor Naval Ship R & D
Center, Bethesda, USA
E. N. Comstock, Naval Ship Engineering Center,
Washington, USA
Genevieve Comte-Bellot, Ecole Centrale de Lyon,
Ecully, France
Reilley E. Conrad, Naval Ship Engineering Center,
Washington, USA
Ralph D. Cooper, Office of Naval Research,
Washington, USA
Bruce D. Cox, David Taylor Naval Ship R & D Center,
Bethesda, USA
630
William E. Cummins, David Taylor Naval Ship R & D
Center, Bethesda, USA
Douglas J. Dahmer, David Taylor Naval Ship R & D
Center, Bethesda, USA
Tore G. Dalvag, AB Karlstads Mekaniska Werkstad,
Kristinehamn, Sweden
Stephen H. Davis, John Hopkins University,
Baltimore, USA
Charles W. Dawson, David Taylor Naval Ship R & D
Center, Bethesda, USA
William G. Day, Jr., David Taylor Naval Ship R & D
Center, Bethesda, USA
Jean-Francois M. Demanche, Bassin d'Essais des
Carenes, Paris, France
Jean-Claude Dern, Bassin d'Essais des Carenes,
Paris, France
William K. Dewar, Woods Hole Oceanographic Institu-
tion, Woods Hole, USA
Warren C. Dietz, U. S. Coast Guard, Washington, USA
Richard C. DiPrima, Rensselaer Polytechnic
Institute, Troy, USA
Jan M. Dirkzwager, Ministry of Defence, The Hague,
Netherlands
Stanley W. Doroff, Office of Naval Research,
Washington, USA
Phillip Eisenberg, Hydronautics, Inc., Laurel,
USA
N. M. El-Hady, Virginia Polytechnic Institute,
Blacksburg, USA
J. W. English, National Maritime Institute,
Feltham, England
Robert Falls, Maritime Administration, Washington,
USA
Hermann F. Fasel, University of Stuttgart, Stuttgart,
Federal Republic of Germany
Archibald M. Ferguson, University of Glasgow,
Glasgow, Scotland ;
Peter D. Fitzgerald, Exxon International, Florham
Park, USA
Francois N. Frenkiel, David Taylor Naval Ship R & D
Center, Bethesda, USA
Daniel H. Fruman, Laboratoire d'Aerodynamique,
Orsay, France
Donald Fuhs, David Taylor Naval Ship R & D Center,
Bethesda, USA
Michael Gaster, National Maritime Institute,
Middlesex, England
Edward M. Gates, University of Alberta, Edmonton,
Canada
Carl Gazley, Jr., The Rand Corporation, Santa
Monica, USA
William K. George, State University of New York,
Buffalo, USA
Robert K. Geiger, Office of Naval Research,
Washington, USA
Douglas L. Gile, Boulder, USA
Alex Goodman, Hydronautics, Inc., Laurel, USA
Stephan At. Goranov, Bulgarian Ship Hydrodynamics
Center, Varna, Bulgaria
Paul S. Granville, David Taylor Naval Ship R & D
Center, Bethesda, USA
Richard A. Griffiths, U. S. Coast Guard, Washington,
USA
William L. Haberman, Rockville, USA
Jacques B. Halder, David Taylor Naval Ship R & D
Center, Bethesda, USA
Francis R. Hama, Princeton University, Princeton,
USA
Henry J. Haussling, David Taylor Naval Ship R & D
Center, Bethesda, USA
Grant E. Hearn, British Ship Research Association,
Wailsend, England
Harold I. Heaton, Johns Hopkins University, Applied
Physics Laboratory, Laurel, USA
Isom H. Herron, Howard University, Washington, USA
Leo H. Holthuijsen, Delft University of Technology,
Delft, Netherlands
Max G. A. Honkanen, Engineering Company M. G.
Honkanen, Helsinki, Finland
Louis N. Howard, Massachusetts Institute of
Technology, Cambridge, USA
Chun-Che Hsu, Hydronautics, Inc., Laurel, USA
Thomas T. Huang, David Taylor Naval Ship R & D
Center, Bethesda, USA
Lee M. Hunt, National Academy of Sciences-National
Research Council, Washington, USA
Stephen J. Hunter, Admiralty Marine Technology
Establishment, Haslar, England
Erling Huse, Ship Research Institute of Norway,
Trondheim, Norway
Takao Inui, University of Tokyo, Tokyo, Japan
Shunichi Ishida, Ishikawajima-Harima Heavy
Industries Co., Ltd., Yokohama, Japan
Gerald S. Janowitz, North Carolina State University,
Raleigh, USA
Stuart D. Jessup, David Taylor Naval Ship R & D
Center, Bethesda, USA
Bruce Johnson, U. S. Naval Academy, Annapolis,
USA
Virgil E. Johnson, Hydronautics, Inc., Laurel, USA
Francois J. Jouaillec, French Ministry of Defence,
Paris, France
Peter Numa Joubert, University of Melbourne,
Melbourne, Australia
Vijay K. Jyoti, Dominion Engineering Works, Ltd.,
Montreal, Canada
Lakshmi H. Kantha, Johns Hopkins University,
Baltimore, USA
Paul Kaplan, Oceanics, Inc., Plainview, USA
George M. Kapsilis, M. Rosenblatt & Son, Inc.,
Gaithersburg, USA
Hiroharu Kato, University of Tokyo, Tokyo, Japan
R. G. Keane, Jr., Naval Ship Engineering Center,
Washington, USA
Andreas P. Keller, Technical University of Munich,
Munich, Federal Republic of Germany
Colen G. Kennell, Naval Ship Engineering Center,
Washington, USA
Philip S. Klebanoff, National Bureau of Standards
Washington, USA
Leslie S. G. Kovasznay, Johns Hopkins University,
Baltimore, USA
Ruby E. Krishnamurti, Florida State University,
Tallahassee, USA
Gert Kuiper, Netherlands Ship Model Basin,
Wageningen, Netherlands
Jurgen H. Kux, University of Hamburg, Hamburg,
West Germany
Louis Landweber, University of Iowa, Iowa City, USA
Arie J. W. Lap, Royal Netherlands Naval College,
Dan Helder, Netherlands
“Jochen Lauden, Hamburgische Shiffbau-Versuchsanstalt,
Hamburg, Federal Republic of Germany
George K. Lea, National Science Foundation,
Washington, USA
Yves Lecoffre, Alsthom Atlantique, Grenoble Cedex,
France
Choung M. Lee, David Taylor Naval Ship R & D
Center, Bethesda, USA
Yu-Tai Lee, University of Iowa, Iowa City, USA
Lennox J. Leggat, Defence Research Establishment
Atlantic, Nova Scotia, Canada
Spiros G. Lekoudis, Lockheed-Georgia Company,
Marietta, USA
John A. LeRoy, Australian Naval Attache Office,
Washington, USA
Wen-Chin Lin, David Taylor Naval Ship R & D
Center, Bethesda, USA
Robert R. Long, Johns Hopkins University, Baltimore,
USA
Hans J. Lugt, David Taylor Naval Ship R & D Center,
Bethesda, USA
Justin McCarthy, David Taylor Naval Ship R & D
Center, Bethesda, USA
John M. Macha, Texas A & M University, College
Station, USA
Leslie Mack, California Institute of Technology,
Pasadena, USA
Toshio Maeda, Mitsubishi Heavy Industries, Ltd.,
Kobe, Japan
Allen H. Magnuson, Virginia Polytechnic Institute,
Blacksburg, USA
Robert W. Manning, Naval Sea Systems Command,
Washington, USA
Chiang C. Mei, Massachusetts Institute of Technology,
Cambridge, USA
Kenneth R. Meldahl, The Boeing Company, Seattle,
USA
John W. Miles, University of California, San Diego,
USA
Robert J. Mindak, Office of Naval Research,
Washington, USA
Erik Mollo-Christensen, Massachusetts Institute
of Technology, Cambridge, USA
Vincent Monacella, David Taylor Naval Ship R & D
Center, Bethesda, USA
Alan W. Moore, Admiralty Marine Technology
Establishment, Teddington, England
David D. Moran, David Taylor Naval Ship R & D
Center, Bethesda, USA
William B. Morgan, David Taylor Naval Ship R & D
Center, Bethesda, USA
Kazuhiro Mori, Hiroshima University, Hiroshima,
Japan
Parma Mungur, Lockheed-Georgia Company, Marietta,
USA
Walter H. Munk, University of California, San
Diego, USA
Hitoshi Murai, Tohoku University, Sendai, Japan
Paul M. Naghdi, University of California, Berkeley,
USA
Ali H. Nayfeh, Virginia Polytechnic Institute,
Blacksburg, USA
J. Nicholas Newman, Massachusetts Institute of
Technology, Cambridge, USA
Francis Noblesse, Massachusetts Institute of
Technology, Cambridge, USA
David J. Norton, Texas A & M University, College
Station, USA
John A. Norton, Bird-Johnson Company, Walpole, USA
John F. O'Dea, David Taylor Naval Ship R & D
Center, Bethesda, USA
Denis C. O'Neill, Ministry of Defence, Bath,
England
Marinus W. C. Oosterveld, Netherlands Ship Model
Basin, Wageningen, Netherlands
Blaine R. Parkin, Pennsylvania State University,
State College, USA
Virendra C. Patel, University of Iowa, Iowa City,
USA
631
Mariano Perez, Canal de Experiencias Hidrodinamicas,
Madrid, Spain
Gonzalo Perez Gomez, Spanish Society of Naval
Architects, Madrid, Spain
Frank B. Peterson, David Taylor Naval Ship R & D
Center, Bethesda, USA
Owen M. Phillips, Johns Hopkins University,
Baltimore, USA
Ennio Piantini, Ministero Difesa Marina, Rome,
Italy
Pao C. Pien, David Taylor Naval Ship R & D Center,
Bethesda, USA
Leonard J. Pietrafesa, North Carolina State
University, Raleigh, USA
Gregory Platzer, David Taylor Naval Ship R & D
Center, Bethesda, USA
Allen Plotkin, University of Maryland, College
Park, USA
Alan Powell, David Taylor Naval Ship R & D Center,
Bethesda, USA
Jaakko V. Pylkkanen, Helsinki University of
Technology, Helsinki, Finland
Arthur M. Reed, David Taylor Ship R & D Center,
Bethesda, USA
Sidney R. Reed, Office of Naval Research,
Washington, USA
Bernd Remmers, Kempf & Remmers, Hamburg, Federal
Republic of Germany
Eli Reshotko, Case Western Reserve University,
Cleveland, USA
Wolfgang Reuter, Naval Ship Engineering Center,
Washington, USA
M. B. Ricketts, David Taylor Naval Ship R & D
Center, Bethesda, USA
Joel C. W. Rogers, Johns Hopkins University,
Applied Physics Laboratory, Laurel, USA
Richard R. Rojas, Naval Research Laboratory,
Washington, USA
Olle G. A. Rutgersson, Swedish State Shipbuilding
Experimental Tank, Goteborg, Sweden
Manley St. Denis, U. S. Naval Academy, Annapolis,
USA
Nils Salvesen, David Taylor Naval Ship R & D
Center, Bethesda, USA
Geert H. Schmidt. University of Technology, Delft,
Netherlands
Michael Schmiechen, VWS Berlin Model Basin, Berlin,
Federal Republic of Germany
Joanna W. Schot, David Taylor Naval Ship R & D
Center, USA
Paul Sclavounous, Massachusetts Institute of
Technology, Cambridge, USA
Carl A. Scragg, Science Applications Inc., San
Diego, USA
Som D. Sharma, Massachusetts Institute of Tech-
nology, Cambridge, USA
Young T. Shen, David Taylor Naval Ship R & D
Center, Bethesda, USA
Vincent G. Sigillito, Johns Hopkins University,
Applied Physics Laboratory, Laurel, USA
Leslie Sinclair, Stone Manganese Marine Ltd.,
London, England
Olav H. Slaattelid, Ship Research Institute of
Norway, Trondheim, Norway
Neill S. Smith, Naval Coastal Systems Center,
Panama City, USA
J. A. Sparenberg, University of Groningen,
Groningen, Netherlands
Nicholas R. Stark, Beltsville, USA
Frank X. Stora, U. S. Army, Fort Belvoir, USA
632
Albert M. Sturrman, Royal Netherlands Navy, The
Hague, Netherlands
Ming-Yang Su, U. S. Navy, NORDA, Bay St. Louis,
USA
Hiraku Tanaka, Ship Research Institution, Tokyo,
Japan
Stephen A. Thorpe, Institute of Oceanographic
Sciences, Surrey, England
Yoshiaki Toba, Tohoku University, Sendai, Japan
Ernest O. Tuck, University of Adelaide, Adelaide,
Australia
Marshall P. Tulin, Hydronautics, Inc., Laurel,
USA
Ka-Kit Tung, Dynatech, Torrance, USA
J. Stewart Turner, Australian National University,
Canberra, Australia
Willem van Berlekom, Swedish State Shipbuilding
Experimental Tank, Goteborg, Sweden
Jan D. van Manen, Netherlands Ship Model Basin,
Wageningen, Netherlands
Pieter van Oossanen, Netherlands Ship Model Basin,
Wageningen, Netherlands
Jan van der Meulen, Netherlands Ship Model Basin,
Wageningen, Netherlands
Christian von Kerczek, David Taylor Naval Ship
R & D Center, Bethesda, USA
Alice Vucinic, Rijeka University, Rijeka, Yugoslavia
Nicholas Vytlacil, Westinghouse Electric COED,
Annapolis, USA
David A. Walden, U. S. Coast Guard, Washington,
USA
Lawrence W. Ward, Webb Institute of Naval Archi-
tecture, Glen Cove, USA
Richard M. Wargelin, U. S. Navy, Suitland, USA
John V. Wehausen, University of California,
Berkeley, USA
Michael A. Weissman, Flow Industries, Inc., Kent,
USA
Ernst-August Weitendorf, University of Hamburg,
Hamburg, Federal Republic of Germany
John R. Weske, University of Maryland, College
Park, USA
Robert E. Whitehead, Office of Naval Research,
Washington, USA
Sheila Widnall, Massachusetts Institute of
Technology, Cambridge, USA
Karl Wieghardt, University of Hamburg, Hamburg,
Federal Republic of Germany
Colin B. Wills, Admiralty Marine Technology
Establishment, Haslar, England
Theodore Y. Wu, California Institute of Technology,
Pasadena, USA
Chia-Shun Yih, University of Michigan, Ann Arbor,
USA
Bohyun Yim, David Taylor Naval Ship R & D Center,
Bethesda, USA
Hajime Yuasa, Mitsui Engineering & Shipbuilding
Co., Ltd., Tokyo, Japan
P. Richard Zarda, David Taylor Naval Ship R & D
Center, Bethesda, USA
PREVIOUS BOOKS IN THE NAVAL HYDRODYNAMICS SERIES
"First Symposium on Naval Hydrodynamics." National
Academy of Science-Nation Research Council,
Publication 515, 1957. Washington, D. C.;
PB133732.
"Second Symposium on Naval Hydrodynamics: Hydro-
dynamic Noise and Cavity Flow," Office of Naval
Research, Department of the Navy, ACR-38, 1958;
PB157668.
"Third Symposium on Naval Hydrodynamics: High
Performance Ships," Office of Naval Research,
Department of the Navy, ACR-65, 1960; AD430729.
"Fourth Symposium on Naval Hydrodynamics: Propul-
sion and Hydroelasticity," Office of Naval
Research, Department of the Navy, ACR-92,
AD447732.
"The Collected Papers of Sir Thomas Havelock on
Hydrodynamics," Office of Naval Research,
Department of the Navy, ACR-103, 1963; AD623589.
"Fifth Symposium on Naval Hydrodynamics: Ship
Motions and Drag Reduction," Office of Naval
Research, Department of the Navy, ACR-112, 1964;
AD640539.
"Sixth Symposium on Naval Hydrodynamics: Physics
of Fluids, Maneuverability and Ocean Platforms,
Ocean Waves, and Ship-Generated Waves and Wave
Resistance," Office of Naval Research, Depart-
ment of the Navy, ACR-136, 1966; AD676079.
"Seventh Symposium on Naval Hydrodynamics: Unsteady
Propeller Forces, Fundamental Hydrodynamics,
1962;
633
Unconventional Propulsion," Office of Naval
Research, Department of the Navy, DR-148, 1968;
AD721180.
"Eighth Symposium on Naval Hydrodynamics: Hydro-
dynamics in the Ocean Environment," Office of
Naval Research, Department of the Navy, ACR-179,
1970; AD748721.
"Ninth Symposium on Naval Hydrodynamics: Unconven-
tional Ships, Ocean Engineering, Frontier Problems,"
Office of Naval Research, Department of the
Navy, ACR-203, 1972; Two Volumes; Vol. 1, ADA-
010505; Vol 2, ADAO10506.
"Tenth Symposium on Naval Hydrodynamics: Hydrody-
namics for Safety, Fundamental Hydrodynamics,"
Office of Naval Research, Department of the Navy,
ACR-204, 1974; ADA0O22379.
"Eleventh Symposium on Naval Hydrodynamics: Unsteady
Hydrodynamics of Marine Vehicles," Office of
Naval Research, Department of the Navy. Also
available from Mechanical Engineering Publications
Limited, London and New York.
The above books are avilable on microfilm
and microfiche from the National Technical
Information Service, U. S. Department of
Commerce, Springfield, Virginia 22151.
Some early issues are also available in paper
copies. The catalog numbers, as of the
date of this issue, are shown for each book.
NOTE:
i
i mal
f
i
te a
a “
é Ty
P \
* Fi i
« 3
=
ie e
{
y
3
Mee =
= ‘ os
ii wi Ste |
iT > y
Sill ri
SMa ff
> wish
aaa
i nt
Ag D ie ce ies
' re
1
Panipat
ae
Pat tle
i
i
Ma
on i
Gye)
(ae As
} iy
hee Wit
iy) 14
we rei
1uly
i
oe
ptt
4
ie
l—
Ailis
inl
‘ f
i “
1 atl
rey
lu
Bia
With
nn, r i vt With r
PORN Nae Ne
}
ie.
Fi
\
Ve ANF
AMR). Mere na ;
lity 4 x
yee
an |
PA Pits of 7
P Nai) id
th " t
ivr un
th
Mites
i |
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11