Hydrodynamics in the ocean environment

Eighth Symposium

NAVAL
HYDRODYNAMICS

~HYDRODYNAMICS IN THE
OCEAN ENVIRONMENT

ARC-179

Office of Naval Research
Department of the Navy _

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Eighth Symposium
NAVAL HYDRODYNAMICS

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

sponsored by the
OFFICE OF NAVAL RESEARCH
the
NAVAL UNDERSEA RESEARCH AND DEVELOPMENT CENTER
and the
CALIFORNIA INSTITUTE OF TECHNOLOGY

August 24-28, 1970
Rome, Italy

MILTON S. PLESSET
T. YAO-TSU WU
STANLEY W. DOROFF

MARINE

Editors ,noGr, BIOLOGICAL
ae bn LABORATORY
4 Pe ADE TL LE ELITE LLYN
3 é LIBRARY
AC ay $

eins VBOBS HOLE, MASS.
OFFICE OF NAVAL RESEARCH— DEPARTMENT OF THE} NAWYH. © I.

Arlington, Va.

PREVIOUS BOOKS IN THE NAVAL HYDRODYNAMICS SERIES

“First Symposium on Naval Hydrodynamics,’’ National Academy of Sciences—National
Research Council, Publication 515, 1957, Washington, D.C.; PB133732, paper copy
$6.00, 35-mm microfilm 95¢.

“Second Symposium on Naval Hydrodynamics: Hydrodynamic Noise and Cavity Flow,”
Office of Naval Research, Department of the Navy, ACR-38, 1958; PB157668, paper
copy $10.00, 35-mm microfilm 95¢.

“Third Symposium on Naval Hydrodynamics: High-Performance Ships,”’ Office of Naval
Research, Department of the Navy, ACR-65, 1960; AD430729, paper copy $6.00,
35-mm microfilm 95¢.

“Fourth Symposium on Naval Hydrodynamics: Propulsion and Hydroelasticity,’’ Office
of Naval Research, Department of the Navy, ACR-92, 1962; AD447732, paper copy
$9.00, 35-mm microfilm 95¢.

“The Collected Papers of Sir Thomas Havelock on Hydrodynamics,” Office of Naval
Research, Department of the Navy, ACR-103, 1963; AD623589, paper copy $6.00,
microfiche 95¢.

“Fifth Symposium on Naval Hydrodynamics: Ship Motions and Drag Reduction,’
Office of Naval Research, Department of the Navy, ACR-112, 1964; AD640539, paper
copy $15.00, microfiche 95¢.

“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, Department of the Navy, ACR-136, 1966; AD676079, paper copy
$6.00, microfiche 95¢.

“Seventh Symposium on Naval Hydrodynamics: Unsteady Propeller Forces, Funda-
mental Hydrodynamics, Unconventional Propulsion,”’ Office of Naval Research, Depart-
ment of the Navy, DR-148, 1968; AD721180; Available from Superintendent of Docu-
ments, U.S. Government Printing Office, Washington, D.C. 20402, Clothbound, 1690
pages, illustrated (Catalog No. D 210.15:DR-148; Stock No. 0851-0049); $13.00.

NOTE: The above books, except for the last, are available from the National Technical
Information Service, U.S. Department of Commerce, Springfield, Virginia 22151. The
catalog number and the price for paper copy and for microform copy are shown for
each book.

Statements and opinions contained herein are those of the authors

and are not to be construed as official or reflecting the views of
the Navy Department or of the naval service at large.

For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D.C. 20402 - Price $10
Stock Number 0851-0056

li

PREFACE

Continuing in an uninterrupted manner since 1956, the biennial symposia on naval
hydrodynamics convened for its Eighth Symposium, August 24-28, 1970 at Pasadena,
California. This conference was jointly sponsored by the Office of Naval Research, the
Naval Undersea Research and Development Center, and the California Institute of
Technology.

The technical program in this series is traditionally structured about a limited num-
ber of topics of current interest in naval hydrodynamics. In the case of the Eighth
Symposium, “‘Hydrodynamics in the Ocean Environment’’ was selected as the focal theme
not only because of the present widespread research interest and activity in this subject
but also in recognition of 1970 as the inaugural year of the ‘International Decade of
Ocean Exploration.” This motif for the Eighth Symposium was also aptly reflected in
the banquet address to the participants by Rear Admiral O.D. Waters, USN, then Ocean-
ographer of the Navy.

The organization and management of a meeting of this magnitude requires the atten-
tion and energy of a large number of people over a long period of time. To Dr. Harold
Brown, President of the California Institute of Technology, to Captain Charles Bishop,
Commander, Naval Undersea Research and Development Center, and to all the various
members of their organizations who contributed in many different ways to the success
of the Eighth Symposium, the Office of Naval Research is deeply indebted, and to them
we extend our heartfelt gratitude and appreciation for a job well done. It is particularly
appropriate, however, to acknowledge the specific roles of Professor Milton S. Plesset and
Professor T.Y. Wu of the California Institute of Technology and Dr. J. Hoyt of the Naval
Undersea Research and Development Center who as a group carried the lion’s share of
the responsibility for the detailed planning and day-to-day management of the Eighth
Symposium. We take special pleasure in acknowledging the invaluable assistance of Mrs.
Barbara Hawk, secretary to Professor Plesset, who in a most gracious and efficient man-
ner carried out a multitude of important tasks in support of the Symposium. In addition,
Mrs. Hawk, together with Mrs. Alrae Tingley, were responsible for the preparation of the
typescript which was used in the publication of these proceedings. Mr. Stanley Doroff of
the Office of Naval Research played his usual critical role, participating actively in every
aspect of the planning and execution of the arrangements for the Eighth Symposium.

feb bg u~

RALPH D. COOPER
Director,

Fluid Dynamics Program
Office of Naval Research

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CONTENTS

Page

EC PAC Ce ar ei arya aaday Me) te ees ue gin ore ea ee oe PAL
Address of Welcome , wie xi

Rear Admiral C. O. Holmquist, "Chief of ‘Naval

Research and Assistant Oceanographer for

Ocean Science
Address at the Symposium Banquet - xiv

Rear Admiral O. D. Waters, Jr., "Oceanographer

of the Navy

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

TSUNAMIS. ... .- . oe 3

G. F. Carrier, Beeacd Gaiversity: iGa abridge,

Massachusetts

LABORATORY INVESTIGATIONS ON AIR-SEA
INTERACTIONS ~ «. «°'s : arenes 14
BE. ¥.+Hsw and He -<Y. vate iStantard: University,
Stanford, California

AIR-SEA INTERACTIONS; RESEARCH PROGRAM AND

PAGCILET IG: AT IMS. ccs Geocs ee Si ign ee 3¢
M. Coantic and A. Favre, IMST, Macaeitles
France
EXPLOSION-GENERATED WATER WAVES ......-. 74
Bernard LeMéhauté, Tetra Tech, Inc. , Pasadena,
California

RESONANT RESPONSE OF HARBORS (THE HARBOR

PARADOX REVISITED)... Fs A 95
John W. Miles, University of California, San Diego,
California

UNSTEADY, FREE SURFACE FLOWS: SOLUTIONS
EMPLOYING THE LAGRANGIAN DESCRIPTION OF
THE MOTION « « « « « . » 117
C. Brennen and A. K. Whitney, Galifouata Institute
of Technology, Pasadena, California

TWO METHODS FOR THE COMPUTATION OF THE
MOTION OF LONG WATER WAVES —- A REVIEW
AND APPLICATIONS . « <« 5 ele ey etne

Robert L. Street and Pobert oe C. [Gian
Stanford University, Stanford, California and
Jacob E. Fromm, IBM Corporation. San Jose,
California

AN UNSTEADY CAVITY FLOW .s « a ea
D. P. Wang, The Catholic University ere
America, Washington, D.C.

DEEP-SEA TIDES. ..... ¢ se esiperte
Walter H. Munk, University of California,
San Diego, California

STABILITY OF AND WAVES IN STRATIFIED FLOWS. .
Chia-Shun Yih, University of Michigan,
Ann Arbor, Michigan

DISCUSSION 4 Adepie elie. « °
L. van Wijngaarden, Tvente Institue
of Technology, Enschede,

The Netherlands

REPLY TO DISCUSSION ... aoe aera
Chia-Shun Yih, University ae
Michigan, Ann Arbor, Michigan

ON THE PREDICTION OF IMPULSIVELY GENERATED
WAVES PROPAGATING INTO SHALLOW WATER . .
Paul R. Van Mater, Jr., United States Naval
Academy, Annapolis, Maryland, and Eddie Neal,
Naval Ship Research and Development Center,
Washington, D.C.

THREE DIMENSIONAL INSTABILITIES AND VORTICES
BETWEEN TWO ROTATING SHPERES ... .. « « -
J. Zierep and O. Sawatski, Universitat Karlsruhe,
Karlsruhe, West Germany
DISCUSSION e e e e e e e e e e e e e
L. van Wijngaarden, Twente Institute
of Technology, Enschede,
The Netherlands

REPLY TO DISCUSSION ...« «2 © © + » *
J. Zierep, Universitat Karlsruhe,
Karlsruhe, West Germany

vi

Page

147

189

217

219

235

236

239

Z15

287

288

Page

ON THE TRANSITION TO TURBULENT CONVECTION. . 289
Ruby Krishnamurti, Florida State University,
Tallahassee, Florida

TURBULENT DIFFUSION OF TEMPERATURE AND
SALINITY: - AN EXPERIMENTAL STUDY ..... 3414
Allen H. Schooley, U.S. Naval Research
Laboratory, Washington, D.C.

SELF-CONVECTING FLOWS..... : ete é we 322
Marshall P. .Tulin, Hydronautics, incorporated.
Laurel, Maryland, and Josef Shwartz,
Hydronautics-Israel, Ltd. and Israel Institute
of Technology

RADAR BACK-SCATTER FROM THE SEA SURFACE. . 361
K. Hasselmann and M. Schieler, Institut fur
Geophysik, University of Hamburg

INTERACTION BETWEEN GRAVITY WAVES AND
FINITE TURBULENT FLOW FIELDS... . : 389
Daniel Savitsky, Stevens Institute of Technology,
Hoboken, New Jersey

DISCUSSION .. . $48 446
Dr. N. Hogben, National Paysites,
Laboratory, Ship Division, Feltham,
Middlesex, England

REPEY?..O DISCUSSION... 20: : 447
Daniel Savitsky, Stevens Institute of
Technology, Hoboken, New Jersey

CHARACTERISTICS OF SHIP BOUNDARY LAYERS... 449
L. Landweber, University of Iowa, lowa City, Iowa

DISCUSSION .. .- wee 473
Dr. N. Ho gber: National eieieal
Laboratory, Ship Division, Feltham,
Middlesex, England

DISCUSSION .. . ar is fe 474
H. Lackenby, The Britigh Ship Research
Association, Northumberland, England

REPLY TO DISCUSSION |». % . ue 475
L. Landweber, Univer siuy of towels
Iowa City, lowa

vii

Page

STUDY OF THE RESPONSE OF A VIBRATING PLATE
IMMERSED INA FLUID... . Daal hha 477
L. Maestrello, NASA Langley Ropeaneh Genter’
Hampton, Virginia, and T. L. J. Linden,
European Space Operations Center, Darmstadt,
Germany

RECENT RESEARCH ON SHIP WAVES. «.'« «© « «+6 e@ « 519 —
J. N. Newman, Massachusetts Institute of
Technology, Cambridge, Massachusetts

DISCUSSION .. . eo oh pA ere, 540
N. Hogben, National Physical .
Laboratory, Ship Division,
Feltham, Middlesex, England

VARIATIONAL APPROACHES TO STEADY SHIP WAVE
PROB LE MSr sie. c's, oot 8. ° ear ea ek 5AaT
Masatoshi Bessho, The Defense Agademy ,
Yokosuka, Japan

WAVEMAKING RESISTANCE OF SHIPS WITH TRANSOM
SDRINe hs ae . = Tne 573
Boy int, Naval Ship ‘Res Astin ane De einen
Center, Washington, D. C.

DISCUSSION® iii. \ S) wets 599
Georg PF. Weinblim. ieatignt ee
Schiffbau, Hamburg, Germany

DISGUSSION® (a... +. = ; ‘ 3+ ohn euMerng shoe 601
S. De. Sharma aC aor ao Pre coe
University of Michigan, Ann Arbor,
Michigan

REPLY TO DISCUSSION) & “4 '. s ethene 604

B. Yim, Naval Ship Research saya
Development Center, Washington, D.C.

REPLY TO DISCUSSION .°. . . Sie ate 604
B. Yim, Naval Ship Research aad
Development Center, Washington, D.C.

BOW WAVES BEFORE BLUNT SHIPS AND OTHER
NON-LINEAR SHIP WAVE PROBLEMS ...... . 607
Gedeon Dagan, Technion-Israel Institute of
Technology, Haifa, Israel and Marshall P. Tulin,
Hydronautics, Inc., Laurel, Maryland

DISCUSSION e e e e e e e e e e ° ° e e e 622
L. van Wijngaarden, Twente Institute of
Technology, Enschede, The Netherlands

viii

DISCUSSION e e e e e e e e e e e e e e e e 624
Prof. Hajime Maruo, Yokohama
National University, Yokohama, Japan

REPLY TO DISCUSSION ... Sete s eOeS
Gedeon Dagan, Technion- arene
Institute of Technology, Haifa, Israel

REPEY TO -DIsCusSsION "3" *ie 2. Sr scieeien O20
M. P. Tulin, Hydronautics, rae ;
Laurel, Maryland

SHALLOW WAVE PROBLEMS IN SHIP HYDRODYNAMICS 627
E. O. Tuck and P. J. Taylor, University of
Adelaide, Adelaide, South Australia

DISCUSSION e s e es e e e e e e e e e e e 658
Prof. Hajime Maruo, Wekohen
National University, Yokohama, Japan

REPLY TO DISCUSSION... . tutte « & “O59
Ee O. Tuck and P, J. Toe
University of Adelaide, Adelaide,
South Australia

SINGULAR PERTURBATION PROBLEMS IN SHIP
HYDRODYNAMICS.... + ese O05
T. Francis Ogilvie, Galera, on Michigan,
Ann Arbor, Michigan

THEORY AND OBSERVATIONS ON THE USE OF A
MATHEMATICAL MODEL FOR SHIP MANEUVERING
IN DEEP AND CONFINED WATERS ......... 807
Nils H. Norrbin, Statens Skeppsprovningsanstalt,
Sweden

THE SECOND-ORDER THEORY FOR NONSINUSOIDAL
OSCILLATIONS OF A CYLINDER IN A FREE
SURFACE... . oe o 8 e ‘ne -s “JOS

Choung Mook ee Neal Ship Reccanen Aa
Development Center, Washington, D.C.

DISCUSSION 4s a6 6 3 Jer (95d
Edwin C. James, Galtiornta Tnetitite
of Technology, Pasadena, California

REPLY &@© DISCUSSION 2 a. ‘ ‘ e. 250
Choung Mook Lee, Naval Ship Ricsoareu
and Development Center, Washington D.C.

ix

THE DRIFTING FORCE ON A FLOATING BODY IN
IRREGULAR WAVES... - “ aves 955
Je H.. Gog Verhagen; Netherioede Ship Model
Basin, The Netherlands

DYNAMICS OF SUBMERGED TOWED CYLINDERS... 981
M. P. Paidoussis, McGill University,
Montreal, P.Q., Canada

HYDRODYNAMIC ANALYSES APPLIED TO A MOORING
AND POSITIONING OF VEHICLES AND SYSTEMS

IN A SEAWAY e es 2 ° e 9 ° e ° e e e e e e e e e 1017
Paul Kaplan, Oceanics, Inc., Plainview,
New York

WAVE INDUCED FORCES AND MOTIONS OF
TUBULAR STRUCTURES 2". ss © 2) et Ne 1083
J. R. Paulling, University of Caltiornta,
Berkeley, California

SIMULATION OF THE ENVIRONMENT AND OF THE
VEHICLE DYNAMICS ASSOCIATED WITH
SUBMARINE RESCUE ... ate" somes ee
H. G. Schreibes:, i Jrs,<d- Bontkowele: onal
K. P. Kerr, Lockheed Missiles and Space Co.,
Sunnyvale, California

AUTHORS INDE ZG the. sh air whee ee) sR eee es 6 5 eG 1185

ADDRESS OF WELCOME

Rear Admiral C, O. Holmquist
Chtef of Naval Research and Assistant
Oceanographer for Ocean Science

I am pleased to welcome you to the Eighth International
Symposium in Naval Hydrodynamics. This is a symposium sponsored
every other year by the Office of Naval Research, with the objective
of bringing together the leading investigators in the field of hydro-
dynamics research throughout the world.

ONR has held many international meetings during the nearly
quarter-century of its existence. This is in line with its charter
issued by Congress, which includes the responsibility to disseminate
information on world-wide trends in research and development. It
is for this reason, for example, that we have a branch office in
London.

This series of meetings, however, has a unique characteristic.
Every other meeting is held outside the United States. Two years
ago we met in Rome, and two years from now we plan to hold this
symposium in another country. This stimulates attendance by non-
U.S. participants.

This year we welcome to the United States a number of dis-
tinguished researchers in the field of hydrodynamics. As you have
noted in your program, you will hear papers read by scientists
from institutions as far away as Australia. The information made
available through this international meeting will not only provide the
U.S. Navy with new ideas for significantly improving its ship designs
but also a pool of knowledge that will stimulate international coopera-
tion in science, The Navy has already received important benefits
from an exchange of research data with other countries.

In regard to this symposium, ONR owes a great deal both to
the Naval Undersea Research and Development Center and the
California Institute of Technology, who have joined together to serve
as hosts. We appreciate very much their efforts in arranging this
meeting and providing the excellent facilities.

I might add as a personal note that I am delighted to have this

opportunity to return to the Cal Tech campus, where I studied for
my doctorate in the early 1950's. Since my field is aeronautics, I

xi

cannot pose as an expert in hydrodynamics, although I am sure you
recognize that the two fields have similar and related problems. In
fact, in ONR we label our program Fluid Dynamics, with part of this
program dealing with hydrodynamics and part with aerodynamics.

Aside from serving as co-hosts, both Cal Tech and NURDC
have made major contributions to the work in hydrodynamics, some
under ONR sponsorship. For some time Cal Tech has been studying
a problem of critical concern to the Navy. This is the damage
caused to propellers and other vital components by cavitation.
Theoretical and experimental investigations on basic problems in
fluid mechanics conducted here are assisting naval engineers in
solving cavitation damage problems. At the same time, this work
is adding to our knowledge of the phenomenon known as supercavita-
tion, which has led to the development of supercavitating propellers
and hydrofoils resulting in increased speed of specialized naval
vehicles.

A major program at NURDC sponsorship promises not only
to reduce drastically drag resistance during turbulent flow but also
to reduce the flow noise which frequently interferes with sonar
operations. I am referring to the use of polymer additives which
when injected into the boundary layer of water promises to give naval
vehicles the capability of burst speed.

At present NURDC is engaged in achieving a complete under-
standing of the mechanism of the drag and noise reduction properties
of dilute solutions of polymer additives. This will give us a firm
technical basis for predicting what extent we can achieve drag re-
duction and flow-noise suppression on Navy vehicles.

Research in hydrodynamics is carried out under contract to
ONR at a variety of academic and at industrial organizations and at
naval laboratories and field stations. Typical of the universities
participating in the program are Stevens Institute, the Massachusetts
Institute of Technology, Stanford, University of California, Harvard,
Florida State, and Michigan in addition to Cal Tech. Industrial
organizations include Hydronautics, Inc., and LTV Research Center
and the Ampex Corp. Our in-house work in addition to NURDC is
performed at the Naval Ship Research and Development Center, the
Naval Research Laboratory, the Naval Ordnance Laboratory and the
Naval Postgraduate School in Monterey, California.

Each of these three elements -- the university, industry and
the Navy laboratory -- have a unique contribution not only to the
Navy's fluid dynamics program but to Navy research and development
in general, Universities provide us with the more fundamental data
on which all good technology is based. Industry has special know-
how in producing test beds and experimental hardward needed to
prove our theories. The Navy laboratory provides in one location
theoretical scientists working with naval engineers and naval officers

si

who have an intimate understanding of the Navy's operational prob-
lems.

As an example of what this combination can produce, we have
developed computer programs to predict the coupled motions of
heave and pitch for surface ships operating in a seaway. The input
information that is used consists of ship geometry, forward speed,
and a stochastic description of the sea state. Another computer
program simulates the launch perturbations of a torpedo leaving a
moving submarine. This provides a relatively inexpensive method
for determining the operational limitations during launch, an insight
into how launch problems can be solved, and tool for the design of
future submarine weapon systems.

The research process is continuous and complex, and it is
rarely, if ever possible, to label a new discovery as the product of
one individual or even one institution. Research has to be coopera-
tive, and we can achieve the most by cooperating on an international
scale. As this meeting indicates, ONR and the Navy subscribes to
that objective. Iam sure that all of us are faced with the problem
of producing the maximum amount of significant research results
with a minimum of funds and manpower, so that we should all benefit
from a mutual sharing of our knowledge.

xili

ADDRESS AT THE SYMPOSIUM BANQUET

Rear Admiral O. D. Waters, Jr.
Oceanographer of the Navy

Mr. Chairman, distinguished foreign guests, geniuses in
residence, Ladies and Gentlemen:

It is both an honor and a pleasure to be given an opportunity
to speak here tonight to the delegates to the 8th Symposium on Naval
Hydrodynamics.

It is obviously an honor for a mere sailor to be invited to
talk to so erudite an audience and under such distinguished sponsor-
ship as the California Institute of Technology, the Office of Naval
Research and the Naval Undersea Research and Development Center.

It is a particular pleasure since it is not often the wheel of
fortune stops right on your number and you get invited to speak just
fifty miles from the birthplace of a brand new grandchild.

I believe it's customary about here for a visiting speaker to
tell a condescending joke about California smog but since most of you
read the newspapers you know that we on the East Coast are now
living in a glass house where that subject is concerned. After all,
when it gets to the point where you can no longer see the National
Capital from the top of the Washington Monument you can't pass it off
any longer as a morning haze.

In any case I arrived here by way of Alaska where most of
the country's current supply of fresh air seems to be stockpiled so
my lungs are back in pretty good condition.

This subject of smog and pollution in general reminds me that
an acquaintance recently told me of an opinion poll he claimed had
been taken among American Indians. Only 12% of them, he said,
felt we should get out of Vietnam, but 88% thought we should get out
of North America.

I originally intended to say a few kind words about the sponsors
of this annual event but changed my mind. Anything about the valuable
work that has been done in oceanography and many related fields by
the Office of Naval Research and the Naval Undersea Research and

xiv

Development Center (our chief semanticist had us remove the nasty
word warfare from their title) would come under the heading of
bragging about a relative. And after bringing myself up-to-date on
the history of the California Institute of Technology I felt there was
just nothing I could say. Even an amateur of science who walks
across a campus where such men as Millikan and Michaelson once
tarried to think, feels as an art lover must feel when he walks ona
stone bridge across the Arno where Leonardo once set his mighty
sandal. The debt the nation and the Navy owe this Institute is beyond
all calculation.

The point was adroitly made, I thought in a booklet about Cal
Tech that Dr. Plesset was kind enough to send me. The booklet
contained a picture of a man on a bicycle as an illustration of the
Institutes recreational opportunities. The man on the bicycle who
was unidentified in the caption, was Einstein.

Dr. Plesset also provided me with a program of Symposium
events and I ran through it looking for a possible clue as to what I
should choose as atopic. Several arresting items caught my eye.
Listed was a paper on "The Second-Order Theory for Nonsinusoidal
Oscillations of a Cylinder in a Free Surface." Another was on
"Three Dimensional Instabilities and Vortices between two Rotating
Spheres," and another on "Interaction between Gravity Waves and
Finite Turbulent Flow Fields."

Well, I know when I'm out of my league so I decided to just
make a few First Order remarks on the mission of Navy Oceano-
graphy and how it is organized.

First a definition. Hydrodynamics is not generally considered
to be oceanography but then neither specifically is anything else.
Oceanography as we use it is just an omnibus word for any scientific
or engineering discipline as it applies to the oceans.

It is nothing new. In the American Navy it goes back at least
to our pre-civil war patron saint, Lieutenant Matthew Fontaine
Maury, who used his knowledge of winds and currents to help the
clipper ships set their famous world speed records. In Great Britain
it goes back to the famous voyage of the HMS CHALLENGER.
Benjamin Franklin took a lively interest in it and so did Aristotle.

But modern oceanography in the Navy dates from the christen-
ing of the NAUTILUS and the nuclear missile submarines that followed
it. Warfare had suddenly become truly three-dimensional. The new
mission of Navy oceanography was to see to it that the Fleet was
given the information it had to have to insure its ability to operate
efficiently in this new and deadly area of underseas warfare.

Before I tell you how we went about this let me say a few

words about the broader aspects of oceanography. In the Navy we
consider it as our field of special competence and we are entrusted
with close to half the Federal budget -- or about 210 million dollars
in this current year of fiscal austerity.

Work in the entire field however is carried on at three levels.

First there is the National effort. This involves industries
like the oil business -- 15% of our oil already comes from under-
water -- and the fishing industry where our annual catch can be
greatly increased with a better understanding of the ocean currents
and temperatures, which influence the distribution of fish -- and the
growing aquatic recreation field where beach erosion, the character
of marine life, the most efficient design of boat hulls and other
oceanographic factors are most important.

The next area is the Federal effort. There are close to
thirty major Federal agencies concerned with oceanography to some
degree. The Department of the Interior in fisheries. The Food and
Drug Administration in medicine from the sea. And the Coast Guard
with a variety of oceanographic interests -- to mention just a few.
This Federal effort is now being examined from an organizational
standpoint. The President has recommended and the Congress is
considering a broad new plan to streamline this effort under unified
executive direction.

This brings me to the third area of oceanography, the military
aspects, for which the Navy, quite logically, is the Defense Depart-
ment agent.

Our job of controlling the seas for defense requires that Navy
have the broadest program in scope in the Federal Government.

To avoid duplication of effort and give us a clear-cut chain
of command we put all of our efforts under the technical direction of
the Oceanographer of the Navy.

For organizational efficiency the program was set up in four
divisions. These divisions are Ocean Engineering and Development,
Ocean Science; Operations, and Environmental Prediction Services.

Our newest and fastest developing area is Ocean Engineering
and Development. Seven major efforts are included here: undersea
search and location; submarine rescue and escape; salvage and
recovery; diving; instruments for survey and environmental pre-
diction; and underwater construction. We have allocated 57 million
dollars for these programs this year.

Our first rescue vessel, the DSRV-I , which can be equipped

also for survey work was launched recently at San Diego. Our first
nuclear deep submersible, the NR-I, has already completed its

xvi

early tests, and is currently undergoing some changes including
improvements to its main propulsion system. The DSRV-II will be
ready soon for launching. Our goal with these vehicles and their
attendant systems is a capability of rescuing personnel down to
submarine crush depth. They will be made available on request to
other governments and some are already making the necessary modi-
fications required for utilizing their services.

Our first nuclear propelled deep sea vehicle the NR-I, has
done some bathymetric work during her sea trials and is undergoing
continuing tests to determine the limits of her capabilities.

We are working on a Large Object Salvage System (LOSS).
The goal is to develop a capability of bringing up a submarine intact
down to a depth of 850 feet.

An extension of the engineering effort is our Deep Ocean
Technology or DOT program designed to anticipate the multiplying
requirements of the pioneering technology.

For instance we are well past the blue print stage on our
proposed Deep Submergence Search Vehicle (DSSV) designed to
operate to a depth of 20,000 feet -- a depth that accounts for 98%
of the ocean floor.

An immediate concern is with new power packs. The old
style batteries just can't give us either the speed, power or endurance
now required. We need electrical systems that will operate in salt
water and we are working on thermochemical power sources. We
are currently sponsoring a design competititon between two firms in
this area. It is a long range item that already shows promise.

Our new machines with all of the improvements we are
achieving are no better than the skills of the men who operate them.
To make the point by hyperbole, if I had only a dollar to spend I would
spend 95 cents on training and equipping men and 5 cents on the hard-
ware. So the whole engineering effort is concerned with extensive
bio-medical work, particularly in relation to deep saturation diving.

We are already working deeper than 600 feet in the open sea
and 1,000 feet experimentally. We are hoping to go to 2,000 feet,
perhaps 3,000 feet before we are through.

This means we need more and more bio-medical data for
equipment design and for shaping the selection, training, operational
use and health care of our aquanauts and undersea vehicle pilots.

We are taking the field of underwater medicine from its
rather narrow corner as an occupational sub-specialty, for its
scope transcends its size in at least three important ways. First,
it has forced us to study the effects of pressure on living systems,

xvii

a study neglected in biology as compared with other fields of science
and one which promises to advance the understanding of normal pro-
cesses. Second, it is an important confluence of the rapidly mixing
disciplines of biology and engineering. And finally, it is the keystone
to safe and effective utilization of a growing number of underwater
systems.

Our second field, first really in long-range importance, is
science.

The primary objective is to provide the basic knowledge
needed in all our programs. About 75% of this effort is directed
toward anti-submarine warfare particularly in studies of the behavior
of sound underwater, as sound is our only practical method of de-
tecting a potential underwater enemy. Much of this work is done
under contract with academic and non-profit institutions such as
Cal Tech where we can pick the brains of hundreds of the nations top
scientists. Engineering, of course, comes in here to provide and
equip the platforms that our research scientists need to work from
-- ships, deep submersibles, flip type vessels that can stand on
their head, surface and subsurface buoys, airplanes, satellites,
even a floating ice island.

Next our Operations effort. It functions in direct support of
the Fleet. In addition to much else, including various world-wide
surveys, it carries out duties imposed on us long ago by law to pre-
pare and disseminate charts and publications necessary for naviga-
tional safety both for Navy ships and for the Merchant Marine.

In support of this program is our Environmental Prediction
section which operates as an undersea weather bureau to forecast
those changes within the waters of the ocean that affect our opera-
tions, In this field we work very closely with the Fleet Numerical
Weather Center.

Despite our concentration on our primary defense mission,

the Navy program is necessarily a broad one -- the broadest in the
Federal program -- for the seas are our domain -- and we must
know and understand everything we can about them -- the animal

life that abounds in them, the nature of the ocean waters, their
circulation, the character of the bottom, and much else.

Thus many of the things we must learn and study are of
interest to others, in the government, including foreign governments,
in the academic world, in industry. I am proud to report that the
Navy takes part in many cooperative programs in such fields as
fisheries, oil and minerals from the sea, wave predictions, and
others. We strongly support this phase of our program because as
taxpayers it gives us a feeling of accomplishment to see federal tax
dollars doing double duty. I will give just two examples.

xviii

A friendly neighbor, Iceland, asked for help when they
realized that herring, which make up 90% of their export products,
were going to be difficult to find this last season. The herring
migrate from Norway and stop off at the East Coast of Iceland when
they reach the cold edge of the Greenland current. When this current
meanders or changes its location, as these ocean currents are likely
to do, it may divert the fish away from their normal grounds near
Iceland, as happened recently. We diverted an ice patrol plane with
a heat measuring sensor long enough to find the cold wall of the
current. And sure enough there were the herring. We are planning
now to help Iceland develop its own capability for this kind of work.

We are also providing technical help in harbor improvement
programs for several South American countries, and we are running
annual courses on oceanography and hydrography for foreign
students.

We opened our files on ice reconnaissance and trained some
people and also provided an on-board oceanographer to the owners
of the great new tanker the MANHATTAN, which has recently
successfully navigated the Northwest Passage. Free passage of this
once impassable channel should prove to be an invaluable national
asset both from an industrial and a strategic viewpoint.

Recognition of the importance of oceanography and hydrography
to present day and future naval operations, coupled with a concern
for the availability of technically competent naval officers within
these areas, has caused us to establish a new Special Duty Officer
category. It will consist of approximately 140 officers of ranks
Ensign through Captain. Promotion opportunities are equal to that
of'an Unrestricted Line Officer.

Inputs to the specialty at the Ensign level will come from the
Naval Officer Candidate School at Newport, R. I., and the Naval
Reserve Officer Training Corps Contract Units. Applicants must
have a degree in oceanography, or in another field of earth science,
physical science, marine science or engineering (with emphasis
on survey engineering for hydrography or ocean engineering for
oceanography); must have completed mathematics through calculus
plus one year of college physics and chemistry; and should have a
B average or better in mathematics, physical science and engineer-
ing courses. Graduates of the U. S. Naval Academy and the Naval
Reserve Officer Training Corps Units (regular students) may apply
after approximately three years active duty.

The first three years of commissioned service will consist of
a tour of sea duty on an oceanographic or hydrographic survey vessel
and a shore duty tour at a naval facility involving application of
oceanographic information to naval operations. Subsequent tours
may include management of research and development projects,
oceanographic forecasting, mapping, charting and geodesy, instructor

duty, and administration of various areas of the Navy's Oceanographic
program.

Turning to the Federal scene, the big push now would seem
to be with the war on pollution and, certainly we need to fight it.
Oceanography of course is involved here, particularly in the coastal
zones, estuaries and lakes.

Thousands of words are being written on the pollution of our
environment and the new "in" word is ecology. My daughter heard
it and read it so often she decided to look it up. The dictionary told
her it meant "the relationship between living organisms and their
environment.” "Here," she said, "I was wondering what it was and
I've been right in the middle of it all the time."

Our new consciousness of our total ecology and the drive
against pollution are going to lead to some complex conflicts -- such
as between the off-shore oil interests and the conservationists --
the real estate developers and the fishing industry. When you drain
a salt marsh, for instance, you interfere with the food chain that
supports the fish we need for human food. Involved also is the huge
and growing water recreation industry.

In solving our problems we have to be sure not to throw out
the baby with the polluted bath water.

Has oceanography got an assured future? Yes of course. We
are going to have to turn more and more to the inexhaustible seas
for the food and the minerals we will need for the worlds exploding
population.

But I don't see that future going up in a near vertical line as
it did, for instance, in the space business. In the first place there
are none of those big hunks of development money lying around these
days.

But I do see it going up steadily in a much more gently rising
curve.

But go up it will and as it goes we will need more and more
sophisticated equipment and techniques to gather and evaluate infor-
mation and ever smarter and better educated men to program and
run them.

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

Monday, August 24, 1970

Morning Session

Chairman: F. H. Clauser
California Institute of Technology

Page
Tsunamis 3
G. F. Carrier, Harvard University
Laboratory Investigations on Air-Sea Interactions a

EH Y. Hsu, H.- Y¥~ Yu, Stanford University

Air-Sea Interactions; Research Program and Facilities
at IMST Siti
A. Favre, M. Coantic, IMST Marseille, France
Presented by: A. Ramamonjiarisoa

Explosion-Generated Water Waves 71
B. Le Méhauté, Tetra Tech, Inc.

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TSUNAMIS

G. F. Carrier
Harvard Untverstty
Cambrtdge, Massachusetts

I. INTRODUCTION

An understanding of the coastal inundation caused by Tsunamis
requires the piecing together of several studies. Among the poten-
tially important characterizing features of the phenomenon are: the
temporal and spatial distribution of the ground motion which initiates
the Tsunami, the distance from the source to the target area in ques-
tion, the bottom topography of the intervening ocean, and the topag-
raphy of the coastal area itself. These are all discussed in some
detail in [1], a manuscript which was prepared in conjunction with
a longer series of lectures than this one. In order to avoid excessive
duplication of publication, we content ourselves here with a brief
summary of that material. As will be evident, many details remain
to be explored; unfortunately, there is no evidence to suggest that
even a more comprehensive understanding of the phenomena will sug-
gest procedures for alleviating the intensity of Tsunami inundation.

Il. INITIATION AND DEEP WATER PROPAGATION

The wave generated by a submarine earthquake is large
enough in lateral extent and small enough in amplitude so that a
linear theory is completely adequate for an analysis of the propaga-
tion over deep water. However, the propagation path is so long that
dispersion and its attendant changes in wave shape cannot be ignored.
Accordingly one can adopt either the classical linear theory of
gravity waves or the Boussinesque formalism to study the early
stages of the wave propagation. When either is done, for a basin of
constant depth, H, it is convenient to present the results in terms
of a particular family of initial ground motions. We discuss here
the waves which result when the ground motion is given by

F, (x,t) = ier. [ - tal 5(t).

When the half width, L, of the distrubed region is "small," the
wave which arrives at a distance x, will have been greatly affected
by dispersion; when the width of the generating ground motion is

Carrter

longer, smaller distortions of the wave will be apparent at xp».
Figures ita, 1b, and ic illustrate the quantitative aspects of the
foregoing statement. In the notation of those figures,

l= 4en

When the depth of the water is 3 miles and x,= 3000 miles, the
three cases shown represent ground motions whose half widths are
0, 19 and 33 miles. Figure id indicates the wave which ensues
when the ground displacement is given by

F= Fy (x,t) - F) (x +520 5t)

with a = 10. That is, the ground motion has a dipole character
rather than a general subsidence or elevation. The persistent lore
that the second or third crest of the Tsunami penetrates more than
the first makes it interesting to speculate (in view of Figs. 1) that
many initiating ground motions may be of dipole form.

III RUN-UP ON A PLANE BEACH

When the wave encounters a sloping shelf along which the
water depth generally goes to zero, the wave steepens and becomes
greater in amplitude. Accordingly one no longer can rely on a linear
theory. However, the shelves of real interest are such that the
distance along the wave trajectory above sucha shelf is short enough
so that dispersion in this region is not of any real importance.

There is a non-iinear, non-dispersive shallow water theory
which leads to tractable problems when the depth of the basin is
linear in one horizontal coordinate and when the entire phenomenon
is independent of the other. Thus, we can regard the results re-
ferred to in Section 2 as the input information for a study in which we
ask how such waves climb up a sloping shelf. The analysis which
accompanies such a study involves only the solution of a linear
equation whose interpretation in the non-linear context is explicit
and accurate.

The result of interest is the ratio of the run-up, No, (the
vertical distance above sea level to which water encroaches) to the
wave height, 1,, at the edge of the shelf. One interesting result is
this:

Hor @ = 10

- -176
qe we hot cay,

Tsunamts

. qt *31a

OStt ——OblL wait Oell Ol 0011 0601 0801 0201 0901 0S0i Ov0l 0¢0l 0201 0101 0001 066 086

OOl= VHdIv 09

ey "Sha

x
Ostt Obl l O¢ll raul OL OO 0601 0801 0201 0901 0S0L OvOl O¢0l 0201 O10! 0001 066 086

O'O= VHdIV og

Carrter

Oflt

x

zi

oll

DOL

0601

0801

0201

x
ool 0601

0801

0201

0901

0901

0s01

0S01

pol

PT °Sta

Ob0l O¢0l 0201

OT °StaT

O¢0l 0201

0101

0101

000!

0001

066

066 086 016 096

O'Ol = VHd 1V

086 06 096

O'O€ = VHd1V

Tsunamts

where A = 4.2 if the ground motion is upwardor A= 5.6 if it is
downward. The corresponding results for other values of L can
easily be found (the calculation requires only the use of the method
of stationary phase). The dependence on the shelf slope, 0, is that
which would be found for monochromatic waves whereas the depen-
dence on x_ is a consequence of the dispersion during the deep
water propagation.

IV. DEEP OCEAN TOPOGRAPHY

If there were systematic variations in the water depth between,
say, the Aleutians and the equatorial Pacific, one might expect that
the relative intensities of the Tsunamis (with Aleutian source) which
were incident on different Pacific islands might differ because of
mid-ocean refractive effects. Exhaustive studies of this effect have
certainly not been completed but the indications are that this is not
a major reason for the different response at (for example) Wake and
Hawaii. One might also anticipate that the irregular deep ocean
topographical variations could seriously modify the wave which
propagates across the ocean. This possibllity has been analyzed
treating each event as a member of an ensemble of phenomena each
of which take place over a topography which is itself a member of a
stochastically described collection of random topographies. This is
motivated loosely by the fact that the one-dimensional topography
between any given source and any given target differ from that associ-
ated with any other source-target pair, and the fact that the topog-
raphies are so poorly known that little else can be done. The result
of this study indicates that the ratio of intensity at x, of the wave
over the irregular bottom to that over constant depth is characterized

by
2

el
ee iar
ana

where € is the ratio of the average irregularity height to the average
depth and L = 2nN where N is the number of wave lengths of the
monochromatic wave whose scattering is being studied. For wave
lengths in the spectral region of major interest, the effect of this
facet of the wave propagation seems to be of relatively small im-
portance too.

V. ISLAND TOPOGRAPHY

When the wave encounters an island, the lateral scale of that
island has the same order of magnitude as much of the important
part of the wave length spectrum. Thus, the pretense that the wave
climbs a plane shelf must be corrected. The refractive effects so

Carrier

implied cannot be estimated readily py geometric optics methods at
such wave lengths and one must resort to numerical procedures.

The results of such studies are depicted in Figs. 2 and 3, taken from
Lauterbacher [2]. Figure 2 indicates the variations of intensity
with position on a given island and Fig. 3 indicates the extent of this
effect for different ratios of wave length to island size.

<j
N
fi
+ —_
W all
<< i]
— a
z 10 >
= WAVE INCIDENT AT 0° 9 2
< q
O 8S
iL —=
ar 7
S Y 6a
=
< 5 <¢
Ww 7 Ww
> 4s
< <
= [33
= 2s
= =
= 15
x< =
aq 0 0 xX

8 & 8 2 8 4 8
COAST POSITION (MEASURED IN RADIANS AROUND ISLAND CENTRE)

Fig. 2. Maximum wave amplification at coast (OAHU)

Tsunamts

L/R = 1.67

Patel
BEACH MAXIMUM

"op! iP)
BEACH MAXIMUM

WAVELENGTH )/L

Fig. 3. Ratio of two-dimensional to one-dimensional maximum wave
amplitude on beach. L, island diameter at ocean floor; R,
island diameter at beach.

ACKNOW LEDGMENT
This work was supported in part by the Office of Naval

Research under contract N00014-67-0298-0002 and in part by the
Division of Engineering and Applied Physics, Harvard University.

REFERENCES

1. Carrier, G. F., "The Dynamics of Tsunamis," to appear in the
Proceedings of the Summer Symposium on Mathematic Prob-
lems in Geophysics, 1970.

2. Lauterbacher, C. C., "Gravity Wave Refraction by Islands,"
J. Fluid Mechs., Vol. 41, Part 3, pp. 655-672, April 1970.

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LABORATORY INVESTIGATIONS ON
AIR-SEA INTERACTIONS

Ee Yo Hsu and H. Y. Yu
Stanford Untversity
Stanford, California

I. INTRODUCTION

Since the comprehensive review on wind wave generation by
Ursell [1956], there have been renewed, intensive studies, theoreti-
cal as well as experimental, on the subject. Although significant
contributions have been made by many investigators, the final goal
of achieving a basic understanding of the fundamental mechanism of
energy transfer between a turbulent air stream and the ocean has not
been realized. A unified, comprehensive theory of wind wave gener-
ation must provide adequate explanation of the energy transfer
between the two media at all stages of wave growth from capillary
waves to sea swell. In the absence of sucha unified theory, a
convenient classification of various flow regimes in wind-wave
generation may be made by use of the ratio of water wave celerity
C and the air shear velocity u* at the interface. When C®# ats
the dominant mechanism of energy transfer between air and water
is the "viscous mechanism," characterized by the critical layer
being within the laminar sublayer and treated by Miles [ 196 at.
When C¥% 10 u’, the critical layer is outside the laminar sublayer
and the dominant mechanism of energy transfer is the "inviscid
mechanism" (Miles [1957, 1967] and Benjamin [ 1959]) with transfer
arising from the normal pressure acting on the interface and the neces-
sary phase angle between the pressure distribution and the progres-
sive wave. As pointed out by Longuet-Higgins [ 1969], neither of the
above theories accounts for two well-established features of wave
generation: (1) the existence of some wave energy in a frequency
range corresponding to waves traveling faster than the mean free-
stream velocity and (2) the damping of a swell by an adverse wind.

The experimental investigations of Sutherland [1967], Hires
[1968], and Chang [1968] and many others are limited to the viscous
range. Because of the high Reynolds number in a typical wind blowing
over the ocean surface, the viscous mechanism can be safely neg-
lected as irrelevant to full-scale wave energy transfer. Hence,
Miles' inviscid model has received most attention and been widely
employed in comparisons with experimental data obtained in full
scale (ocean) and laboratory simulations.

11

Hsu and Yu

The dearth of systematic measurements taken under controlled
conditions closely comparable to those of Miles' model was a moti-
vation for our research program at Stanford University. In order
to examine the applicability of Miles! inviscid theory, experiments
were designed for measuring the wave induced perturbation pressure
or inviscid Reynolds stress under steady-state and unsteady-state
conditions. Other experiments were also devised for measuring the
growth of mechanically generated waves subjected to wind action.
From these measured wave growths, the growth factor of Miles was
calculated. The objectives of this paper are to present a summary
of our experimental data in the inviscid range, to compare our data
and other existing data to the theory and the ocean observations,
and to suggest specific and fruitful avenues for further study.

Il. A BRIEF REVIEW OF THE THEORY

To facilitate presentation and discussion of the experimental
data, a brief outline of the assumptions, key equations and results
of Miles' inviscid, shear-flow theory are presented below.

The deep-water, wave profile is assumed to be a progressive,
sinusoidal wave, expressed as

n ='avexp! hik(x:- Gt)], kay<< 1 (1)

where a is the amplitude, k= 2n/L is the wave number, lL is the
wave length, and C is the wave celerity. The assumptions of irro-
tational, incompressible water motion lead to the existence of
velocity potential. By substituting the velocity potential in the
linearized Bernoulli equation and evaluating the result at the free
surface, one obtains the equation of motion governing the propaga-
tion of a small amplitude, surface wave

2
Py OMe ©
Pele ey, age Pa (2)

where g is the acceleration caused by gravity, p, is the mass
density of water, and p, is the aerodynamic pressure caused by
the wind stream.

Miles [1957] assumed the aerodynamic pressure p, has
the form

p, = (2 + iB)p,U, kn (3)

where pg is the mass density of the air, U, is a reference speed
for the air, and @ and f are, respectively, the in-phase and

12

Laboratory Investtgattons on Atr-Sea Interactions

out-of-phase non-dimensional-pressure-coefficients. The phase
angle @ is

g = tan’! (B) (4)

The constants @ and 8 were determined by solving an inviscid Orr-
Sommerfeld equation which represents the perturbations (caused

by the water wave at the interface) to a wind shear-flow described

by an assumed logarithmic, mean velocity distribution

Uly) = U, In = (5)

fe)

where y is the vertical distance from the mean water surface and
Y, is the roughness height.

The effect of the impressed aerodynamic pressure P, on
the surface wave can be evaluated by solving Eq. (2). It follows
that the complex wave celerity

2
1 p fU
a : pesslipe gd © 1 Bonet
C ei ae (2 +i6)(ct) J (6)
where C, = (g/k)’?. Substituting Eq. (6) into Eq. (1) yields
4 p thins
a= a, exp ls kC, co (<4) Bt] (7)
where a, is the amplitude at t=0.
It is convenient to measure the growth of wave amplitude as

a function of fetch x in a wind-wave channel. The dynamic equi-
valence, valid for x >>L, is given by Phillips [1958] as

x= Sot, me

where C,/2 is the group velocity of a deep-water wave. Conse-
quently, the fetch-dependent amplitude growth a is

2
Ke are
a= a, exp [£2 © utpal o
w
where a, is now the wave amplitude that would exist without wind

13

Hsu and Yu

action ior-at. x = 01.

The total energy per unit of surface area E of a small-
amplitude, sinusoidal, progressive wave is

If the energy corresponding to a, is E,, Eq. (8) may be rewritten
as

2. 2

E = E. exp [££ x’u’ px] (9)
° EP, |

The out-of-phase pressure component BP is responsible for the energy

transfer from the air stream to the wave. Experimental results are
presented in the non-dimensional form

log, tie = AF (10)

where F is a non-dimensional fetch

k?2
HPS ete

and
A= (2 £2) log ee toa x10.
Py fe) :
for pes 1 gm/cm°* and Sp 0.00118 gm/cm’*.

III. LABORATORY INVESTIGATIONS
3.1. Techniques of Simulation
3.41.1. Moving wavy-boundary (steady-state)

In attempts to verify Jeffreys' sheltering hypothesis, Stanton,
et al. [1932], Motzfeld [1937], Thijsse [1951], and Larras and
Claris [1960] measured pressure distributions over stationary,
solid, and two-dimensional sinusoidal boundaries in either a wind
tunnel or a water channel. In the light of the critical-layer mecha-
nism proposed by Miles, these stationary wavy-boundary experiments

14

Laboratory Investtgattons on Atr-Sea Interacttons

cannot be regarded as an adequate, steady-state simulation of the
wind-generated wave problem, because the critical layer in the
experiments is of zero thickness and, hence, the critical level lies
on the stationary boundary. All of the above experiments, with the
exception of Thijsse's indicated a smaller sheltering coefficient than
that anticipated by Jeffreys, who expected the pressure distribution
to be out-of-phase with the wave (in accordance with Miles' inviscid
theory). The resulting small sheltering coefficient may be attributed
to either viscous or finite wave-amplitude effects.

For amore realistic steady-state simulation of wind-generated
waves and demonstration of the importance of the critical-layer
mechanism of energy transfer, the wavy boundary must be moving
with a speed equal to the wave celerity and opposite to the direction
of mean free-stream. An important advantage in this simulation is
that the flow field is steady. Consequently, measuring techniques
are greatly simplified.

The first successful moving, wavy boundary experiment and
its resultant presentation of the normal pressure distribution on the
boundary were reported by Zagustin, et al. [1966, 1968]. Subse-
quently, Ott, et al. [1968] extended the Zagustin investigation and
used refined experimental procedures to achieve better experimental
accuracy. Small amplitude waves with a length of 3 ft and amplitude
of 0.65 in. were used. Because of the limited capability of the
experimental facility, G/U. was limited to approximately 0.75 (U,
is the air velocity at the edge of the boundary layer).

3.1.2. Flexible wall with progressive waves (unsteady- state)

Kendall [1970] described a series of experiments on wind-
wave simulation in a low turbulence wind tunnel. The wavy wall was
the floor of the constant pressure test section of the tunnel. The
surface of the wavy wall was composed of neoprene rubber sheet
which was constrained to form a series of sinusoidal waves (length =
4 in. and height = 0.25 in.). The rubber sheet was supported from
beneath by a series of ribs which were connected to individual. circu-
lar eccentric cams. Each cam was positioned with proper phase
difference on a common cam shaft expending the length of test section.
Rotation of the cam shaft caused each rib to execute a reciprocating
vertical motion and thus a progressive wave form was produced.
Reversing the direction of rotation of the cam shaft produced waves
traveling in the opposite direction, giving - 0.5 < C/U, <0.5.

The boundary conditions for the two methods of wind- generated
wave simulation described above deviate slightly from those of a true
air-water interface. If the fluid particle velocity in a wave motion
is small compared to the wave celerity (true for small amplitude
waves), the moving wavy boundary simulation approximately satisfied
the boundary conditions. In the flexible wall experiment the surface

15

Hsu and Yu

particle motion resulting from the flexure of the rubber sheet was a
backward-rotating 3:1 ellipse as compared with the forward- rotating
circle of deep water waves. Again, the boundary condition was
approximately satisfied for small amplitude waves.

3.1.3. Wind-wave research channel (unsteady-state, true air-
water interface)

The physical features of the Stanford facility were reported
by Hsu [1965]. The channel is approximately 6 ft high and 3 ft
wide and has a usable test section length of 75 ft. At the downwind
end, there are a beach to absorb wave energy and a centrifugal fan
to produce the wind in the channel. At the other end, the air is
drawn vertically through a system of filters and then carried hori-
zontally on to the water surface at the beginning of the test section
by a converging elbow. A hydraulically-driven, horizontal-displace-
ment, wave-generating plate is located 17 ft upstream of the test
section. This distance is sufficient to allow generated waves to
become fully established prior to being subjected to wind action.
Sinusoidal waves, ranging in frequency from 0.2 to 4 cps, can be
generated. The maximum wind speed is approximately 70 fps with
a nominal water depth of 3 ft inthe channel. A limitation of the
present facility is that the wind and the propagating waves move in
the same direction.

3.2. Measurements of Wave-Induced Perturbation Pressures

Because the flow field was steady in the moving wavy-boundary
experiment (Sec. 3.1.1), two conventional, but small (1/32 in. O.D),
pitot-static probes were used for all the velocity and pressure measure-
ments. The reference probe was located in the free-stream while the
other probe could be moved to any distance from the moving boundary
by atraversing mechanism. Realizing that the traversing probe
must be aligned with the local flow direction, we mounted this probe
in a special rotating device. These probes were connected to a
Pace P-90 differential pressure transducer through a manifold
system which provided selective readings of dynamic pressure or
pressure differential between the two probes.

The pressure measurements in the flexible wall experiments
(Sec. 3.1.2) were made through static holes in the flexible surface.
Essentially, a length of metal tubing in the form of a loop was used
to connect the static hole and the pressure transducer. The loop
served to cancel the unwanted pressure gradient generated by the
motion of the tubing.

Because the thickness of critical layer was small in the wind-
wave channel experiment (Sec. 3.1.3), the measurements of pertur-
bation pressure were obtained by use of a specially-designed wave
following system. Again, the perturbation pressure is the difference
between the pressure at the air-water interface and that in the free

16

Laboratory Investtgattons on Atr-Sea Interacttons

stream and was monitored by two identical pressure sensors, one
at the interface and the other in the free stream, through a Pace
P-90 differential pressure transducer. The whole system was
allowed to follow the wave motion so that the lower pressure sensor
was kept a fixed distance from the instantaneous air-water interface
and inside the critical layer. The unwanted pressure signal caused
by the motion of the system was determined by calibration tests and
removed in the final data reduction.

3.3. Measurement of Wave Growth

A series of experiments was run to measure wave growth
rate in the Stanford wind-wave channel. Small-amplitude, deep-
water waves with frequencies varying from 0.9 to 1.4 cps were used
with the maximum wind speed ranging from 12 to 44 fps (fan speed of
100-300 rpm). Time records of wave profiles were obtained with
capacitance-wire sensors at seven locations spaced at 10 ft intervals
along the centerline of the test section. Air velocity distributions
were taken at six intermediate locations with a conventional pitot-
static probe.

Although the mechanically-generated waves were initially of
small amplitude and closely sinusoidal, they become steep and some-
what non-sinusoidal with increasing fetch in response to the wind
action. The true wave profile could be viewed as a superposition of
a mean wave and a spectrum of ripples. Therefore, a phase averag-
ing procedure was adopted to determine the mean wave profile at
each fetch and fan speed. The mean wave profile at each phase angle
was the result of averaging 35 waves in the time series. The stream
function fitting technique introduced by Dean [1965] and outlined for
this application by Bole and Hsu [1967] was used for evaluating the
kinetic and potential energy of each mean wave profile. Finally, the
total wave energy at each location of the test section was adjusted
for wave energy dissipation due to viscous action. The dissipation
was determined experimentally for conditions without wind.

Along with the mean wave profile, the ripple variance of the
water surface about the mean wave profile at each phase angle of the
wave and the mean ripple variance and standard deviation for all the
phase angles were calculated. The ripple variance is, of course,
proportional to the potential energy contained in the ripple.

IV. RESULTS AND DISCUSSION

4.1. Water Surface Roughness (Unsteady-State, True Air- Water
Interface)

When mechanically generated waves were subjected to wind
action, ripples were always present and were superposed on the
waves. Thus, the water surface can no longer be regarded as smooth

17

Hsu and Yu

and its roughness can be described by the ripple standard deviation
o of the water surface elevation about the mean-wave profile. It
was observed that o increased with wind speed at the same fetch.
In general, o increasedas y,u /v increased. The values of o
are listed in Table 1 and vary from about 0.001 to 0.039 ft.

From a least-square fit of the velocity profile Eq. (5) to the
measured data, values of U, and y, can be obtained and hence the
values of y,, ky,, and B (see Sec. 2). The values of yg are com-
piled in Table 2 and vary from 0.004 to 0.011 ft. The values in
Table 1 and Table 2 show that the ripple standard deviation is larger
then the critical layer thickness in all cases. It seems that the
surface roughness or ripples should destroy the organized actions
of vorticity which Lighthill [1962] presented as the physical expla-
nation of Miles' instability mechanism. Thus, Miles' interpretation
of the energy transfer mechanism -(adopted from Lin [1955]) as the
perturbation Reynolds stress working against the mean velocity pro-
file at the critical layer is severely strained by the existence ofa
ripple layer large enough to obliterate the critical layer.

The potential energies of wind+generated ripples with and
without mechanically generated waves are presented in Table 3.
The presence of the generated waves decreases ripple energy sig-
nificantly. Although there are many irregularities, ripple energy
generally decreases as wave frequency increases. Exceptions occur
at 300 rpm and 60 ft fetch where the 1.2 and 1.4 cps waves are
breaking and ripple energy is sharply decreased. Sample power
spectra of the ripples superposed on a 1.1 cps wave were obtained
by subtracting the mean 1.1 cps wave profile from the original water
surface elevation time series, The remaining time series, which
contains only ripple variation, was then spectral analyzed. The
resulting power spectra of wind- generated ripple with and without
mechanically generated waves is exhibited in Fig. 1. Spectral
peaks for the two cases appear at about the same frequency, but
spectral density is drastically reduced when waves are present.

The two possible reasons for ripple attenuation in the
presence of waves are |
a. sheltering effects retard ripple generation by the wind,

b. non-linear wave-wave interactions cause ripple energy
to be dissipated and to be transferred to the waves as
suggested by Longuet- Higgins [1969] (see later discussion
on wave energy).

4,2. Mean Wave Profiles
Mean wave profiles were determined by phase-averaging

over records of 35 waves. A sample of corresponding pairs of
mean wave profiles and their corresponding original recordings for

18

Laboratory Investigations on Atr-Sea Interactions

TABLE 1. RIPPLE SPECTRUM STANDARD DEVIATION

o x 10° (ft)

Fetch Mechanical Wave Frequency ( ee eee ee

NououwnrF
Noo pre

3 3
6 5
6 8
8 9
8 0
8 2

*
Waves breaking.

19

Hsu and Yu

CRITICAL LAYER THICKNESS

TABLE 2 e

y, xX 10° (ft)

cps)

“~
>
13)
i=)
o
S
iow
oO
u

fy
7)
>
wo
=
ec
@
13)
cial
S
wo
G
io)
Oo
=

20

Laboratory Investigattons on Atr-Sea Interacttons

TABLE 3. RIPPLE SPECTRUM POTENTIAL ENERGY
Values x 10° (£t-1b/£t7)

Fetch Mechanical Wave Frequency (cps)

NFrFO FrFRrRFOOO

*
Waves breaking.

Zi

Hsu and Yu

200 RPM

——— I. cps wave
No wave

Frequency (cps)

Fig. 1. Influence of 1.1-cps wave on 200-rpm ripple
spectra

1 cps and 1.3 cps waves at 300 rpm is given in Figs. 2 to 5. A close
examination of these figures (wave motion is toward the left and the
usual Sanborn attenuation scales are marked) reveals that the posi-
tions assumed by the ripples influence the mean wave profile. For
example, ripple superposition in the 1.0 cps wave caused the mean
wave crest to become flattened. However, ripple superposition on
the 1.3 cps wave did not cause mean wave profile distortion. Many
of the records show ripples superposed in such a way that the crest
was sharply peaked. The existence of such conditions may enhance
separation of the air flow over the wave surface.

A source of error arises from the fact that mean wave pro-
files were distorted, and yet their energy was compared with that of
Miles' pure sinusoids. An expression consisting of a cosine and sine
plus their two higher harmonics was least-squre fitted to each of the
mean wave profiles (see below). Results indicated that the total

22

°

Laboratory Investigattons on Atr-Sea Interacttons

(99S 7°Q = UOTSTATP
yejuoztz1oy J) wida QoE te
s8ujpiooer uroques sdo-Q*t

"€ *S1q

(ft)

Surface Elevation

Water

uidi go¢g 7e SoTtjord oaem ueourt sdo-g*y “7 *3Iq

(Bop) ejbuy esoyg
0 06 os! O12 ce) 06 osi O12 ose

Wd OOF

(49) 49004

23

Hsu and Yu

(298 Z°O
= UOTS]A]Tp Teqwuoztoazy JT {dur
-yeoiq soAaeMm = *q*m) urdi QOE

ye sSujpIooer uroques sdo-¢*], “Gc *3Tq

yoyey _ 81D9S

(ft)

Surface Elevation

Water

fo}

°o

widi go¢€ ve soTyyord ovaem uvout sdo-¢*y, *F “817

(Sep) ejBuy esoud

te) 06 os! ol2 fo) 06 os! ol2 o9¢

Wd OO€
sdo ¢-|=43

(43) Ydbed

24

Laboratory Investigattons on Air-Sea Interactions

error introduced by representing the mean wave by these five terms
was not greater than 5 per cent in any case.

4.3. Wave Energy

The total wave energy (potential plus kinetic) was calculated
by the method developed by Dean [1965] through least-square- fitting
an analytic stream function to the mean measured wave profiles.
Details of the procedure were presented by Bole and Hsu [1967]. In
order to compare the measured wave energy with Miles' prediction,
wave dissipation in the channel, determined experimentally under
the condition of no wind, as a function of fetch was added to the
measured wave energy. Figure 6 shows the energy ratio E(x)/E(0),
as a function of downstream distance for a 1.4 cps wave subjected to
various wind speeds, while the results in Fig. 7 are for a constant
wind speed (300 rpm) acting on waves of various frequencies. The
data was then reduced to the non-dimensional fetch F defined in
Eq. (10). The final results, compared with Miles' inviscid theory,

40 50

Oo 10 20

30
x--Fetch (ft)

Fig. 6. 1.4-cps wave growth vs. x

AD

Hsu and Yu

300 RPM

(e} 10 20 30 40 50
x-- Fetch (ft)

Fig. 7. 300-rpm_wave growth vs. ȴ

are exhibited in Figs. 8 and 9. In nearly every case, experimental
values fall well above the theoretical line. By assuming the growth
to be totally dependent on F inthe form of Eq. (9), we calculated
a ratio of the experimental and theoretical B values and present the
results in Table 4. The f-ratios vary from about 1 to 10. The
mean ratio for all frequencies and rpm's is about 3.

The total spectral energy of the ripples in most of the experi-
mental cases was no more than about 20 per cent of the mechanically-
generated wave energy. Phillips [1966] argued that non-linear inter-
actions between waves should be weak. Hence, our procedure of
measuring the growth of a single wave within a spectrum should be a
valid means of evaluating the parameters necessary for comparisons
with Miles' theory.

26

Laboratory Investtgattons on Atr-Sea Interacttons

6.0

5.0

40

E(F) 30
E(0)

2.0 0.04

0.03

fe) 100 200 300 400 500 600

Fig. 8. 1.4-cps wave growthvs. F

0.06
E(F)

r(x

0.04

300 RPM

(wb)- Wave Breaking

0.02
te) 200 400 600 800 1000 1200 :

Fig. 9. 300-rpm wave growthvs. F

24

Hsu and Yu

TABLE 4, RATIO OF EXPERIMENTAL TO THEORETICAL B

Mechanical Wave Frequency (cps)

(ey [oopaofia [ie [aa | a)

—
|
on,

WNIOROC TAMHREUYN AUARNOS YAWWON AAPA N

c=

OorwWwWn sb

-—_
1
~~

°

OrFwWrN Oo RNY W OF

*

*

°

AIRO’ OROWN OMDMDUID WHWWNEH

TONWO CHOMOhHO NOWNNY NONNOS WEEN O
PRE Ne PP PND WHWNR YYNWEH PNW

2
9
9
-8
3
2
3
7
8
oak
.8
4
5
5
6
2
al
1
1
8

FPrRFoONNDND FPRNYNDY UWWPRQY FPR RH Ww Ww
OFrPrFr FrFrRFOrF NNWNHBR OF KF O

ee
2.
Ie
3

3.
us
Ue
@s
4,
6

3

3.
2.
3.
2.
Us
4,
2.
2.
is

NWR wD WwW

PRR PN FPR OFR BWNYHO WHWHWWH DOANE
>

*
Waves breaking.

On the other hand, our experimental evidence indicated that
the wind- generated ripples riding on the mechanically- generated
waves had a tendency to break on the crests rather than in the
troughs. Longuet-Higgins [1969] showed that, in breaking, the
ripples may impart a significant portion of their momentum to the
longer waves in a strong non-linear interaction. Mollo-Christensen's
[1970] field observations showed that there were relatively high
peaks of energy in a high frequency band located near the crest of
the main wave. It is difficult to conclude that Mollo-Christensen's
data, taken in a confused sea, whether these high frequency peaks
are produced as a result of the breaking waves, caused by wave

28

Laboratory Investigations on Atr-Sea Interactions

groups of different frequency overtaking one another, or partly by
the generation of high frequency waves on the wave crest. To fully
investigate the non-linear, wave-wave interactions and to establish
the role of ripples in the transfer process, measurements similar
to those of Mollo- Christensen and additional detail measurements
of the velocity field below the air-water interface should be carried
out under the controlled conditions of a laboratory simulation.

The incompatibilities of the Miles’ mathematical model with
the natural wave-growth environment, as discussed in the previous
section, were anticipated by Miles. He stated in this 1957 paper
that "our model cannot be expected to have more than qualitative
significance for rough flow." It would appear that a more realistic
model and an improved theory of energy transfer cannot be formu-
lated until detailed studies of the structure of air flow near the air-
water interface are carried out.

4.4. Non-dimensional Pressure Coefficients

The non-dimensional pressure coefficients @ and B ob-
tained from the various techniques of laboratory simulation are
exhibited in Figs. 10 and 11 as functions of ky,. Comparison between
the measured values of the in-phase pressure coefficient @ (steady-
state, moving-wavy-boundary; unsteady-state, wind-wave channel)
and the Miles' theory is shown in Fig. 10. The experimental values
of the out-of-phase pressure coefficient B, evaluated from wave
growth measurements, are shown in Fig. 11. Although there is
considerable scatter in the experimental data, the deviation from
the inviscid theory is clearly evident and is consistent with the
results of the wave growth measurements. Because of the limited
capability of the experimental facility in the steady-state, moving-
wavy-boundary experiment, experimental values were limited to
ky, = 0.1.

The experimentally determined phase angle g obtained from
the three different methods of laboratory simulation -- moving-wavy-
boundary, flexible boundary with progressive waves, and wind-wave
channel -- is shown in Fig. 12 as a function of C/u’. In view of
the uncertainties among investigators in determining u values,
the experimental phase angle as a function of C/Ug@ and their cor-
responding theoretical values are shown in Fig. 13. The JPL-data
includes negative values of C. Because the measured velocity pro-
files varied to some extent with Ug and C as discussed by Kendall
[1970] , theoretical values of g for the case in which Um = 5.5
in./sec and C=0 were calculated.

In an attempt to detect flow separation in the region near the
air-water interface in the wind-wave channel experiments, pressure
measurements over waves of various amplitudes with constant fre-
quency were made. The measured phase angles for two wave
frequencies, 0.6 and 0.78 cps, are shown in Fig. 13. The scatter

29

Hsu and Yu

Stanford Data: |. Pressure Measurement
f=0.4cps C=8.64fps a: & 2.5"
f=O.5cps C=8.25fps a: wIl.15"
f=O0.6cps C=7.55 fps a: o1.15" 01.5" 62.10" $3.10
f=O0.78cps C=6.40fps a: 01.25" 92.12" 063.15"

Il.Steady State Moving Belt Simulation
asO;75'% -X#3,0" \[Casi92 tps = x
a

Inviscid Theory >

io > 2 4” 26" 8 1072 2 4 6 810"! 2 4.) BR

ky,

Fig. 10. Comparison between measured and theoretical values of
a vs. ky,

30

Laboratory Investtgattons on Air-Sea Interacttons

°° SA g jO Son[ea [ed]},ie1090y} pue perznseou usemjeq uosjzedurog “Ty *8qq

%y 18 9 »& 2 pUlg= 9. 8

2 2.018 9 » 2 ¢0ls 9 » 2

el

91
e: jUaWaINSDAW ABsauzy 2ADM ‘III
= $d} 26°€=9 oe =X 1o2:0= oz
uol,OjnWIS 419g Bulaow a404S Apoays
,she® ,220@ ,GZ1%:0 sdjyOv'9=9 sd9g2'0=3
,Olee ,0OV'2@ ,GI® ,Stl® :0 sdj GGz=9 sdd9Q9'O=43
,Gli@ :0 sdj Gz°e=9 sd9Go=43 b2

1g cg :0

sd} ~9'8=9 sd9 pO =3
sUaWaINSDEW a4nssald °| :040Q psoyudss

82

|

Hsu and Yu

xD *sA & JO SONTeA Ted]}Je1O9Y4} pue pernseos UseMjoq uosjaeduioy °7y *8tq

ty
Ol-
oo!
sd) G¢G‘2 =) sd99°O=3

8d} GZ°g=9 sd9G‘O=3 as
Ov!
sdsp9'@=9 Sdd p'O=3 09!

Ksoay, ~~~ ' 7
os!

x = sdj 26'¢€=9 O'S=X 1G2°0=0
uolyOINWwIS 412g Bulaow a4yn0is Apoays ‘ji
sale ® ,el2oO {Sele :0 sd} Ob'9=9 sd981°0=5

,Oree® ,O1z2e Sie ,SIi® :0 sdy GG2=d sd9 9°0 =5
,stl@ ;0 sd$ G27e=9 sdd9 G°O =3
829 :0 $d$ 9'B=D $49 H'O =3

JURZWAINSDAW AINSS9Jg “| ;04DGQ P4sojudyS
2998/4, 9OIY 228%, G50 -04100 df

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Laboratory Investigations on Atir-Sea Interactions

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33

Hsu and Yu

of the experimental data precludes any definite conclusion about
possible flow separation. Although a unified theory is needed to
describe the relationship between the phase angle » and + C/U F
the experimentally determined phase angles in the inviscid range do
indicate a correct trend compared with Miles' theory.

V. CONCLUSIONS

The accumulated laboratory experimental evidence obtained
at Stanford and elsewhere indicates general support for Miles'
inviscid theory of energy transfer between air stream and progressive
waves through the phase shift of the aerodynamic pressure at the
interface. However, the experimental growth rate is considerably
in excess of Miles' prediction, being approximately three times
larger. The most fruitful avenue for further study would appear to
be to reexamine the necessary simplifying assumptions in the Miles'
inviscid model. The incompatibilities near the air-water interface
suggest that detailed experimental investigations of this region are
essential before an understanding of the energy transfer mechanisms
and the conditions under which they occur can be fully established.
The effects of turbulence, possible flow separation, ripple super-
position and boundary layer development are complex, but could be
modelled and fruitfully studied in laboratory simulations.

REFERENCES

Benjamin, T. B., "Shearing Flow over a Wavy Boundary,’ J. Fluid
Mech., 6, 161-205, 1959.

Bole, J. B. and Hsu, E. Y., "Response of Gravity Water Waves to
Wind Excitation," Stanford Univ. Dept. of Civil Engineering
Téch. Rep.“Nos 794, (1967 «

Chang, P. C., "Laboratory Measurements of Air Flow over Wind
Waves Following the Moving Water Surface," CERv8-
69PcC1i8, Colorado State Univ. , 1968.

Dean, R. G., "Stream Function Representation of Non-Linear Ocean
Waves," J. Geophys. Res., 70, (18), 4651-72, 1965.

Hires, R. I., "An Experimental Study of Wind Wave Interactions,"
Tech. Rep. No. 37, Chesapeake Bay Inst., Johns Hopkins
Univ. , 1968.

Hsu, E. Y., "A Wind, Water-Wave Research Facility," Stanford
Univ., Dept. of Civil Engineering Tech. Rep. No. 57, 1965.

Kendall, J. M. Jr., "The Turbulent Boundary Layer Over a Wall

With Progressing Surface Waves," J. Fluid Mech., 41,
Pt. 2, 13 April 1970, pp. 259-282.

34

Laboratory Investtgattions on Atr-Sea Interacttons

Larras, H. and Claria, W., "Recherches en Souffleries sur L'Action
Relative de la Houle et du Vent," La Houille Blanche, 6,
647-677, 1960. =

Lighthill, M. J., "Physical Interpretation of the Mathematical Theory
of Wave Generation by Wind," J. Fluid Mech., 14, 385-398,
1962. ~~

Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge Univ.
Press, London, 1955.

Longuet- Higgins, M. S., "A Non-Linear Mechanism for Generation
of Sea Waves," Proc. Roy. Soc. A, 1969.

Miles, J. W., "On the Generation of Surface Waves by Shear Flow,"
J. Fluid Mech., 3; 185-204, 1957.

Miles, J. W., "On the Generation of Surface Waves by Shear Flow,
Part 4," J. Fluid Mech., 13, 433-477, 1962.

Miles, J. W., "On the Generation of Surface Waves by Shear Flow,
Part 5," J. Fluid Mech., 30, 163-175, 1967.

Mollo-Christensen, E., "Observations and Speculations on Mechanisms
of Wave Generation by Wind," Dept. of Meteorology, MIT,
1970.

Motzfeld, H., "Die Turbulent Stromung an Welligen Wanden,"
£4. Angew. Math. Mech, ; 17, 193-212, 1937.

Ott, R., Hsu, E. Y. and Street, R. L., "A Steady-State Simulation
of Small Amplitude Wind-Generated Waves," Stanford Univ.
Dept. of Civil Engineering Tech. Rep. No. 94, 1968.

Phillips, O. M., "Wave Generation by Turbulent Wind Over a Finite
Fetch," Proc. 3rd Natl. Congr. Appl. Mech., pp. 785-789,
1958.

Phillips, O. M., The Dynamics of the Upper Ocean, Cambridge
Univ. Press, New York, .

Stanton, T. E., Marshall, D., and Houghton, R., "The Growth of
Waves on Water Due to the Action of Wind," Proc. Roy. Soc.,
Ser. A., MSs Pp. 283-293, 1932.

Sutherland, A. S., "Spectral Measurements and Growth Rates of

Wind-Generated Water Waves," Stanford Univ. Dept. of
Civil Engineering Tech. Rep. No. 84, 1967.

35

Hsu and Yu

Thijsse, J. T., "Growth of Wind-Generated Waves and Energy
Transfer," National Bureau of Standards, Washington, D.C.,
Circular No. 512, 281-287, 1951.

Ursell, F., "Wave Generation by Wind," Survey in Mechanics,
Cambridge Univ. Press, 1956.

Zagustin, K., Hsu, E. Y., Street, R. L., "Turbulent Flow Over
Moving Boundary," J. of the Waterways and Harbor Div.,
Proc. ASCE, 397-414, 1968,

Zagustin, K., Hsu, E. Y., Street, R. L., and Perry, B.; "Flow
over a Moving Boundary in Relation to Wind-Generated
Waves," Stanford Univ. Dept. of Civil Engineering Tech.
Rep. No. 60, 1966.

36

AIR-SEA INTERACTIONS: RESEARCH PROGRAM
AND FACILITIES AT IMST

M. Coantic and A. Favre
IMST
Marseille, France

ABSTRACT

This research concerns the small-scale physical pro-
cesses responsible for mass, momentum and energy
exchanges between the atmospheric surface layer and
the oceans.

Their theoretical study has been undertaken. It outlines
the importance of turbulence and the influence of recip-
rocal interactions between the various transfer pro-
cesses. It has led to the design of an experiment where
the natural phenomena shall be partially simulated, ina
large laboratory facility.

This one combines a micrometeorological wind tunnel
with a 40 meters long wave tank, under controlled tem-
perature and humidity conditions. It has been extensively
tested with a one-fifth scale model. It is presently under
construction, and wiil be operative by 1971.

Instrumental studies have also been undertaken, and
results obtained in the measurement of turbulence in
water flows.

I. INTRODUCTION

The knowledge of energy exchange processes between atmos-
phere and oceans appears of major interest for oceanography as well
as for meteorology. These two media have indeed to be considered
as elements of a single system, for the dynamical and thermodynam-
ical evolution of each of them largely depends on interactions through
their common boundary.

37

Favre and Coanttie

One of the essential steps in the solution of the air-sea inter-
action problem lies in the understanding of small-scale processes
in the air and water layers adjacent to the interface, where the
various forms of energy are either transferred or converted, while
going from one medium to the other. The experimental study of
these phenomena involves a detailed and delicate exploration ofa
region whose thickness is of the order of the wave height. Now,
experiments performed at sea are subjected to such environmental
constraints that the accuracy and repeatability of measurements
seems necessarily limited. It has therefore appeared useful to
complement field studies by laboratory experiments, where an ex-
tensive investigation is feasible under exactly repeatable conditions
and with the possibility to control independently each of the govern-
ing parameters.

This is the program which has been undertaken atI.M.S.T.,
and which is described in the present paper. The preliminary steps
of this program have included: collection of information about cur-
rent research; attempt of a critical survey of existing knowledge,
in order to find out definite research objectives; and a first theoreti-
cal study of the physical mechanisms of air-sea interactions, and
of their governing parameters. These studies have led to the con-
clusion that it would be feasible to obtain, in the laboratory, a partial
simulation of the atmospheric-oceanic energy exchange processes,
provided that a sufficiently large facility could be realized.

The following steps of the program have then comprised:
the preliminary design of this facility, combining a micrometeorolo-
gical wind tunnel with a 40 meters long wave tank; the realization of
a one-fifth scale model, and its use for various preliminary tests
and experiments; the detailed design and the building of the large
wind-wave facility; and, last but not least, the development of various
theoretical and instrumental researches.

The purpose of the present paper is to introduce the various
objectives and results of our research program, and to describe the
facilities which have been, or are being, realized. Due to space
limitation, that presentation will be limited to a rather short account
referring to previous publications for more details, when possible.
The plan adopted is logical rather than chronological:

- Theoretical studies;

- Setting up the characteristics and design of the large air-

sea facility;

- Model tests;

- Building of the air-sea facility;

- Studies of measuring instruments and methods.

At last, we shall try to draw some preliminary conclusions
about this program, the prospects it opens, and its possible appli-
cations. We shall also have the pleasure to express our thanks to
the many individuals and organizations who have contributed to its
realization.

38

Atr-Sea Interactions; Program at IMST

Il. THEORETICAL STUDIES

1. The Physical Mechanisms of the Ocean-Atmosphere Interaction

As it is well known, the small-scale transfer of energy be-
tween atmosphere and oceans occurs following four various mecha-
nisms, sketched by Fig. 1:

a) Radiation, including: i) short-wave radiation from the sun on the
sea surface, which is partially reflected and absorbed over a
more or less large depth under the interface; ii) long-wave radi-
ation coming from the atmosphere and from the sea, and involv-
ing a radiative transfer process between the interface itself and
the atmospheric layers (see II.3).

b) Evaporation (or condensation), and turbulent convection of water
vapor, which, due to the very high latent heat of vaporization of
water, leads to a turbulent latent enthalpy transfer from the sea
surface to the atmosphere.

c) Turbulent convection of sensible enthalpy, resulting from tem-
perature differences between adjoining points of the system.

RADIATION TURBULENT ENTHALPY KINETIC ENERGY
TRANSFER TRANSFER
Incident Reflected

Z WY] j Yj
j ay YY YY YY YY Yyy / Yy YY YY

: : ; ' Gaseous phase
4 4. L S Interface
Q $

Liquid phase
Zz

__ RESULTING ENERGY
TRANSFER = S+L+0:S'+Q'

Short wave Long wave

Received § Emitted
Sensible

Fig. 1. Schematic display of energy transfers in the vicinity of the
ocean-atmosphere interface.

39

Favre and Coantie

d) Transfer of kinetic energy across the turbulent boundary layers
on both sides of the interface, of which the most obvious effect
is the generation of waves.

Information on these mechanisms can be gathered in many books,
ranging from meteorology (e.g. Brunt [ 1939], Haltiner and Martin
[1967] , Roll [1965]) to oceanography (e.g. Lacombe [ 1965],

Phillips [1966], Sverdrup [1957]), or devoted to atmospheric tur-
bulence (e.g. Lumley and Panofsky [1964], Monin and Yaglom [1966],
Priestley [1959]). One of the first steps of our program has been to
attempt to review the physical laws and equations governing the
ensemble of these phenomena (see Coantic [ 1968]).

The main conclusion that can be reached is that, although the
above types of transfer have been analyzed and listed separately,
they are not independent, and the key of the problem lies in their
reciprocal interactions. For instance, processes a) and b) set in
action very large amounts of energy, whereas c) and chiefly d) are
responsible for much smaller exchanges. However, the kinetic
energy transfer, which enters as the smallest term in the energy
balance, strongly influences the turbulent evaporation and convection
processes. Infact, except for certain radiation effects, air-sea
interactions are essentially governed by turbulence.

This is only one aspect of the aforementioned reciprocal inter-
actions. Other ones will appear, for instance, when considering the
boundary conditions for the various variables at the interface, or
when expressing the conservation of the different energy fluxes, as
schematized on the lower part of Fig. 1. Furthermore, two most
important peculiarities are displayed when comparing the present
case to the more classical problem of simultaneous heat, mass and
momentum exchange between a fluid flow and a more or less rough
surface. In the latter case, the turbulent convective processes,
if not completely understood, are sufficiently well known to allow a
good estimate of the various transfer rates. However, the methods
of computation therein developed are not applicable here for two
main reasons:

- On one hand, because the boundary is no longer static, and pos-
sesses "dynamic rugosities" capable of yielding and absorbing
momentum with large variations of the ratio of the tangential shear
stresses to the normal pressure forces. This fact will have conse-
quences difficult to ascertain, not only upon the dynamical exchange
mechanism but also upon the degree of "Reynolds analogy" be-
tween this process and those concerning exchange of scalar vari-
ables.

On the other hand, because, due to the well known stratification
effects in the atmosphere, heat and humidity can no longer be con-
sidered as "passive scalar containments." This means that the
turbulent structure of the boundary layer, and the transfer rates
themselves, are strongly modified by the direction and intensity
of the vertical heat and humidity gradients.

40

Atr-Sea Interacttons; Program at IMST

As discussed in our previous publications (Coantic [ 1968], Coantic

et al. [1969]), and in Part III of the present paper, the preceeding
considerations have been the basis for the settling of our research
program and the design of our simulation facility. Some aspects of
the problem are already being the subject of theoretical investigations,
which we shall now mention shortly.

2. Wave and Current Generation by Wind

The transfer of mechanical energy from air to sea has two
main consequences: the development of currents and turbulence in
the upper ocean, and the generation and amplification of waves. This
latter process can be broadly described as follows: the turbulent
atmospheric boundary layer exerts on the water surface normal and
tangential stresses, with steady, periodic and random components.
As a consequence of these stresses and of the gravity and capillarity
restoring forces, motions of a wavy character are generated at the
interface. As soon as their amplitude becomes appreciable, non-
linear effects are developed, which result in a modification of the
airflow structure and, hence, of the applied stresses, the existence
of a continuous wave spectrum and the production of turbulent energy
in the sea. The wave amplitude is then limited by the dissipative
action of turbulence and viscosity.

As mentioned earlier, the understanding of this complex
mechanism is essential to elucidate, not only the dynamical, but
also the thermodynamical aspects of air-sea interactions. A careful
study has, accordingly, been undertaken (Ramamonjiarisoa [ 1969,
1970]), first of existing theories (based on models proposed by
Miles and Phillips) and later on of more recent developments in the
researches of Stewart, Mollo-Christensen, Longuet-Higgins,
Hasselmann and Reynolds, among others. This helped us in identify-
ing some points that have to be subjected to experimental study,
namely: the existence of separation after the wave crests, the phase
shift between surface pressure and elevation, the spatial and temporal
variations of Reynolds stresses, and of the turbulent structure of the
flow in general. Our future measurement program has been estab-
lished in consequence, taking advantage of the possible use of the
space-time correlation technique, and of numerical data processing
methods to separate the "mean," the "phase average" and the "tur-
bulent" parts of each variable.

3. Interaction of Turbulent and Radiative Transfers

Another typical example of reciprocal interactions between
the various modes of energy transfer near the air-sea interface is
the simultaneous transport of sensible enthalpy by turbulent con-
vection, and by infrared radiation. The turbulent heat flux is
usually assumed constant with height in the atmospheric surface
layer. However, the validity of this hypothesis is known to be
questionable, due to a possible vertical variation of the infrared

41

Favre and Coantte

radiative flux (see e.g. Munn [ 1967]).

This problem has been approached theoretically, using semi-
empirical expressions fitted to the emissivity curves, and assuming
logarithmic temperature and humidity profiles. A first approxima-
tion of the radiative heat flux divergence is thus obtained analytically,
as a function of the surface layer parameters (Coantic and Seguin
[1970]). Numerical values of the infrared flux gradient, dq,/dz,
in the first ten meters of the marine atmosphere are shown in Fig. 2,
for two different wind velocities (A: Uj) =3 m/s; B: Ujo = 9 m/s);
two sea surface temperatures (cases 1,2: 8, = +5 CG; cases 3,4:

8, = + 20 °C); and two temperature differences (cases 1,3: 0,9 - 89 =
- 5 °C; cases 2,4: O19 - 0g = +1 °C). The resulting vertical vari-
ations of the turbulent heat flux, shown by Fig. 3, are seen to attain
unexpectedly large values, of the order of 30 to 40%, when wind
velocity is low and humidity is high.

These results are considered as preliminary. If confirmed,
they could lead to a reinterpretation of some experimental data,
and should appeal to an extension of turbulent transfer theories to
the case of a variable heat flux.

-08 -04 (a) : +04 a mw/em¥m

Fig. 2. Computed vertical variations of radiative flux divergence
for various atmospheric situations

42

Atr-Sea Interacttons; Program at IMST

d~)

WV Za |

V SS,
OQ 6./ Sy)
O Qi Q2 Q3 04

Fig, 3. *Relative vertical variation of the turbulent heat flux, for
various atmospheric situations

4. Water Vapor Turbulence and Its Measurement

The turbulent transfer of humidity in the lowest atmospheric
layers is, as mentioned earlier, one of the principal mechanisms
for the exchange of energy between air and sea. In addition, this
process governs the mean distribution and turbulent structure of
specific humidity in the lower levels, and thus exerts an essential
influence on electromagnetic wave propagation. Therefore, the
contemplated studies require the measurement of humidity fluctu-
ations, whose levels and scales have to be estimated to delineate
suitable measuring devices.

This estimation has been obtained by: a) studying the equa-
tions governing the mean distributions, turbulent fluxes and levels
of fluctuations of humidity b) examining the known experimental
data; and c) predicting the form of the spectrum from the Kolmogorov-
Obukhov theory (Coantic and Leducq [1969]). Figure 4 compares the
predicted spectral behavior of turbulent humidity fluctuations (after
some shift towards lower frequencies), and recent measurements by
Miyake and McBean [1970]. Considering the experimental under-
estimation of the high frequency part of the spectrum, the overall
agreement is not too bad.

Once the estimate is made, it is then possible to define

specifications for devices measuring humidity turbulence, for use
either in the field or in the laboratory (see VI. 3).

43

Favre and Coantte

LOG(FX SPECT)

© DEW POINT
@ LYMAN ALPHA

ry
©°, LOG(FZ/U)

-$0 ~20! -90-x0,0 4 4014 [20

Fig. 4. Comparison between predicted spectral behavior of turbu-

lent humidity fluctuations, and measurements by Miyake
and Mc Bean [ 1970]

5. Two-Phase Processes in the Vicinity of Air-Sea Interface

The equations governing the mean properties of air-sea inter-
actions are usually written in an earth fixed Eulerian frame of refer-
ence, and different sets of equations have to be used in the gaseous
and in the liquid phase. Due to the unsteady random character of the
interface, this means that an appreciable part of the system has,
strictly speaking, to be treated as a two-phase flow.

If one wants to take into account the obviously important
effects of sea spray in the lower atmosphere, and of air bubbles in
the upper ocean, the necessity of considering two-phase effects is
still more clear. Prompted by chemical and nuclear engineering
problems, notable progress has been gained these last years in the
analytical and empirical description of such processes. We plan to
apply the methods therein developed to the study of the two-phase
portion of the ocean-atmosphere system.

Ill. STUDY OF CHARACTERISTICS AND DESIGN OF THE SIMU-
LATING FACILITY

14. Conditions for Modelling Small-Scale Air-Sea Interactions
In consequence of the physical mechanisms of air-sea energy

exchanges, the planned laboratory experiments will concern the
structure of turbulent velocity, temperature and humidity boundary

44

Air-Sea Interacttons; Program at IMST

layers obtained at the interface between an airflow and a water mass.
The three basic processes of momentum, heat and mass transfer will
be effectively realized by controlling air velocity, temperature and
humidity, and water velocity and temperature. Furthermore,
appropriate heating or cooling will provide an approximate repre-
sentation of the most important radiation effects.

However, such experiments will be really useful in modelling
the atmospheric-oceanic phenomenon, only if the three aforementioned
specific features: turbulent atmospheric structure, stratification
effects, and interface motion, are at least partially reproduced.

This seems feasible, provided that a sufficiently large facility can
be realized.

2. Simulation of the Atmospheric Dynamical Structure

It is well known that the atmospheric surface layer motions
can be simulated in the laboratory, in so-called "micrometeorologi-
cal wind-tunnels" (see e.g. Pocock [ 1960], Cermak et al. [ 1966],
McVehil et al. [1967], Mery [1968]). In short, these motions are
characterized, on one hand by extremely high values of Reynolds
number, and on the other hand, by stratification effects corresponding
to appreciable values of Richardson number. For a good modelling,
these dimensionless numbers have to keep significant values in the
laboratory flow. For the latter, this implies rather large tempera-
ture differences, and low wind velocities. In consequence, to pre-
serve sufficiently high Reynolds numbers while observing cumulative
stratification effects, it is necessary to build large facilities. Simi-
lar conclusions are reached if one considers the problem of main-
taining the ratio between the roughness height at the surface and the
boundary layer thickness or the Monin-Obukhov length, or if one
requires the reproduction of an appreciable Kolmogorov inertial
range.

For these reasons, the test section length of micrometeoro-
logical wind tunnels reaches several tens of meters and the velocity
range is of the order of a few meters per second, while provision
is made for creating temperature differences of several tens of
degrees centigrade. The main characteristics of our project are as
follows:

- Length of the water surface forming the interface in the test
section: 40 meters.

- Air velocity range: 0.5 to 14 meters per second.

- Maximum temperature and specific humidity differences: 30 °C,
and 25.10°° Kg water by Kg air.

The estimated performance of the facility is sketched by Fig. 5,
which shows the rather wide range of dimensionless parameters that
should be covered. The general scheme of the tunnel is given in
Fig. 6. It is a closed-circuit wind-tunnel, with several rather

45

Favre and Coantie

unusual dispositions, dictated by specific requirements. For
instance, to obtain stable functioning at the lowest velocities, the
return circuit's area has been purposely reduced; the total head

loss has been increased by tightly finned heat exchangers acting as
flow equalizers just upstream the settling chamber; and the diffusors
have been fitted with vortex generators and stabilizing vanes. The
test section's area is 3.20 by 1.45 meter, and the overall size of the
facility 61 by 7.50 meters. The wind velocity can be continuously
varied from 0.5 to 14 meters per second, with a relative accuracy
of 2.10°°, using a helicoidal fan driven by a variable speed D.C.
motor with electronic regulation.

ENE Ae

‘el a
FAS SRE

0 10 20 30 40 Xm oy 0 500 i000 1500 ~~ 2000
=—=——{) = "ims
SESSA U = 4mis (a) ( b )

a =: 20 30 40 Xm
Cd)

Fig. 5. Estimated performance of the facility: a) Reynolds number,
and boundary layer thickness; b) Sensible and latent en-
thalpy fluxes; c) Wind-Waves' age and significant héight;

d) Richardson number.

46

Air-Sea Interactions; Program at IMST

D, SECTION A

FPP ERTTOOOT TTT TOTS TTT
Parmer eae SP se eC Lr, SE ie ee Nn gl, eee)
oO—L-O

SECTION B

Fig. 6. General scheme of the wind-water tunnel

3. Reproduction of Heat and Mass Transfer Processes

Supposing a convenient flow structure has been obtained, the
existence of nonzero temperature and partial water vapor pressure
differences between the water surface and the incoming air flow will
be sufficient to cause turbulent convective processes of mass and
sensible and latent enthalpy similar to those encountered in the
atmospheric boundary layer.

The equations governing these transfers being linear with
respect to temperature and humidity, these last variables can be
fixed on grounds of experimental convenience, as long as stratifi-
cation effects do not arise. The estimated values of flux Richardson
number, computed at one-quarter boundary layer thickness and for
a temperture difference amounting to 25 °C, are displayed in Fig.
2(d) as a function of longitudinal abscissa and velocity. At the
highest velocities, Richardson number is clearly negligible, and
temperature and humidity differences will be chosen, to improve
experimental accuracy, at the highest levels authorized by the
equipment's capabilities (see Fig. 2(b), and below). On the other
hand, at the lowest velocities, temperature and humidity can no
longer be considered as scalar passive contaminants, and their
differences will be chosen in order to obtain a given Richardson
number, i.e. a given effect on the dynamical structure.

47

Favre and Coantte

A first approximation representation of radiative beat ex-
changes seems also to be feasible in the laboratory. At the small
scale we are interested in, the main effect of short wave solar radi-
ation is a global elevation of the oceanic temperature, that can be
reproduced by heating the water mass. Due to the radiative trans-
fer process mentioned in II.3, the reproduction of infrared heat
exchange is more delicate, but its primary effect yet remains the
cooling (or occasionally heating) of the interface itself. This
localized heat sink shall be simulated, either by increasing the
cooling produced at the same place by evaporation, or by lowering
the temperature of the ceiling of the test section, and thus con-
trolling the radiative heat exchange between this wall and the water
surface.

The designed facility will allow independent control of air
and water temperatures in the 5 °C - 35 °C range, with an accuracy
of the order of 0.1 °C. The relative humidity of air entering the test
section will be varied from 60% to 100%. Fig. 7 schematizes the
main components of temperature and humidity control system: cooling
and drying (by condensing) coils, heating coils and vapor injectors
in the air circuit; cooling and heating heat exchangers in the water
circuit; heat generator, frigorific unit with cooling tower, steam
boiler, regulating system. The working principle is represented
using the temperature-mixing ratio diagram.

=e

HEATER 160000 cu

HEATER

90000 cal /b

at
if

Fig. 7. Schematic diagram of temperature and humidity control
systems

48

Atr-Sea Interactions; Program at IMST

4. Reproduction of Interfacial Motions

The problem of obtaining laboratory waves statistically
similar to those encountered over the oceans has been thoroughly
studied these last years, with the view, either to perform more
realistic structural tests, or to experimentally investigate the
mechanism of wave generation by wind. The works of Veras [| 1963],
Hidy and Plate [ 1965], Hsu [1965], Gupta [1966], and of the
Waterloopkundig Laboratorium [ 1966a,b] can be cited among many
others. The unsymmetrical, randomly varying and three-dimen-
sional waves existing in nature can be simulated only at the cost of
building large laboratory facilities. The so-called "wind-wave
tunnels" reach one hundred meters in length and several meters in
width, with smooth and parallel side walls and an efficient absorbing
beach at the end.

The main characteristics of waves naturally generated by
wind along our 40 meters long tank have been forecast from the
preceeding references and are shown by Fig. 2(c). It is clearly
possible to generate gravity waves of appreciable amplitude, and
thus to cover a nonnegligible range of Froude numbers. However,
the "wave age," i.e. the ratio of the celerity of propagation, C,
of dominant waves to the wind velocity, U, remains low, especially
at the highest velocities. The same is true of the ratio C/U~ (where
U+ is the friction velocity in the boundary layer), which is known
as an important parameter in the wave generation process. Asa
matter of fact, these two ratios control the relative magnitude of
normal and tangential stresses exerted by wind on water, with im-
portant consequences upon the various energy exchange mechanisms
(see II.1). It is therefore necessary to have the possibility to act
on these parameters, by controlling the wave height and celerity
independently of wind velocity. This will be done by means ofa
wavemaker set at the beginning of the water channel and conveniently
randomly actuated. It is known that the combined action of sucha
device and of wind blowing will result, after some distance, ina
satisfying wave pattern.

The details of that part of the equipment are sketched by
Fig. 8. A new type of wavemaker, comprising a fully submerged
wave plate connected to the tank by means of bellows, has been
imagined. This arrangement allows to realize a fairly smooth
joining of air and water flows, even in the presence of waves. The
end of the channel will be equipped with an absorbing beach made
from parallel tubes with a 7° slope. A slight water movement
(between 0.1 and 0.01 m/s) necessary for cleaning and temperature
controlling purposes, will be insured by«a recirculating 35 HP heli-
coidal pump.

*
By A. Ramamonjiarisoa.

49

Favre and Coantte

SMOOTH
AIR JOINING

CONTRACTION
KOO OL eT

eke) — 2: eC ar
NOES Va a PAE TNy, gt SO PAU ACS
xr $3,0¢ Waistes =f IE 2
a ASS 8

WATER

aie

Se ogo
BS s

Fig. 8. Details of junction between air and water flows, and arrange-
ment of the submerged wavemaker.

5. Further Details and Conclusions

At each step in the design of the facility, we endeavored to
improve the simulation of the natural phenomenon and some of the
arrangements taken to this end have just been described. A peculiar
problem was set by the parasitic boundary layers which unavoidably
originate along the side walls and ceiling of an elongated working
section, and which are known to result in cumbersome secondary
motions. The dispositions adopted to reduce these effects, thereby
improving the representation of the unlimited atmospheric-oceanic
system, are represented by Fig. 9. The cross sectional shape of
the working section (see Fig. 9(a)) has been designed with a height/
width ratio of 1 to 2.2, and furthermore fitted, like in the Water-
loopkundig Laboratorium [1966a,b] design, with vertical plates
restricting the span of the useful water surface to 2.62 meters.

The lateral quays thus realized will limit the parasitic dynamical as
well as thermodynamical effects in the central part of the working
section, where the measurements will be performed. To further
improve the two-dimensionality of the flow, and to prevent the inter-
action of the studied boundary layer with that one developing on the
working section's ceiling, boundary layer control devices will be
used. As shown by Fig. 9 (b), they combine boundary layer suction
(by means of slots or porous walls) and blowing (through slots),
taking advantage of the possibilities of tangential blowers.

At last, it will be necessary, specially at the lowest wind
velocities, to artificially trigger transition, and eventually to in-
crease the boundary layer thickness, by means of devices similar to
those studied by Counihan [1969] or Campbell and Staden [ 1969].

50

Atr-Sea Interacttons; Program at IMST

Suction through porous wall

Suction and blowing
by slots

(b)

Fig. 9. Details of test section: a) Section view, showing the lateral
quays; b) Boundary layer control devices.

All the foregoing will make clear that we have tried to insure
an acceptable simulation of the main aspects of air-sea interactions.
Entire modelling, with full similarity, cannot, of course, be attained.
We believe, however, that experiments where the various physical
mechanisms are effectively put in action, and where basic parameters
possess significant values, will realize a partial simulation of natural
exchanges, thus affording the possibility of interesting investigations.

51

Favre and Coantte

IV. PRELIMINARY ONE-FIFTH SCALE MODEL TESTS

A one-fifth scale model of the large air-sea interaction
facility has been built. The primary object was to check and im-
prove various design characteristics; altogether it was also planned
to perform instrumental studies, and to execute preliminary small-
scale scientific experiments.

This scale model is a detailed reproduction of all parts of
the large facility, including not only the aerodynamic and hydraulic
elements, but also the equipment controlling heat and humidity ex-
changes. A view of the wind-water tunnel, and of the control console,
is given in Fig. 10.

Fig. 10. General picture of the model

52

Atr-Sea Interactions; Program at IMST

1. Overall Aerodynamic Tests

The first use of the model has been to test the global aero-
dynamic performance of the facility. The initial design, represented
by the upper part of Fig. 11, suffered from several imperfections
resulting in a low power factor and in an inadequate working stability.
Detailed flow explorations have led to successive amendments in the
geometry of the model (see Pouchain [1970] and Coantic et al. [1969]).
The final design, shown in the lower part of Fig. 11, offers satis-
factory performance, and has therefore been adopted in the later
building of the large facility, the aerodynamic characteristics of
which have been predicted from the model tests.

SECTION A

SECTION B,C

INITIAL
SECTION D

ee — hea

SECTION E,F

0 In FINAL

Fig. 11. Improvements in aerodynamic design of the model

2. Flow Exploration and Improvement in the Working Section

A deeper flow study of the working section has then been per-
formed, of which typical results, obtained for a wind velocity of
8.3 m/s, are displayed by Fig. 12. It can be seen that the situation
is good in the entrance section, with a very flat velocity profile,
and a turbulence intensity below 0.002 (the effects that can be dis-
cerned near the water surface are the consequence of artificial
boundary layer thickening in the final part of the contraction). The
further growth of the various boundary layers, and the fact that they

53

Favre and Coantte

©1=280m

° 4 e
121 Um/s X= 160 mm X=2800mm X=5600 mm

ae | ein

ee e—e— is e—e—2— o— 0 —0 —-0 —0— 0— O— 9 __

| | lyr Oss
Lam dL ~L—h—A—A DD hb — a—a— b—D—A—A
OO 0 OO Ome a OOO OO 0 0 OO Om XN =

- 300 - 200 - 100 0 100 200 300

Fig. 12. Flow characteristics in the test section: a) Vertical
velocity profiles; b) Horizontal velocity profiles;
c) Vertical distributions of turbulence intensity.

54

Atr-Sea Interacttons; Program at IMST

begin to join together towards the end of the working section, leading
to a fully developed channel flow, are also apparent.

As already mentioned (see III.5), it is therefore necessary to
take steps to restrict the development of the lateral and upper bound-
ary layers. Tests performed under different conditions have proven
that the contemplated control method was efficient in this respect
(Pouchain [1970]). Fig. 13 illustrates typical results obtained for
various blowing rates (i.e. ratios of jet velocity to mean flow veloc-
ity), while sucking through a porous wall and blowing across a 15
millimeters height slot. The improvement in velocity distribution
is clear. As shown by Fig. 14, the turbulent intensity distribution is
also ameliorated. Some problems related to pressure perturbations
still have to be solved, but, on the whole, the method appears as
promising.

No blowing ese
With blowing Uj /U, = 105 ------ -

Fig. 13. Improvement in test section's flow by means of boundary
layer control: a) Flow configurations for three rates of
blowing; b) Variation of boundary layers! thickness.

55:

Favre and Coanttie

X=5750 X=157 =0

e= 15mm

Fig. 14. Effects of blowing on turbulence intensity distributions.

he Hydraulic Tests

The hydraulic performance of the facility has also been sub-
jected to various tests. The functioning of the water recirculating
circuit has been controlled. The working of the new submerged
wavemaker, and of the absorbing beach, has also been found satis-
factory.

Observations of waves generated by wind in the model tank,
such as those shown by Fig. 15, suggests they qualitatively possess
the three-dimensional random structure typical of oceanic wind
waves. Measurements of wave spectrum at different fetches along
the working section have just been done, and the results displayed
in Fig. 16 compare favorably with those of previous studies (see
the references in III.4). The spectral shape and evolution strongly
suggests the existence of nonlinear effects transferring energy from
higher to smaller frequencies, as recently postulated by Longuet-
Higgins and Mollo-Christensen.

56

Aitr-Sea Interactions; Program at IMST

Fig. 15. Sample view of wind-waves obtained for a 4 m/s velocity
and a 4 m fetch.

Fetch

e F=840 mm
Vv F=2340 mm
° F=3840 mm
°o F=5340 mm
5 F=6840 mm

10°

10°

Fig. 16. Evolution of wind-waves' spectra as a function of fetch
along the model's test section.

Mm
~]

Favre and Coantie

4. Tests of Temperature and Humidity Control Systems

Various working tests of that part of the equipment have been
executed. The validity of the previously chosen control methods
has been checked, the obtainable temperature and humidity range
has been controlled, and the stability of regulating loops has been
tested. After some improvements, the overall thermodynamic per-
formance of the model has been correct.

A further study of temperature repartitions upstream and
inside the working section has then been undertaken, for different
flow and thermal conditions. Typical results are shown by Fig. 17,
where an initially isothermal airflow, and the development of thermal
boundary layers can be observed. The temperature distribution in
the entrance section is usually good; except in extreme cases of
large heating and velocities of the order of one meter per second,
where parasitic stratification effects appear.

U.=260ms 8,=52°C — Twnt27 °C ee
papers eras ues PLIPIIG We Att / f yyy / , mae ++ | 300
200
X=0
100
seer = A
30 20 10 Cc
30 20 10 0 “
eee)
30 20 10 0 8-8, C
ee)
30 20 10 0

Fig. 17. Temperature Distribution in the Model's Test Section

V. CONSTRUCTION OF THE LARGE AIR-SEA INTERACTION
FACILITY

In view of the rather considerable size of the designed wind-
wave tunnel, its erection was not possible inside I.M.S.T. 's main
building. It was therefore decided to build a new laboratory, includ-
ing the large air-sea facility, its auxiliary equipments and a group
of offices, workshops and laboratories, and located in the new
Marseille-Luminy Campus. Its floor plan is shown by Fig. 18.

58

Air-Sea Interacttons; Program at IMST

Echelle (Ale Eee Metres

Seufflerie SE 16
a

a a |

Fig. 18. Air~Sea Interactions Laboratory Floor Plan

The preliminary design of the facility has been determined by
I.M.S.T., and its detail drawings set up with the aid of architect
and engineering offices. The construction works have been planned
in three stages: a) Erection of buildings, concrete structural parts
of the tunnel, electric equipments, temperature and humidity con-
trol systems; b) Fitting up of the main elements of air and water
circuits, including static parts (tunnel walls etc.) as well as pump
and fan; c) Completion and equipment of the facility, placing control
apparatus and such parts as wave-maker and boundary layer control
devices in position.

Works have been started in January 1969, and step a) is
fully completed from several months (see Fig. 19). Step b) is now
nearly achieved, and the first run of the wind-water tunnel is planned
for the end of the present year.” Our program forecasts about one
more year for the execution of step c), and the beginning of strictly
scientific experiments by the end of 1971. These experiments will
concern: first the dynamic exchange process alone; then, the heat
and mass transfer processes; and later on the effects of stratifica-
tion upon these three mechanisms. The execution of this program
will clearly take several years.

VI. RESEARCHES RELATED TO INSTRUMENTATION PROBLEMS

Some of the anticipated experiments obviously necessitate,
either the development of new measuring instruments and methods,
or the adaptation of existing ones. Corresponding researches have
been undertaken since the early stages of our program.

¥
Note added in proof: this has been achieved by November 1970.

59

Favre and Coantie

a
o Bees
ae
a'e's's|

ae

ae
ee@dexs: “te 8

‘ ’
SKRSSSSSREOHTS oR,

1
>
ae
:
e ‘
~ >

Fig. 19. Constructions! progress by May 1970: view of refrigerating
coils, and of concrete contraction and test section.

1. Turbulence Measurements in Water Flows

A first work has been devoted to the measurement of velocity
and temperature mean and fluctuating values in water flows, and of
the associated turbulent momentum and heat fluxes. The adaptation
to this problem of the well known hot-wire technique has been studied
experimentally. An apparatus including a tubular water channel was
constructed to that end, and hot-wire sensors were manufactured.

The theoretical and experimental study of the performance of various
types of wires and films has resolved satisfactory methods of cali-
bration and measurement, particularly for commercially manufactured
conical hot films for which dimensionless cooling laws have been pro-
posed, and for slanting wedge-shaped films. The intensity and the
spectrum of turbulence, and the Reynolds stresses themselves, have
been determined inside a circular conduit, with a comparable degree
of accuracy to that attainable in air flows (see Fig. 20). Later on,

the effects of water temperature variations upon the hot-film response

60

Atr-Sea Interacttons; Program at IMST

|
io-8- COANTIC AIR
© D=- 765mm
fe) D= 257,8mm
10°’-
RESCH WATER

@ D- 44 mm

Fig. 20. Turbulence measurements in water flows: a) Comparison
of turbulence spectra measured in dynamically similar
air and water flows.

61

Favre and Coantte

e LAUFER © MARTIN
@ BALOWIN © MICKELSEN COA NTIC AIR
© SANOBORN Temperature cste © WEISSBERG ° ei ay" => mm
© SANDBORN Intensite cste © ASHKENAS » D= 76mm Gg D=258
@ NEWMANN et LEARY @ GAVIGLIO RESCH WA TER
e

o Re 39000
« Re 83000

— Computed

Fig. 20. Turbulence measurements in water flows: b) Comparison
of turbulence intensities measured in dynamically
similar air and water flows; c) accuracy checking
of measured shear stresses.

62

Atr-Sea Interacttons; Program at IMST

have been thoroughly studied (see Fig. 21), and measurements of the
intensity of temperature turbulence have been executed.

These results are given in a number of publications: Resch
[ 1968, 1970], Resch and Coantic [1969], Ezraty and Coantic [1970],
Ezraty [1970]. They show conclusively that, subject to some pre-
cautions, turbulence measurements can be quite accurately per-
formed in water flows, using hot-film anemothemometers.

2. Measurements of Turbulent Fluctuations of Humidity

After the theoretical study reported in II.4, the development
of a water vapor turbulence measuring technique has been undertaken.
Various methods have been considered: psychrometry, dew-point
measurement, use of hot-wire, absorption of Lyman alpha or infrared

No eG
os, Sg
a A ©, -17
34 NuzNu-=> -B (a Tape” ae
Ke !
— git 4
gf 059<V.<33
- gf + é py
| Oo c'O,
B --62.69 ar ae , 795 = 43,71
ir 39,28
30 eee? <a 36,57
30,64
22,50 7,93
4371
39,28
34,57
25,95 51,86
47,93
43,71
39,28
3175 5736

51,86
47,93

L

32

22 co

28 ae
26 a ee

26 nee

ec @«eepee7 & ¢— @ ®& 46 ¥ 6

4371

Fig. 21. Dimensionless Cooling Law for a Conical Hot-Film.

63

Favre and Coantie

radiation, measurement of refractive index, use of the ionic probe.
These techniques which depend on the thermodynamic properties of
moist air present advantages from the point of view of miniaturi-
zation, whereas those based on electromagnetic properties are
advantageous from the point of view of bandwidth. Psychrometry
and Lyman alpha absorption seem to be the most promising ones,
the latter appearing to have the best chance for adaptation to parti-
cularly difficult measuring conditions (Coantic and Leducq [ 1969]).

For practical reasons, psychrometry has been chosen for a
first deeper investigation. A small calibrating tunnel has been built
and various types of miniaturized small time constant psychrometers
have been manufactured and tested. A prototype psychrometer is
used with wet and dry balsa fibers, of which surface temperature is
measured by platinum resistors (5 microns in diameter), the result-
ing electrical signals being fed into a small analogue computer,
whose output is directly proportional to specific humidity fluctuations.
The physical size of the sensing head is a few millimeters and the
bandwidth several cycles per second (Leducq[1970]). These results
are only preliminary, and further studies are now being undertaken.

3. Data Proces sing

Considering on one hand the volume of measurements to be
taken, and on the other hand the necessity, mentioned in II.2, to
separate any variable in its "mean," "phase average" and "turbu-
lent" parts, the use of digital data acquisition and processing methods
seem unavoidable.

Preliminary studies have been done of a small data acquisition
system for continuous digitizing and numerical tape recording of
turbulent variables. The recorded data could be later, pre-pro-
cessed on the system itself and eventually transferred to a large
computer for a comprehensive treatment. This data acquisition
system could also be useful for control process of the tunnel and of
the measuring equipments, thereby largely increasing the efficiency
of the facility.

VII. CONCLUSIONS AND PROSPECTS

The research program which has just been introduced is
obviously a long-term one. It is therefore too soon to state any
definitive conclusion, and we can only try to survey some preliminary
results, and to think about the probable prospects of the research
we have undertaken.

The main result of the work performed till now is the detailed
definition of a scientific experiment which, although done in the
laboratory, seems capable to give results applicable to the natural
processes occurring near the ocean-atmosphere interface. The

64

Atr-Sea Interacttons; Program at IMST

interest of this original approach is clear, but its success will be
ascertainable only after fulfillment of the anticipated experimental
program. Indeed, in spite of the growing evidence in favor of the
laboratory simulation of such geophysical phenomena, it is only by
direct comparison with field data that the exact degree of similarity
achieved with the natural process will be known. It is, however,
reasonable to expect, at worst, a partial success of this approach,
namely the elucidation of some among the many unsolved aspects of
small scale air-sea interactions.

Besides this main point, a few results have been, to date,
obtained in various related domains. We shall quote: some progress
in the understanding of effects of water vapor transfer and long-wave
radiation upon turbulent heat transfer in the atmospheric surface
layer; some improvements in the technique of micrometeorological
and wind-wave facilities; results on the measurement of turbulent
processes in water flows, with the aid of hot-film anemothermometers.

It is clear that much work still remains to be done, when the
present program is necessarily limited in scope. On the other hand,
the described facilities will offer possibilities that will not be fully
exploited by only one scientific group. Therefore, it should be
possible in the future to consider some cooperation with other groups,
in order to take full advantage of the capabilities of that tunnel.

The researches that can be performed with such an equipment
have multiple fields of application. The results of the present pro-
gram could obviously be useful for oceanography and naval hydro-
dynamics (wave forecasting, shiprouting, fishing, pollution, oceano-
graphic methods, etc.), as well as for meteorolory (long range fore-
casting, air pollution, future climatic control, etc.), and seem also
to be applicable in other domains: chemical engineering, air con-
ditioning, heliotechnique, and even biological and agricultural
micrometeorology. Furthermore, new experiments could be planned
in the fields of acoustic wave propagation, light transmission or
reflexion, electromagnetic propagation, structural mechanics, and
soon. In short, this facility ought to be a kind of "pocket ocean-
atmosphere interface," where basic as well as applied experiments
could be done more easily and economically than in the field.

VIII ACKNOWLEDGMENTS

This research program would have never been undertaken
without the insight of a number of scientists who have encouraged us
to make a start in such an enterprise. In the first place, we have to
cite the name of President Maurice Roy who, after the I[UGG-IUTAM
Symposium on Turbulence in Geophysics, held at IMST in September
1961, advised us to engage ourselves in some kind of laboratory
experiments related to air-sea turbulent interactions. Then, the
Presidents and the members of the "Comité de Recherches

65

Favre and Coantie

Atmosph€ériques de la Délégation Générale & la Recherche Scientifique
et Technique" induced us to go further in that way, and secured the
financial support necessary to start the program. We are happy to
specially mention here the name of Professor H. Lacombe, who

has been, from the very beginning, the adviser and ardent supporter
of the project. We have also received much interest and helpful
advice from Ingénieur Général R. Legendre, Professor L. Malavard,
and from numerous French and foreign scientists whose names cannot,
for lack of space, be mentioned here,

The program is being supported by several French govern-
mental Agencies, to which we are happy to have here an opportunity
to give due acknowledgments. We already mentioned the primary
intervention of the "Délégation Générale & la Recherche Scientifique
et Technique" which supported the preliminary research program,
and granted the major part of funds necessary for the building of the
large air-sea interaction facility. The "Direction des Enseignements
Supérieurs du Ministére de 1'Education Nationale," through the
"Université d'Aix-Marseille," has provided the building site and
financed the construction of offices and laboratories. The operational
cost of the research is presently supported by the "Centre National
pour 1'Exploitation des Océans." The program also benefits of the
direct or indirect aid of several other Agencies, chiefly the "Centre
National de la Recherche Scientifique," the "Direction des Recherches
et Moyens d'Essais," and the "Office National d'Etudes et Recherches
Aérospatiales."

The results presented in this paper are the consequence of
the cooperative work of many individuals. The scientific and tech-
nical team in charge of the program includes a number of research
workers, doctoral students and technicians, to each of whom a
definite part has been attributed in the project management. We
shall mention: Dr. P. Bonmarin (Engineer in charge), Dr. A.
Ramamonjiarisoa (Study of dynamical interactions), Dr. F. Resch
(Turbulence measurements in water flows), Mr. B. Pouchain (Model
tests), Mr. D. Leducq (Humidity measurements), Mr. R. Ezraty
(Measurement of turbulent fluxes in water), Mr. B. Seguin (Turbu-
lent and radiative transfer computations), Mr. J. Quaccia (Designer),
Mrs. F. Laugier, and MM. P. Chambaud, B. Bacuez, M. Bourguel,
B. Zucchini and A. Laurence (Technical assistants having contributed
to the program). Last, but not least, we should like to acknowledge
the cooperation of the various contractors who have been, or still
are, contributing materially to the construction of the large wind-
wave facility, and of the associated equipments.

66

Atr-Sea Interactions; Program at IMST

REFERENCES

Brunt, D., Physical and dynamical meteorolo » Cambridge University
Press, Cambridge, 1939.

Campbell, G. S. and Standen, M. M. ‘ "Progress report II on simu-
lation of earth's surface winds by artificially thickened wind
tunnel boundary layers," Laboratory Technical Report LA-37,
National Aeronautical Establishment, National Research
Council of Canada, 1969,

Cermak, J. E., Sandborn, V. A. » Plate, E. J., Binder, G. K.,
Chuang, K., Meroney, R. N., and Ito, S., "Simulation of
atmospheric motion by wind tunnel flows »" Technical Report
Colorado State University, A.D. 633 584, 1966.

Coantic, M., "Les interactions atmosphtre-océans, les processus
physiques et les €quations quiles gouvernent,"1I.M.S.T.
Report, Cahiers Oceanographiques Vol. XXI, N° 1 pp. 17-
46; N° 2 pp. 105-143; NO 3, pp. 223-249, 1969.

Coantic, M.,Bonmarin, P., Pouchain, B., and Favre, A., "Etude
d'une soufflerie pour recherches sur les changes atmosphtre-
océans," A.G.A.R.D. Conference Proceedings N° 48, 1969.

Coantic, M. and Leducq, D., "Turbulent fluctuations of humidity
and their measurement," Radio-science, Vol. 4, N° 12,

Coantic, M. and Seguin, B., "On the interaction of turbulent and
radiative transfers in the surface layer," IUGG-IAMAP-
AMS Conference on Planetary Boundary Layers, Boundary
Layer Meteorology, Vol. 1, pp. 245-263, 1971.

Counihan, J., "An improved method of simulating an atmospheric
boundary layer in a wind tunnel, Atmospheric Environment,
Vol. 3, pp. 197-214, Pergamon Press, 1969.

Ezraty, R. and Coantic, M., "Sur la mesure des tensions de
frottement turbulent dans un écoulement d'eau," C. R.
Académie des Sc. Paris, T., 270, pp. 613-616, 1970.

Ezraty, R., "Sur la mesure des caracteristiques turbulentes dans
de €coulement d'eau," Tht®se Doct. Eng. Marseille, 1970.

Gupta, A. K., "An experimental investigation of the generation of
water waves by air shear flows," ASRL TR 116-3, Dept. off
Aeronautics and Astronautics, Massachussets Institute of
Technology, 1966.

67

Favre and Coantte

Haltiner, C. J., and Martin, F. L. Dynamical and physical meteoro-
logy, McGraw Hill, New York, 1967.
Hidy, G. M. and Plate, E. J., "Wind action on water standing in a

laboratory channel, Report P.M. 135- NCAR, Boulder,
Colorado, 1965.

Hsu, E. Y., "A wind, water-wave research facility," Dept. of Civil
Eng. Tech. Report 57, Stanford University, 1965.

Lacombe, H., Cours d'Océanographie Physique, Gauthier- Villars-
Paris, 1965,

Leducq, D., "Recherches sur un hygrométre adapté & la measure
des fluctuations turbulentes," Thése Doct. Ing. Marseille,
1970.

Lumley, J. L. and Panofsky, H. A., The structure of atmospheric
turbulence, Interscience, New York, 1964.

Mc Vehil, G. E., Ludwig, G. R., and Sundaram, T. R., "On the
feasibility of modeling small-scale atmospheric motions,"
Cornell Aeronautical Laboratory Report N° Z B -2328-P-1,
1967

Mery, P., "Reproduction en similitude de la diffusion dans la couche

limite atmosphérique," Communication Comité Technique
N° 86, Sté Hydrotechnique de France, Paris, 1968.

Miyake, M. and Mc Bean, C., "On the measurement of vertical
humidity transport over land," Boundary Layer Meteorology,
Vol. 1, NO 1, pp. 88-101, 1970.

Monin, A. S. and Yaglom, A. M., Statistical Hydromechanics,
Nauka, Moscou (English translation J.P.R.S. 37, 763), 1966.

Munn, R. E., Descriptive micrometeorology, Academic Press,
New York, 1966.

Phillips, O. M., The dynamics of the upper ocean, Cambridge
University Press, Cambridge, T300-

Pocock, P. J., "Non-aeronautical applications of low-speed wind
tunnel techniques," AGARD Report 313, 1960.

Pouchain, B., "Contribution & l'étude sur maquette d'une soufflerie
de simulation des interactions océans-atmosphére, "
Thése Doct. Ing. Marseille, 1970.

Priestley, C. H. B., Turbulent transfer in the lower atmosphere,
University of cago Press, cago, °

68

Air-Sea Interacttons; Program at IMST

Ramamonjiarisoa, A., "Théories modernes de la génération des
vagues par le vent," I,M.S.T. Report (unpublished), 1969.

Ramamonjiarisoa, A., "Rapport sur les missions d'études au
Canada et aux Etats Unis en 1969," I.M.S.T. Report
(unpublished), 1970.

Resch, F., "Etudies sur le fil chaud et le film chaud dans l'eau,"
These Doct. Ing. Marseille, 1968.

Resch, F., and Coantic, M., "Etudie sur le fil chaud et le film
chaud dans l'eau," Le Houille Blanche, N° 2, pp. 151-161,
1969.

Resch, F., "Hot-film turbulence measurements in water flow,"
J. Hydr. Div. Proceedings ASCE, pp. 787-800, 1970.

Roll, H. U., Physics of the marine atmosphere, Academic Press,
New York, 1965.

Sverdrup, H. U., Oceanography, Handbuch der Physik, vol. XLVIII,
Springer, Verlag, Berlin, 1957.

Veras, M. S. Jr., "Etude expérimentale de la houle dans un canal,"
These Doct. d'Université, Paris (Division d'Etudes Maritimes
duCREC), 1963.

Waterloopknndic Laboratorium, Delft, a Windgoot van het Laboratorium

"de Voorst" Detailmeting aan de Luchterstroming S. 72-I, 1966;
Varmegeving Luchtciculatie-system windgoot, S. 72-II, 1966b.

69

EXPLOSION-GENERATED WATER WAVES

Bernard LeMéhauté
Tetra Tech, Ine.
Pasadena, California

I. INTRODUCTION: A review of the state of the art

This paper reviews recent developments concerning water
waves generated by underwater explosions, with particular emphasis
on the wave generation mechanism. Analytic models for predicting
water waves from explosions in deep water are presented together
with methods for improving our understanding of the wave generation
mechanism.

A submerged detonation almost instantaneously produces hot
gas or plasma within a limited volume. High temperatures and
pressures result in two disturbances of the ambient fluid: emission
of a shock wave traveling outward, which vaporizes a mass of water;
and radial motion of the fluid, so that the "bubble," consisting of
explosive debris and water vapor, begins to expand. At the same
time, if undisturbed by bounding surfaces, it begins to rise due to
its buoyancy.

During the expansion phase, pressure within the bubble falls
considerably below the ambient hydrostatic pressure, owing to its
outward momentum acquired by the water. The motion then re-
verses; the bubble contracts under hydrostatic pressure, [| acquiring
inward momentum and adiabatically compressing the central gas
volume to a second -- but lower -- pressure.] Upon reaching its
minimum diameter, several phenomena may occur: energy is radi-
ated by the emission of a second shock wave; if near the free surface,
the contracted bubble will not be spherical but may evert, the bottom
rushing up and passing through the top; lastly, the surface of the
contracting bubble is extremely unstable -- it may break up irregu-
larly, forming a spray within the bubble.

The bubble may lose enough energy through repetitive expan-
sions and contractions so that it collapses entirely, leaving a mass
of turbulent warm water and explosion debris, and no waves of conse-
quence will be generated. This case is typical for large, deeply
submerged detonations, and will not be further discussed here.

For shallower explosions, the nature of ensuing surface

re

Le Méhauté

motions depends upon the phase with which the bubble reaches the
surface. Depending upon the depth of the burst, these may include

a well-formed hollow column, a very high, narrow jet, or a low,
turbulent mound, followed by development of a prominent base surge.
The duration of this "initial" disturbance may be quite great, starting
from the first appearance of a mound as the bubble nears the free
surface, to the collapse, under gravity, of the water thrown upwards.
Base surge of plumes are not related to shock interaction. These
phenomena are ultimately manifested by a system of water waves
radiating from the location of the explosion.

If the water is of uniform depth, the wave system will have
circular symmetry. Most of the energy of the explosion goes into
the shock wave and local turbulence; only a small amount (10% at
the most) actually appears in the ensuing water wave system.

As a result of its complicated origin, the initial disturbance
comprises a broad, irregular spectrum. Since deep water is a
dispersive medium, as the waves travel outwards, they will become
sorted according to frequency; the longer waves running ahead, and
the shorter waves trailing behind. A curve of wave period versus
time at any distant location, therefore, will monotonically decrease,

In general, the energy distribution among the frequencies
generated will not be uniform; the spectrum will be peaked near a
frequency corresponding to a wave length which is a small multiple
of the radius of the central disturbance.

At a fixed distance r from the explosion, if one measures
the maximum amplitude 4,,,, of the envelope of the water waves
generated by a given yield W as a function of burst depth, curves
having the general shape shown in Fig. 1 will be obtained. These
curves have two characteristic peaks.

UPPER CRITICAL DEPTH

LOWER CRITICAL DEPTH

DEPTH OF DETONATION

Fig. 1 A schematic illustration of the relationship
between wave amplitude, range, depth-of-
burst, and yield

We

Exploston-Generated Water Waves

The first peak is called upper critical depth (u.c.d.); the
second one is the lower critical depth (l.c.d.). While there is so
far no; adequate theoretical explanation for the u.c.d.,.the l.c.d.
is clearly analogous to the influence of burst depth on crater dimen-
sions in solid materials, and is related to the balance between ex-
plosion energy going into cratering and that vented to the atmosphere,

Another interesting feature which has been experimentall ob-
served is a change of phase between corresponding waves of trains
generated by explosions above and below the upper critical depth.
Such a change, in fact, is predicted between theoretical models of
wave trains generated by an initial impulse acting on the surface and
an initial surface elevation, respectively, suggesting that the impulse
model may be more appropriate for explosions above the upper criti-
cal depth. The u.c.d. is a rather puzzling aspect of explosive wave
generation. Abundant experimental data with HE charges within the
range 0.5 - 300 lbs exhibit a large scatter under presumably identi-
cal conditions, mq, varying between 0.5 - 2 times that at the l.c.d.
Moreover, the scaled wave frequency at "ma, is uniformly higher,
indicating a smaller effective source radius. Lastly, the existence
of the u.c.d. is still somewhat in question for large explosions,
since several attempts to reproduce it with 10,000 lb HE charges
have been unsuccessful. It has been suggested (Kriebel [| 1968]) that
the upper critical depth effect is obtained from interference between
the direct incident shock wave and its reflected waves, resulting in
more effective containment and greater cavity expansion than from
deeper or shallower charges. As the detonation depth increases,
the pressure impulse on the free surface has less and less effect
on the cavity formation and ultimately becomes negligible. This
undoubtedly influences the shape of a theoretical cavity, which pro-
duces an equivalent system of water waves, since the dimension of
this cavity is closely related to both frequency and amplitude of the
first envelope. Nevertheless, it appears that the large data scatter
obtained under fixed experimental conditions at the upper critical
depth are largely due to Taylor instability of the collapsing cavity.

The mathematical model described later can be adjusted to
produce practically any type of wave train desired, by assuming
various shapes for the initial cavity.

i. . INPUT CONDITION

The theoretical formulation of an overall mathematical model
for simulating the time history resulting from an underwater detona-
tion is an extremely complicated task. However, keeping in mind
the main objective of our problem -- the generation process of water
waves -- many detailed phenomena, chemical or nuclear, can be ig-
nored, retaining only the kinematic and dynamic features.

For example, one can consider only the following phases,

73

Le Méhaute

both of which are mathematically tractible.

The Compressible Hydrodynamic Phase
The Incompressible Phase

The first is arrived at by defining the conditions which prevail at the
location of the explosion and in its immediate neighborhood. The
second makes use of this input as an initial condition to calculate the
water waves at a distance far from the explosion.

2.1 The Compressible Hydrodynamic Phase

The compressible phase of an underwater detonation lasts
for a relatively short time. Initially, upon detonation, there is im-
mediate vaporization of water around the weapon due to intense
radiation. This takes place in a time scale of microseconds. As
the shock wave propagates outward, the bubble front initally coin-
cides with the shock front (Fig. 2). Only the initial phase of the
formation of the bubble need be considered compressible. Bubble
migration, expansion, and collapse can be treated as incompressible,

BUBBLE FRONT j SHOCK FRONT

TIME AFTER DETONATION, t

r
b
DISTANCE FROM EXPLOSION ,r

Fig. 2 Qualitative relationship between shock front
and bubble front

or, perhaps more correctly, as quasi-incompressible, since subse-
quent shock waves are emitted at each minimum of the bubble history.
It is obvious that, from the point of view of wave generation, the only
problems of importance during the compressible phase concern the
fluid motions generated in the vicinity of the bubble and the effect

of reflected shocks. The early, compressible phase of bubble (cavity)
expansion can be treated by the particle-in-cell technique, which has
been successfully employed in many similar cases of compressible
flow, both in fluids and in solids (Mader [ 1967]).

74

Exploston-Generated Water Waves

Again, it should be emphasized that the objective is to obtain
an input condition for wave generation rather than to solve the many
associated problems of bubble dynamics, shock propagation, and
radioactive debris distribution. However, these problems cannot be
ignored, inasmuch as they affect the wave generation process.
Extensive work has been performed on bubble dynamics of both con-
ventional and nuclear explosives (Snay [1966]). The energy partition-
ing between radioactive potential energy, thermal energy to heated
and vaporized water, shock energy, and kinetic and potential bubble
energy has been investigated by a number of authors (see, for ex-
ample, DASIAC Special Report 104 - Secret). Detailed studies of
shock propagation and pressure fields have been performed by many
investigators. Some of these studies neglect the effect of gravity;
others make gross assumptions about the thermodynamic properties
of the bubble. But all of these effects should be included in an
effective model for analyzing the wave generation process, unless
it can be shown that they can be neglected because they do not affect
the wave characteristics.

2.2 The Incompressible Phase

The input condition being defined by the compressible phase,
the subsequent cavity behavior may be treated as incompressible
flow. It is tentatively proposed to analyze the wave generation pro-
cess through a numerical solution to the time-dependent, viscous,
incompressible flow of a fluid with a free surface. Figure 3 is an
experimental example of near-burst free surface history. The most

CENTERLINE s 55 FRAME 10 10 15 15 25 25
45
75
—w~rt
SX ey.
a Nese SENSE 105
Wk) FSS} —

POL

M
)
‘
)

2 64 FRAMES /ssec.
WATER DEPTH - 8ft.
DEPTH OF BURST -6 in.

Fig. 3 Cavitv shape versus time - shot 11 (Courtesy of URS)

15

Le Méhauté

promising method applicable to this problem seems to be the MAC
Method developed by Harlow and his group at the Los Alamos
Scientific Laboratory. Further study to improve both accuracy and
efficiency (with respect to computer time) has led to development

of other techniques, such as SUMMAC (Chan, et al. [1969]). Despite
the degree of sophistication that has been achieved for treating free
surface flow problems by numerical techniques, problems still re-
main which require some approximation. These are related to the
amount of energy dissipated by viscous turbulence associated with
the plume and base-surge radiating from the explosion. Energy
dissipated by the radiating surge is similar to that in a tidal bore,
except for the difference in water depth. The choice of a suitable
viscosity coefficient that realistically accounts for turbulent dissi-
pation can only be made empirically, and will be related to the mesh
size of the numerical model. This choice is also subject to the con-
straint of numerical stability.

Ill WATER WAVE FORMULATION

3.1 General Analytical Generation Model

Using the initial conditions obtained by the above methods,
one can determine the water waves generated by such disturbance
analytically.

The problem of surface waves generated by an arbitrary --
but localized -- disturbance of the free surface has been investigated
by Kajiura [ 1963], who has derived very general solutions incor-
porating the effects of initial displacement, velocity, pressure, and
bottom motion. Kranzer and Keller [ 1959] present a simplified
approach through the assumption of radial symmetry. The two solu-
tions are equivalent under appropriate conditions, but, because the
former permits utilization of the previous methods in the form of a
time-dependent input condition, the approach of Kajiura [ 1963] will
be adopted here.

The problem may be formulated as follows. In water of
constant depth D, the coordinate system is established with x and
y in the horizontal pone of the undisturbed surface and z* taken
eames 34 upward; a is the time, n'(x™ wt yt ) the cee ge eleva-
tion, cosy eect at *) the particle Beas and p*(x*,y ee
the pressure. The motion is assumed pe ooh iniphyiny the
existence of a potential function ®(x* vw ,z*,t*). Dimensionless

quantities are introduced as follows:
x = x*/D y=y*/D Z= z*/D
t = t*Vg/D n= */D v*/Vg/D
& = 6*/(D/ gD) p = p*/pgD

<
I

76

Exploston-Generated Water Waves

where g is the gravitational acceleration.

Making use of Green's formula, Kajiura [1963] gives a solu-
tion of V@=0 satisfying bottom conditions (6, =0, z2=-1) and
the linear free surface condition:

@.2(x;y.Z:T) = 1/an\ (( Go, ~ BG,,), 2945
S
: 1/an\ (ce, : 5,G,,), _1 1S, (1)
P 0 °

where S denotes the source region, initial conditions are denoted
by subscript zero and G is the appropriate Green's function; 7 is
a time associated with the generation period. The Green's function
is:

00 ze
G(x9sVo2Z93T |X» Ys Z5t) -{ —olk=) [sinh k} 4 = |z-zolt
fe)

2 sinh k }1 + (z +2)

cosh k(1 +z) cosh k(1 +20) dk

(2)

where r is the horizontal distance between the source point (x9, yo)
and the point under consideration (x,y) i.e.,

Za k
: =z}! ~ cosw(t-7){ coshk

—2 2 2
re = (x -. xg)” ty = yo) (3)
and
w* = k tanh k. (4)

Clearly, G is symmetric with respect to t and 7, and with respect
to source and field points. The quantities w and k are dimension-
less frequency and wave number, respectively.

Applying the given boundary conditions and integrating Eq. (1)
with 0<7<t gives, after some calculation

ntp= (lum +R as, (5)
5)

where

fies

Le Méhauté

F = (Cr. i Gamo.g Zj=z=0 (6)

and

|
"

t
2 i PG, a7 + (PGee) eat

t
ee if p,G,; dt + (pG,+)-20 Zy=z=0 (7)

It can be seen that F, contains the contribution to n from
the initial velocity and surface deformation of the source while Fy,
represents the influence of initial pressure. Therefore, the model
is very general; in fact, Kajiura[ 1963] gives an additional term repre-
senting the contribution from an arbitrary bottom disturbance, which
is ignored here. It has been found, however, that such generality is
not necessary in order to make practical predictions of water wave
production. Instead, it is possible, for example, to absorb the
effects of initial velocity and pressure into a fictitious initial surface
deformation, chosen in such a way that the predicted waves are
essentially the same as would be found using actual velocity, defor-
mation and pressure.

3.2 Simplified Approach

The advantage of this approach in practical work will be
apparent later in discussing the correlation between theory and
experiment. The essential point, however, is that, instead of
needing to predict the complicated phenomena leading to initial
deformation, velocity and pressure, it may be sufficient to utilize
easily measurable quantities to calibrate a simplified source model.

With this in mind, we rewrite Eq. (5) as a Green's function
of time only:

1
n= rai (iGsin) 2gdS5 . zgez= 0 (8)

Furthermore, it is reasonable to assume that, for a single
explosion in water of constant depth, the problem is symmetric about
the z-axis passing through the source. When the appropriate opera-
tions [ in Eq. (8)] are performed, and the transformation to cylindri-
cal coordinates (r,®) is made, one obtains the time-and-space
dependent surface elevatior™

00

{ 277 00 Ss
n(r,t) = ao k cos ut{ i" Neto) Seater ite dr, ae,} dk (9)

78

Exploston-Generated Water Waves

where 1(ro) is the initial deformation. Noting that
ree x? +r7- 2rr cos (8. - 6) (10)
) ) fr)

the Bessel function J)(kr) may be rewritten according to Graf's
addition theorem as

Jo(kr) = Ig(kr)Ig(kr,) + 2) J,(kr)J,(kr,) cos n(®,- 8). (14)

n=l

Integration with respect to 0, from zeroto 2m deletes the sum-
mation so that

ro) 00
H(t) = k cos wtJg(kr) {\ Mg Fo) Jol Kt) Zo arg} dk. (42)
C

The same result was obtained previously by Kranzer and Keller
[1959] using integral transforms; in the literature dealing with radial
dispersive waves, Eq. (12) is generally referred to as the Kranzer-
Keller solution.

Equation (12) is a double integral solution that can be con-
siderably simplified by additional approximations. In particular,
for large r and t, J,.(kr) may be replaced by an asymptotic cosine
function, and the resulting integral approximated by the method of
stationary phase (Stoker [ 1965]), to obtain

~ 1 AcV(k)
Wr.t)* — ZV")

m(d) cos (Ar - tv tanh 2) (13)

k=

where
4 00
US { NolTo) IJo(Kro) ro dro (14)
fe)

is the zero-order Hankel transform of the initial elevation (ro) and

k i

— (15)
2cosh k yk tanhk

V(k) == cana

is the wave group velocity, and \ is the particular value of k fora
given r and t found from:

79

Le Méhauté

Wiair/t. (16)

The problem now is to choose 1,(r,) in such a way, depending on
water depth and explosion characteristics, that Eq. (13) best fits
an observed wave train.

3.3 Time-Dependent Free Surface Deformation

Another possible mathematical model includes the kinetic
and potential energy transmitted to the water by both the atmospheric
overpressure and the gaseous expansion of the bubble. The initial
conditions are now time-dependent, and at least one additional
parameter (time) is added to the initial conditions.

While Eq. (13) gives the total energy partitioned to water
waves as a function of detonation depth, as well as the distribution of
energy amongst frequencies, the introduction of time-dependence,
if properly used, permits a better fit to observations, not only for
the first maximum of the wave envelope mg, (and its corresponding
wave number k,,), but also to the whole shape of the wave envelope.

As an example of a time-dependent input condition, consider
for example,

Wig (Fos T) = No (Zo) sin or (17)

where 7, can be considered as the dimensionless period of first
expansion of the crater cavity from an explosion (Whalin [ 1965]).

The initial surface velocity at time 7 =0 is

Zy = 0
d T
,, | 2=0 =G8) = see A(rd) (18)
=0
T= t =— O .

and the resulting wave train is given by

_ mw _ md) /kV(k) y
n(r,t) = i Hier =yMny oc sin (Xr - ty\ tanh 2). (19)

It is interesting to note that n(r,t) is independent of real-time history
of the free surface deformation and depends only upon its time deriva-
tive at time 7 =0.

Exploston-Generated Water Waves

3.4 Main Features of the Mathematical Models

[ Based on Eqs. (13) and (14)], typical examples of various
models for initial surface deformations No (r ro) are given in Table I.
The first case, a parabolic water crater, is that proposed by Kranzer
and Keller [1959].

The general features of traveling wave trains given by the
equations presented in Table I are:

ae The waves travel radially from the explosion.

b. The leading free ee disturbance or leading wave
travels at velocity ygD.

c. Ata given location, the frequency of individual waves
increases monotomically.

d. The amplitudes of individual waves (cosine function in
Eq. (12)) are modulated into groups of successively
smaller amplitude by the slowly varying Bessel function
J,(kr,) in Eq. (14).

e. The number of waves ina given group increases with
time or distance traveled.

f. The length of a group increases linearly with time or
distance traveled.

ge. The frequency associated with a specific crest decreases
with time or distance traveled (equivalently, a given
crest moves forward within a group).

he The frequency associated with the maximum amplitude
of a given group is constant.

i. The maximum height of a given group decreases as the
inverse of time or distance traveled.

je The maximum height of successive groups passing a
given point decreases with time,

These features are partly illustrated in Fig. 4, which shows a com-
puted wave train at three different locations. The general decay of
wave height with distance is the result of both radial dispersion and
circular spreading. This radial dispersion is characterized by a
general increase in the wave length of individual waves with distance.

Wave crests occur when cos (wt - kr) = 1 (Eq. (13)), and
crest order numbers are given by wt - kr = n(2n-1) where n is an
integer. It is also interesting to note that in the case of deep water,
the trajectories of individual waves inthe r-t plane are defined
by parabolae: n(2n-1) = gt @/4r, See consecutive arrival times at
any point r will be inthe ratios t: t/V3 3:t/¥5, etc. Similarly, at any
instant of time t, the consecutive crest radii will have the ratios
rir/3:r/5, etc.

81

qusurad etdstq sUMIOA 42eN O17
dry] yim s1joqereg
(Gre cal ear
2/1

(q4veyy/Y37 - AY) 809 “Vv

qusulsd e[dsiq sUMIOA 49N O197
drq yim 991990q -Yy}41NO J
= 44
( om EP) ca gf ¥p 2 — = Py
2/1 y) AX XOW G
lp

( yyueiy /* 3-4 )so09 ty = (7 ‘a)u

yuoursd eTdsiq BsUIN[OA JeN O197-U0N
odTjoqeieg

Le Méhauté

MP/A P-\ (2, 2
( (4) A / Me XDW Gg
Z

( 4 4yuey x / 3-414) SOO ly = (3 ‘ajlu

(sSo[UoTsUusUIIG) uotssaadxq Teuotjoun J uo1ze14SNIT]

apnytydury oaeM ButzNsoy UOTIJELUIIOJOG IoeJANG [eIWIUT

2/1

NOILVYWYOTAd AOVAUNS TVILINI 40 NOILONNGA SV NOILVINW YOU AAVM

I 2TavL

82

Exploston-Generated Water Waves

CREST LOCUS DIMENSIONLESS FREE SURFACE ELEVATION

ENVELOPE

LINE OF MAX CREST
(CONSTANT PERIOD)

DIMENSIONLESS DISTANCE, r =r°/D

DIMENSIONLESS TIME t=t*/g/pd

Fig. 4 Schematic drawing of wave trains as function of time at
three different locations (Van Dorn, personal communi-
cation)

Based on this formulation, an equivalent crater size defined
by its maximum depth ‘omax and radius R can be empirically re-
lated to yield W and detonation depth as shown in the following.

However, the present formulation is oversimplified. The
cavity shape (and not only its overall dimension) is a function of
submergence depth and charge weight, and the phenomena of the
upper critical depth is very sensitive to the manner in which the
cavity is formed.

IV. CORRELATION WITH EXPERIMENTS
4,1 Practical Formulation
In Eq. (13) the cosine term represents the individual waves,

while the remainder of the expression gives their varying amplitude
or envelope which we shall call A:

83

Le Méhauté

A =1 SVK) (20)

in which it is understood that k is the root of Eq. (16)

It can be seen from Eq. (20) that for any fixed value of r,
the least nonzero value of k for which dA/dk = 0 is independent
of r; this means that the maximum of the first wave envelope (where
dA /dk = = 0) is associated with a constant value of k (and, therefore,
wavelength and period) throughout its propagation. This constant
value of k at the first envelope maximum, k,,, depends only on
the nature of the source disturbance 1,(ro) cones the factor 7(k).

Evaluating A at I ax? we can write

= Constant (2 1)

Anat = {Fite (MH) }

Keke

for a particular source deformation 1,(rog), which means that the
amplitude of the maximum waves is inversely proportional to r.

Before we can proceed with the quantification of the theoretical
model, we must select an appropriate form of ,(r,). The two con-
straints on this choice are, first, that the resulting wave envelope
shape be sufficiently similar to observed shapes that some manipu-
lation of numerical coefficients will give an accurate fit; and, second,
that the Hankel transform of 1)(ro9) be within our power to obtain in
a closed form.

In addition, it would be nice -- although it is not really neces-
sary -- to have 1,(r,) intuitively resemble the effective surface
deformation due to an explosion. For all these reasons, we are led
to try simple polynomials in ms for 1,(r,) with crater-like shapes.

Of the three forms which have been used in practical work
(Table I), the last has been tentatively established as most suitable:

Talo) = Tomax 2 ( (2) -1]r, =

o>)

toc R

where ‘oma is a coefficient which, for the sake of simplicity, will
be written as ‘oq in the following.

The wave amplitude is then given by

84

Hxploston-Generated Water Waves

2

/
ment) les (4s) J,(kR) cos (kr - tVk tanhk), — (22)

ag

the two "cavity parameters" n, and R being embodied within our
previous expression for the envelope amplitude, A. It is through
empirical determination of these two parameters that we hope to
correlate theory and experiment.

4.2 Experimental Correlation

While y. and R cannot be experimentally measured, they
can be determined indirectly from Rees and eats which are
characteristic of the source disturbance, and also measurable.
Hence, we seek to relate k,,, and n,,,r to the characteristics of
the explosion by experiment, and n, and R to ky, and Nmar
by theory. The expression giving kg, in terms of nN, and R is

GA
ie a 0 (23)
since this expression defines the maxima of the wave envelope; the
least non-zero value of k for which the above expression holds is

K ax?
For k__ >3 (relatively deep water) vi = /2 = const
max be > TdV/dk !
and
V2 mg J,(kR) . (24)
mheretore, Kk can be determined from the first turning value of

max
the Bessel function J3(kR); viz for

Kay = 4.20. (25)

Our other measurable, mgr, may now be related to 1) and
R by evaluating Ang, (OF Tmax) at k = Kkmqe When this is done and
the resulting expression is simplified, we have

Wott = 10ST ae (26)

All that remains now is to relate "mgr and Kg, to the
characteristics of the explosion; these are W, explosive yield in
pounds of TNT, Z, detonation depth in feet, and D, the water depth

85

Le Mehauté

in feet. A large volume of experimental data (with small chemical
changes in relatively deep water) has been obtained at the Waterways
Experimental Station, Vicksburg, Mississippi, from which the
following empirical relations were deduced:

* at 0.54
i limagt oO. Wi

WeCeds Pie O> Fos =~ 0-25 (27a)
K riage) BeOS ae Ow
ieee =10 w?4
L. Ges -0.25> oo32-7.5 (27b)
-0.3
* ft
ee Ss soo Ww
at Z
Insufficient Data wos <7 gees, (2-7)

The products Teer given above were determined from the empiri-
cal data of Fig. 5. Corresponding data for shallow water explosions
and other aspects of explosion-generated waves in shallow water are
beyond the scope of this presentation, and the reader is referred to
LeMéhauté [ 19714].

YIELD
O W =0.50 Ibs TNT
fe) = 2.00 lbs TNT
A 10.00 Ibs TNT
+ 125.00 Ibs TNT
e 385.00 Ibs TNT
MONO 1966 ©
HYDRA IL -A a

=9,250.00 Ibs TNT
= 14,500.00 Ibs TNT

0 ay 2. = -4 -5 a6 =7 -8
z/w2:3

Fig. 5 An empirical scaling fit relating the maximum wave height
“max With distance from explosion r*, yield and depth of
explosion (data provided by Waterways Experimental Station)

86

Exploston-Generated Water Waves

Figure 6 presents examples of the matching between theoreti-
cal wave envelope and wave records due to a 9, 620 lb TNT explosion.
The slight irregularities in the symmetry of the recorded wave trains
are attributed to partial shoreline reflection interferring with the
radiating wave trains (Hwang et al. [1969]. But, in general, the
computed wave envelopes agree fairly closely with the observed
amplitudes.

4.3 Limitiations of the Model Due to Scale Effects

An examination of Fig. 5 reveals that the bulk of data upon
which predictions are based are restricted to yields from one-half to
a few hundred pounds of TNT. One wonders then just how reliable
extrapolation to very large yield (say, 10'9 pounds of TNT) would be.
The limited data available from nuclear explosions is insufficient to
resolve this problem. Comparison between crater data in soft
materials for both nuclear and TNT explosions suggest that the laws
of similitude may be applied to contained explosions but may not
apply over a large yield range for venting detonations. In particular,
the shock wave from a nuclear explosion travels much faster in air
than in water, which is not the case for a TNT explosion.

We may infer several things, however, just from the nature
of the scaling parameters given by Eq. (27). Consider, for example,
the groups n*,,.r*/W°54 and Z/W®3. In each, the exponent of W
was chosen to best compress the data of Fig. 5 into a single curve,
since W represents an energy, dimensional analysis suggests that
Nmaxt/(W/pg)'/2 and Z/(W/pg)!/4 are appropriate scaling parameters,
although similar conditions also require that other parameters, such
as atmospheric pressure and sonic velocity in water, should also be
scaled with yield. These conditions are never satisfied experimental-
ly, and it is therefore not surprising that exponential scaling alone is
not satisfactory. Moreover, the fact that the parametric coefficients
vary with Z means that the phenomena are not simply scalable
(Pace et al. | 1969]). Lastly, the lack of evidence for an u.c.d. at
large yields suggests that the generation process is fundamentally
different.

For small yields (and subsequent small depth at burst) hydro-
static pressure is small compared to atmospheric pressure; for
large yields the reverse is true. In the former extreme, dimensional
analysis suggests 1/3 power scaling; in the latter, 1/4 power scaling.
In an analogous review of earth crater scaling, (Chabai [1965]) has
proposed an "overburden scaling law" in which the scaling exponent
varies between these two extremes, but without convincing improve-
ment in agreement to the experimental data.

87

Distance Between
Gage and SZ 3,600 ft.

lll Het 568
neem

ee ll
sare ST a

7 (feet)

iil i mci

: Bi

t (seconds =
Fig. 6 Compari of OSI 1966 Mono Lake experiments with
theory

88

Exploston-Generated Water Waves

4.4 Energy Coupling

The deficiencies of simple exponential scaling are more appar-
ent when considering the efficiency of energy coupling into water
waves. The analytic source models discussed above are linear, and
thus the total wave energy is equal to ns potential energy of the
source model; i.e., proportional to 16 2@R2, But, in view, of Eq. (26),
the empirical relations given in Eq. (27) imply that 16 2R2 ~ 8
which obviously cannot be true for all yields, since it ‘states that
wave energy increases faster than explosion energy under geometri-
cally similar conditions. It is also pertinent to recall that the calcu-
lating of energy based on the theoretical source model may lead toa
significant error; since only the first wave train has been watched
with experiments, it may happen that the following wave train contains
less energy than the theoretical model, as due the dissipative
mechanism which influences the high frequency waves. Keeping in
mind these reservations, it is found that the energy in the wave train
is

2
Ey = 126(n 2) ft=Lb,

Then, inserting the value of n,,,r in terms of yield and water depth,
it is found that at lower critical depth, the efficiency e is

e = 0.0074 WO-%E 1% (W is in pounds).

At upper critical depth, the increase of efficiency with yield
within the range of available experiments is even more pronounced,
For example, e which is 1% in the case of 0.5 lbs of TNT has been
found to be 6% in the case of an explosion of 375 pounds, which implies
that n° = w-6! at upper critical depth. Such results cannot, of
course, be extrapolated to atomic yield.

Since the fraction of yield energy appearing as waves is only
a few per cent for the largest tests so far conducted, we are faced
with the problem of trying to distinguish very small energy differ-
ences in normalizing analytic models to actual experiments. While
the present models provide adequate predictions for the largest
waves over an impressive range of yields (0.5 - 64,000,000 lbs TNT
equivalent), it is recognized that important phenomenological factors,
such as atmospheric pressure, shock interaction, and cavity stability
have been neglected, each of which can reasonably be expected to
influence wave formation to some extent. What is really surprising
is that such simple models work as well as they do, considering the
great complexity of the process of explosive wave generation.

89

Le Méhauteé

ACKNOWLEDGMENT

The writer has had the opportunity of collaborating with a
number of researchers who have deeply contributed to establishing
the present state of the art. The original contributions of Dr. Li-San
Hwang, Manager of the Hydrodynamics Group at Tetra Tech, Inc.
and Mr. David Divoky have been of invaluable assistance in assembling
and editing this material and verifying formulation and notation.

Dr. William Van Dorn of Scripps Institution of Oceanography and

Mr. Robert Whalin of the Waterways Experiment Station have also
significantly contributed to the contents. Dr. Van Dorn revised this
manuscript and made many most pertinent suggestions. Mr. John
Strange provided the writer with the set of experimental data on wave
generation obtained by the Waterways Experiment Station. This study
was sponsored by the Office of Naval Research, Contract No.
N00014-68-C-0227, under the technical management of Mr. Jacob L.
Warner.

REFERENCES

Chabai, A. J., "On scaling dimensions of craters produced by buried
explosives," J. Geophys. Res., vol. 70, no. 20, pp. 5075-
5098, 1965.

Chan, R. K. C., Street, R. L. and Strelkoff, T., "Computer studies
of finite amplitude water waves," Tech. Report No. 104,
Stanford University, ONR Contract No. Non 255(71)NR-62-320,
June, 1969.

Hwang, L.-S., Fersht, S. and Le Méhauté, B., "Transformation and
run-up of tsunami type wave trains on a sloping beach," Proc.
13th Congress I.A.H.R., vol. 3, pp. 131-140, 1969.

Kajiura, K., "The leading waves of tsunami," Bull. of Earthquake
Res. Inst., vol. 41, pp. 535-571, 1963.

Kranzer, H. C. and Keller, J. B., "Water waves produced by
explosions," J. App. Physics, vol. 30, no. 3, 1959.

Kreibel, A. R., “Cavities and waves from explosions in shallow
water," URS Research Co., Report No. URS-679-5, DASA
Contract No. N0014-67-C-045, 1968.

Mader, C. L.., "Fortran BKW: a code for computing the detonation
properties of explosives," Los Alamos Scientific Laboratory
of the University of California Report No. LA-3704 under
Atomic Energy Commission Contract No. W-7405-ENG. 36,
1967.

90

Exploston-Generated Water Waves

Le Méhauté, B., "Explosion-generated water waves," Advances in
Hydrosciences, Academic Press, New York (publication
pending), 1971.

Pace, C. E., Whalin, R. W., Sakurai, A. and Strange, J. N.,
"Surface waves resulting from explosions in deep water,"
Report No. 4, Waterways Experiment Station, Vicksburg,
Mississippi, 1969.

Snay, H. G., "Hydrodynamic concepts selected topics for underwater
nuclear explosions," NOL TR 65-52, DASA-1240-1(2) U.S.
Naval Ordnance Laboratory, September 15 (AD-803-113), 1966.

Stoker, J. J., "Water waves," Interscience Publishers, Inc.,
New York, New York, 1957.

Whalin, R. W., "Contributions to the Mono Lake Experiments,"
NESCO Report S 256-2, ONR Contract No. Nonr-5006(00),
1965.

Whalin, R. W., "Research on the generation and propagation of water
waves produced by underwater explosions (U)," National
Marine Consultants Report NMC-ONR-64, Part II: A Prediction
method (CONFIDENTIAL), 1965.

91

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pe

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

Monday, August 24, 1970

Afternoon Session

Chairman: J. Wehausen
University of California, Berkeley

Page

Resonant Response of Harbors (The Harbor Paradox
Revisited) 95
Jie We Miles, University of California, San Diego

Unsteady, Free Surface Flows; Solutions Employing
the Lagrangian Description of the Motion 117
C. Brennen, A. K. Whitney, California Institute
of Technology

Two Methods for the Computation of the Motion of
Long Water Waves -- A Review and Applications 147
R. L. Street, R. K. C. Chan, Stanford University,
and J. E. Fromm, IBM Corporation

An Unsteady Cavity Flow 189
D. P. Wang, The Catholic University of America

93

RESONANT RESPONSE OF HARBORS
(THE HARBOR PARADOX REVISITED)

John W. Miles
Untversity of Californta
San Dtego, California

I. INTRODUCTION

We consider the surface-wave response of a harbor to a
prescribed, incident wave in an exterior half-space on the hypo-
thesis of linearized, shallow-water theory, an ideal fluid, anda
narrow mouth, invoking the equivalent-circuit techniques that have
proved so efficient in attacking analogous problems in acoustics
and electromagnetic theory. These techniques offer significant
advantages in practice: (i) the sub-problems of external radiation,
channel coupling, and internal resonance may be attacked separately;
(ii) the equivalent-circuit parameters may be expressed as homo-
geneous, quadratic forms that may be simply approximated without
solving the complete boundary-value problem; (iii) observed values
(including those from model experiments) of dominant parameters,
such as resonant frequencies, may be incorporated in preference
to, or in place of, theoretical values; (iv) empirically determined
dissipation parameters (resistances) may be incorporated; (v) ana-
log computation, both conceptual and electrical, may be invoked to
expedite understanding of the resonant response.

Referring to Fig. 1, we consider a harbor H that opens to
the sea through a narrow mouth M ina straight coastline, x =0.
Let

Ci(x;y) =5V, exp {- jk (x cos 0, + y sin 6; )} (1.4)

br =¢; (- x,y) (1.2)

be the complex amplitudes of the incident and specularly reflected

A more detailed version of this work has been published elsewhere
[ Miles, 1971].

95

Miles

Fig. 1. Schematic diagram of harbor opening on straight
coast line; ¢€;, €; and O, are, respectively, the
incident, specularly reflected, and scattered
waves

(from x = 0) waves on the hypothesis of the monochromatic time
dependence exp (jwt), where € denotes free-surface displacement
(we omit the modifier complex amplitude of throughout the subse-
quent development), k is the wave number, and Vj = 2¢;(0,0) is
a measure of the excitation of the harbor through M. By narrow,
we imply

a/R << 1 and ka<<1i, (2. 32,6)

where a is the width of M, and R is a characteristic dimension
of H. These restrictions imply that the motion within H is small,
and that the energy of the motion induced by V; (or, more precisely,
by the pressure pgV,) is dominantly kinetic and concentrated near
M (the narrowness of which implies locally high velocities), except
in the spectral neighborhoods of the resonant frequencies of the
harbor. An appropriate measure of this dominant motion is the flow
through M, say I, which, by hypothesis (linearized theory), must
be simply proportional to V;. We regard Vj; and I as the voltage
and current at the input terminals of an equivalent circuit and seek
a description of the resonant response of the harbor in terms of the
voltages induced in this equivalent circuit.

The input impedance, Z; = V,/I, for the configuration of
Fig. 1 may be resolved (see Fig. 2a) into a series combination of a
radiation impedance, Zu= Rat jXy- and a harbor impedance,
Zy= jXy» where 1 i XI tl? /o, and X,|I|*/w are respectively
proportional to the power radiated from i through M (in the form

96

Resonant Response of Harbors (The Harbor Paradox Revistted)

of a scattered wave, (,), the non-radiated energy stored in the ex-
terior half-space, and the energy stored in the harbor (we also could
incorporate an empirical, resistive component in Z,, say Ry, to
account for an energy dissipation proportional to Ry tls). We infer
from the solution of the corresponding acoustical radiation problem

[ Miles 1948; §3 below] that both Ry and Xy are bounded, positive-
definite functions of w, by virtue of which we may regard them as
single resistive and inductive elements, respectively (although neither
Ry nor Xy has the same frequency dependence as its elementary,
electrical counterpart). We infer from the analogy with the corre-
sponding acoustical resonator [ Morse 1948, §23] that Zy comprises
an infinite sequence of parallel combinations of inductance L, and
capacitance C,, which bear a one-to-one correspondence to the
natural modes of the clos, harbor and resonate at the corresponding
frequencies, w, = (L,C,) , together with a single capacitor Cg,
which corresponds to the degenerate mode of uniform displacement,
for which wo= 0. The solution within H may be expanded in this
infinite set of modes, with the root-mean-square displacement and
the kinetic and potential energies in the n'th mode being proportional
to the voltage across C, and the energies stored in Ly and Cp,
respectively. The arguments of the preceding paragraph suggest
that the individual modal impedances are important only in the
neighborhoods of their respective resonant frequencies, and hence
that Z,may be approximated in the neighborhood of w=wW, bya
lumped inductance, say Ly, in series with either Cg or the single,
parallel combination of L, and C,, such that the energy in all modes
but the n'th is proportional to L [I “. The corresponding equivalent
circuit is shown in Fig. 2b (we give a quantitative derivation of this
equivalent circuit in §§2 and 3).

Fig. 2. Equivalent circuit for harbor opening directly at coastline:
(a) implied by (3.2); (b) implied by (3.2) and (4.6).

97

Miles

The voltage-amplification ratio, « |v,/V,|, provides a
measure of the resonant response in the cee ere of w = w,.
The zero'th mode, in which the harbor acts like a Helmholtz reso-
nator, is unique in that the equivalent circuit reduces to a series
combination of Ray Lyt ly, and C, and exhibits a simple, series-
resonant behavior path a resonant frequency, say Wp, that is deter-
waar by a balance between the potential energy stored in H,

2 ce es the kinetic energy stored in the vicinity of M,

= f° + ‘( The results for the rectangular harbor Lites and
Munk 19 peaoce that the sharpness of the Helmholtz resonance
is ec me by

= {log (R/a)} (1.4)
and that
oe o(s!/%y, ae ="O(1/- 6): and Qo= O(1/5) (1.5a,b,¢)}

as a/R > 0, where Gq. is the peak value of @ ae and Q, is the
ratio of the resonant Goceeney, to the half- -~power bandwidth of the
resonance curve for the n'th mode.

The resonant response of the harbor in the higher modes is
strikingly different than that of a simple, series-resonant circuit in
consequence of the proximity of the parallel-resonant frequency,
Wp, at which Z; = oo, and the series-resonant frequency, w,, at
which |Z, | has a minimum and @, = >> 1. We show in §4 that

@,= , + O(8), G,= O(1/8), and Q,=0(1/8) (n# 0) (1.6a,b,¢)

It follows from (1.5) and (1.6) that narrowing the harbor
mouth does not affect the mean-square response to a random excitation
in the spectral neighborhood of w= w, (which response is proportional
to @, to afar if the bandwidth of the random input is large compared
with ” Bt_) "except in the akg We mode, but that the response in that
mode increases inversely as 6! Miles and Munk [1961] overlooked
the proximity of parallel and series resonance in the higher modes and
arrived at the erroneous conclusion that narrowing the harbor mouth
would increase wan /O,, for all modes, rather than only the Helmholtz
mode, and designated the phenomenon as "the harbor paradox." In
fact, as pointed out by Garrett [1970], this qualitative conclusion is
inconsistent with their quantitative results, which actually imply
(1.6) for the higher meds in a narrow rectangular harbor. Garrett
also showed that Ga 2/6) is similarly invariant for excitation of a
circular harbor through an open bottom and correctly conjectured
that the result holds generally for the higher modes in any harbor.
In brief, the harbor paradox originally stated by Miles and Munk

98

Resonant Response of Harbors (The Harbor Paradox Revisited)

holds only for the Helmholtz mode and otherwise must be replaced by
the weaker paradox that narrowing the harbor mouth has no effect

on the mean-square response of the higher modes to a random input
in the absence of friction (narrowing the mouth increases friction,
thereby decreasing the response, in a real harbor). It follows that
the higher modes are not likely to be strongly excited, but that the
Helmholtz mode may dominate the response of a harbor to an exterior
disturbance that has significant energy in the spectral neighborhood
Of Wo.

Carrier, Shaw and Miyata [1970] consider a harbor that
communicates with the coast through a narrow canal and find that
both @, and Qo are significantly increased (as might be inferred
from the analogy with the classical Helmholtz resonator; cf. Rayleigh
[1945], §307). We show in §5 that such a canal is analogous to an
electrical transmission line and may be replaced by a symmetrical,
four-terminal network for the calculation of V, (see Fig. 3). The
analogy with the transmission line rests on the hypothesis that only
plane waves are excited in the canal. An examination of the effects of
higher modes shows that the elements of the four-terminal network
may be appropriately generalized, but that the plane-wave approxi-
mation is likely to be adequate if the breadth of the channel is less
than a half-wavelength.

Fig. 3. Canal and equivalent circuit for the plane-wave
approximation. The impedances Z,, = Z,, and

Z,, are given by (5.4)

The precise determination of Z, and Z,, requires the solu-
tion of an integral equation for the normal velocity in M (or, in the
case of an intervening canal, a pair of integral equations for the
normal velocities across the terminal sections of the canal). The
formulation of §§2 and 3 yields variational approximations to Zy
and Z, that are invariant under a scale transformation (i.e. a

99

Miles

change in the mean value) of the velocity in M and stationary with
respect to first-order variations of this velocity about the true solu-
tion to the integral equation (cf. Miles and Munk [1961] and Miles

[ 1946, 1948, 1967]; we omit the explicit formulation of the integral
equation and further discussion of the variational principle in the
present development). The resulting representation of Zy is rela-
tively insensitive to the geometry of H and yields a simple, ex-
plicit approximation that depends essentially only on ka. The cor-
responding representation of Z, requires Green's function (subject
to a Neumann boundary condition) for the closed harbor, the explicit,
analytical construction of which is possible only for those boundaries
(rectangular, circular or circular-sector, and elliptic or elliptic-
hyperbolic sector) that permit separation of variables; however, we
may infer the matrix representation of this Green's function for a
polygonal approximation to an arbitrarily shaped harbor from Lee's
[1971] collocation solution of the general problem. We give explicit
results for a circular harbor in §6 with special emphasis on the
Helmholtz mode. It appears from these results that a large harbor
with a short entrance or a small harbor with an entry canal of length
comparable with R may resonate in the Helmholtz mode under
tsunami excitation.

II HARBOR IMPEDANCE

Let x and y be the Cartesian coordinates in the free sur-
face, t the time, w the angular frequency,’ h the depth,

1/2

c = (gh) and k =" (2.1a,b)

the wave speed and wave number, ¢€ the free-surface displacement,
a the x-component of the particle velocity, 6 and u the corre-
sponding complex amplitudes, such that

{£(x,y,t),i(x,y.t)} = @[{0(x,y),ulx,y)}e”’], (2.2)

where ® implies the real part of and j=y-1,
r= udS (dS = h dy) (2.3)
M
the flow through M,

Vee a u* av ) ae dy (2.4)

*
a weighted measure of the displacement in M, where u_ is the

100

Resonant Response of Harbors (The Harbor Paradox Revistted)

complex conjugate of u,

Zy= V/L= n| i‘ u ay|* i tu* dy (225)

the harbor impedance, and

* *
P= ZR (ogn J tu dy} =z pgR(VI ) (2. 6)

the rate at which energy flows through M. We may regard eV, 6I,
(a/B)Z,,, and aPR(VI") as the voltage, current, impedance, and
power in an equivalent electrical circuit, where the constants of
proportionality, @ and £6, may be chosen to obtain convenient
electrical units. The choice a=86=1 is implicit in the discussion
in §1, but not in what follows except as noted.

Solving the shallow-water equations (Lamb [1932], §189) for
an assumed velocity in M, subject to the boundary condition that the
normal derivative of €, n° V6, vanish on B, the lateral boundary
of the free surface in H, we obtain

&(x,y) = (jw/g) \ G(x,y30,n)u(0,n) dy (2:7)

where

G(x,y3,7) = » (ke - Ky! Walxs yal >7)> (2.8)

is the point-source Green's function for H, the w, are the nor-
malized eigenfunctions for the closed harbor, and the summation is
over the complete set of these functions. The wy, are real and satisfy

(v2 + 1) Up 0) (x,y in Hi); (2. 9a)
(n> V)b, = 0 on B, (2. 9b)

and
i. Yn, dA = 6, (2. 9c)

where k, are the eigenvalues (resonant wave numbers), and 6mn
is the Kronecker delta. We designate the degenerate (but non-trivial)

Miles

solution corresponding to w= const. by n= 0:

me eO) oe Ae (2. 10)

where A is the area of H. We also note that more explicit results
may require the use of two indices to count off the individual modes,

The exact determination of the assumed velocity u(0,y)
requires u and ¢ to be matched across M to the corresponding
solution of the exterior boundary-value problem (see §3 below).

This matching condition yields an integral equation for u(0,y), the
exact solution of which in finite terms does not appear to be possible;
however, simple approximations to u(0,y) are capable of yielding
excellent approximations to Zy and Z, by virtue of the associated
variational principle (cf. Miles [1946, 1948, 1967] and Miles and
Munk [1961]). We proceed directly to such approximations by intro-
ducing the normalized trial function f(y), such that

HO ,Alasell /AY Ely), (sy) dgtayits (2.14a,b)
M

In the subsequent development, we neglect the dependence of f(y) on
k and assume that it depends only on the geometry of M. The
validity of this approximation, which also implies that f(y) is real,
depends essentially on the antecedent approximation ka << 1.

Substituting (2.11) into (2.4) and (2.7), combining the results
in (2.5), and invoking (2.8), we obtain

v=! tf * ay (2.12)
M
and
Zi » Lins (2.13)
n
where
. 2 . p-
Ww a w n
a (geallf, wel = (gaa). ee

is the modal impedance, and pp is a dimensionless measure of the
excitation of the n'th mode through M (note that p,o=1 and

Z, = 1/jwA). The Z, in the equivalent circuit appear in series, Z,
as a capacitor, and each of the remaining Z, asa parallel combina-

102

Resonant Response of Harbors (The Harbor Paradox Revisited)

tion of an inductor and capacitor, L,= tin/(weA) and C,= A/\in. The

dominant terms in Zy as w—~0O are Zo and the sum of the inductive
reactances obtained by neglecting w* relative to wa in the remaining

Lane

III. RADIATION IMPEDANCE

The solution of the shallow-water equations in the exterior
half-space (x <0) for a prescribed incident wave, say Ci(x,y), and
the assumed velocity u(0,y) in the harbor mouth is given by
[ Miles and Munk 1961]

C(x,y) = O(x,y) + O)(-x,y) + O.(x.y), (3. 1a)

where

t.(x,y) = - 3 (w/g) | He [klx? + fy-nl?)”?Juio,n) an (x = 0),
(3. 1b)

He is a Hankel function, the first two terms on the right-hand side

of (3.1a) give the solution for total reflection from the plane x = 0
(as would occur if M were closed), and ¢, is the scattered wave.
Substituting u into (3.1) from (2.11), setting x = 0, and then sub-
stituting the result into (2.12), we obtain

Vee Vi 2/23 (3.2)
where
Vi = 2{ t.f* dy (3. 3a)
M
= 26.(0,0) (ka << 4) (3. 3b)!

is the equivalent exciting voltage of the incident wave, and

Za $ (w/e) | Hy (kly-n|)£ (y(n) dn dy (3.4)
M“M

The definition of Vi implicit in (1.1) corresponds to the approxi-

mation (3.3b).

103

Miles

is the radiation impedance of the harbor mouth. The equivalent
circuit corresponding to (3.2) is sketched in Fig. 2a.
The velocity distribution in M for ka<< i corresponds to

that for potential flow. Normalizing this distribution according to
(2.11b), we obtain

fly) = 0 [ay -yV* (ly| <a). (3.5)
Substituting (3.5) into (3.4) and invoking ka << 1, we obtain
Zy= (w/c )[3 + jAyka)] (ka << 1), (3.6)

where

wAy = in [ 8/(yka)] , (3.7)

and In y = 0.577... is Euler's constant.

IV. RESONANT RESPONSE

An appropriate measure of the response of the harbor to a
prescribed incident wave is the mean-square elevation, say o%, as
determined averaging over both space and time (the temporal
average of (° is 3|¢]|*):

c*=ta'( Iz |? da. (4.1)
H

Substituting { into (4.1) from (2.7), invoking (2.8) for G and (2.11)
for u, carrying out the integration over A with the aid of (2.9c),
and invoking (2.14) for Zn= Vn/I, where V, is the voltage induced
across Zp by I, we obtain

ot= 3) wily =slviF > Oe, (4.2)
n n
where
Gul) = pa [Va/Vi | = wae |Zn/(Zyt Z| (4.3)

is the amplification factor for the n'th mode, and

Kk = k°A = w(A/gh) (4.4)

104

Resonant Response of Harbors (The Harbor Paradox Revisited)

a a dimensionless measure of (the square of) the frequency (similarly,
BA). 2 Invoking (3.3b) on the hypothesis a PK << 1, we obtain
of = mraty | for the (temporal) mean-square elevation of 2Ci, by
vintue of le (4.2) reduces to

=o?) Qk). (4.5)
2

The hypotheses (1.3a,b) imply |Z,| << + Z| for each
of the modal impedances in the summation of (2.13) except in the
neighborhood of kK = K,, where the sum may be approximated by

ZyF (jo /c*) [A +p,(x, - «71, (4. 6a)
where

A, = y Barlen? (4. 6b)

m=0 being excluded from the summation. Invoking (2.14), (3.6),
and (4.6) in (4.3), we obtain

Gol) = fh u? + [ado(a) - 1]*5°V? (4. 7a)
and
Gn(ic) = wl? $2 (ie = wal +L - tg - wel f?, (4. 7b)
where
A(x) = Ay + Aj(ka), (4. 8a)
A, = A,+Aj(k.a) (n #0). (4. 8b)

The peak values of Gy, are given by
mel -I/2
Go = 2Ko and G,= 2n, A, (n # 0), (4. 9ayvb)
where K = he is the series-resonant point determined by
~ ~ ~ -|
KyA(K,) = 1 and Ky, = K, + pA, (n # 1). (4.10a,b)

~

The amplification factor drops off sharply on both sides of K = K,

105

Miles

andis O(1/A,) for |K - k,| >> 41/A,. The point kK =k, corresponds
to parallel resonance (Zp =o), for which the total flow through M
vanishes (I = 0) whilst o* remains of the same order as o;. We
define the Q of the resonant response near K = Kp as the ratio of

the resonant frequency to the half-power bandwidth, such that [ the
frequencies at the half-power points are proportional to fra (143 Q:')]

G[xK,(1 + Gr )] = GL (4.11)

Substituting (4.7) into (4.11) and invoking (4.10), we obtain the first
approximations

~-| ~

Qo = 2k =G, (4. 12a)
and
il 2 4~-,2
Q, = 2h, KA, = 2 KAG,- (4. 12b)

Now suppose that the incident wave is random with the power
spectral density Sj(f), such that

©
oe - S| (£) df (w = 2rf), (4.13)
0
where f is the frequency. Generalizing (4.5), we obtain

2
c=) f S,(£) |G,(«) | af (4.14)
fe]
n

for the power spectral density in the harbor. Substituting (4.7) into
(4.14), invoking w= ck/VJA, and calculating the contribution of the
resonant peaks at ™ = w, on the hypothesis that their bandwidths are
small compared with those of S,(f), we obtain

o° = (gh/a)'” », PS) (Eq) (4.15)
n
where
Be ee a Ghd 2 ye aol
P, = (4m) Ky an [it (O,/Ka) (kh - Re dd (4. 16a)
0)
- 5 Pol Gr (Qn/ ha oO) (4. 16b)

106

Resonant Response of Harbors (The Harbor Paradox Revisited)

is the power-spectrum-amplification factor for the n'th mode. Sub-

stituting (4.9), (4.10) and (4.12) into (4.16b), we obtain

Pp =4Ki”, (4.17)

from which we infer that narrowing the harbor mouth does not affect
the mean response to a random input except in the Helmholtz mode,
but that it does increase significantly the response in that mode [ this
conclusion ignores the increase in viscous dissipation that would be
associated with narrowing the mouth].

V. EQUIVALENT CIRCUIT FOR CANAL

We now interpose a cana of breadth b and length £ between
the harbor and the coast, as shown in Fig. 3, and obtain the equivalent
circuit on the assumption that only plane waves need be considered
in the canal. This approximation is strictly valid only for kb << 1,
but a more complete analysis shows that the effects of the cross- waves

(y-dependent modes) are not likely to be significant for kb <r.

Invoking the plane-wave approximation, u = u(x) and ¢ = C(x),
in (2.3) and (2.4), we obtain

I(x) = bhu(x) and V(x) = C(x). (5.1 a,b)

Assuming 1(0) =I, and I(£) = Ig, we obtain the transmission-line
solution

I(x) = csc k£[ I, sin k(£-x) + I, sin kx] (5. 2a)
and
V(x) = (jbc sin kf)! EI, cos k(£-x) - Ip cos kx]. (5. 2b)

Setting V(0) = V, and V(£) = Vp in (5.2b), we obtain the matrix
equation

Vv _ Zi; 22 I (5.3)

V2 Z i220 -T,

Twe use canal in the same sense as Lamb [ 1932, §169ff] - Some might

regard the synonym channel as more appropriate in the present context.

107

Miles

where
Z 1; = 'Zoo— = (j/be) cotyiel, Zio = - (j/bc) csc kf,
and (5.4)

Zi - Zip = Zop- Zio = (j/bc) tang kf.

The four-terminal network implied by (5.3) and (5.4) is sketched in

Fig. 3, wherein the arms (Z,, - Z,) and pillar (Z, ) are inductive

and capacitative, respectively, for kl<qm (£ less fran a half-wave-
length).

The preceding results remain valid for a canal of arbitrary
(but constant) cross section S if h=S/b, where b is the breadth
of the canal of variable depth in the sense that the effects of the cross
waves (y-dependent modes) that are generated by a change in depth
are negligible in the shallow-water approximation (see Lamb, §176
for a qualitative argument and Bartholomeusz [ 1958] for a proof).

Inserting the equivalent circuit for the canal between the
equivalent circuits for the harbor mouth (at x = 0) and the harbor
(at x = £), we obtain the equivalent circuit shown in Fig. 4a. Cal-
culating I, and the corresponding voltage drop across Z, and
invoking (4. 3a) for the modal amplification factor, we obtain

Rp Le gelMytse) Le te(Ayte fl

Fig. 4. Equivalent circuit for harbor connected to coast through
canal: (a) general case; (b) Helmholtz mode (kA << 1,
kf << 1).

108

Resonant Response of Harbors (The Harbor Paradox Revtstted)

ZI
p¥2Gi(x)= [Vil [Zale|

2 /-l
[(Z,, + ZiV(Z,, + Zs) Zs | Ze Zao

ol -\
[(Zy + Z) cos kf +j{(bc) +bcZZ} sin ké| |Z, |,

(5. 5c)

where (5.5c) follows from (5.5b) through (5.4). The frequency de-
pendence of G,(x) is qualitatively similar to that established in §4,
but K,- K, may not be small. The values of G, and Qn may be
substantially larger than those given by (4.9) and (4.12); however,
(4.17) remains valid for n#0, andthe results therefore are of
limited interest. There also exist modes that correspond to reso-
nance of the canal itself, for which x= is approximately a node
and the motion excited in H is small, but these, too, are governed
by (4.17) in the sense that decreasing the channel width does not
affect the mean response of the canal to a random input except in
the Helmholtz mode.

We consider further the gpecial case of Helmholtz resonance,
assuming kf << 1 as wellas k A<<i. The equivalent circuit then
reduces to that of Fig. 4b. Calculating |V,/V,| in this circuit
and neglecting terms of O(k*b£) relative to unity, we obtain

Gol) = {5 (1 +a) e+ [KA(K) - 1, (5.6)
where
a=bl/A (5.7)
is the ratio of the canal and harbor areas, and
A(x) = Ay t+ (1 + a)Ay(ka) + (1 + Za)(£/b). (5.8)
Resonance is determined by K,A(kK) = 1 and yields
-l~-l

G,=Q,= 2(1 ta) Ko (5.9)

and
(5.10)

in place of (4.9a), (4.12a), and (4.17).

109

Miles

VI. CIRCULAR HARBOR

The eigenfunctions determined by (2.9) for a circular harbor
of radius R are given by

Vogt 10) = A” [ I,(ide)T' Jolidgr/R) (m= 0), (6. 1a)
-/2
(r,e) = (2) |1 - (@) 7 GER) |e
vod) = AY | - GE) | a

sin m8

(ni = £5, (6. 1b)
and

In(ims) = 9 (m= 0,152,257 B'S 0, ae eee)

where r is the polar radius measured from the center of the harbor,
68 is the polar angle measured from the midplane of the mouth, we
write Wmdr,9) in place of u,(x,y), the indices m (the number of
azimuthal nodes) and s (the number of radial nodes) jointly replace
the single index n in §2, and the eigenfunctions obtained by choosing
the alternatives cos m@ and sin m@ are distinct. The eigenvalues
are given by

Kms = lig) « (6. 2)

The zero'th mode of (2.10) corresponds to m =s = 0, for which
tga Or

We specify M by R=1 and -20y< 0 < 20,4, where

0,= a/R << 1, (6.3)

by virtue of which we may neglect the curvature of the harbor
boundary over its intersection with the straight coastline. The
essential approximation is sin 270, = 204; which is in error by less
than 5% for a/R<i1. Carrying out the calculation of p,, (2.15),
and A, (4.6), on the basis of the approximations (6.3) and (3.5),
we obtain

tims= (2 - Sol 1 - (m/jt,) 1 [4 + Olm'6)] (6.4)

for the cos m@ modes and pms=0 for the sin m@ modes [the
approximation (6.4) is not uniformly valid as m—~ o, but it suffices
for all but the calculation of A,] and

110

Resonant Response of Harbors (The Harbor Paradox Revisited)

nA, = Es + In (4R/a) (Oy << 1). (6.5)

Combining (3.7) and (6.5) in (4.8), we obtain
wA,. = 3.0135 + 24n(R/a) - £n(kR), (6. 6)

wherein k= kp, for n# 0.

The resonant wavelength, = 2m/ky, G) = Q), and Py for
the Helmholtz mode, as determined by (4. 9a), (4.10a), (4.12a), and
(4.17) in conjunction with (6.6) are given by the lowest curves in
each of Figs. 5-7. The higher curves in Figs. 5-7 are based on
(5.8) - (5.10) and illustrate the striking effects of an intervening
canal on Helmholtz resonance. Q,, as determined by (4.12), is
plotted in Fig. 8 for the first five modes. The remarkable sharpness
of the higher modes, vis-a-vis the Helmholtz mode, is borne out by
Lee's [1971] experiments.

b/R

Fig. 5. Wavelength for Helmholtz resonance of circular harbor plus
canal (b= a for £=0). The results are strictly valid only
for b/R << 1 and kof << 1, but the corresponding errors
are not likely to exceed 5 - 10% for b/R< 1 and kof <3

Miles

0.01 0.03 0.1 b/R 0.3 1.0

Fig. 6. Resonant amplification factor, G, = Q,, for Helmholtz mode
in circular harbor. kof > 3 to the right of the dashed line.

Resonant Response of Harbors (The Harbor Paradox Revisited)

Fig. 7. Power-spectrum-amplification factor for Helmholtz mode
in circular harbor. Kol > $ to the right of the dashed line.

Miles

1000

(m,s) = (0,1)

300 PF

100

Qms

30

Fig. 8. Q,. for the first five modes ina circular harbor. The dashed
portions of the curves correspond to ka> 1

114

Resonant Response of Harbors (The Harbor Paradox Revisited)

The period for the Helmholtz mode is given by

T, =r,/¢ = an(A/gh)? KY? (6.7)

Choosing R=1000' and h = 20', we obtain Ty = 2\,/rR minutes,
which approximates typical tsunami periods (20 - 40 minutes) for
\,/2mR in the range of 5-10 (see Fig. 5). We infer that a large
harbor with a short entrance (£/R << 1) or asmall harbor witha
canal (£/R ~ 0.3-3) may act as a Helmholtz resonator under
tsunami excitation.

REFERENCES

Bartholomeusz, E. F., "The reflexion of long waves at a step,"
Proc. Camb. Phil. Soc., 54, 106-18, 1958.

Carrier, G. F., Shaw, R. P. and Miyata, M., "The response of
narrow mouthed harbors ina straight coastline to periodic

incident waves," J. Appl. Mech. (in press), 1970.

Garrett, C. J. R., "Bottomless harbors," J. Fluid Mech., 43,
443-49, 1970.

Lamb, H., Hydrodynamics, Cambridge University Press, 1932.

Lee, J. J., “Wave induced oscillations in harbors of arbitrary
shape," J. Fluid Mech., Boy (2-93, 1975.

Miles, J. W., "The analysis of plane discontinuities in cylindrical
tubes," Parts Tand II. J. Acoust. Soc. Am., 17, 259-71,
272-84, 1946.

Miles, J. W., "The coupling of a cylindrical tube to a half-infinite
space," J. Acoust. Soc. Am., 20, 652-64, 1948.

Miles, J. W., "Surface-wave scattering matrix for a shelf,"
J. Fluid Mech.., 23, 755-67, 1967.

Miles, J. W., "Resonant response of harbors: an equivalent-circuit
analysis," J. Fluid Mech., 46, 241-65, 1971.

Miles, J. W. and Munk, W. H., "Harbor paradox," J. Waterways
Harb. Div., Am. Soc. Civ. Engrs, 87, 111-30, 1961.

Morse, P. M., Vibration and Sound, New York: McGraw-Hill, 1948.

Rayleigh, Lord, Theory of Sound, New York: Dover, 1945.

pa

‘ dy Dilacas
its ail

on ia
eS a

rikoonw

UNSTEADY, FREE SURFACE FLOWS;
SOLUTIONS EMPLOYING THE LAGRANGIAN
DESCRIPTION OF THE MOTION

Christopher Brennen, Arthur K. Whitney
Caltfornta Institute of Technology
Pasadena, Caltfornia

ABSTRACT

Numerical techniques for the solution of unsteady free
surface flows are briefly reviewed and consideration
is given to the feasibility of methods involving param-
etric planes where the position and shape of the free
surface are known in advance. A method for inviscid
flows which uses the Lagrangian description of the
motion is developed. This exploits the flexibility in
the choice of Lagrangian reference coordinates and is
readily adapted to include terms due to inhomogeneity
of the fluid. Numerical results are compared in two
cases of irrotational flow of a homogeneous fluid for
which Lagrangian linearized solutions can be con-
structed. Some examples of wave run-up on a beach
and a shelf are then computed.

I. INTRODUCTION

There are many instances of unsteady flows in which analytic
solutions , even approximate ones, are not available. This is par-
ticularly true of free surface flows when, for example, non-linear
waves or even slightly complicated boundaries are involved. Though
analytical methods are progressing, especially through the use of
variational principles (Whitham [1965]) and, in some cases, the
non-linear shallow water wave equations yield important results
(Carrier and Greenspan [ 1958]) there is still a need for numerical
methods. Indeed, numerical "experiments" can be used to comple-
ment actual experiments.

Until very recently numerical solutions in two dimensions

£17

Brennen and Whitney

invariably seemed to employ the Eulerian description of the motion
though the Lagrangian concept has been used for some time in the
much simpler one-dimensional case (e.g. , Heitner [ 1969], Brode

[ 1969] ) and to make small time expansions (Pohle [1952]). Perhaps
the best known of these Eulerian methods is the Marker-and-Cell
technique (MAC) begun by Fromm and Harlow [ 1963] and further
refined by Welch, et al. [1966], Hirt [1968] , Amsden and Harlow
[1970] , Chan, Street and Strelkoff [1969] and others. The most
difficult problem arises in attempting to reconcile the initially un-
known shape and position of a free surface with a finite difference
scheme and the necessity of determining derivatives at that surface.
In the same way, few solutions exist with curved or irregular solid
boundaries. In steady flows, mapping techniques have been em-
ployed to transform the free surface to a known position (e.g. ,
Brennen [ 1969]). It would therefore seem useful to examine the use
of parametric planes for unsteady flows. The Lagrangian description
in its most general form (Lamb [ 1932]) involves such a plane and by
suitable choice of the reference coordinates, the free surface can
be reduced to a known and fixed straight line. However a discussion
of other parametric planes and mapping techniques is included in
Section 3.

The major part of this paper is devoted to the development
of a numerical method for the solution of the Lagrangian equations
of motion in which full use is made of the flexibility allowed in the
choice of reference coordinates. For the moment, we have restricted
ourselves to cases of inviscid flow. Very recently, Hirt, Cook and
Butler [1970] published details of a method which employs a
Lagrangian tagging space but is otherwise similar to the MAC tech-
nique. This is further discussed in Section 4B.

Il. LAGRANGIAN EQUATIONS OF MOTION

The general inviscid dynamical equations of motion in
Lagrangian form are (Lamb [ 1932]):

< Ya Ze P,
1
(Xy-F) | Xbp +(%y-G) P Yep (2-H) Zoe +S 7 Pep =o (1)
pal Y, Ze P,

where X,Y,Z are the Cartesian coordinates of a fluid particle at
time t, F, G, H are the components of extraneous force acting upon
it, P is the pressure, p the density and a, b, c are any three
quantities which serve to identify the particle and which vary con-
tinuously from one particle to the next. For ease of reference
(X,Y,Z) are termed Eulerian coordinates, (a,b,c) Lagrangian co-
ordinates. Suffices a,b,c,t denote differentiation.

Lagrangtan Solutions of Unsteady Free Surface Flows

If Xo, Y,, Z, is the position of a particle at some reference
time ty (when the density is p,) then the equation of continuity is
simply

Q(X,Y,Z) _ , (Xp, Yo, Zo) (2)

Ae eter,

Frequently it is convenient to define a, b, c as identical to Xp», Yo, Zo;
thus reducing the R.H.S. of (2) to p 9; however it will be seen in the
following sections that flexibility in the definition of a, b, c is of
considerable value when designing numerical methods of solution.

If the extraneous forces, F, G, H, have a potential 922 and
p, if not uniform, is a function only of P then, eliminating 2 + P/p
from (1):

0 ee
ar (UpX, - U.X, + V,Y, - V-¥, + W,Z, - W,Z,) =—b= 0

) oT:
Br (UcXq- UgXe +t VeY, - Va¥_ + WyZ, - U,Z,) = ae = 0 (3)

) oT
5-H (UgX, - UpX, a Vows = Vii%e Le Wo2p - W, 2Z,) i “5 = 0

where, for convenience, the velocities Xy> Y,> Z, are denoted by
U, V, W. The quantities [,, [,, [3 are related to the Eulerian
vorticity components, 6,, 5, 63 by

PD, = 6,(¥,2, - ¥,2,) + (2.x, - ZX) + O,(X,¥, - X,¥,)

Ty = 6)(%Zq - YgZ_) + SA(Z,Xq - ZX.) + 05(X, ¥, - X,Y.) (4)

r; 7 CY oZp ~ YpZ) C(2,X, - Z,X,) " C3(X,Y, e X,Y)

(Thus, of course, vorticity changes with time are due solely to
changes in the coefficients of the L.H.S. of (4) which, in turn,
represents stretching and twisting of the vortex line.) Given the
vorticity distribution €(X,Y,Z) at some initial time, t,, T(a,b,c)
(which is independent of time) may be obtained through Eqs. (4) and
used in the final form of the dynamical equations of motion, namely
Eqs. (3) integrated with respect to time.

ts

Brennen and Whitney

For incompressible, planar flow the equations reduce to
Continuity: XqYp- YgXp = F(a,b) (5)

(or differentiated w.r.t. t):
U,Y,- U,Yy ete VaX, = Vix. = 0 (6)
Motion: UX, - U,X, + UA Ss - Vee =) = ia.) (7)

By introducing the vectors Z=X+t+iY and W= U - iV, (6) and (7)
conveniently combine to:

ZW», = Zp Wa i I(a,b). (8)

Other types of flow have also been investigated. For example,
in the case of a heterogeneous, or non-dispersive stratified liquid in
which p is a function of (a,b), Eq. (8) becomes:

t
! |
ZaWy~ ZpWq =~ (T(asiTaey, ~ 5) (Xp - FMP, ~ PG)
te]

+ (4, - G)(p,¥, - p,¥,) odtee M9)

The integral term therefore manufactures vorticity. The methods
developed for a homogeneous fluid in Sections 4A to D are modified
in Section 4E to include such effects.

III. OTHER PARAMETRIC PLANES

It may be of interest to digress at this point to consider other
parametric planes (a,b), which are not necessarily Lagrangian.
That is to say the restrictions X,(a,b,t)=U, Y,(a,b,t) = V are
abandoned so that U,V are no longer either Eulerian or Lagrangian
velocities. Provided J = 8(X,Y)/8(a,b) # 0, or ow, the equation for
incompressible and irrotational planar flow remains

ZW, - Z,W, = 0- (10)

To incorporate one of the advantages of the Lagrangian system, it

is required that the free surface be fixed and known, say on a line

of constant b. Then the kinematic and dynamic free surface conditions
are respectively

120

Lagrangian Solutions of Unsteady Free Surface Flows

(U =. X%,)¥,-.(V - YK 50 (11)

(U, + F)Xq + (Vz; + G)¥, + (U - X,)U,g - (V - ¥4)Vq = 0. (12)

Now a useful choice concerning the (a,b) plane would be to
require the mapping from (X,Y) to be conformal. Then, of course,
(10) simply reduces to the Cauchy-Riemann conditions Ug, = - Vp,
U,p= V, sothat W =U - iV is an analytic function of c=a+tib or
of Z.

In this way, John [1953] has constructed some special, exact
analytic solutions. The kinematic condition, (11), has the particular
solution W(a,t) = Z,fa,t) on.the free surface, which implies W(c,t) =
Z,(c,t) by analytic contingation. If, in addition,

+ (F +iG) = iZ,K(c,t) (13)

where K is real on the free surface, then the dynamic condition
thereon is also satisfied. John discusses several examples for various
choices of the function K.

The potential of such methods may not have been fully realized
either analytically or numerically. In the latter case, however, the
conformality of the (X,Y) to (a,b) mapping is not necessarily a
great advantage, whereas a fixed and known free surface position
most certainly is.

The digression ends here and the following sections develop
a Lagrangian numerical method from the equations of Section 2.

IV. A NUMERICAL METHOD EMPLOYING LAGRANGIAN
COORDINATES

A method for the numerical solution of incompressible,
planar flows is now described. It attempts to take full advantage of
the flexibility in the choice of Lagrangian coordinates.

A. Time Variant Part

The method uses an impiicit scheme with central differencing
overtime, t. Thus Z Pia, b) is determined at a series of stations
ab riee: distinguished by the integer, p- Knowledge of velocity values,
EY , at a midway station p +2 enables y Ated (a,b) to be found from
Z’ through the numerical approximation

p+l pri

Zs, Zt TZy (error order TZeee) (14)

121

Brennen and Whitney

where 7 is the time interval. Acceleration values, Zi , needed in
the free surface condition (Section 4C) are approximated by

(Zo - Zz CAWG (error order 7T4Z 4444). Thus the main part of the
solution involves finding P, ‘. knowing Z? Z; and their previous
values.

The first time step (from p=0 to p=t1) requires a little
special attention. Clearly Z °(a,b) is chosen to fit the required
initial conditions. But further information is required on a free
surface which will enable the accelerations in that condition to be
found (see Section 4C).

B. Spatial Solution

A method of the present type is restricted to a finite body of
fluid, S. However, S, could be part of a larger or infinite mass of
fluid if an "outer" approximate solution of sufficient accuracy was
available to provide the necessary matching boundary conditions at
the interface. The region, S, need not be fixed intime. It would
indeed be desirable, for example, to "follow" a bore.

In a great number of cases of widely different physical ge-
ometry including all the examples of Section 6, it is convenient to
choose S to be rectangular inthe (a,b) plane. This rectangle
(ABCD, Fig. 1) is then divided into a set of elemental rectangles.
The motion of each of these cells of fluid is to be followed by deter-
mining the Z values at all the nodes.

NODE NUMBERING IN FREE
SURFACE CONDITION:

D a

Fig. 1. The Rectangular Lagrangian Space, S, Showing the
Numbering Conventions Used

122

Lagrangian Soluttons of Unsteady Free Surface Flows

Making the assumption of straight sides the actual area ofa
cell in the physical plane is

A= 3[(X, - X)(¥Y, - ¥,) - (X,- X,)(¥, - Y3)] (15)

Number suffices refer to the four vertices, numbered anticlockwise;
other node numbering conventions are shown in Fig. 1. If this area
is to remain unaltered after proceeding in time from station p to
pti through Eq. (14) then

Imag {(Zo - Za)’(W, - We) - (Z, - Zg) (Wp - Wa) }

pti/2 Pp. ae
+ r{(U, - U,V, - V4) - (Up - U,V, - Vz} + 2A AD

=0=R, (16)

where the terms on the L.H.S., second line are numerical cor-
rections required to preserve continuity more exactly and prevent
accumulation of error over a large number of time steps. The nu-
merical value of the L.H.S. at some point in the iterative solution
is termed the continuity residual, Rg.

Assuming linear variation in velocity along each side of the
cell, evaluating the circulation around 1234 and setting this equal to
the known, initial circulation, I, yields (in the case of a homo-
geneous fluid):

Real {(Z, - Z,)(W, - Ws) - (Z, - Z3)(Wp - W,)} - 20, =0= Ry (17)

Slight hesitation is required here since, for validity, the Z and W
values in this equation should relate to the same station in time.

But by choosing to apply it at the midway stations and substituting
zPrve Ze + (r/2)ze" the T terms are found to cancel and (17) per-
sists when the values referred to are Z and Ww?*!# R, is the circu-
lation residual. The modification of (17) in the case of a hetero-
geneous fluid is delayed until section 4E.

Combining (16) and (17) produces the cell equation:
(Z, - Z,)(W, - W,) - (Z,, - Z,)(W, - W,) Main Part

Zar - Aa i Continuity
T

t+ ir {((U,- U,)(V, - Vay - (U,- U,)(V, - V,)} + Corrections

- 21, Permanent Cell Circulation Term

123

Brennen and Whitney

1
+ Te {(Wig Wier a)(Z, - Z,) Higher
= (Wis ots Wio = W3 - Wz) (Z4 - Z3) Order
+ (W, + Wi. - We - 3)(Z, - Zs) Correction
- (W, + W,, - W, - W,)(Z, - 2,)} if required
=0=R,+iR, =R, the cell residual. (18)

The higher order correction, included for completeness, allows the
shape of the cell sides and the variations in velocity along them to be
of cubic form. Without it the neglected terms are of order Z Wppp»
ZabWap, etc., with it they are of order Za Wbppbppp etc. Values
referred to are Z? and Wt [Ptl2 vPtv2,

Though this derivation of the cell equation is instructive, it
can be obtained more directly (except for the continuity correction)
by integration of (8) over the area of the cell in the (a,b) plane
(using Taylor expansions about the center of the cell).

p The cell equations must now be solved for weve (W'F="PV);
Z being known, in order to proceed in time.

In a recently published paper, Hirt, Cook and Butler [ 1970]
take a rather different approach in which the (a,b) plane is employed
merely as a tagging space. The equations are written in essentially
Eulerian terms, no derivatives with respect to a,b appearing. The
numerical method (LINC) is similar to that of the MAC technique
(Fromm and Harlow [1963], Welch, et al. [1966] , Chan, Street
and Strelkoff [1969], etc.) and involves solving for the pressure at
the center of a cell as well as for the vertex velocities. Advantages
of the method described in the present paper are: the pressure has
been eliminated (though this may be disadvantageous in compressible
flows); no special treatment is required for cells adjacent to bound-
aries; inhomogeneous density terms are relatively easily included.
However, since the LINC system is based on the Eulerian equations
of motion, the inclusion of viscous terms is more easily accomplished
than in the present method where such an attempt leads to horrendous
difficulties.

C. Boundary Conditions

To complete the specifications, a condition upon waits is
required at each of the boundary nodes. ot Bis usually takes the form
of an expression connecting U and V °. For example, solid
boundaries, whether fixed or moving in time, may be prescribed by
a function, F(X,Y,t) =0. Then the required relation is

124

Lagrangtan Solutions of Unsteady Free Surface Flows

p+v2

F(x? + tu y+ ey 4) = 0 (19)

Dynamic free surface conditions are simply constructed from
Eqs. (1). If, for example, the only extraneous force is that due to
gravity, g, inthe negative Y direction, the condition on a free
surface suchas AB, Fig. 1, is

XyXq t (Yes + g)Y, =

a a 7 (20)

a” Nea 3/2
(Xq + Ya)

where T is the surface tension if this is required.

Unlike the field Eqs. (8) or (18) these boundary conditions
may not be homogeneous in all the variables. In a given problem
only the boundary conditions are altered by different choices of
typical length, h (perhaps an initial water depth), and typical time,
say yh/g inthe above example. Then, using the same letters for
the dimensionless variables, g and T/p in Eq. (20) would be re-
placed by 1 and S = T/pgh’. The numerical form of that condition
used at a free surface node suchas 0 (Fig. 1) is:

¥ 2 +1/ =
a x)"(Ue i Us ip FAY ay (Ve ies Vo on 7)
= 18(R - 5) (24)

where F is assessed at each node as

_ LOK, - XY, + ¥, - 2¥,) - (, - ¥,MX, + X, - 2X0]

ae ee ee ee e E / Cae R e
: [oxesany ty wey

and the accelerations have been replaced by the expressions given
in Section 4A. Again, Eq. (21) relates Up, to Shae since all
other quantities are known.

If the liquid starts from rest at t =0 (as in the examples of
Section 6) then difficulties at the singular point t = 0 can be avoided
by choosing to apply the condition at t= 7/4 rather than t =0.

Using Zy, = 22)"/r and Z=Z°+ (7/4)zV2 at that station the special
boundary condition becomes

1/2 \/2

Us {(X, - Xs)" +F(U, - Uy} + (v, - va)” (vy? + 7g /2) = 0 (22)

125

Brennen and Whitney

in the case of zero surface tension.
D. Method of Solution

It remains to discuss how the equations may be solved to find
at every node. Due to the non-linear terms in (18) and some
boundary conditions as well as to the fact that a good estimate of
w?’*'2 can be made from values at previous time stations, a simple
iterative or relaxation scheme was employed. Such a method in-
volves visiting each cell in turn and adjusting the W values at its
vertices in such a way that repetition of the process reduces the

cell residuals, R, to negligible proportions. But, on arrival ata
particular cell, there are an infinite number of ways in which its
four vertex values can be altered in order to dissipate the single cell
residual. However, experience demonstrated that a procedure based
on the following changes (AW, 9 3anq4) was superior in convergence
and stability to any of the others tested:

wti72

AW, = - AW; = wiR(Z, - Z3)/8A
(23)
AW, = - AW, = wiR(Z, - Z,)/8A

Here w is an overrelaxation factor and A is the area of the cell,
which is unchanged with time and given by the expression (15). These
incremental changes have a simple and meaningful physical inter-
pretation. As can be seen from Fig. 2, they are a combination of two
changes, one representing pure stretching and the other pure rotation,
which dissipate respectively the continuity and circulation components
of the residual.

Having visited each and every cell, the boundary conditions
were then imposed. Where these were Sen in the form A.U?*'/#+
B.V?*!2+ C=0=R,, A,B,C being constants and Rgthe residual,
the following ehanges" were made, the choice being based upon experi-
ence:

p+i/2
AU A| Re
se op te (24)
Ayetv2 al (A® + B*)

The whole process was then repeated to convergence,

E,. Inhomogeneous Fluid

In a non-dispersive, inhomogeneous fluid, p(a,b), which is
independent of time, will be prescribed through the initial choice of
Z (a,b). Indeed in many cases it will be convenient to choose Z° in

126

Lagrangian Soluttons of Unsteady Free Surface Flows

3 4

Fig. 2(a). The Cell in the Reference Plane (a,b)

ee
MQ2=- gq (22-24)

4Q niet (Zaza)
2°! Ga '427 44

R. —_———
AQ, = - gh (Z,-Z3)

INCREMENTAL VELOCITY CHANGES INCREMENTAL VELOCITY CHANGES
WHICH DISPERSE THE CONTINUITY WHICH DISPERSE THE CIRCULATION
RESIDUAL, R_ (PURE STRETCHING) RESIDUAL, R,; (PURE ROTATION)

Fig. 2(b). The Cell in the Physical Plane (X,Y)

such a way that p is some simple analytic function of (a,b). This
is particularly desirable because by substituting for p, pg, py in

Eq. (9), this can then be integrated over a cell area (as in Section 4B)
to produce a convenient additionalterm, 9 on the L.H.S. of the
cell Eq. (18). Since the expression for gre will depend upon that
choice of p(a,b) an example will illustrate this.

If p is to be constant along the free surface, AB, Fig. 1,
and along the bed, CD, it may be possible to choose Z° such that
p is a linear function of b, say p= Ppcop(1 + yb) where
Y= Pap/Pep - 1 and b=1 on AB. Then,

127

Brennen and Whitney

p+i2 p-V/2
Qio34 a G i234
+1/2 a

ve! p
= -31n (1 - p) [{(y, FULUSSU) ° = 40,40, +0, Fu)

p
x {X,- X,-X,+X,}

+ -\/2 Pp
+{V, +Vp tV, +¥,)°"'7-(V, +Vp V5 4Vg) ? + amg} {¥, -¥,-¥, +¥%} ]
Lt p pri/2 p 2
t+ 2-5 in (t-w} [(x,-x)°(u, 40,0" - (u, tu)? 7}
p p+i/2 p-i/2
- (X,- X3) {(U, + U,) a (U;+U,) }

+ =F

+ (¥, -¥,) {(v, tv.) = (vy, - Vv, + 27g}
+1/ p-i/2

~ (Xg- Ya) {(Vp + Va) = (Vg +V_) + 278}

where p = yAb/(1 + yb3,), b3, being the b value on side 34 of the
cell and Ab the difference across each and every cell. The first

term is of order p, the second order p*. The boundary conditions
are usually identical to the homogeneous case.

V. ACCURACY, STABILITY, CONVERGENCE AND SINGULARITIES

A. Accuracy

If the cell equation, (18), is used without the higher order
spatial correction, an indication of the errors due to neglected higher
order spatial derivatives can be obtained by assessing the value of
that correction and inferring its effect upon the final values of W.
Unfortunately, the mesh distribution and mesh size required for a
solution of given accuracy will not be known a priori and can only be
arrived at either by trial and error or by using some technique of
rezoning. The latter method in which cells are subdivided where and
when the violence of the motion demands it, can be difficult to pro-
gram satisfactorily and has not been attempted thus far.

Errors due to higher order temporal derivatives are most
fa regulated by ensuring that, for each cell, both
T| W, "Wel /IZ, - Z| and T|W,- W,|/|Z,- Z, are comfortably less
than unity. A workable rule of thumb can be devised in which a
suitable Tt for a particular time step is determined from the W and
Z values of the preceding step.

128

Lagrangian Soluttons of Unsteady Free Surface Flows

By Stability of Cell Relaxation

Suppose the central member, cell A, of the group of cells
shown in Fig. 3 contained a residual Ry which was then dissipated
according to the relations (23). Transfer functions, Dag, Dg, etc.,
will describe the residual changes, AR,, etc., in the surrounding
cells where

AR, = wD,,R,, etc. (24)

Fig. 3. Z-Plane

For example
Dag = {(Z, - Zis\(Zp- Z4) - (Zig - Z4(Z, - Zs) }i/8A,

where A, is the area of cell A. For convergence of the relaxation
method it is clearly necessary that the w for each cell be chosen
so that all w|D| are significantly less than unity. It is instructive
to inspect the case in which all the cells are roughly geometrically
similar inthe Z plane. Then

ld? - a3
Dial = [Daal = Dae = iD ~ ors vam =i

129

Brennen and Whitney

dd 4
[Dacl = [Dye] = |Dacl = |Dasl * oa = ao ye

where d,, dp are the lengths of the cell diagonals. For square cells,
Y, =Y,=0 and the situation is stable. However difficulties may
arise when the cells are very skewed or elongated and it is in such
situations, in general, that care has to be taken with the relaxation
technique.

C. Observation on the Cell Equation

One feature of the basic cell equation, (18), itself demands
attention. Note that without the higher order spatial correction, the
residuals, R in all of the cells (of Fig. 3) remain unaltered when
the W or Z values at alternating points (say the odd numbered
points of Fig. 3) are changed by the same amount. Such alternating
"errors" must be suppressed. Some damping is provided by the
higher order spatial correction since it is not insensitive to these
changes. But experience showed this to be insufficient unless all
the boundary conditions also inhibited such alternating "errors."
Solid boundaries usually provide adquate damping. For instance,
in Fig. 4(a) fluctuations in U on BC, DA andin V on BC are
obviously barred. But the free surface provides little or no such
suppression and as will be seen in the next section this can lead to
difficulties. It is of interest to note that some of the solutions of
Hirt, Cook and Butler [1970] exhibit the same kind of alternating
errors.

In the MAC technique, neglected higher order derivatives of
the diffusion type and with negative coefficients (a "numerical"
viscosity) can lead to a numerical instability if not counteracted by
the introduction of sufficient real viscosity. In the present method,
as with that of Hirt, Cook and Butler [1970], the convection terms
which cause that problem are not present. The higher order spatial
correction does contain terms of diffusion order, but it cannot be
directly correlated with a viscosity since viscous terms are ofa

2
different form (i.e., like f vVxy I dt). Also, the higher order spatial
correction has a beneficial rather than a destabilizing effect.

D. The Free Surface

By including previously neglected derivatives, the numerical
free surface condition (without surface tension) is found to correspond
more precisely to:

2
A
{XqXt+ * YolYr+ t 1)} + ier { XeaaX t+ + YoaalYt+ t 1)}

2
4
be 7 {XX t YaYererd = 0 (26)

130

Lagrangtan Solutions of Unsteady Free Surface Flows

where Aa isthe a difference across a cell and the second and
third terms constitute truncation errors. Inspect this in the light of
a linearized standing wave solution (see Section 6A), i.e.,

X=a-Mcocos Jkt sin ka’e*”

iT}

Y = bikM cos vik t cos ka ef?

where the variables are non-dimensionalized as in Section 4C and
k is the non-dimensional wave number in the a,b plane. Then,
the second and third terms of Eq. (26) will be insignificant provided

2 2
k (Aa)

respectively. Or, in terms of a wavelength, \ = 21/k:

R. >> 2 and T<< 45 AX (27)

since Aa® AX, the X difference between points on the free surface.
The first condition states the inevitable; namely, that the solution
will be hopelessly inaccurate for (a,b) plane wavelengths comparable
with the mesh-length Aa. Given that the first condition holds then
the second says that 7 << 84AX. For a travelling wave system the
same condition states that T should be less than the time taken for
a wave to travel one mesh length. This constitutes a restriction on
T which is usually more stringent than that of Section 5A. If, for
example, the depth of the fluid is divided into N intervals andthe X
difference across each cell is of the same order as the Y difference
then T should be less than 8/N.

A more difficult problem arises when the first condition is
considered alongside the fact, ascertained in the previous section,
that the field equation provides little or no resistance to disturbances
whose wavelength is equal to Aa. The only resort would seem to be
to some artificial damping technique which would eliminate or sup-
press these small wavelengths. The technique used in the examples
to follow was to relax the W values on the free surface such that
w = pw'?° + (1 - B)W* where W'S° was the value indicated by the
free surface condition, W* the value which would make the numeri-
cal equivalent of Woagg, be zero at that point and B was slightly less
than one half,

1314

Brennen and Whitney

E. Singularities

Successful numerical treatments of singularities depend upon
the availability of analytic solutions to the flow in the neighborhood
of that point. For example, at a corner between solid walls the
velocity varies as the (1 - $)/B power of distance from that junction
where f is the included angle. If this is /2 (as at points C or
D, Fig. 4(a)) the variation is linear and thus the numerical estimate
of the circulation around the cell (see Section 4) in such a corner is
a good one. Where the angle is not 1/2 (D, Fig. 4(c)) errors will
occur due to the non-linear variation of velocity, but corrective pro-
cedures are easily devised.

A great deal less is known about the singularities at a junction
of a free surface and a solid boundary. If the wall is static and verti-
cal (A, Fig. 4(a)) so that X,, = X,= Xp;= 0, etc., it follows from the
equation of motion that if Y, =0 at t=0 then it is always zero for
irrotational flow; the tangent to the free surface at the wall is always
horizontal. Thus the free surface condition without surface tension
is automatically satisfied at such a junction and only weak singular
behavior is expected. But a similar analysis of the case when the
wall begins to move at t=0 (remaining vertical) indicates that Yt
must be infinite at the junction (B, Fig. 4(a)) at t = 0, the singularity
being logarithmic in space. An extension to t#0 has not so far been
obtained. One approach might be a Fourier analysis of the step in
X;; 80 that the steadily oscillating solutions of Fontanet [1961] could
be used. These suggest that Y,, becomes finite for t>0.

SLOPING
BEACH

X

Mactth Xl amexe

FIG. 4(b) FIG. 4(d)

132

Lagrangtan Solutions of Unsteady Free Surface Flows

In the examples to follow (see Figs. 4(a) to (d)) satisfactory
numerical solutions could be obtained by ignoring all but one of the
singularities. The exception was the shoreline, point A, Fig. 4(c).
If B is the angle between the tangent to the free surface at A and
the horizontal then correlating the two boundary conditions yields:

(Zia = - el? /(cot B cos a + sin a) (28)

Thus the sign of B determines the direction of the acceleration up
or down the beach, If the fluid starts from rest at t=0, B=0,
then (Z;+;+)t:0 = 0 and successive differentiation of the basic equation
(8) and the boundary conditions yields (for irrotational motion):

TT.
- ee i
Zrists Zate? Spire 2 tate. =O OF ».unless o@ = 3 (29)
Z Z Z. =0oro, unless a= "or =
ttittt *“atttt > “brttt ? 4 6

These relations suggest a behavior which is logarithmically singular
in time at t=0 unless @=n/2n, n integer. Roseau [1958] found
similar logarithmic singularities in periodic solutions for the general
case which excluded @=1/2n and another set of particular angles
(see also Lewy [1946]). But a systematic analysis of the singular
behavior (especially for t# 0) has not as yet been completed. Rather,
since the relations (29) no longer necessarily hold if the condition
of irrotationality near that point is relaxed, the problem was circum-
vented numerically by replacing the circulation condition on the single
cell in that corner by the condition (28) at the point A and the time

= 0 was avoided by applying (28) at t = 7/4 just as was done with
the general free surface condition (Section 4C).

Note that strong singularities could be introduced by unsuitable
mapping to the (a,b) plane.

VI. SOME RESULTS INCLUDING COMPARISONS WITH LINEAR
SOLUTIONS

A. Lagrangian Linearized Solutions

Linearized solutions to the Lagrangian equations are obtained
by substituting K=at6&, Y=B +n into the equations of continuity
and motion and neglecting all multiples of derivatives of § and n.
For incompressible and irrotational planar flow the Cauchy-Riemann
conditions 6 = - nb, &=Nq result so that € tin, and therefore

Z-c (where c=atib) is an analytic function of c. In the absence

Brennen and Whttney

of surface tension the free surface condition reduces to
Ey ten, =0 (g = 1 in the dimensionless variables) (30)

only when the additional assumption that 1+ << gis made. In this
way harmonic solutions can be obtained for some simple problems.

In passing, it may be of interest to compare Lagrangian
linearization with the more common Eulerian type, at least in some
simple cases. For travelling waves on an infinite ocean the first
order Lagrangian terms are precisely those of Gerstner's waves.
The Eulerian solution must be taken to the third order to achieve this
waveform. On the other hand, while the Eulerian solution is always
irrotational the Lagrangian only approaches it. Thus the comparitive
accuracy of the two methods depends upon what particular feature of
the flow is under scrutiny. A comparison of the works of Zen'kovich
[1947] and Penney and Price [1952] for standing waves on an infinite
ocean demonstrates the same features.

B. Example One, Figs. 4(a),, 551627 9:85 9% and 10

In the example of Fig. 4(a), the liquid is initially at rest in
the rectangular vessel BCDA; between t=0 and t=T the side BC
moves inward according to

2
X,dt) = M sin nt /2T for Ot <RE
=M for C2ck

With a suitable choice of M and T this creates a wave which
travels along the box, builds up on and is reflected by the opposite
wall, AD. The linearized solution (which requires a Fourier
analysis of the free surface boundary condition) is

(os)
Z- c= Xpdt) [1 : a + » R,B,(t) sin ae (31)
k=
where
ve T k
T
R, = M/nk (“4 - 1) cosh (3
B(ticiees ale Gicosn O< <7
k k as

= cos vt + cos y(t - T)} t> Tt

eh
_ | kr kr)“
Meek ee tank |

134

Eigse 6s

Lagrangtan Soluttons of Unsteady Free Surface Flows

t/Y¥ = 0.0 8.0 12.0 16.0 20.0 2U.0 26.0

+ X o ¢

SYMBOL © © 4

90.0 1.0 2.0 3.0

4.0 5.0 6.0 7.0 8.0 9.0 10.0 X
Linearized solution to example 1: M=0,53, T= 327, T=0.53,
showing free surface position at a selection of times, t.

%/¥ = 0.0 8.0 12.0 16.0 20.0 24.0 26.0

+ xX o

SYMBOL © O 4
MESH 65X9 POINTS

0.0 1.0

2.0 3.0 4.0

5.0 6.0 7.0 8.0 9.0 10.0 Xx
Numerical solution to example 1: M=0.53, T=327T, 7 =0.53,

showing free surface position at a selection of times, t

£35

VAS)

1.4

Fig. 7.

1.0

o2
0.0

Bigs: 8.

Brennen and Whitney

t/Y¥ = 0.0 4.0 8.0 12.0 16.0 20.0 23.0 26.0 30.0
SYMBOL © © 4 + X © # X& Z

—~——---

Se ee ee ee

1.0 2.0 3.0 4.0 5.0 6.0 a0 8.0 9.0 10.0: X

Linearized solution to.example 1:.M = 1.16, T = 167, = C.60;
showing free surface position at a selection of times, t

t/Y = 0.0 4.0 8.0 12.0 16.0 20.0 23.0 26.0 30.0
SYMBOL © © 4 + X © # XX Z

MESH 65 X 9 POINTS

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

Numerical solution to example 1: M= 1.16, T = 167, T=0.60,
showing free surface position at a selection of times, t

136

Lagrangtan Solutions of Unsteady Free Surface Flows

=
N
t/Y¥ = 0.0 4.0 8.0 12.0 16.0 20.0 22.0
© A + X © 4

SYMBOL
>| ks

oOo
N

[0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 X
Fig. 9. Linearized solution to example i: M = 2.00, T= 167, T=0.48,
showing free surface position at a selection of times, t

pease aa | a | iE
t/y = 0.0 4.0 8.0 12.0 16.0 20.0 22.0

SYMBOL 0 10) a ate x © Gr
MESH 65X9 POINTS

2.4

2.0

20.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 xX

Fig. 10. Numerical solution to example 1: M=2.00, T=167,7 =0.48,
showing free surface position at a selection of times, t

ve Wg

Brennen and Whttney

and £ is the a difference between the walls AD and BC. In

Figs. 5 and 6, 7 and 8, 9 and 10 the numerical and linearized free
surface shapes are compared for three cases of increasing wave
amplitude. As the amplitude increases the similarity between the

two diverges; both the wave velocity and the build up on the wall
become progressively greater in the numerical solution. Note also
that, especially in Fig. 10, the peak of the wave is much sharper than
in the linearized solution. For amplitudes less than that of Figs. 5
and 6 the results were almost identical.

GC. Example Two, Figs. 4(b), 11, and 12

The second example, Fig. 4(b), introduces moving and curved
solid boundaries; the liquid is disturbed from rest by a bed uplift of
the form:

2 2 =
For Xi <X<X2, ¥,=Msin |S] sin [RAH] for 0<t<T

2k
= 2 nx |
= M sin | 7X- Xt for t> 2
For. Xi Xi wo uke’, Yop = 0 ali at

Within certain extreme limits on M and T this causes a surface
wave immediately above the bed disturbance which then spreads out
to each side and is followed by a depression wave over the bed uplift.
The linearized solution is

00
Vee eS e) +) R _ [itann (75) awe coat ete
k=l

+ Bt) ain (SEE | (32)

where

R, = M {sin (5X2 : in (GX) 1/20? ce (Bis e(x2 = X1))"}

Vy Ee tanh Ens

2sin a for 0 <<" T., =(Z2 for. t > T

=
my
rT

B,(t) =o, cos vtti-(1+o,) cost for O<t<T

2 t g (cos vt + cos v(t - T) foraysh Sn

138

Lagrangian Solutions of Unsteady Free Surface Flows

t/Y¥ = 0.0 3.0 5.0 7.0 10.0 13.0 16.0 18.0 20.0
SYM80L © > x x Zz

90.0 1.0 2.0 3.0 4.0 5.0 X

Fig. 11. Linearized solution to example 2: M=0.344, Xi = 0.75, X2=2.12,
T = 67, T=0.35, free surface positions at selection of times, t

t/Y¥ = 0.0/3.0 5.0 7.0 10.0 13.0 16.0 16.0 20.0
SYMBOL © © 4 + X © # X& Z

MESH 40 X9 POINTS

50.0 1.0 2.0 3.0 4.0 5.0 X

Fig. 12, Numerical solution to example 2: M=0.344, X1 = 0.75, X2= 2.12,
T =67, T=0.35, free surface positions at selection of times, t

13g.

Brennen and Whitney

r, = sech® (3£)/[E% - 4]

and £ is the a difference between the vertical walls. For T of
the order of 2 or 3 and for values of M upto0.3, at least, there
was virtually no difference between the numerical and linearized
solutions. Figures 10 and 11 in which M = 0.344 demonstrate this.

D. Example Three, Figs. 4(c), 13, 14, 15. A Sloping Beach

By altering the condition on the boundary AB of example one
and employing the shoreline treatment of Section 5E, the interaction
of the waves with a sloping beach could be studied. In Fig. 13 a
small wave appraches a 27° beach. As the horizontal inclination of
the tangent to the free surface at the shoreline (B) decreases, the
shoreline (A) accelerates up the beach until B becomes positive.
The acceleration then reverses (as in Eq. (28)) and the wave reaches
maximum run up. The backwash is extremely rapid and positions
t/r = 21, 22 suggest that this causes the small wave which is follow-
ing the main one to break. By this time the cells have become very
distorted and the mesh points excessively widely spaced to allow
further progress. A similar succession of events takes place with
the larger wave and smaller beach angle (18°) of Fig. 14. Note in
this case,the large run-up to wave-height ratio. In neither of these
cases does there appear to be any tendency for the main wave to
break on its approach run. Indeed the reaction with the beach is
similar to the behavior predicted by Carrier and Greenspan [ 1958]
in their non-linear shallow water wave analysis. The wave amplitude
was further increased and the beach slope decreased to 9° in an
attempt to produce breaking on the approach run. A preliminary
result is shown in Fig. 15. Variations in the application of the free
surface condition and in the shoreline treatment have, as yet, failed
to remove the irregularities in that solution. A stronger shoreline
singularity coupled with an insufficiently rigorous treatment of it may
be to blame. An optimistic viewer might detect a breaking tendency.

E. Example Four, Figs. 4(d), 16. A Shelf

One final example is shown in Figs. 4(d) and 16 where the wave
travels up a shelf, created by changing the boundary condition on CD,
Fig. 1. Excessive vertical elongation of the cells on top of the shelf
caused this computation to be stopped at the last time shown. (At
this point the wave height/water depth ratio on the shelf is of the order
of 2.) However, one can detect a splitting of the wave into two waves
as might be expected from the theory of Lax [1968].

140

Lagrangian Soluttons of Unsteady Free Surface Flows

t/r = 8.0 11.0 14.0 17.0 19.0 21.0 22.0
SYMBOL oO Oo & + x Oo

eS MESH 30X9 POINTS

20.0 1.0 2.0 3.0 U.0 5.0 6.0 7.0 X

Fig. 13. Example 3 with M=0.30 and T=67T, T=0,.571. The beach slope
is 27°. Free surface positions at selection of times, t

ce
| t/y = 8.0 12.0 14.0 17.0 19.0 21.0 23.0 25.0
EK SYMBOL © © 4 + x © # ®&
i}
MESH 30X9 POINTS
©
we

20.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 X
Fig. 14. Example 3 with M=0.60 and T=87, 7T=0.571. The beach
slope is 189, Free surface positions at selection of times ,t

141

Brennen and Whitney

=e Je ee
N
th = 12.0 16.0 20.0 21.0 22.0 22.8 23.7 2U.5

| SYMBOL oO © -A + x © a: x
| MESH 51 X9 POINTS :

oO

20.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 X

Fig. 15. Example 3 with M=2.00, T=167, T=0,481. The beach slope
is 9°. Free surface positions at selection of times, t

2.0

2.4

t/Y = 8.0 10.0 11.4 12.2 13.1 13.9 14.3
SYMBOL CROP ay et a xX nO a 4

MESH 80X9 POINTS

2.0

50.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 X

Fig. 16. Example 4 with M =2.00, T=167T, T=0.481. Shelf defined by
X1=4.41, X2=5.16, HR=0.3. Free surface positions at
selection of times, t

142

Lagrangtan Soluttons of Unsteady Free Surface Flows

VII. CONCLUDING REMARKS

Rather severe examples were taken in order to test the
limiting characteristics of the method developed. Provided the
various interval limitations were adhered to only two problems arose
which could prematurely conclude a computation. First, excessive
elongation of the cells in regions of the most violent motion could
cause the mesh points to be excessively widely spaced; rezoning
could, however, make it possible to continue. Secondly, it would
appear that a more detailed knowledge and treatment of some singu-
larities is required. Work on this, and especially on the shoreline
singularity of example three, is in progress at the moment.

Other types of examples which have been only briefly investi-
gated thus far are: the matching with a semi-infinite region in which
some analytic solution is used; the inclusion of surface tension; the
extension of the method to three dimensions; examples in which the
fluid is inhomogeneous. It is hoped to present such results in the
near future.

The authors are deeply appreciative of the kind and considerate
help given by Professor T. Y. Wu.

This work was partially sponsored by the National Science
Foundation under grant GK 2370 and by the Office of Naval Research
under contract N00014-67-A-0094-0012.

REFERENCES

Amsden, A. A. and Harlow, F. H., The S.MAC method: A numerical
technique for calculating incompressible fluid flows, Los
Alamos Scientific Laboratory Report LA-4370, 1970.

Biesel, F., "Study of wave propagation in water of gradually varying

depth," in Gravity Waves, U.S. National Bureau of Standards
NBS Circular 521, 1952.

Brennen, C., "A numerical solution of axisymmetric cavity flows, au
J. Fluid Mech., 37, 4, 1969.

Brode, H. L., "Gas dynamic motion with radiation: a general numeri-
cal method," Astronautica Acta, 14, 1969.

Carrier, G. F. and Greenspan, H. P., "Water waves of finite
amplitude on a sloping beach," J. Fluid Mech., 4, 1958.

143

Brennen and Whitney

Chan, R. K-C., Street, R. L. and Strelkoff, T., Computer studies

of finite amplitude water waves, Stanford University Civil
Engineering Technical Report No. 104, 1969.

Fontanet, P., Théorie de la génération de la houle cylindrique par
un batteau plan, Thesis, University of Grenoble, 1961.

Fromm, J. E. and Harlow, F. H., "Numerical solution of the prob-
lem of vortex street development," Physics of Fluids, 6,

1963.

Heitner, K. L., A mathematical model for the calculation of the
run-up of tsunamis, Thesis, California Institute of Technology,

1969.

Hirt, C. W., The numerical simulation of viscous incompressible
fluid flows, Proceedings of the 7th ONR Symposium, Rome,
1968.

Hirt, C. W., Cook, J. L. and Butler, T. D., "A Lagrangian
method for calculating the dynamics of an incompressible

fluid with free surface," J. of Computational Physics, 5,
1970.

John, F., "Two-dimensional potential flows with free boundaries ,"
Communs. Pure and Appl. Math, 6, 1953.

Lamb, H., Hydrodynamics (6th Ed.), Cambridge University Press,
1932.

Lax, P. D., "Integrals of non-linear equations of evolution and
solitary waves," Communs. Pure and Appl. Math., 21, 1968.

Lewy, H., "Water waves on sloping beaches," Bull. Amer. Math.
Soc. , 52, 1946.

Penney, W. G. and Price, A. T., "Finite periodic stationary gravity
waves ina perfect liquid," Phil. Trans. Roy. Soc., A, 244,
1952.

Pohle, F., "Motions of water due to breaking of a dam, and related
problems," in Gravity Waves, U.S. National Bureau of
Standards Circular 521, 1952.

Roseau, M., "Short waves parallel to the shore over a sloping beach,"
Communs. Pure and Appl. Math. , ii, 1958.

Sekerz-Zen'kovich, Ya. I., "On the theory of standing waves of finite

amplitude on the surface of a heavy fluid," (R) Dokl. Akad.
Nauk. SSSR, (N.S.), 58, 1947.

144

Lagrangtan Solutions of Unsteady Free Surface Flows

Wehausen, J. V. and Laitone, E. V., "Surface Waves," Handbuch
der Physik, Vol. IX, Fluid Dynamics III, 1960.

Welch, J. E., Harlow, F. H., Shannon, J. P. and Daly Be Js

The MAC method, Los Alamos Scientific Laboratory Report
No. LA-3425, 1966.

Whitham, G. B., "A general approach to linear and non-linear

dispersive waves using a Lagrangian," J. Fluid Mech., 22,
2, 1965. _

145

‘
; - = x |
~ . | its) , \
[
. ¥

TWO METHODS FOR THE COMPUTATION OF
THE MOTION OF LONG WATER WAVES —
A REVIEW AND APPLICATIONS

Robert L. Street
Robert K. C. Chan
Stanford University
Stanford, California
and
Jacob E. Fromm
IBM Corporatton
San Jose, Caltfornia

I. INTRODUCTION

The continuing evolution in speed and capacity of digital com-
puters has encouraged the development of many computationally
oriented methods for analysis of the movement of waves over the
surface of the ocean and onto the shore. Carrier [1966] gave analy-
tical techniques requiring numerical evaluation for the propagation
of tsunamis over the deep ocean and for the run-up on a sloping beach
of periodic waves that do not break. He noted that linear theory is
valid in the deep ocean and over much of the sloping shelf; thus, non-
linear theory is needed only in specific regions where the nonlinear
contributions to the dynamics are important. However, his nonlinear,
approximate theory was developed only for the plane flows. An ex-
tension and application of Carrier was made by Hwang, et al. [1969].
They studied the transformation of non-periodic wave trains ona
uniformly sloping beach using the nonlinear shallow water wave
equation and the Carrier-Greenspan transform. This transform
fixes the moving, instantaneous shoreline of the physical plane to a
single point in the transformed plane. Although the analysis deals
only with plane flows and does not handle breaking waves, it does
predict wave run-up and reveals a significant beat phenomenon.

To study nonlinear effects and/or to account more completely
for the waves' reaction to arbitrary ocean topography and boundaries,
it is natural to turn to numerical methods and their accompanying
computer codes. The simplest of the numerical methods are repre-
sented by the refraction techniques of Keulegan and Harrison [ 1970]

147

Street, Chan and Fromm

and Mogel, et al. [1970]. These methods, based on linear, geo-
metric-optics theory, are applicable to arbitrary bottom topography
but can predict neither breaking nor run-up. They also neglect
reflection and diffraction effects. The computer code is very simple,
requiring step-by-step solution of Snell's law and a wave intensity
equation over a grid of bottom depths.

Vastano and Reid [1967] described a procedure employing a
numerical integration of the linearized long wave equation to study
tsunami response at islands. They concluded, by comparison with
analytic solutions for special cases, that their numerical model gave
an adequate representation of the solution to the linearized equations.
Their paper indicated that the work was a lead-in to a more general
treatment of arbitrary bottom topography. At the island they useda
vertical cylinder that penetrates the surface and thereby restricts
the movement of the instantaneous shoreline. No run-up can be cal-
culated in the usual sense.

In another approach, Lautenbacher [1970] used linear,
shallow water theory to study run-up and refraction of oscillating
waves of tsunami-like character on islands. His method allows for
the moving, instantaneous shoreline and, of course, for superposi-
tion of individual results from monochromatic waves. Working from
an integral equation formulation and employing a Hankel function
representation of the far-field radiation condition similar to that
used by Vastano and Reid [1967] , Lautenbacher used a grid of dis-
crete points to numerically integrate the integral equation of his
model. Combining his work with Carrier [1966], Lautenbacher was
able to estimate total tsunami run-up from a distant source. He also
emphasized the importance of refractive focusing effects.

The numerical methods aimed specifically at modelling non-
linear effects take three forms:

a. Approximate, plane-flow models for arbitrary or sloping
beaches and based on approximate equations.

b. Exact plane-flow models.

c. Quasi-three-dimensional models,

The approximate, plane-flow models are represented by the
work of Freeman and LeMéhauté [1964], Peregrine [1967], Heitner
[1969], Street, et al. [1969], Camfield and Street [1969] , and Madsen
and Mei [1969a, 1969b]. With the exception of Heitner [1969], the
authors used Eulerian coordinates. Freeman and LeMéhaute [ 1964]
applied the method of characteristics to the nonlinear, shallow water
wave equations for plane flow. They described a method for com-
puting the shoaling of a limit-height solitary wave on a plane beach,
predicting the point of breaking inception by the crossing of character-
istic lines and computing the subsequent bore development and run-up
at the shoreline. A term was added to the equations to correct the

148

Computatton of the Motion of Long Water Waves

assumed hydrostatic pressure distribution beneath the wave, the
assumption not being valid for finite amplitude waves. Camfield and
Street [1969] used a refined correction term, but the results were
not entirely satisfactory in either case.

Peregrine [1967] and Madsen and Mei[1969a] derived approxi-
mate, nonlinear, governing equations for the propagation of long waves
over slowly varying bottom topography. In these equations the verti-
cal component of motion is integrated out of the computation so a
single-space-dimension problem is all that remains. Madsen and
Mei [1969a] showed that, while they and Peregrine used different
approaches and solution methods, the equations are the same when
presented in the same variables. Furthermore, Madsen and Mei
[1969a] explained that these nonlinear equations, obtained under
assumptions similar to those leading to cnoidal waves in the case of
horizontal bottoms, give a uniformly valid description of long wave
problems as long as breaking does not occur. In particular, their
equations were derived under the condition that the Ursell parameter

2
U, = ye = O(1) (1)
ie)

where 1 is a measure of wave amplitude, L, is a characteristic
wave length and do is the water depth. Thus, the nonlinear, govern-
ing equations are of the Korteweg-deVries type (KdV) that have per-
manent solutions, e.g., cnoidal waves, for the case of horizontal
bottoms. Madsen and Mei[1969a] demonstrated that, although the
equations pertinent to each of the three groups of long waves (Airy
where U,>>1, KdV where U, = O(1) and Linear where UU, << 1)
have different mathematical solutions, the features characterizing
each group are all contained in the equations derived under the
assumption of waves of the KdV type.

The method of characteristics was applied by Madsen and Mei
[1969a, 196 9b] to solve their equations for initial, boundary value
problems involving solitary waves and periodic waves on plane slopes
and a shelf; their equations, like those of Peregrine [1967], are
applicable to general, uneven bottoms. Street, et al. [1969] pre-
sented a numerical model APPSIM and results based on the Peregrine
[1967] method, but employing initial and boundary conditions similar
to those of Madsen and Mei[1969b]. These methods reproduced the
nonlinear breakdown on a shelf of a solitary wave, the breakdown
having previously been observed only in experiments [Street, et al.,
1968]. Furthermore, comparison shows quantitative agreement _
amongst these methods and relevant experiments. Run-up cannot
be calculated with these models which employ the vertical beach (or
island) face that was used by Vastano and Reid [1967]. However,
Peregrine's [1967] derivation included the two horizontal space
dimensions, while Madsen and Mei [1969a] did not. Accordingly,
as an extension of APPSIM, a quasi-three-dimensional model

149

Street, Chan and Fromm

APPSIM2 is based on Peregrine's equations and is described in detail
in Section III below.

Heitner [ 1969] presented a nonlinear method based on
Lagrangian coordinates and a finite element representation of the
fluids. His theory retains terms representing the kinetic energy of
the vertical motion; thus, like the methods described just above,
Heitner's approximate method permits permanent waves to propa-
gate. Unlike those methods, Heitner's formulation gives a repre-
sentation of wave breaking inception, bore formation and run-up
on a linear beach for plane flows.

The exact, plane-flow models are represented by the work of
Brennen [1970], Hirt, et al. [1970] , and Chan and Street [1970].
All are based on the exact equations of motion; however, Chan and
Street [1970] work in Eulerian coordinates, Brennen [1970] uses
Lagrangian coordinates and equations, and Hirt, et al. [1970] em-
ploy Lagrangian coordinates but retain the Eulerian form of the

overning equations. Based on the Marker-and-Cell (MAC) method

Welch, et al. , 1966], Chan and Street's model for water waves is
called SUMMAC and is discussed in Section IVofthis paper. Brennen
has applied his technique to tsunami generation by ocean floor move-
ments and to run-up on abeach. Hirt, et al., made applications to
wave sloshing in a tank by way of verification of their method, called
LINC, which has wide application in ocean wave analyses as well.
All the exact, plane-flow methods mentioned employ some form of
finite- difference or cell representation of a discrete grid of points.

Finally, the quasi-three-dimensional methods are represented
by the work of Pritchett [1970] and Leendertse [1967] and by the
extension to two horizontal dimensions of the APPSIM program of
Street, et al. [1969] discussed above. Pritchett presents a code
for solving incompressible, two-dimensional, axisymmetric, time-
dependent, viscous fluid flow problems involving up to two free
surfaces. The basic equations are exact; heuristic models for tur-
bulence simulation are used. Scalar quantities such as heat and
solute concentration can be traced, and the fluid may be slightly non-
homogeneous. Most of the variables, their placement in the compu-
tational mesh, and the free surface treatment are those used in MAC
[ Welch, et al., 1966].

Leendertse [1967] developed a computational model for the
calculation of long-period water waves in which the effects of bottom
topography, bottom roughness and the earth's rotation were included.
The equations of motion are vertically integrated so only the two
horizontal space-dimensions remain (much in the manner of Peregrine,
[1967]), but while Peregrine [1967] , Madsen and Mei[1969a, 1969b]
and Street et al. [1969] retain terms to account for the vertical
accelerations of the fluid, Leendertse [1967] does not. His equations
become the usual nonlinear shallow water wave equations with added
terms to account for the bottom roughness and earth's rotation. He

150

Computatton of the Motion of Long Water Waves

focuses his attention on the modelling of long waves such as tsunamis
in areas with irregular bottom topography and complicated ocean
boundaries. His computation uses a space-staggered scheme (where
velocities, water levels, and depth are described at different grid
points) and a double time step operation in which the time integral

is considered over two successive operations in a manner designed
to make effective use of the space-staggered scheme. Among the
papers mentioned in this Introduction, only Leendertse [1967],
Welch, et al. [1966] and Chan and Street used computer graphic
display for output of results. The value of graphic display is illus-
trated in the results presented in the remainder of this work.

II. THE PRESENT WORK

Street, et al. [1969] gave a progress report on the develop-
ment of computer programs fortwo numerical, finite-difference
models for the study of long water waves. These models and their
accompanying programs were, as noted above, given the acronyms
APPSIM and SUMMAC. Both were based on representation of the
motion of inviscid, incompressible fluids in terms of the Euler
equations of motion in Eulerian coordinates. Flow boundary con-
ditions were derived from physical requirements and the governing
equations at the boundaries. The mathematical models thus obtained
were then transformed to numerical, finite difference models for the
purposes of computation. In 1969 the study had been confined to
plane flows, but the numerical results had been verified by compari-
son with experiments and the work of others. The models were to
provide detailed flow field data in the portion of the wave shoaling
process where nonlinear effects are significant, but breaking has not
occurred.

Our approximate simulation (APPSIM) is based on the method
of Peregrine [1967] and supplemented by the work of Madsen and Mei
[1969a, 1969b]. APPSIM was implemented for quasi-two-dimensional,
plane flows (vertical motion integrated out). For the purpose of
implementing, testing, and verifying the program and method, we
simulated the propagation of solitary waves on a stepped slope which
represents the configuration of the continental slope and shelf, i.e.,
we examined long waves in moderately shallow water. The key
criteria to be satisfied were

a. Solitary waves propagate stably on a horizontal bottom.

b. Solitary waves decompose into undular bores when the
waves propagate onto a stepped slope [ Street, et al.,
1968]. —

c. Wave heights must be in good quantitative agreement with
available experimental data.

As reported by Street, et al. [1969] APPSIM met these criteria.

Street, Chan and Fromm

The successful application of APPSIM to examples of plane
motion of long waves indicated that the method could be applied to a
quasi-three dimensional simulation (two horizontal space dimensions
with vertical effects represented in once integrated equations) of the
motion of waves over arbitrary bottom topography. We have ex-
tended APPSIM to handle general bottom topography and both solitary
and oscillatory wave inputs. The new method is called APPSIM2 and
presented in SectionIII below.

An objective of our exact simulation was to provide detailed
information about wave processes near the shore and at the ocean-
structures interface. The Stanford- University- Modified Marker-and-
Cell (SUMMAC) method computes time-dependent, inviscid, incom-
pressible fluid flows with a free surface; the method is suitable for
analyzing two-dimensional flows. Initially, we simulated the propa-
gation of solitary waves in a horizontal channel filled with fluid to
unit depth and with vertical end walls, The solitary wave propagation
problem possessed several key features:

a. The theories for the wave motion against the wall were
not in agreement with experiments.

b. The solitary wave should propagate stably (without change
of form) in zones not near the channel walls.

c. Perfect reflection from the walls should occur.

We undertook a significant modification of the MAC method to create
a numerical scheme suitable for water wave simulation. As reported
in 1969, the resulting SUMMAC simulation met the criteria of stable
propagation and perfect reflection of solitary waves and resolved the
disagreement between theory and experiment for motion against a
vertical wall.

The successful application of SUMMAC to the initial example
indicated the possibility of employing a modification of the same
technique to attack a variety of other problems. We have subsequently
studied the generation of water waves by a periodic pressure pulse
and the shoaling and run-up of solitary waves on a stepped slope and
on plane beaches. A summary of the presently implemented SUMMACG
method and results for the periodic pressure pulse problem are pre-
sented in Section IV of this paper. An evaluation of the numerical
qualities of SUMMAC and a report of the shoaling and run-up studies
are given in Chan, et al. [1970] and Chan and Street [1970b].

152

Computatton of the Motion of Long Water Waves

Ill. APPSIM2

3.14. The Governing Equations and Auxiliary Conditions

Dimensionless variables are defined below and in Fig. 1 where
the physical domains considered are also illustrated. The variables
are

* * -!
(x sy en d*)d,

(Xess od)
/2
t—1 (e/a)

(u*, v*) (gd) /?

(u,v)

where those on the left-hand sides are dimensionless and d, is the
depth in the deepest part of the simulated wave tank. Here, x and
y are fixed, Cartesian coordinates in the horizontal plane; u(x,y,t)
and v(x,y,t) are the corresponding mean (vertically-averaged)
horizontal fluid velocities; n(x,y,t) is the free surface shape,
measured from the still or undisturbed water line in the tank, and
d(x,y) is the depth of the still water.

de (ER
oma Aeromax (ae),

b. Submerged Seamount (Plan View)

Fig. 1. Definition of symbols and simulated wave tanks for APPSIM2

153

Street, Chan and Fromm

For waves of the KdV type that we wish to study, U, = O(1).
If

€ = No/do (2)

and
o = dg/L. <1 (3)

for long waves, then from Eq. (1), Uy=e€/o*=1 and €=0% This
is the relation on which Peregrine [1967] based his expansion-in-a-
parameter analysis. He pointed out also that d(x,y) = O(1) and the
derivatives of d equal O(c) are necessary restrictions; otherwise,
the variations in the depth of water are shorter than the incident
waves and tend to generate shorter waves, thus upsetting the scheme
of the approximations.

Under the above conditions Peregrine [1967] obtained the
momentum equations

1 1 .2
u, tuu, +vu, tn Fs d[ (du),, + (dv) yy] =i bug Vaylt

(4)
OSS yg Oy <i 0<t

1 Ake
Mea We le ae d[ (du),y + (dv)yy | aid ees Vole
(5)
0 < x < By ~0-<-y<- Lo, — 0X

and the continuity equation

n, +[(d + nul, +[(d + nv], = 0

(6)
O<—x <i e, Os y's bo, 11.0. <t

These equations and the appropriate auxiliary conditions described
below are solved numerically by a straightforward finite difference
scheme that is presented in Section 3.2.

The auxiliary conditions are the boundary and initial condi-
tions appropriate to Eqs. (4-6) for motion in a vertical-walled tank
(Fig. 1). For solid, vertical walls the velocity perpendicular to the
wall is zero at the wall and non-breaking waves reflect perfectly
from the wall. If n is the normal coordinate and U and V are

154

Computatton of the Motton of Long Water Waves

typical normal and tangential fluid velocities at the wall, then in
inviscid flow

1, = 0 (7)
a) (8)
Vi= 0 (9)

Un (10)

Accordingly, for Wall 1 shown in Fig. 1 we have for 0<y<L,,
Or<i t;

n,(0 ry »t) =0, v,(0,y,t) = 0, u(0 ryt) =0, u,,(0,y,t) = 0

Similar conditions hold on the other walls.

If we choose to prescribe n=n(t) along Wall 1 where x=0,
then a special set of conditions must be derived (cf., Madsen and Mei
[1969a] who point out that this prescription corresponds physically
to the situation of having measured the incoming wave height at a
particular station), Peregrine [1967] gave the irrotationality con-
dition

wy-% =Z4lV > (dull, -5alV- (du)l
-Faa[V-U] +z aalV- Ul (14)

where u = (u,v) and V = (8/8x, 8/8y). Ifthe bottom is flat in the
neighborhood of Wall 1, d,=d,y=0 and Eq. (11) becomes

(6 teas i (12)

But, if n(0,y,t) = n(t), then until some reflection from a shoal or
beach in the tank returns to x =0,

uy(0,y,t) = 0 (13)
and, hence,

v,(0,y >t) = 0 (14)

155

Street, Chan and Fromm

Given (0,y,t) = n(t) and Eqs. (13) and (14) and using Eqs. (4-6)

at Wall 1, one can uniquely determine u, v and 7 there for the
case of a prescribed incoming wave. The conditions at the remaining
walls are, of course, not changed.

The appropriate initial conditions for Eqs. (4-6) are the
initial values of the dependent variables, viz.,

n(x,y,0) = ni(x,y); ulx,y,0) = uj(x,y)s = v(x,y,0) = vj(x,y) (15)

3.2. The Difference Representation and Computation Scheme

For numerical computation the region 0Sx=L,,0Sy=1,
is covered by a grid of discrete mesh points with a spacing Ax= Ay =
4 and calculations are carried forth with time steps At. To allow
for proper representation of the boundary conditions the grid indices
(i,j) run over the intervals (1,M) and (1,N) respectively and
points (1,j), (Mj), (i,1) and (i,N) lie outside the tank walls, e.g.,
x=0 is equivalent to (2,j), etc. Consequently, L, = (M-2)A and
Le=(N-2)A (see Fig. 1).

In the finite difference representation of the differential
equations and auxiliary conditions, central space-differences are
always used; both forward and time-splitting schemes are used for
time-differences. The differential equations of motion, eqs. (4-6),
lead to a highly nonlinear and coupled set of difference equations.
These are solved iteratively, using a predictor-corrector method.
If u, v and n are known at all the grid points at t!.e n' time
level, the following scheme leads to calculation of u, v and 7 at
the nti!? level.

First, u, v and 1 are predicted at the nti level by use
of the nonlinear, shallow water wave equations (Eqs. (4-6) with their
right-hand sides set to zero). With the superscript P indicating
the predicted value, the difference equations are, after rearrange-
ment for computation,

P At
i) aaa er 2 (aj jng My) F Vly — yj) F Tiny > m1) (16)
py Ale aCAt oe joo" \+ E “a

At
Mig = Mig ~ Ty Cig (Giely + Niet - Gi-ny - Tieng) F diy F TiMCGiedy ~ Bi-1j

PN inc Milas i Moiel ce dtiel es Silas nij-0)) (18)

156

Computatton of the Motton of Long Water Waves

t
where variables without superscript are known at the nf time level.

Second, the x-momentum, Eq. (4) is used to obtain a difference
equation for u values at the nti™ Jevel. Central space-differences
and time-splitting are used about the point (i,j,n+43); this leads to
a difference equation that is implicit. Now we gather the u™! terms
from the j'? row on the left-hand side of the equation and all other
terms on the right-hand side. The heart of our procedure is to use
uP, vP and 7? values in lieu of the u™*!, vy"! and n*! terms which
appear on the right-hand side of the equation, these terms being
mostly inthe j-1 and jt1 rows. The result, when rearranged for
computation is, for each j,

n+l n+l nel _
Au + BM + Cull =H, 15 3,4,060,M-2 (19)
where
2
AE 1 di
= - pelle ie age
A 4A bea af 2A* ( 3 dijdj.1,)
adj,
B= Ll + aSu,
3A
At 4 dy
aa Ms | en eS 2 dij diey )
: At
E = ig’ BA ij iat Uj _4;)
At P
AGA Vip lijet = Mifare Bie

At P P
— ZA Miatj - Nien t Niet) - Ti-g)

dij
+ eae (- Fig Piany * 24 j4iy - Fi)

4 P P P
FZ (distjet Vietiet ~ FietjetVi-tjet ~— Fiety r%i6 jet
p
midis ip ¥ - dj a

faljel i+ljel Vieljel i-ljal Vi-tjet

IOS tei peljat daar qae)

157

Street, Chan and Fromm

‘ Visi) T 20;j - Yi-tj

Pe

64

1, P P P P
FAV atjeh 7 Vic ljeht Viatjat PoVi-ajat
5 Vixtfel S:Viedjon tT Viste bY intjt ))

Equation (19) must be solved for all i simultaneously for a given j.
The matrix of coefficients of the unknowns in Eq. (19) is tridiagonal
and is quickly and easily solved by a tridigonal-matrix equation
solver employing Gaussian Elimination. The process is repeated for
each j until the ee are known. Appropriate boundary conditions
are introduced at the ends of the j'® row in each computation.

Third, the y-momentum Eq. (5) is used to obtain a difference
equation for the v values at the nti™ Jevel. The result is entirely
equivalent to that for the u values, viz., the third-order terms on
the right-hand side create a naturally implicit system so time-splitting
is used. Now, however, we use vP and nP values for all i-1 and
iti points along a given column of implicit equations (i = constant,

3 <= j= N-2) and we use the u™! values just computed. The result,
when rearranged for computation, is

~ nel ~ nel n+l .
AVii + Bj, + Cv iin =£ , j= 3,4,.-.,N-2 (20)

where

~ 2dij
B=1 + Sy
34
At 1 / dj
an Vi ae ae dys dij)
At e P

E=Vijy - ZA ay (Via een a Vii io) )
ot rail aia)
Pega, Vise Vise, & Vij-1

At Ee esl ic
PE Aa Bijeii Myel = ja) t

158

Computatton of the Motton of Long Water Waves

dij 1 n+l n+l nel
2ae \ 4 (diggjat Ciatjel ~ Fienjet Bi-tjet — Tietj-1 Uistj-1

n+l
a Gi ype Pie aj ~ Gigtjetietjel + Gj-tjet Uj - Ijel
Be Gi gtjet ia tj-t zs dij. 4i-tj-1) a Fijat Vijel

+ 2dijvij - dij-1Vij-t)

2

dj; 1 ,_nel nel n+] + n+|
= an (5 (Wis ijel ~ Upetjet ~ Vie tye 7 Bprj-r
~ Uistjel " Uj ijel t Gietj-1 Uj-1j-1)

- vijr + 2vij - vij-t)

Now, Eq. (20) must be solved for all j simultaneously for a given i.
Again we use the tridiagonal-matrix equation solver. The process
is repeated for each i until the vjj are known.

Fourth, the continuity Eq. (6) is used to obtain 3 difference
equation for the 7 values at the nti™ jevel. The u™ and v"*
values are used in an equation that, as suggested by Peregrine [1967],
uses an average for u and v values, but forward differences for
"1 values. The result, rearranged for computation, is the explicit
equation

nel At ul t+ uij
= 1 i}
Bip. Ty xa ( 2 (di.iy + Misty - Fi-tj - Ning)

4 n+l nel
FS (dij t mig hlinng + Yisty - inj — Mn
n-l ne
TVileh oc Vilas “iclre wat
vit + vij
2. (dijet i Nijel ~ dij et ie ni} (21)

This equation is used to compute nit! for 2512 M-1., 2-2 j
and then the first iteration, the predictor iteration, at the n
time level is complete.

= N-1,
th

If now the second through fourth steps above are repeated with

59

Street, Chan and Fromm

P values replaced by the nt 11” values just calculated, the accuracy
of the solution is increased; this is the corrector iteration. Numeri-
cal tests showed that the computations remained stable if at least
two iterations were used (one predictor, one corrector). The u™,
v +l and n+! values obtained in the second and the third iteration
agreed to at least four significant figures after several hundred At
steps in simulation of solitary wave motion onto a shelf (Fig. fia).

Boundary conditions in difference form were derived from
Eqs. (7-10) in the case of solitary wave simulation. The wave was
started well inside the tank walls which were held rigid. For ex-
ample, for Wall 1 in Fig. 1 we have from Eqs. (7-10) at any time
level 1), = Ngjr Vij = V3j> 4ej= 0, and Wyj = - Ug} Other walls have
similar conditions.

For input of an oscillatory wave propagating in the positive
x-direction at x = 0 we prescribe

2nto.- At « (nti) (22)
fo)

where ‘to is the amplitude (usually small) and Lo is the wave
length. The celerity Cy is taken to be unity in nondimensional
terms. From Eq. (14) we have v,, = v3). Because n3* and 74;
are computed explicitly in the tank region, ni; is obtained by
polynomial interpolation according to the second-order formula

n+l n+l n+l n+l
Me ig ee Ee, (23)
Finally, the continuity difference Eq. (21) is used for points (2))
where it has not been previously employed to relate uf! to the
values inthe interlor, j= 2. With u i known as a function of |
Ugj» Uzi, etc., Eq. (19) can be used in 2S i= M-2 and the ui
found; vi values are replaced by v{, values in the first iteration.

Figure 2 is a flow chart for the APPSIM2 computations.
These were performed on an IBM 360/91 system. For a typical
computation with A =0.25, At=0.22, M= 154, N = 54 and 126
time steps the program required about 360K bytes (90K words) of
core storage and 4 minutes of CPU time (about 1/30 minute per
time step). The stability of the method is discussed in Sec. 3.4
after presentation of computational results.

3.3. Results and Discussion
To illustrate the focusing effect of wave refraction and the
reaction of waves to a shelf geometry, solitary and oscillatory waves

were shoaled over the bottom topography shown in Fig. la. The
water depth in the tank was 1.0 while the depth of the shelf was 0.4.

160

Computation of the Motton of Long Water Waves

REMARKS

BOTTOM, dj;

SOLITARY OR
OSCILLATORY
WAVE IN

+X- DIRECTION

WAVE IN
CALC u,v,7

(shallow water eqns)

CALC PREDICTED VALUES uP? y? pP EXPLICIT
KOUNT = 0
use uP PnP
CALC u®*!, KOUNT = KOUNT +1 IMPEICLT. IN
X- DIRECTION
use unt! yP nP
IMPLICIT IN

Y-DIRECTION

EXPLICIT
(uses advanced
velocities, however)

SET uP sult! pps noth yPaynrl

KOUNT 2 2

WRITE/TAPE DUMP

T= TOP;STOP

Fig. 2. APPSIM2 flow chart

The deep and shallow portions are connected smoothly by a cosine
curve.

For the solitary wave simulation, the pertinent parameters
were L, = 38, Lg= 13, x,= 14.5, X%,= 19.5, yy= 7.75, Yg= 2-75,
A=0.25, At=0.2165, and no= 0.1. The wave was started with its
crest lying along x,= 8.0 and propagated in the positive x-direction
toward the shelf. Chan and Street [1970a] showed that the effective
half-length of a solitary wave of amplitude No = 0.1 is about 11 so
it is necessary to correct the initial Boussinesq [ Wiegel, 1964]
wave profile for the influence of the wall at x = 0; however, it was
unnecessary to correct the leading portion of the wave for the bottom
influence. The initial u-velocity distribution was calculated from
Eq. (6) under the assumptions that v=0, n = n(x,t) and the wave is
moving at a constant speed Co=1+0.5 7 [Wiegel, 1964] with
constant form. In addition At was selected in accordance with the
Courant-type condition

A
Ats=—
Co

Results of the solitary wave computation are shown in Figs. 3-5.

164

Chan and Fromm

Street,

HH

Hr Ht

,
wry

4H

arr ae
StH HHH

xt ao
a

oH
oa

HHH
tHE
HH

+H
SR,

oper

poccths

sposcpeseccpe.

HHH
HEHEHE

Yt
Se

HE

HEHEHE

HEHE EERE EEE HE

EEE
OHHEE

T=213.0
T=19.5

HOHE tt

i
Et
Ap

:
E
:
E
E

EEE EEE EEE HE

HE EEEEEEHEE EE EEE EEE EEE HEEEEEEEHEEHE EEE HEE EEE
EEEEHEEEEEE HELE EEEEEEHEEEEEEHEEEEEE HEEL HEHEHE

HE

——
oo

EEEEEEEEHEEEEEEE
HEHEHE

SHEEEEEEE HH
Het —
tHE —H4-H

AHH
HE

OHH HHEHE

bhatenee

26.0
162

T=
Free surface maps for a solitary wave on the shelf

6)

Fig.

Computatton of the Motion of Long Water Waves

r= 19-5

Fig. 4. Velocity map (v) for t= 19.5

0.2

0.1 7

0.0

18 20 22 24 26 28 30 32 34 36
“O52 x

LEGEND: - SCALE EXAGGERATED FOR ILLUSTRATION
---APPSIM (2-DIMENSIONAL SIMULATION)
——APPSIM2(3-DIMENSIONAL SIMULATION )

0.1
hea x ”
= 0.0

2 4 6 8 10 12 14 #16 18 20 22 24 26 28 30 32 34 36
x

Fig. 5. Wave profiles for two- and three-dimensional propagation

163

Street, Chan and Fromm

The evolution of the free surface is illustrated by free sur-

face maps in Fig, 3, while Fig. 4 shows a v-velocity map for

= 19.5. The maps are printed during program execution by a sub-
routine that scales the variable values on a range running from a
minimum of zero to a maximum of ten. Only odd numbers are
printed at their corresponding node points. These maps are ex-
tremely useful for initial interpretation of the data, Later, quanti-
tative studies of results can be made because the u, v, n fields
are stored on tape after every five time steps and maps are made
after every 20 to 40 steps. Thus Fig. 5 illustrates a quantitative
comparison between the two-dimensional results and the three-
dimensional simulation at t= 19.5 for y=0 and y=L»2. Both
the effect of wave refraction and the nonlinear response of the flow
are evident.

As another example, Fig. 6 contains pictures of the develop-
ment of the yn, u, and v fields for oscillatory waves shoaling ona
shelf. The pertinent parameters were L, = 66, L,= 41, x;= 25.5,
X5= 45.5, yr = 25.5, ye = 15.5, 4=0.5, At=0.5, and 2) = Hy =
0.05. The input wave at Wall 1 (Fig. 1) had eerie length Ly '=(20
and speed C,=1.0 sothe period Ty = 20. The actual computed
length was essentially. 20 also. In this case we sought to simulate a
large region so A was large; even so 458K bytes of core storage
were required for the program. In spite of the rather coarse grid
the computed properties of the waves were smooth and well behaved.

T= 21.5
T= 44.5
T=68.5
| i
7-CONTOURS U-CONTOURS V-CONTOURS

Fig. 6. Shoaling on a shelf (oscillatory waves)

164

Computation of the Motton of Long Water Waves

The contour lines in Fig. 6 were computed by a plotting pro-
gram developed by Schreiber [1968]. The facility used was an IBM
2250 graphic display unit in which the computed contour lines are
projected on a TV screen. The contour plots were recorded by
photographing the surface of the screen. Several motion pictures
have also been made with this apparatus.

Two 2250 units are used when films are made. First, as
noted above the computed field values are stored on tape during the
APPSIM2 computer run. Later, a special program calls up the
tapes and transforms the field data to contour lines. These are
transmitted to the 2250 units. One is used as a control console to
monitor picture quality and to set the movie camera speed. The
second unit has a 16 mm movie camera mounted on it and focused on
the screen. The camera operation is synchronized with the suc-
cession of contour plots flashed on the TV screen. Titles are also
constructed on the screen and filmed. Judicious editing transforms
the 16 mm film into a useful and interesting movie. As the sequence
of Fig. 6 shows, the evolution of the flow fields is particularly
instructive. Because all the pertinent parameters are usually shown
simultaneously on the screen with the contour plots, quantitative
interpretations of the contour information can be made directly
from the graphic display. For example, at t = 68.5 the maximum
wave height along the line y =0 is about 0.06, while along y=Lap,
the height is 0.05 (i.e. , the wave is unaffected by the shelf at this
time). In Fig. 6, the n-plot increment for t = 68.5 is 0.01, the
maximum value is n = 0.064 and the minimum is n = - 0.026. For
oscillatory waves it is necessary to have some detailed printout in
addition to the graphic display for quantitative analyses because the
contours are not marked with their contour level values.

Finally, we simulated long wave amplification by a circular
submarine seamount and compared our results with the experimental
values of Williams and Kartha [1966]. The following pertinent param-
eters exactly match one of their experimental runs for a half-seamount
(Fig. 1b) with non-dimensional parameter X = 2mb/L,= 3.0: xc= 56.7
is the distance from the wave generator to the peak of the island;

b = 7.73 is the radius of the base of the seamount, T = 16.6 is the
wave period, No= 0.0082, d=1.0 beyond the seamount base,

€ = 0.116 is the submergence of the seamount at its peak and

L,= 23.2. The tank length L, = 116 was selected to prevent re-
flections from reaching the seamount. The amplification ratio
H;/H, = Ay was calculated where H,= 2n, and Hj = the trough-to-
crest distance on waves at the island peak where d=e€e.

The experimental Af = 2.42, while A; = 2.46 according to
the refraction theory of Mogel, et al. [1970] and As = 2.70 accord-
ing to APPSIM2 for a seamount whose shape was given by

d(x,y) = (41.0 - aS) +€

165

Street, Chan and Fromm

with

eres re x)" eon?

and q=1.0 (the shape factor). This was a linear seamount with a
sharp peak where the first derivative of d(x,y) is discontinuous and
the higher derivatives are undefined (cf., Sec. 3.1). Unfortunately,
the difficulty of resolution of the island features near the peak was
compounded by the fact that to simulate the experimental conditions
within our nominal core allotment of 500K bytes we had to use A =
0.5 where 4 =0.25 would have been preferable. As a consequence,
we believe, of the coarse grid the short waves generated by the
island wave response are not properly resolved, being of the order
of one or two grid divisions. The solution, therefore, while not
unbounded, appears unstable.

On the other hand, Williams and Kartha [1966] did not report
on the sea-state near the islands and our results might be physically
reasonable. Atest using q = 2.0 which gives a dome-like island
produced an A; = 4.65 for X= 2mb/L, = 3.0 which is slightly
beyond the range of the experiment. However, this A¢ value lies,
as does our result for q = 1.0, within the uncertainty band of re-
sults presented by Williams and Kartha.

3.4. Stability Analys is

Initial calculations with APPSIM2 and no iteration, viz.,
operating with only a single predictor/corrector step produced some-
what ragged results after several tens of time steps. Accordingly,

a linear stability analysis was made to examine the amplitude pro-
perties of the computational scheme.

For the stability analysis we set dj; = 1 and defined the
constant parameters

B= At+ v;/4

(24)
y=At+n/4
R= At/A

The equations defining the computational scheme were linearized by
considering only difference quantities as variables and treating the
remaining terms as the constants of Eq. (24). Thus, the prediction
Eqs. (16-18) become

166

Computation of the Motion of Long Water Waves

p 1 {
Uy = 4a - FU - Bien) - Play - Bij.)

- $ Renietj - Nj-1)) (25)

=) 4 |
Mp a) Pipe Mijn)

ini nij-0 (26)
4 1

nig = Nii - > eM - Ni-tj) - > PCnijed - Nij-1)
- S(vARMuieyy - vig) - GOYFR Mv qa - Ye) (27)

The remaining Eqs. (19-21) of the corrector step were treated ina
similar manner; where P values appeared, the values from Eqs.
(25-27) were introduced. To test the resulting linearized form of
Eqs. (19-21) we introduced the Fourier component solution (or error)

—ex ilo, x + oy) iwt
e

W-=w

where we seek to determine if w values, either real or complex,
exist such that W is a solution of the difference equation and where

*
u
—> te *
= Vv = constant
nN
Ole 2m/ry

for representative wave lengths \, and XX» in the x- and y-directions
respectively. Now, let p = e!¥4! so

—> — KK n i(o,x+o,y)
(=

W=Wu (28)

at any point in time and space. Thus, we insert Eq. (28) in the
linearized u, v, 1 equations and obtain

167

Street, Chan and Fromm

Aiy AQ yg] U
x *
[A] W = |ag, agp - ail v =) (29)
*
ay 7930. 331)"

where the aij = ajj(%,Bsy,R,As0, s05,p). Because Woe OF Sin
general, a Fourier component solution can exist and Eq. (29) can be
satisfied only if |A| = 0, viz., only if a set of eigenvalues exists for
the matrix A. The condition |A| = 0 leads to a determination of bs
linear stability depends on the amplitude of p. If |p| 1 the solu-
tion by Eqs. (16-21) would be termed linearly stable.

Analysis of the coefficient matrix A in Eq. (29) is complex
and is not reproduced here, but the key results are as follows.

First, if 6, = 20A/)d, and 85 = 2nA/r», then for finite At, \, and
Nos

lim |p. | = il
A—=0

Similarly, for finite A, \, and \3,

lien, |e =
At—0

The solution scheme is stable in these limit cases.

Second, the case 6,=0, namely, \,= 00, was investigated.
Then,

ln | = f(a, y,9,)

for assumed A and R. This approach leads to a complex, cubic
equation with a possible root

Ju,| = 1 + Oe) (30)

and a possible root pair with maximum modulus equal to or slightly
exceeding unity, i.e.,

lu2,3lmax = 1 (34)

Specifically, in a solitary wave computation with A= 0. 5“and
Mt-=\0525, we have for n= 0.1, y =0.05, a= 0.05, p=0, R=Oe

168

Computatton of the Motton of Long Water Waves

and
-7
Ips] e 1+7xX10 <1 +o0(a)
lo3lmox © 1.005 <1 + O(a)

for \,;> 2A. For i, = 24, |p| =1 exactly. Forsythe and Wasow
[ 1960] suggest that the errors may be controllable and lead to a
stable computation when

[els 2 O(At) (32)

In the present case a@< At because a= uAt/A and u/A <1;
accordingly, the condition (32) can be written as |p. | = 1+ O(a).

Our computational experience with APPSIM which has 0,=0 always
and does not use iteration also suggests that the computation is stable,
at least for several hundred time steps. APPSIMZ2, however, because
of its coupled (u,v, 7) equations and the propagation of error from
the u-field where |ujj | is relatively large compared to the v-field
and n-field, does require at least one iteration (which tends to make
the computation more like an implicit scheme) to retain a smoothly
varying solution on a smoothly varying bottom topography.

3.5% Prognosis

The computational results indicate that APPSIM2 is a useful
means of studying the evolution of flow fields in wave shoaling over
smoothly varying bottom topography. However, the method requires
considerable computer storage and moderate execution time. Thus,
APPSIM2Z, which models nonlinear processes in nonbreaking waves,
should be used only when nonlinear effects are expected to be sig-
nificant, other methods (cf., SectionI) being appropriate otherwise.
As Madsen and Mei [1969a] indicate, the equations of KdV type used
in APPSIM2 should make the method applicable to a wide range of
long wave problems.

Two futher steps should be made in the development of the
method. First, the linear stability property could be improved by
introduction of a second-order central difference method for the
convective terms in Eqs. (18) and (21). This central difference
[ Fromm, 1968] leads to a modification of Eq. (31) such that

l2,3 lear =1

for most components of interest. Alternatively, Eq. (21) can be
made an entirely implicit equation for ni; [cf., Eq. (19)]; this
will eliminate the growing contribution represented by Eq. (31) and

169

Street, Chan and Fromm

caused by the explicit nature of the ate equation. Second, a series
of simulations of specific hydraulic models should be made to deter-
mine the grid size A required to resolve the smallest significant
feature of a problem and to determine the sensitivity of the simula-
tion to discontinuous bottom topography.

IV. SUMMAC

Chan and Street [1970a] proposed the SUMMAC computing
technique as a tool for analyzing two-dimensional finite-amplitude
water waves under transient conditions. The method is, as noted
above, a modified version of MAC which was developed by Welch,
et al. [1966]. The essence of the initial modifications consisted of
a rigorous application of the pressure boundary condition at the free
surface and extrapolation of velocity components from the fluid
interior so that inaccuracy in shifting the surface boundary is kept
at a minimum.

The objective of this section is to provide a summary of the
SUMMAC method, of its application to water wave problems and of
a number of new improvements added to SUMMAC since Chan and
Street [1970a] was written.

4.1. Summary of the Method

The fluid is regarded as incompressible and the effect of
viscosity on the macroscopic behavior of flow is considered to be
negligible. The entire flow field is covered with a rectangular mesh
of cells, eachof dimensions 6x and dy. The center of each cell
is numbered by the indices i and j, with i counting the columns
in the x-direction and j counting the rows in the y-direction ofa
fixed Cartesian coordinate system (Fig. 7). The field-variable
values describing the flow are directly associated with these cells
[ Welch, et al. 1966]. The fluid velocity components u and v and
the pressure p are the dependent variables while the independent
variables are x, y and the time variable t.

In addition to the cell system which represents the flow field
by a finite number of data points, there is a line of marker particles
whose sole purpose is to indicate where the free surface is located.
These hypothetical particles may or may not represent the actual
fluid particles at the free surface, depending on whether one chooses
the Lagrangian or the Eulerian point of view to calculate the motion of
free surface,

The marker-and-cell system provides an instantaneous repre-
sentation of the flow field for any particular time. When an initial
set of conditions is given, the entire fluid configuration can be ad-
vanced through a small but finite increment of time 6t. First, the
pressure for each cell is obtained by solving a finite-difference

170

Computatton of the Motion of Long Water Waves

j= JMAX- ~-|~ --

Fig. 7. Cell setup and position of variables

Poisson's equation, whose source term is a function of the velocities.
This equation was derived subject to the requirement that the resulting
finite- difference momentum equations should produce a new velocity
field that satisfies the continuity equation (conservation of mass). The
finite- difference equations of motion are then used to compute the new
velocities throughout the mesh. Finally, the marker particles are
moved to their new positions, their velocities being interpolated from
the nearby cells. The new flow configuration now serves as the initial
condition for the next time step and the foregoing procedure is repeated
as many times as necessary for the investigation. With proper choice
of 6x, dy and 6t, the SUMMAC algorithm is capable of yielding so-
lutions that are computationally stable and also reasonably faithful in
simulating the physical phenomena.

Dimensionless variables are used throughout (cf., Sec. 3.1).

The governing equations for an incompressible, inviscid fluid are

Pa Tb t+vyz—=-y t+ By, (33)

171

Street, Chan and Fromm

dv dv dv _ dp
and
du BV _ 9 (35)

ox By

Here, p is the pressure; gy, and gy are the x and y components
of the gravity acceleration whose absolute value is g and t is the
time variable. Also, if the direction of gravity is the same as the
-y direction, then g,=0 and gy=- 1.

Boundary conditions are easily derived for the fluid motion at
the solid walls of the tank (cf. , Chan and Street [1970a]). For incom-
pressible fluids with very low viscosity, such as water, it is suffi-
ciently accurate to use at the free surface the single condition

p= Pg (x; t) (36)

where pg is the externally applied pressure at the free surface.
Under usual circumstances p,= 0, but it can also be prescribed as
a function of x and t for some problems.

As shown in Fig. 7 the computation region is divided into a
number of rectangular cells. The fluid pressure p is evaluated at
the cell centers, while u is defined at the mid-point of the right-hand
and left-hand sides of the cell and v is defined at the midepoint of
the upper and lower sides. Then, for the cell (i,j) the following set
of equations are derived from Eqs. (33) and (34):

rei 7 ied t dt gy + (Pij - Piety) (37a)
Tuite 7 oar t-5t gy ot ot (Pi.y, - Pij) (37b)
“el = Viieg + OB, +e (Pij - Pijst) (37¢)
Viel = Vijek + 6t gy + s (Pij-1 - Pij) (37d)

In the above equations, variables with the superscript nt1
are related tothe nti‘ time step. Variables lacking a superscript
are evaluated at the n't step. Thus, Eqs. (37) are suitable for
updating the values of u and v about the cell (i,j). The "eonvective
contributions" u* and v* are

Computatton of the Motion of Long Water Waves

* du du
Wish) = Yiedj SE iis By itd (38)
* av ov
Vijed = Vijes + St ¢ - Ret ES 5 V By Dlist (39)
* * n+l n+
where Vid) and Vij contribute to uj. j and vjjsk » respectively,

through the convection process and ( ) represents Me as yet un-
specified finite difference approximation of the enclosed terms.

Before Eqs. (37) can be employed to compute the new velocities,
the p field must be obtained. Consider the finite-difference conti-
nuity equation [see Eq. (35)]

we Fe adele ied
n+l Ey ie j Bites j ij+ ~ Yij- _
Dij = a Pe ee (40)

Substituting Eqs. (37) into Eq. (40) and requiring Dj, = 0 leads to
the pressure equation

a 21: $DO: 1; as + pi
—— Pi+tj Pi-lj + Pij+l Pij-! + -
Pij =( 5x2 Sy 2 Rij) (41)
Here

Z= 2(<5 + 52) (42)

sj 1 ee aE 4 L

ee ee ij iied - Vij-d
a a ptt ot : ee M2)

Near the free surface "irregular stars" (Fig. 8) must be used to de-
rive an appropriate pressure equation so that, in the discrete sense,
the free surface condition p =p, is applied to the exact location.

Let ), No, Na, Ng be the lengths of the four legs of the irregular
star (Fig. 8) and p,, Py», P3: Pg be the value of p at the ends of these
legs. Then, it can be shown the irregular-star pressure equation is

= — 179730 Pi 7 TPs + "4P2 + Pa yp. 44
Pi ~ 2mang t 17 Fes ns nn, wea peg

Equation (44) reduces to Eq. (41) when Eq. (44) is applied to an
interior cell.

Street, Chan and Fromm

Fig. 8. Irregular star for p calculations

The hypothetical particles that mark the free surface are
moved to their new locations according to their locally interpolated
values of u and v. Fora given particle k we find the velocity
component u, for the particle by making a Taylor series expansion
about the nearest data point of the u field. Similarly, a series
expansion about the nearest data point of the v field gives v,, the
y-component of the particle velocity. With u, and v, available,
each free surface marker particle is advanced by the following
formulas:

ri
—s
WW
=]
+
_
=
[o4]
ct

(45)
n n+l
yy + ve dt

—-
Hl

where x and Yk refer to the position of the yh particle at the
nt" time step. Also, the particle velocities are evaluated at the
advanced, i.e., nti , time step.

The quantities u and v are not defined outside the fluid
domain, but they are needed to carry out the computations using
Eqs. (37) and (43) and the particle velocities near the free surface.
We calculate these undefined u and v values by a simple linear
extrapolation from the fluid interior.

A complete set of initial data -- the u and v fields and the
position of a line of particles depicting the free surface, are needed
to start the computation. The initial pressure p needs to be known
only approximately, such as a hydrostatic distribution, because the
p field is solvable if u and v are given.

A704

Computatton of the Motton of Long Water Waves

The evolution of fluid dynamics is calculated in "cycles," or
time steps. At the start of each cycle the source term Rjj for
each cell is evaluated by Eq. (43). The pressure p is computed
only for those cells whose centers fall in the fluid region; either
Eq. (41) or Eq. (44) is used as appropriate. The successive-over-
relaxation method is used to solve the p field. The iteration is
terminated when

(m) (m-1)
i CP <€ (46)

for every cell, where (m) means the m'" iteration and eP isa
predetermined small positive number. The accuracy in solving pij
at the n~ time step has a direct bearing on the accuracy of satisfy-
ing the continuity equation Di; nl 0 [Eq. (41)] at the nti" step.
Smaller values of Dij We result len smalles €p are used, However,
there is little improvement in reducing Di; for €,< 10°” because
the round-off level of the computer has been reached.

Now Eqs. (37) yield the new velocities. Then each marker
particle is advanced to its new position by Eqs. (45). Thus a cycle
is completed and the next one can be started immediately.

The convective contributions given by Eqs. (38) and (39) can
be approximated by a wide variety of finite difference formulas.
Chan and Street [1970c] show that, while the original MAC and
early SUMMAC equations used a first-order explicit method, second-
order explicit methods are better. Of the two second-order explicit
schemes studied, the so-called "upstream" difference alone rather
than in a "phase-averaged" procedure yields better results in prob-
lems where free surface waves are present. In,this upstream dif-
ference, if Whim represents either Vises jp OF wii) > then for the

case when Um ™0 and Vom >0 Use

Won Gi Wm. l +E ens Wy)
4 (8-1) co ; Da + Fen) (47)
where
W pry = Weim + (wg tm)
+ (a-1) (Wp om - 20 im of W pm) (48)
and

175

Street, Chan and Fromm

n n
ite StU pm , 6 = 5tV em
6x dy

We examined the finite difference convection equations by the
extended von Neumann method in which nonlinear equations are first
linearized. The resulting criteria were

jaj=i1; |pi[ =1 (49)
A second criterion
dt 1
tx < re! (50)

where C =the surface wave celerity, was derived by considering
the propagation of the free surface waves. These were simple linear
analyses and can only be used as guidelines in choosing the time
increment 6t for given 6x and dy. Because numerical dispersion
is quite severe for short wave components, care must be exercised
to provide adequate resolution for all the important features in the
flow. As a rule of thumb, the smallest significant flow feature must
be represented by at least ten cells.

In both the MAC and SUMMAC a line of particles was used to
mark the free surface position. A pair of (xy, y,) values were
associated with the k' particle at the nth time step. Then Eq. (45)
was employed to calculate (xe x.) This procedure is really a
Lagrangian method that tends to be unstable after a large number of
time steps. The problem is not serious for simulating solitary
waves [cf., Chan and Street, 1970a]. But, in calculating periodic
waves a given particle is moved up and down as each wave passes.
In the process a small number is systematically added to and then
subtracted from x, and y, contributing to very large round-off
errors. In addition, there is no restraint on the individual particle
positions because each is calculated independently of the others.

To overcome the difficulty with moving particles, an alterna-
tive approach using the Eulerian point of view can be developed. The
flow region is divided by a number of vertical lines with equal spacing
4 and n is now the height of the free surface measured from the
reference level y = 0 at the channel bottom. The horizontal posi-
tions of these vertical lines are fixed and we only compute the change
in n along each vertical line as time passes.

The kinematic condition at the free surface, from the Eulerian
viewpoint is

176

Computatton of the Motion of Long Water Waves

ee (51)

Many difference schemes may be developed to approximate Eq. (51).
Our tests show that the forward implicit method with the difference
equation

qu _ nr al al th 5 Ue
Be Tek oo ke (Seed ; ) (52)

is one of the best. A stability analysis shows that Eq. (52) leads to
a linearly stable computation with slight dampling.

Numerical tests were carried out in the context of a simple
physical problem whose exact solution was known, viz., a solitary
wave ina horizontal channel. Among five alternative combinations
of surface and correction term treatments tested, that using Eq. (52)
and Eqs. (47) and (48) was the best.

Now consider 6t. The maximum fluid speed in the above
tests was u,,,* 0.30 and 6x =0.5 while 5y = 0.1. According to
Eq. (49)

max

6x _ 0.50 _ 1.67

or 0.30

Umax

The speed of the surface wave is C=1.18. The Courant condition
[ Eq. (50)] would require

ox 0,50 =

But the Courant condition should also be observed in computing
the free surface. Because we used the spacing 4 =0.05 at the free
surface, the condition

A > 0205.
bt < & = pq = 0.0424

must be satisfied. Therefore, the most restrictive condition is

&t < 0.0424. In all the test examples, 6t =0.05 was used. This is
slightly larger than the estimated maximum allowable 6t, but no
distortions or instabilities were noted. However, the result of
seriously violating the Courant condition, i.e., using 6t = 0.10,

was large non-physical distortions that suggest one has to be careful
about the choice of 5t.

Street, Chan and Fromm

Experiments were also performed on the problem of generating
periodic waves by pressure pulse (Sec. 4.2). Instability at the free
surface became explosive after 600 time steps when the particle
method was used. Using the forward implicit method, we were able
to calculate up to more than 3000 steps and there were still no signs
of instability.

The numerical tests described above indicated that it is ad-
vantageous to use the second-order upstream difference method to
compute the convective contributions to a ; and vn - For the
free surface calculations, the forward implicit scheme is best.
However, the particle method of computing the free surface need not
be dismissed altogether. The Eulerian method is restricted to waves
in a channel whose two ends are vertical walls. If the water surface
has an advancing front, such as a solitary wave climbing on a slope
[ Chan and Street, 1970b] , the particle method is the only choice,
When 6t is small enough and the particle velocities are evaluated
at the ntit? time step, the particle method does provide a stable
solution. However, the particle method should not be used in the
simulation of periodic waves over long periods of time.

4.2. Results and Discussion

As an example, periodic pressure pulses were used to
generate a train of oscillatory waves in a channel of constant depth
(Fig. 9). The fluid is entirely at rest at t= 0. Then, the pressure
distribution

on, athe

Xq

R, = F(X,T)

Fig. 9. Setup of pressure pulse problem

178

Computatton of the Motton of Long Water Waves

p, sin (=e) . [eee nea for, OS x= xq

Ps = (53)

0 for x > 4

is applied to the free surface. Here p, is the amplitude of the
pressure pulse, Tp, is its period and xg is the horizontal length of
the surface subject to the prescribed pressure. Equation (53) was
employed by Fangmeier [1967] in solving the same type of problems
using time-dependent potential flow equations.

In the first case, a channel of the length L, = 30.0 was used.
The computation domain consists of 80 X 24 cells, each with 6x =
0.30 and Sy = 0.10. Weused pp=0.10, Tp=7.6 and xq=4.0 in
Eq. (53) to generate the surface disturbances. The development of
the u field is shown in Fig. 10. The plot increment is 0.025 per
contour line with u = 0.0125 on the contours closest to the ends of
the channel. At t = 10.0 the leading wave leaves the generating
area and progresses to the right. At t = 43.493 the first wave runs
up the right-hand wall and reflection begins to interfere with the on-
coming waves. As a result, a standing wave pattern occurs when
t= 72.986 to t= 84. 233.

|
T2#10.000 T= 78.734

T= 72.986 T= 84.233

Fig. 10. Periodic waves (u contours)

179

Street, Chan and Fromm

}
\\

/}
7} \
J

|
|

/ TV. y ZA GINAIAI\\ JIN
Soa | WWIII

T= 10.000 T= 78.734

|
BIAS = 4 LO

T=#30.000 T= 80.984

T= 43.493 T= 62.484

LOI = ROO “A

T= 72.986 T= 84.233

Fig. 11. Periodic waves (v contours)

In Fig. 11, the time history of the v-field is shown. The
plot increment for the contours is also 0.025 per line. On the chan-
nel floor, v=0.0. The first contour above the floor has v= +0.0125
if it is in front of the wave crest, and v = - 0.0125 if it is at the
back. The sparse contours on the right-hand side of the channel at
t = 80.984 and t = 84.233 indicate that when the standing waves
reach their peaks the fluid velocity almost becomes zero temporarily.
This phenomenon is caused by the interaction of the reflected and
incident waves that tend to alternately enhance and cancel each other.

In Fig. 12 we used along channel with L; = 60.0. Thus the
"progressive" wave patterns can be analyzed before the reflection
sets in. The wave train is composed of a group of dispersive waves.
The amplitude increases from the leading wave to the third wave.

It then decreases on the following waves. This observation suggests
that the nonlinear response of the fluid system is somewhat out of
phase with the forcing function at the surface. Therefore, it appears
that pure nondispersive periodic waves cannot be generated by the
disturbance described by Eq. (53) unless the amplitude p, is very
small.

Because of its symmetrical profile, we selected the fourth
wave in Fig. 12 and compare it in Fig. 13 with Stokes' second-order
and third-order theories [ Wiegel, 1964]. Good agreement with the

180

Y/do
0.75

1.25

1.00

0.50

0.25

Computation of the Motton of Long Water Waves

24.00 30.00
X/do

36.00

Fig. 12. A train of nearly periodic waves

T= 7.60
Ls 6.70
H= 0.55

d= 0.9348

23 24

Fig. 13.

25 26
X/do

27

SUMMAC

STOKES' SECOND
ORDER THEORY

STOKES' THIRD
ORDER THEORY

28 29 30

Comparison of wave profiles

181

60.00

Street, Chan and Fromm

third-order theory is found.

To obtain a meaningful comparison with the profiles of the
Stokes' waves, a Fourier analysis was performed on the profile
computed by the SUMMAC method. The SUMMAC wave profile in
Fig. 13 can be expanded in a Fourier series of the Stokes form
[ Wiegel, 1964]. The coefficients can be evaluated by the standard
procedures in calculus. The first ten coefficients have been com-
puted and compared with those for the Stokes' theories. From the
trend of each coefficient, it appeared that as the order of approxi-
mation increases the Stokes' wave converges to our numerical
solution. Also, in comparison of wave speeds we find good agree-
ment with Stokes' third-order theory. The difference is within
0.4 per cent.

In Fig. 14 the distribution of u under the wave crest and the
wave trough is compared with Stokes’ theory. The SUMMAC method
predicts a much lower u velocity under the crest than Stokes' solu-
tions. This discrepancy is probably caused by the fact that the
numerical simulation was made in a channel of finite length which is
a closed system and the waves have not quite reached the steady
state, while the Stokes’ waves hold for an infinitely long channel.
Nevertheless, the slope of the u-distribution (i.e., 8u/8y) is very
close to that of the third-order theory.

(a) (b)

UNDER WAVE UNDER WAVE
TROUGH CREST

Y/ do

° SUMMAC

—— STOKES' SECOND
ORDER THEORY

——— STOKES' THIRD
ORDER THEORY

Fig. 14. Distribution of u under wave crest and trough

182

Computatton of the Motton of Long Water Waves

1.50

DIRECTION OF MASS TRANSPORT

1.25

STILL WATER
LEVEL

1.00

Y/do
0.75

B

3

%

a

8 a et oe

Ti.2s 11.80 11.75 12.00 te.e9 12.50 12.75 13.00 13.25 13,50 13.75 14,00
X/dy

Fig. 15. Motion of fluid particles

The paths of the fluid particles are plotted in Fig. 15. We
selected three fluid particles which lie on the vertical plane x = 12.0
at t=0.0. Their initial vertical positions are y = 6.0, 0.5 and
1.0, respectively. The instantaneous particle positions are plotted
at every 5 &t's (6t = 0.05). Each particle moves in an oscillatory
pattern which completely differs in nature from the translation motion
in a solitary wave. The surface particle travels in a quasi-elliptic
orbit but never returns to its original position. Thus, there is a net
mass transport in the direction of wave propagation near the free
surface. At half water depth the scale of the orbits is smaller and
the current (mass transport) is opposite to the wave direction. On
the channel bottom the particle merely goes back and forth hori-
zontally and the "backward current" is also larger there. Because
the wave channel in our simulation is a closed system, the fluid
carried along by the surface waves must return in the opposite
direction in the lower fluid layers.

Finally, a comparison was made with the numerical solutions
of Fangmeier [1967]. The qualitative agreement was good, as was
the agreement in the wave phase; however, the SUMMAC method
gave a much better treatment of the free surface that markedly re-
duced the height of the largest of the waves as compared to
Fangmeier's simulation.

Street, Chan and Fromm

4.3% Prognosis

The successful application of the SUMMAC technique to
several physical problems indicates its usefulness as an engineering
research tool for analyzing the dynamics of water waves in two space
dimensions. It is capable of providing accurate quantitative results
as well as qualitative descriptions [see, e.g. , Chan and Street,
1970b]. In addition, rapid advance in the design of high-speed com-
puting systems makes numerical modelling economically feasible.

While it is possible to employ the SUMMAC technique to
attack a wide variety of water wave problems, some limitations
inherent in the method must be noted. First, as a result of achieving
a high degree of accuracy in applying the free surface pressure con-
dition by using irregular stars, waves after breaking cannot be simu-
lated. When breaking occurs, the computation must be terminated.
Second, only non-turbulent flows are considered in our model.
Although laminar viscous damping has little effect on large scale
wave motions, energy dissipation due to the turbulence can be sig-
nificant. However, a recent study by Pritchett [1970] shows that it
is feasible to implement a heuristic simulation of turbulence in the
MAC framework.

ACKNOW LEDGMENT

This research was supported in part by the Fluid Dynamics
Branch, Office of Naval Research, through Contract Nonr 225(71),
NR 062-320.

REFERENCES

Brennen, C., "Some Numerical Solutions of Unsteady Free Surface
Wave Problems Using Lagrangian Description of the Flow,"
2nd International Conf. on Numer. Meth. in FluidDyn.,
Berkeley, Calif. , Springer-Verlag, Pub. , September, 1970.

Camfield, F. E., and Street, R. L., "Shoaling of Solitary Waves
on Small Slopes," J. Waterways and Harbors Division,
ASCE, V. 95, No. WW1, Proc. Paper 6380, pp. 1-22,
February, 1969.

Carrier, G. F., "Gravity Waves on Water of Variable Depth," J.
Fluid Mech., Vol. 24, Pt. 4, pp. 641-660, April 1966.

Chan, R. K. C.,*‘ and Street, R. L., "A Computer Study of Finite-

Amplitude Water Waves," J. Compt. Physics, Vol. 6,
No. 1, August 1970 [1970a].

184

Computatton of the Motton of Long Water Waves

Chan, R. K. C., and Street, R. L., "Shoaling of Finite- Amplitude
Waves on Plane Beaches." Proc, 12th Conf. on Coastal Enginer-
ing, Washington, D.C., ASCE, September 1970 [1970b].

Chan, R. K. C., and Street, R. L., "SUMMAC -- A Numerical Model
for Water Waves," Stanford C. E. Dept. T. R. No. 135,
Stanford, Calif., August 1970 [1970c].

Chan, R. K. C., Street, R. L., and From, J. E., "The Digital
Simulation of Water Waves -- An Evaluation of SUMMAC,"
2nd International Conf. on Numer. Meth. in Fluid Dyn.,
Berkeley, Calif. , Springer-Verlag, Pub. , September, 1970.

Fangmeier, D. D., "Steady and Unsteady Potential Flow with Free
Surface and Gravity," Ph.D. Dissertation to U.C. (Davis),
Davis, Calif. , 1967.

Forsythe, G. E., and Wasow, W. R., Finite-Difference Methods

for Partial Differential Equations, J. Wiley and Sons, Inc.,
1960.

Freeman, J. C. and LeMéhauté, B., "Wave Breakers on a Beach
and Surges on a Dry Bed," J. Hydr. Div., ASCE, V. 90,
No. HY2, pp. 187-216, March, 1964.

Fromm, J. E., "Practical Investigation of Convective Difference
Approximation of Reduced Dispersion," IBM Research Report,
RJ531, IBM Res. Lab., San Jose, Calif., 1968.

Heitner, K. L., "A Mathematical Model for Calculation of the Run-
up of Tsunamis," Earthquake Engrg. Res. Lab. Report,
Calif. Inst. Tech., Pasadena, Calif., May 1969.

Hirt, C. W., Cook, J. L., and Butler, T. D., "A Lagrangian
Method for Calculating the Dynamics of an Incompressible
Fluid with Free Surface," J. Compt. Physics, V. 5, No. 1,
April, 1970.

Hwang, L. S., Fersht, S., and LeMehaute, B., "Transformation and
Run-up of Tsunami Type Wave Trains on a Sloping Beach,"
13th Cong. of IAHR, Kyoto, Japan, 31 Aug. to 5 Sept. 1969.

Keulegan, G. H., and Harrison, J., "Tsunami Refraction Diagrams
by Digital Computer," J. Waterways and Harbors Div.,
ASCE, V. 96, No. WW2, Proc. Pap. 7261, pp. 219-233,
May 1970.

Lautenbacher, C. C., "Gravity Wave Refraction by Islands," J.
Fluid Mech., V. 41, Pt. 3, pp. 655-672, 29 April, 1970.

185

Street, Chan and Fromm

Leendertse, J. J., "Aspects of a Computational Model for Long-
Period Water- Wave Propagation," RAND Memo RM-5294-PR,
Santa Monica, Calif., May 1967.

Madsen, O. S., and Mei, C. C., "Dispersive Long Waves of Finite
Amplitude over an Uneven Bottom, MIT Hydro. Lab. Rep.
No. 117, Dept. of Civil Engrg. , Cambridge, Mass. , November
1969 [ 19692] :

Madsen, O. J., and Mei, C. C., "The Transformation of a Solitary
Wave over an Uneven Bottom," J. Fluid Mech., V. 39,
Pt. 4, pp. 781-792, 15 December, 1969 [1969b].

Mogel, T. R., Street, R. L., and Perry, B., "Computation of Along-
shore Energy and Transport," Proc. 12th Conf. on Coastal
Engrg., Wash., D. C., ASCE, September 1970.

Peregrine, D. H., "Long Waves ona Beach," J. Fluid Mech.,
V. 27; Pt. 4, Ppp. 815-827, 1967.

Pritchett, J. W., "The MACYL6 Hydrodynamic Code," Info. Res.
Assoc., IRA-TR-1-70, Berkeley, Calif., 15 May, 1970.

Schreiber, D. E., "A Generalized Equipotential Plotting Routine
for a Scalar Function of Two Variables," IBM Research Rep.
RJ-499 (No. 10673), Computer Applications, New York,
24 May, 1968.

Street, R. L., Burges, S. J., and Whitford, P. W., "The Behavior
of Solitary Waves on a Stepped Slope,"Stanford C. E. Dept,
T.R. No. 93, Stanford, Calif., August 1968.

Street, R. L., Chan, R. K. C., and Fromm, J. E., "The Numerical
Simulation of Long Water Waves -- Progress on Two Fronts,"
Proc, Intl. Symposium on Tsunamis and Tsunami Res.,
East/West Center, Honolulu, Hawaii, October 1969.

Vastano, A. C., and Reid, R. O., "Tsunami Response for Islands;
Verification of a Numerical Procedure," J. Marine Res.,
V. 25, Ne. 2; pp. 129-139, 1967.

Welch, J. E., Harlow, F. H., Shannon, J. P., and Daly, B. J...
"The MAC Method -- A Computing Technique for Solving
Viscous, Incompressible, Transient Fluid- Flow Problems
Involving Free Surfaces," Los Alamos Sci. Lab. Rep.
LA-3425, 1966.

Wiegel, R. L., Oceanographical Engineering, Prentice Hall,
Englewood Cliffs , New Jersey, 1964.

186

Computation of the Motton of Long Water Waves

Williams, J. A., and Kartha,'K. K., "Model Studies of Long Wave
Amplification by Circular Islands and Submarine Seamounts, "
Hawaii Inst. of Geophys., Final Report (HIG-66-19),
November, 1966.

187

wee

AN UNSTEADY CAVITY FLOW

D. P. Wang
The Catholte Untversity of America
Washington, D.C.

I. INTRODUCTION

A perturbation theory for two-dimensional unsteady Cavity
flows has been formulated by the present author and Wu [1965]. In
that formulation we regard the unsteady part of the motion as a
small perturbation of a steady cavity flow already established. This
already established steady cavity flow will be called as the basic
flow. Our perturbation expansion is carried out in terms of a set
of intrinsic coordinates (s,n) of the basic flow. The coordinate s
is the arc length measured along a streamline in the direction of the
basic flow, and n the distance measured normal to a streamline.
An illustration is given in Fig. 1, where the solid lines represent
the basic flow configuration, AB represents the wetted side of the
solid body, AI and BI, the two branches of the cavity wall which
is a free surface. Also shown in Fig. 1 is the unsteady perturbed
flow configuration represented by dotted lines. The unsteady

ee h(s,t)

- |= - =
aa

Fig. 1 Illustration of an unsteady perturbation flow

*
This paper will henceforth be referredtoas W.

189

Wang

displacement of the free surface and the solid body from their cor-
responding locations in the basic flow is denoted by n = h(s,t). If
the wetted side of the solid body and the free surface of the basic
flow are taken to be n=0,h(s,t) is assumed to be a very small
quantity. In W we have found that the linearized kinematic and
dynamic boundary conditions on the free surface for the unsteady
perturbation potential $, are

oe se tage on n=0 (1)

and

0, (2)

Bh vg Stee gn ons

and that the boundary condition on the solid body is

ee = oP + ee (a,b) on n=0., (3)

In the above equations R and q, are respectively the radius of curva-
ture and the constant speed on the free surface of the basic flow, and
q, is the speed of the basic flow. The + (or -) sign onthe right-
hand side of (2) holds for the upper (or lower) branch of the cavity
wall; these signs are necessary to make R a positive quantity. We
should mention here that in obtaining (2) we have assumed that the
cavity pressure remains unchanged during the unsteady perturbation.

If we regard q,/R as an equivalent gravitational acceleration
and the s-coordinate rectilinear, then (1) and (2) are in the same
form as the linearized free surface boundary conditions in water wave
problems. Thus we expect that the centrifugal acceleration q?/R
due to the curvature of the basic flow streamline should play the role
of a restoring force in producing and propagating the surface waves
along the curved cavity wall.

The purpose of the present paper is to use this perturbation
theory to study some unsteady behavior of the Kirchhoff flow when the
solid plate is in small harmonic oscillations.

II. THE BASIC FLOW

In this paper we consider the basic flow to be a flat plate held
normal to an incoming uniform stream of infinite breadth, with-a
cavity formation of infinite length as shown in Fig. 2. This is the
so-called Kirchhoff flow. Both the speed of the incoming stream and
the length of the plate AB are taken to be unity. A set of Cartesian
coordinates (x,y) with its origin at the stagnation point C is chosen
as indicated in Fig. 2, where the point I denotes the point at infinity.

190

An Unsteady Cavity Flow

z— plane

Fig. 2 The basic flow and its conformal mapping planes

The solution of this problem can be obtained by the Levi-Civita
method in terms of a parametric variable ¢ (Gilbarg [ 1960]),
and we simply give it below for subsequent use. The complex
potential f,, velocity w, and the complex variable z=x+iy are

f.-7(t+¢) (4)
w= Sy > (5)

and

£94

Wang

g 1
z=) Wo dc , (6)

where

K=33— (7)

The flow region in various planes can also be found in Fig. 2. For
points on the plate AB, we may deduce from (6)

x= > (sin 20 - 4cos 0~ w~ 20), (8)

where @= Arg, when © is onthe circular arc ACB, and

-nr2Z=0s0. (9)

III UNSTEADY PERTURBED FLOW

It is shown in W that by eliminating h between (1) and (2)
and transforming (s,n) and R tothe variables fy and Wo a
single free surface boundary condition in complex variable form can
be obtained, which is

Re [L(f,)] = 0, (10)
where

f, = >, + ip, (11)

is the complex perturbation potential, and the linear differential
operator L is

L= (s+ wr) -{elSG (Fe tor) - Se or (12)

It is also shown in W that the boundary condition (3) on the
solid body can be transformed into

Im gel = - alte + oe (ach) (13)

192

An Unsteady Cavity Flow

where h is regarded as a given function of s and t. Since along the
solid body df, =q,ds, which is purely real, (13) may be written as

* dh
Im f =- f 37 As = qoh, (14)

where we have set Imf, to zero at s = 0, the stagnation point of the
basic flow. For the basic flow considered and from the definition
adopted for the intrinsic coordinates (s,n), we note that along CB

do = Wo: (s,n) = (x,y), (15)
and along CA
Go = - Wo: (s,n) =- (x,y). (16)

If we denote the prescribed motion of AB as y= 1,(x,t) instead of
n = h(s,t), with the aid of (15) and (16), (14) becomes

x
8
im t= - J) sapl dx - won. (17)

Let us assume that the prescribed motion of AB is given by
n (x,t) = € cos ot, (18)

where e€ is avery small constant quantity and w is the frequency of
oscillation. For convenience¢ in the following analysis, let us intro-
duce an imaginary unit j = a which is regarded as different and
non-interacting with the imaginary unit { used in defining the com-
plex variable z=xtiy. If we agreed that only the real part with
respect to j of a quantity is meaningful to us, we may write (18) as

mi, Goat) = ee). (19)

To avoid any confusion in the notation, from now on when we mention
the real (or imaginary) part of a function ¥, denoted by Re (or
Im 3) as it has been used so far, we mean the real (or imaginary)
part of 3 with respect to i, not with respect to j, evenif 3 con-
tains j. Only when the final result is obtained shall we take the real
part with respect to j as our solution.

If we assume that the disturbance has already been applied for

a long time so that the entire flow is in harmonic oscillation, we may
write the complex velocity potential f(z, t) as

193

Wang

f(z,t) = f(z)e!™’, (20)

If we substitute (20) into (10) we may write the free surface
boundary condition as

Re H=0 (21)
where
2 5
fe a 625, Abd ae Sea at a [o? + jo in (4 awe) ¢, (22)
df df, df,/ df, df, Wo df

which is an analytic function defined in the flow field. We may also
write

2
i df 1 [ dw, ye
H = ag et 2jo- In Wo aa
Wo dz Wo ( She
1 dw
- |w rare aa peattcel |e. (23
[u? +; = CoeerR ) )

Since the differential operator L is purely real on the plate AB,

the boundary condition (17) may be expressed in terms of the analytic
function H. By a straightforward application of L on (17) and by
the use of (19) and (20), the boundary condition on AB may be written
as

Im H= y, (24)
where
2 2
=€ [- ok + jo(w crs ova + [wrx
jo(w, + me, dfy df
; 4 d z
- jo(w, - 3 )] Ge tm wo + Jo atv + 2)I. (25)

The boundary conditions expressed in the forms of (21) and (24) may
be used to determine H for points in the interior of the flow field.
However, to obtain a physically acceptable H, other boundary con-
ditions have to be imposed on H.

We shall assume that the free surface displacement due to the
unsteady disturbance of the plate AB has to be bounded everywhere.
This condition can be satisfied if the free surface displacement is
bounded at the separation points A,B and at the point at infinity.

194

An Unsteady Cavity Flow

It is shown in W that the curvature of the free surface of the basic
flow is

1 . | Ge de : (26)
R Wo df,
where q,= 1 inour problem. From the local mapping behavior near
z| = 00 between the w,-, f,- and z-planes shown in Fig. 2, we see
that
Wo ti” .agr (27)
and
o> = iz as lz| — ©, (28)

where a, is a constant. In the following analysis we always use an
to indicate some constant. The substitution of (27) and (28) into (26)
gives us

{ -3/2
er Olz] ) (29)

on the free surface as |z | — oo. Since we assume that the free sur-
face displacement near the point at infinity has to be bounded, then,
from (2) and (29), we have, near the point at infinity,

spi B=. 00

For harmonic oscillations, (30) suggests that we may write

= Ale )efO as |z| + &, (31)

since along the free surface of the basic flow 0/8@s = 8/@f). The
substitution of (31) into (2) gives

j -f
dA _jwlt-f)

pea as |z| — oo. (32)

is)
In view of (29) and (28), (32) ipplies that along the free surface near
ies

the point at infinity A = O(z , at most, in order that h be bounded
there. With h being bounded at infinity and having a form shown

195

Wang

in (32), we can obtain, from (1), that on the free surface 99,/8n =
o(z7') as |z|-—- oo. These results indicate that along the free sur-
face near the point at infinity both f and df/dz should vanish.
Since the unsteady disturbance is mainly a surface phenomenon, it
is not unreasonable for us to assume that f and df/dz also vanish
in the interior of the flow field near the point at infinity. Therefore,
we assume that

HO as |z| — oo. (33)

This rules out the possibility that there is any induced circulation
around the point at infinity. Based on the assumption that h is
bounded at infinity and the result that on the free surface 909,/8n =
o(z) as |z | —> oo, an integration of (1) will show that if

oo Gel) **#5>0 (34)

in the neighborhood of the points A, B with r the distance from
these points, h will be bounded at A,B. Condition (34) is also
necessary in order that the pressure be integrable over the plate
AB.

To facilitate the determination of H, let us introduce a
transformation

, (35)
G=T- (7? - 1ylf2
where the cut in the LG hep cnaks is taken along the straight line between
-1 and 1, and (72-1)'/* +7 as |t| +m, -aw<ArgtT=m. The
mapping (35) maps the entire basic flow onto the upper-half 7T-plane

as shown in Fig. 3. In terms of the variable 7, the function y
becomes

eile air

y= TR ELT | i (Kw)? 1-7°) far(1-7
+ t2cos'7| + (Ka) [7(1-77)/? (4-579)

£ $2723. = 72 +107 + 4 + (2-372)cos" | +2jKuT {, (36)

where

196

An Unsteady Cavtty Flow

T —plane

(-1,0) (1,0)

Fig. 3 A conformal mapping plane of the basic flow

ay = cos. 7 —=.0¢ (37)

We note that y, shown in (36), has simple poles at 7T=0 and +1,
and therefore, it obviously does not satisfy the Holder condition on
AB, where -1=7=1. Inorder to find H, which is regarded as
a function of T now, let us continue it analytically into the lower-
half 7T-plane by

H(r) = - H(7), (38)
and let us define another analytic function 9(T) by
Q(T) = 7(7? ~ 1)? Hr), (39)

where the branch cut for (7? - 1)!72 has been defined after (35). If
we denote the limiting values of 2 as Im7—~+0 by Qy,, then,
from (39), (21) and (24),

iT]
(o)

on [Re T| > 1
(40)

- 27(1 - reife on |Re T | <i;

With 2,- Q_ given by (40), the function (7) may be determined
(Muskhelishvili [ 1946])

‘ | 21/2
Q(T) = +{) eal do + ss bark, (41)

n=O

where b, are arbitrary constants and N is an arbitrary integer.
From (39), H(T) may be written as

eg

Wang

N
| /
H(1) = een tN o(1-0%)F y(o) 4, +) byr ; (42)

o-T
n=O

However, to satisfy (21), b, have to be purely real. The condition
(33) is equivalent to

H(t) > 0 as |t| > o, (43)
since near | z | = 0
Ta aces (44)
To satisfy (43),
b, = 0 for i= Zs (45)

Due to the symmetry of our problem, which implies
Im f=.0 on Cl, (46)

and due tothe fact that the differential operator L is purely real on
CI, we require that

be O. (47)

This leaves only the constant b,; undetermined. After carrying out
the integrations in (42), we may write

H(t) = ooS95 papery | j(Ku)?M,(7) + (Kul?M, (7)

2 QnjKor + b\7%{(7?-1)'? ], (48)

where

al

M, (7) = 07%(7°-4)(47 +9) +7771)? [ 20- 5 -4(1 tn)7"]

4

Array? in — ore aty os(a), (49)

198

An Unsteady Cavity Flow

M(t) = 2ur(672-5) + w%{572-1) - 2(77-1)/* [4G + (546m) 77]

+ 7(4-572)(72-1)/2 In Zoe + (2-37%)(72-1)'® (7), (50)
G = Catalan's constant = 0. 915965594, (51)

and

| =
-|
or) = WesBe err) 8 ~tw=cos ¢=0, (52)

which cannot be expressed in terms of elementary functions.

It is not difficult to see from (48) that as | 7 | — oo, the
dominating term is the one containing b,, which is of the order eae :
Since the remaining terms in (48) are obtained from the integral
shown in (42), they are, therefore, of the order |7 ee This indi-
cates that the b, term is the most important term for the flow field
near the point at infinity.

With H given by (48), (22) may be regarded as a linear,
second order, ordinary differential equation for f.

If we transform the independent variable from f, to G and
make the following change of dependent variable

1/2

f2 F(t) (S72) acy (53)

The differential equation (22) is readily reduced to

a°*F ae

where
2 1/2
df dw jwfo
G(t) = ($7) (Ge) eH, (55)

f, and wy, as functions of ¢ are given by (4) and (5), and H is
given by (48) with 7 as a function of ¢ given by (35). To help us
to understand the properties of the Eq. (54), let us make the follow-
ing change of variables

199

Wang

ge ioe
(56)
F imei” ,
which transforms (54) into
ar y (1 - 8jK 2p) = = 4ice?* 5
ap? - 8j)Kw cos 2B)F =- 4iGe §. (57)

This is Mathieu's differential equation. The flow region now occupies
in the B-plane a semi-infinite strip shown in Fig. 4. In principle,

a general solution of (57), or (54), can be obtained. If we denote the
solutions of the homogeneous equation of (54) by F,(¢) and F,(),

then a general solution of (54) is

C C
F(¢) = WOR) fF (¢) \ F,(A)G(A) dd - F,(6) \ F, (A) G(A) art,
3 4

(58)

where

dF

W(F|,F,) = art F, - SF, (59)

which is a constant.

B-plane
(7774 40) (37774 ,0)

Fig. 4 <A conformal mapping plane of the basic flow

200

An Unsteady Cavtty Flow

To obtain F explicitly we must first obtain F, and Fj. In
this paper we are not going to obtain the exact forms of F, and Fo,
even though they can be expressed in terms of the solutions of the
Mathieu equation given in (57). We shall, instead, obtain the asymp-
totic representations for F, and Fy as woo. We should mention
here that Langer [ 1934] has developed asymptotic representations
for the solutions of the Mathieu equation with at least one parameter
large; however, instead of modifying his asymptotic representations
to cover the equation shown in (53) with j another imaginary unit,
we shall derive the asymptotic representations for F, and F,
below.

Let us denote

2
OS ie (60)

then, the homogeneous equation of (54) can be written as

2

C

rev
hy

= jKwx (¢)F. (61)

"

Due to the symmetric properties of our problem we need only con-
sider half of the flow region, say the region bounded by ICAI, which
inthe €-plane is a quarter circle as shown in Fig. 2. Inthis region
Xx has a simple zero at € =1 anda pole of order 3 at € =0. If we
make a typical change of the independent variable to €, defined by
(Jeffreys [ 1962])

C
= ae = le () dt, (62)
|
and put
: -1/2
d

ee = { jKwé + r(&)] U, (64)

where

2014

Wang

2 dg Nge
| ove isp eer (65
feleanien w-ne ]

A straightforward expansion of (62) shows that

B= Ot") as 4-0, (66)
and, of course,

r(€) = 0(1) as G1, (67)

Equation (66) indicates that the mapping (62) maps the point I to the
point at infinity in the §-plane. The flow region ICAI inthe §-plane
is shown in Fig. 5. From (65) and (66) we can see that

r(€) = 0(zr) as |E| — o. (68)

Olver [1954] has investigated an equation of the form

a’u

ae [we + r(é)] U, (69)

where ee is a positive large parameter. He shows that for a domain
D, if r(€) is regular in D and r(&) = o(é |!" for some ao > 0

| |
€— plane

7/6

Fig. 5 A conformal mapping plane of the basic flow
region bounded between ICAT

202

An Unsteady Cavity Flow

as |é | — oo, and if the distance between the boundary lines of D
does not tend to zero as |&]|— oo in any subdomain of D, then,

a uniformly valid expansion of U in terms of Airy functions can

be obtained. Olver [ 1957] later extends the result to the case when
yp“ is a large complex parameter. For our problem, all the above
requirements for a uniformly valid expansion in terms of Airy
functions are satisfied except that the parameter pw” in our case is
jKw, where j is an imaginary unit independent of the imaginary unit
i used in the complex variable §. Since i and j do not interact
and j may be regarded as a real quantity so far as the imaginary
unit of i is concerned, we assume that Olver's result is applicable
here and write the two solutions of (64) as

= Ai[(jKu)” é][1 + 0(<)] (70)
and

Algae = Mi +o) ast o-oo, (71)

where Ai(X) is the Airy function with argument X, which may be
expressed either as the sum of two converging series or as an
integral given in the following (Jeffreys & Jeffreys [ 1956])

27i/3
a /

xs- Ls?
Ai(x) = a e * dg. (72)
mi eee

Substituting x =(jKa)'/* into (72) and manipulating the result, we
can show that

AY(JKu)'”¥] = 3{(1-17) ai[(aKey!E) + (1H) Ail(-iKa)”"E]}, (73)

where Ge and (eile will be taken as ewe and aris respec-
tively. Therefore, from (63)
oN
*) =(§ %)
ie 1/3 1/3
~2(F by {(1-ij) Ail (Ko) 6] + (1+ij)Ai[(-iKe) 6]}. (74)

Similarly,

203

Wang

dé -1/2
F, = (e) Us,
dé -1/2 ori/3 ‘
~2 (Se) (1-4) ailGK aye 8] + (1445) Aili a)'3 0? 7},
(75)
In (74) and (75)
dé -1/2 £ 1/4
CaM Ga ae sh

where € is given by (62) and x given by (60). With F, and F,
given in (74) and (75), we may find W(F, ,F,), which is

1/3 jw/6 -iw/3

W(F, , F,) kw s(Ka) e Z (77)
or
WR Fy ays liners Geren, (78)
6)

Let us now express the solution F interms of the 7 variable.
From (4), (5) and (35), we have

fo= KT; (79)
2
(SP) = 4K?7?(7? = PE tS 4)'72}2 ; (80)
and
away". 3 [per Sa = ty (81)
(az ) ~ V2 ~ ~ pts

When we change the variable of integration in (58) from © to T, we
need the quantity d€/d7, which may be obtained from (35) and is

Se - [7 - (r? - 1)? cr? - ty. (82)

F 2
If we denote G(t)d$/dt by g(t)e!“*, then with the aid of (55), (48),

(79), (80), (81) and (82)

204

An Unsteady Cavity Flow

te fitr ty7r&-'4) 34]

(r2_ 1)'72 [j(Kw)°M, (7) +(Ke)*M,(7)

g(T) =

- 2njKwt + b,7%(7? - 1)7]. (83)

In (83) we note that although b, is purely real with respect to i,

it may be complex with respect to j; we also note that the term
2mjKwT is extremely small as compared to the term j(Kew)® M (T) as
w—+ oo, and since both of them are dependent on j, so we may
neglect 2mjKwT from (83) and write

vee | [atest ety tT] [j(K&)°M, (r)

Ag SORE

+(Ka)'M,(7) + b,7%(7?- 1)'7] . (84)

The asymptotic form of F can now be expressed as

1 io WE Fy File] i. F,[ t(o)e**
- Fal oc | F,[t (o)] e(o)eX* ao}, (85)
Gg

where o is the integration variable in the T-plane and F,(¢),
F4(¢), g(7) and W(F,,F,) are given in (74), (75), (84) and (77).

Substituting all the necessary results obtained above into (20);
noting the relations

(1 £ij)> = 2(4 ij)
(86)

(1 +ij)(41 - ij) = 0

when we are taking the real part of (20) with respect to j, we obtain
the complex velocity potential f,; as

ee (eis
eae (reqir®=1 ~ 4)

= Pivolteucskae ;
at mY Meneses) { Aif(iKs)'E] Ail(Ke)"e?"/%] .

me

205

Wang

Ai[iK w)!3 © 77%] Ai[(iK w)'72 ti} EAN g (01+ B,g,o)] de

' Ti _jw(t-Kr7+Ko 7) |
+l ye e* { aif(-iKu)' 6] Ai[(-iKu)”4 @ M/E)

oe

ree -27ri/3

- Ail(-iKw) 6] aif(-ika)’? EEA)! [e,(0)

B g3(c)] dc (87)

where § and x as functions of T are given by

t

2.972 | 3ri/4 do

ee “a
| (2 - 1)'4

and
x= [( BS -r] (89)

€ and X indicate the functional values of & and y when the
variable 7 is replaced by the integration variable o,

g,(7) = Saar pie MoM (7) + Ka)? M7) (90)

g(T) = earn al i(K.w)?M,(T) +(Kw)*M,(7)] , (91)

SA ie Tift + yr? - 1) - 4), (92)

B, is an arbitrary complex constant which is derived from b,, and
B, is the complex conjugate of B,. The lower limits of integration
shown in (87) are chosen for our convenience. Therefore, when
necessary, homogeneous solutions of the form

206

An Unsteady Cavity Flow

A(t) elt) ATK yy! ale

n 2 ‘
Altre Ai[(iKw)'” a ;

95
A(T) eK ail(-iK al] ae

and

-iw(t-Kr2)

-27i/3
A(T)e Ailcikay” e ay

él,

where

1/4
A(t) = —2) (94)
T - V7? -1-i

may be added to (87).

We shall now study the behavior ofthe solution (87) for
|r| > R, >> 1, 0 = Arg? = 1/2. 4A circular arc of radius Ry, 4s
drawn in the o-plane as indicated in Fig. 6. Also shown in that
figure is a hyperbola S representing the equation Re [ i(o# - 7?)] =0.

Fig. 6 Paths of integration for the solution given in
Eq. (87) when 7 is large

207

Wang

If we are travelling along S inthe direction of increasing Imo
and if we restrict ourself to the first, second and fourth quadrants
of the o-plane only, then on the right-hand side of S Re [ i(o?-77)]
<0, and on the left-hand side Re [i(c*-7?)] > 0. When either Im 7
or ReT is zero, S degenerates into the positive real and imagi-
nary o-axes.

From (88), we note that

Scan: (95)
and
1 T
Arg § =zArgTts as |t| + c (96)

Therefore, for 7 and w large, the arguments of the Airy functions
appearing in (87) are large. Their asymptotic representations are
(cf. Jeffreys & Jeffreys [1956]),

AR cep 2
Ll WwW C—O
aw (Kay 274
when ~-carge<Z ;
mig 26 we?
AB 23 i pie
Ai[fikKw) e Era
2m (Kw) €
when - Recargé<=,
Ag 3 3
/s ce ke (97)
|
Ai[(-iKe) §] ~ ———=——px-77a
2Vm (Ku) €
when 3 <Arg & <a
smi 2 salle 771 caws
1/3 - 2mi/3 Aa
Ai[(-iKw) e 6) ~ ———_
ale (xe)! evs
Lin

when - z<Argi<—.

208

An Unsteady Cavtty Flow

A straightforward expansion shows that, as |7T| > R,

-1/4 -| \/4 i/8 -3/4
Bee ema ee Alo any. ee is (98)

And it is easy to see, as some related explanation has been given in
the paragraph after (52), that as |7| > R,

eel 4. 7h, 9/4

Xx g,(T) t B g3(7)] a B, 2

(99)

-|/4 om — _/4, 7i/8 9/4
x [ g,(7) ~ B g,(7)] = Sat oe T :

Let us denote the a of the potential represented by the first inte-
gral in (87) by f, (e

Haleatize ss» lt fey
Js eh Gao Cee er ae

T mi +iw(t-Kr + ko*) -27i/3
xf eee { Ail(axia)”* €] aif (Ku)? 7”)
oe
: Ail (iKo)!> ee Ail(iKw)”° =] \ Creal g (oc) +B, ga(c)] do

(100)

For any 7 with |7| >R, and 0= Arg 7 = 1/2, the path of inte-
gration in (100) = always be chosen to be Lj, which is a path
coming from ooe * to 7, lying completely on the right-hand side
of the hyperbola S and outside the circular arc lo | =R,. A
typical L; is shown in Fig. 6. Since along L; |o| >R,, we may
substitute the expansion given in (97), (98) and (99) into (100) and
obtain

B See Kt 2K)
aS, mae COV8 oi ah @(E, é) do, (101)

1(2Kw)

Ly
where

(6,2) = 2 eA fKag” - PY). (102)

209

Wang

We note that along Li the hyperbolic sine function in (101) may be-
come exponentially large, however, since L,; lies on the right of
S, thefactor exp [iKw(c?-7?)] will be overwhelmingly small there.
Let us now integrate (101) by parts once to get

Tt

2 2
(e) i iw(t-Kt tka ) oy
< Eee 4 fe Bsinh @(€,€)

n(2Kw)/273/4 (2iKw)

Z jw(t-Ke 9k o*) S Ua... ov
= ri/4 © [2° sinh ®(€,&)
008

oss wi/4

+ 04 Se cosh &(E 2) ac} (103)

The integrated part in (103) ig identically zero; when we evaluate it
at the upper limit, sinh ®(€,6) = 0, and at the lower limit, the
factor exp iKwo’) is overwhelmingly small. If we integrate the
integral (103) successively by parts and note the relation that

wi/4 wW/2 ae -/2
2 Bets CK oe pee -* : (104)
37i/4 -yY2

where gue dé /do a a ee o is obtained from (88), we can show
that

(e) i iwt -
fe (105)
41(Kw)
Since: j= O(z'/2) as |z | — oo, from (104), we see that the contri-

bution to the potential due to f/®) is of order |z[’? for |z|

large. This type of potential is acceptable. Now, let us denote the
Bart of the potential represented by the second integral in (87) by
anaes

|

ALL De ZE (g yy ( FB iutt- Kr+Ko*)
2 eee
(Ku)! (T= Vr2-1 =i) a

x {ail (-1Ka) €] ai(-iK we ?”8}

~ Ail(-ika)/Fe°?7”E) aif(-iKw)'7E] | (E/2)'4[ g,(0)+B, g,(0)] ac.

(106)

210

An Unsteady Cavity Flow

To investigate the behavior of ee for |7|>R _, let us divide the

region in the first quadrant of the o-plane outside the circular arc
|o| = Rj into two parts, D; and D3. D,, as shown by the shaded
area in Fig. 6, is bounded by the circular arc |o| = Rj, the imagi-
nary o-axis ye the hyperbola S' representing the equation

Re [ i(o?- R?e””“)] = 0. D, is bounded by the real c-axis, |o| = R,
and S'. For T in Dg, we may choose the path of integration in
(106) to be Ly, whichis similar to L, except now it lies on the
left-hand side of S. Atypical Ly is shown in Fig. 6. Usinga
process similar to that used to obtain (105) from (100), we may
obtain, from (106),

(Ww) eByi i -I
SS a

i
2
4 1(Kw)

(107)

When 7 is in Dj, the hyperbola S will be extremely close to both
positive axes of the o-plane, S degenerates into the axes when T
lies on the imaginary o-axis. For this case we have to deform the
simple path Le into L3+ La, as shown in Fig. 6, in order that the
factor exp [-iKw(o?-77)] will not become oo eee large along
the path. Lz is a path coming from coe ”'’* to 0 on the lower-half
o-plane, from there along the real o-axis towards o = 1, turning a
small circle to the upper side of the real c-axis and along it to 0
on the upper-half o-plane, to circumscribe the cut in the o-plane,
and then leaving 0 to coe3/4, Path La comes from ocoe?™'/4 to
7, lying completely on the left-hand side of the hyperbola S and
outside the circular arc |o | = R,;. The integration along Lg is
convergent; near both ends of the path the integral is exponentially
small, near o=0 M,(c) and M,(c), which appear in g,(c), are
of order o% and o respectively and g3(c) is of order G2: near

o = 1 only Mj(c) contributes to ga(c) a square-root type of, singu-
larity. The latter property of M,(c) is the reason that (Kw) M,(c)
is being kept in the integrand together with the term (Kw)?Ma(c).

We may remark here that if we did not neglect the term 2mjKw7
appearing in g.(c) from g(7) in (83), we would have aterm of the
form 2miKwo. Since the presence of such a term would not affect the
property of the integrand along L, near,o»=0 and 1, and since it
is one order in w smaller than the (Kw) M,(c) term, its neglect is
justified. Let us now denote the contribution to fm from the inte-
gration along L, by Z(7),

2€ (B/x yi Fleiwtt-Ked ps
Z(T) = —= Ai[(-iKw)°~ €
(Ku)? posing a e { i[(-i ]
wer m4 ;
woe™/4 Saray Ai[(-iKu)'” patel (E/x)'4 | (co) +B, g.(o)] do -
[
3

244

Wang

@omi/4
-27i/3 =f
slaiptouna) ee aes Kee yesh rica) PET EAM
3
[e,(o) +B,g,(o)] ao} . (108)

We note that when 7 is in D, the, Air TA Ail (=iKw) Vee
exponentially large and Ail (5 iKw)!/? eat is exponentially small
therefore, in order. that {\" tend to zero as |7| + oo in D,, we
require that the coefficient Ai[(-iKw)'/3&] be zero. This deter-
mines the constant B,, or B,,

: 2
5, - Pi aaes pee on Aaa | (a g(c) av/
L3

; 2
i Ps ateary Per Ey GQ) *g.(o) do. (109)

L
3

With B, given by (109), Z(7) becomes exponentially small when
tT isin D,. We shall not attempt to evaluate By, explicitly in this
paper, however, in view of (91) and (92) we may conclude that

B, = O[(Ku)*]. (110)

Since Ly is outside the circular arc lo | = R,;, we may substitute
the expansions given in (97), (98) and (99) into the integral along Ly

to obtain

a Tt
x= -iw(t- Kr 4K o2)

41
X oM sinh [ 5 eel a Fa ie ae recy do.
(111)

£")

Now, if we apply the method of integration by parts to the integral
along Ly, we can show that for 7 in D,, with Z(T) being expo-

netially small,

Be i ea (112)

AnibK wo)

Summing up all the results obtained in (105), (107) and (112), we

202

An Unsteady Cavtty Flow

have

ei i
f| ~ ——s (B,e

4n(Kw)

T as [il 0. (113)

From (113) we may derive the following results: (i) From (110)
we may conclude that f,* eKw. (ii) For 7 lying on the real 7T-axis,
which corresponds to the cavity wall of the basic flow, f, is purely
imaginary; this indicates that near the point at infinity the perturbation
velocity wy, = 0f,/8z is always perpendicular to the original cavity
wall. (iii) If we recall that 7 = O(z'/#) as | 7 | — oo, we can see that
the perturbation velocity is of order |z lees for large values of |z|;
this, together with the result stated in (ii), implies that the unsteady
free surface displacement tends to zero as |z|—~ oo. (iv) Along the
imaginary T-axis, which corresponds to the line of symmetry of the
flow, f, is purely real; this means that there is no velocity component
normal to the line of symmetry which, of course, is what we should
expect.

It should be pointed out here that the order of magnitude and
the direction of the perturbation velocity on the free surface near
the point at infinity agree with the results obtained by Wang and Wu
[1963] in the study of small-time behavior of unsteady cavity flows.

Finally, we shall investigate the behavior of the solution (87)
near the separation point 7= 1. From (4), (35) and (5), the pertur-
bation velocity w, may be written as

Of Of
Ww, Halas st. (114)

Equation (114) indicates that the singular behavior of w, near 7 = 1
can be studied from that of df /€87 near T= 1. Let us now differ-
entiate f; given by (87) with respect to 7. The differentiation of

f,; with respect to T may be viewed as consisting of four parts; the
differentiation of the 7 appearing in the limits of integration, the
differentiation of the factors exp (+iKwr*), the differentiation of the
Airy functions with respect to 7, and the differentiation of the factor
in front of the curely brackets in (87). Only the latter two parts
produce terms of the form a_(T* - 13" @ near T = 1; all the other
parts either give zero or a finite contribution to w,. Therefore,
condition (34) and the condition that the pressure is integrable over
the plate AB are satisfied.

Since the solution given by (87) behaves properly at infinity and
at the separation point, we conclude that it is the solution of the
problem; no additional solution of the homogeneous equation, as
shown in (93) needs to be added.

243

Wang

ACKNOWLEDGMENTS

I wish to express my appreciation to Professor T. Y. Wu for
useful discussions during this research. I am also indebted to my
wife Yvonne for typing this manuscript.

REFERENCES

Gilbarg, D., Jets and Cavities, Encyclopedia of Physics, Ix,
Berlin: Springer-Verlag, pp. 369-71, 1960.

Jeffreys, H., Asymptotic Approximations , Cambridge University
Press, pp. 52-9, 1962.

Jeffreys, H. & Jeffreys, B. S., Methods of Mathematical Physics,
3rd Ed., Cambridge University Press, pp. 508-11, 1956.

Langer, R. E., "The solution of the Mathieu equation with a com-

plex variable and at least one parameter large," Am. Math.
Soc., Trans. 36, pp. 637-95, 1934.

Muskhelishvili, N. I., Singular Integral Equations, Groningen,
Holland: P. Noosdbett std .',). pp. TOociz: 1946.
Olver, F. W. J., "The asymptotic solution of linear differential

equations of the second order for large values of a parameter,"
Phil. Trans., Roy. Soc. London, 247A, pp. 307-68, 1954.

Olver, F. W. J., "Uniform asymptotic expansions of solutions of
linear second-order differential equations for large values of
a parameter," Phil. Trans., Roy. Soc. London, 250A,
pp. 479-517, 1958. aaa

Wang, D. P. & Wu, T. Y., "Small-time behavior of unsteady cavity
flows," Arch. Rat. Mech. & Analy., 14, pp. 127-52, 1963.

Wane, D. Pr. & Wu, To iv. General formulation of a perturbation

theory for unsteady cavity flows," J. Basic Eng., ASME,
Trans. D; 87, pp. 1006-10, 1965.

214

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

Tuesday, August 25, 1970

Morning Session

Chairman: J. K. Lunde
Skipsmodelltanken, Trondheim, Norway

Page
Deep-Sea Tides 217
W. He. Munk
University of California, San Diego
Stability of and Waves in Stratified Flows (ag he)
C. Yih, University of Michigan
On the Prediction of Impulsively Generated Waves
239

Propagating into Shallow Water
P, van Mater, Jr., U.S. Naval Academy and
E. Neal, Naval Ship Research and Development

Center

215

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iy 7]

DEEP-SEA TIDES

Walter H. Munk
Untversity of Caltfornta
san Diego, Caltfornta

ABSTRACT

The classical Laplace tidal theory, when applied in
numerical form to the world's ocean basins, does not
yield results in good accord with observations. In
part, this may be due to density stratification and in-
ternal tides (coupled to external tides); and in part to
dissipation at the ocean boundaries. At a given port
the spectrum of the observed tides shows a complicated
line structure superimposed over acontinuum. The
continuum rises at the frequencies where the lines are
clustered, probably as a result of internal tides.

Tide dissipation leads to an exchange of angular mo-
mentum between the spin of the earth and the orbit of
the moon. As a result of this spin-orbital coupling,
the length of day and month are both increasing. Ob-
servations of the moon since 1680, of Babylonian
eclipses and of the structure of Devonian tropical coral
(which give the number of Devonian days per year)
confirm these calculations.

To untangle these problems, it is probably necessary
to make observations in the deep sea, relatively re-
moved from the scattering and absorbing boundaries.
Such observations have now been made for the last three
years, and they yield relatively clear pictures of the
deep-sea tidal pattern. The tides in the northeast
Pacific can be roughly accounted for by superposition
of a northward-traveling Kelvin wave (trapped by
rotation to the boundary) and a southward-traveling non-
trapped Poincaré wave.

In order for the calculations to be realistic, they need
take into account the tidal yielding of the sea floor.

rae We |

STABILITY OF AND WAVES IN STRATIFIED FLOWS

Chia-Shun Yih
Universtty of Michigan
Ann Arbor, Michigan

ABSTRACT

A theorem giving sufficient conditions for stability
of stratified flows, which is a natural generalization
of Rayleigh's theorem for shear flows of a homogeneous
fluid, is given. Sufficient conditions for the existence
of singular neutral modes, and consequently of unstable
modes, are also presented, and in the development the
possibility of multi-valued wave number for neutral
stability of the same flow is explained. Finally, neutral
waves with a wave velocity outside of the range of the
velocity of flow (non-singular modes) are studied, and
results concerning the possibility of these waves are
given. In addition, Miles' theorem [ 1961] on the stability
of stratified flows for which the Richardson number is
nowhere less than 1/4, and Howard's semi-circle theorem
[ 1961] are extended to fluids with density discontinuities,

I. INTRODUCTION

The stability of stratified flows of an inviscid fluid has been
studied in a general way, i.e. , without specifying the actual density
and velocity distributions, by Synge [1933], Yih [1957], Drazin
[1958], Miles [1961, 1963], Howard [1961], and others. Of these,
Miles has made particularly substantial contributions to the subject.
However, many questions still remain open. Among these are the
following:

(i) Miles [ 1961] showed that if the Richardson number is
nowhere less than 1/4, the flow must be stable. This
is a sufficient condition for stability. What can one say
regarding the stability of the flow when the Richardson
number is less than 1/4 in part or all of the fluid? Are
there then some sufficient conditions for stability not

219

Yth

covered by Miles' criterion? What, in fact, is the
natural generalization of Rayleigh's theorem on the
sufficient condition for stability of a homogeneous
fluid in shear flow?

(ii) Are there some sufficient conditions for instability?

(iii) Miles [| 1963] has shown that the wave number at neutral
stability can be multi-valued for the same flow, in an
actual calculation for a special density distribution and
a special velocity distribution. Is there an explanation
for this, even if not completely general?

(iv) Do internal waves with a wave velocity outside the
range of the velocity of flow exist? How many modes
are there? What is the character of each mode?

In this paper the questions posed above will be answered in as
general a way as possible. By "general" I mean "without numerical
computation." Although special calculations for special flows,
involving the use of computers, are important because they often
give us insight into and understanding of the subject, and sometimes
are of practical interest, results obtained in a general way are often
more useful. The question naturally arises: Can general results be
continually improved and sharpened, albeit with increasing cost in
labor, but without the use of computers? The answer to this question
necessarily reveals the attitude of the respondent more than anything
else. My answer to it is in the affirmative, and the results contained
in this paper, aside from whatever interest or merit they may have
for those cultivating the subject, are given to substantiate my faith.

In addition, some straightforward extensions of Miles' theorem
mentioned in (i) above, and of Howard's semi-circle theorem [ 1961] ,
are made to make these theorems applicable to fluids with discon-
tinuities in addition to continuous stratification in density.

Il DIFFERENTIAL SYSTEM GOVERNING STABILITY

If U and p denote the velocity (in the x-direction) and the
density, respectively, of the primary flow in the absence of distur-
bances, and u and v denote the components of the perturbation in

velocity in the directions of increasing x and y, the linearized
equations of motion are

ip (uy F Ua +O) = - Bes, (1)
p(v, + Uv,) = - py - BP» (2)

in which subscripts indicate partial differentiation, t denotes time,

220

Stabiltty of and Waves tn Strattftied Flows

p is the deviation of the pressure from the hydrostatic pressure in
the primary flow, p is the density perturbation, g is the gravi-
tational acceleration, and

The equation of continuity

permits the use of a stream function w, in terms of which the velocity
components can be expressed:

se v= a (3)

The linearized form of the equation of incompressibility is

p, + Up, tvp'=0, (4)
in which
plz a,
dy

If n is the vertical displacement of a line of constant density
from its mean position, the kinematic relationship

le ea (5)

holds. All perturbation quantities will be assumed to be periodic in
x and have the exponential factor exp ik(x- ct), so that from (5) and
(3) we have

wes(U=-c)y, u==([(U=c)n]t, v= ik(U-e)7. (6)
Then (1) and (4) give
=p(U-c)’n' and p=-P'n. (7)
Writing

n(x,y st) = Fly)eik- et) (8)

221

Yth

and substituting (6) and (7) into (2), we have, with B denoting -p'/p,
[p(U-c)*F']' + pl Bg-k'(U-c)*]F = 0, (9)

which is the equation used by Miles [1961] and Howard [ 1961] to
study the stability of stratified flows.

Miles [1961] assumed U to be monotonic and U and ) to
be analytic in his studies. Howard [1961] was able to prove Miles'
theorem (on a sufficient condition for stability) and to obtain his own
semi-circle theorem without these hypotheses. But both of them
assumed ‘p to be continuous, and considered the upper boundary to
be fixed as well as the lower one. We shall now show that the
theorems of Miles and Howard can be generalized to allow density
discontinuities. The mean velocity U (though not necessarily U')
will be assumed continuous.

Let there be n surfaces of density discontinuity, and let the
free surface, if there is one, be the first of such surfaces. The
densities above and below the i-th surface of density discontinuity
will be denoted by (p,), and (pg); , respectively, and we shall define
(Ap); by

(Ap), = (py - p,); « (10)

The interfacial condition can be obtained by integrating (9) in the
Stieltjes sense in an arbitrarily small interval containing the discon-
tinuity under consideration, and is, with the accent indicating differ-
entiation with respect to y,

[p(U-c)*F'], - [p(U-c)*F'], = - gApF, (11)

to be applied at any surface of discontinuity. At a free surface Py
vanishes, and (11) becomes

(U-c)*F' = pF, (12)

which is the free-surface condition, to be applied at y=d, d being
the depth. If the upper surface is fixed instead of free, the condition
there is

F(d) = 0. (12a)
The boundary condition at the bottom, where y=0, is

F(0) = 0. (13)

222

Stability of and Waves tn Stratifted Flows

Ill. EXTENSION OF MILES' THEOREM

Following Howard [ 1961], we set
eu w2r,
where W=U- cc. Then (9) can be written as
(pwG')' - [(pU')'/2 + pw + pw'(U'*/4 - gB]G= 0. (14)
The boundary condition at the bottom is
G(0) = 0. (15)
The interfacial conditions (11) become
Py (WG! - U'G/2), - p (WG! - U'G/2), = gApw'G, (16)

to be applied at the surfaces of density discontinuity, and in particu-
lar the upper-surface condition becomes

WG'-U'G/2=gW'G, or G(d)=0, (17)

depending on whether the upper surface is free or fixed.

Multiplying (14) by G", where the asterisk indicates the com-
plex conjugate, and integrating from the bottom to the first surface
of density discontinuity and then from discontinuity to discontinuity
throughout the fluid domain, and utilizing (15), (16), and (17), we
have

(Vewtia'l? +k |G/7] +( Gun |Gl*/2 + (51 u'?/4 - g6] w'|G/w/?

-) gaaw*|c/w|?-) (GU, - @UIJIGP/2=0, U8)

in which each of the integrals is over the entire fluid domain exclusive
of the surfaces of density discontinuity (i.e., it is a summation of
integrals over the layers of continuous density distributions), and

the summation is over the discontinuities, including the free surface
if there is one. If the flow is unstable, c, > 0, and the imaginary
part of (18) is

Valiot? +x%iGiy +f alae - u'7/4] |o/wl* + ) ga,pla/wl? =o,
i (19)

223

Yth

from which it is again evident that if
gb = U'*/4

everywhere in the fluid exclusive of the interfaces and the free surface
(if there is one), the flow must be stable.
IV. EXTENSION OF HOWARD'S SEMI-CIRCLE THEOREM

Equation (9) can be written as
(pW2F')' + p(Bg - k*W*) F = 0.

*

Multiplying this equation by F , the complex conjugate of F, inte-
grating throughout the fluid domain and using the boundary or inter-
facial conditions (11), (12), and (13), we have

(ewilr' P+ lel -(seplFP- Dealer =o, (20)

in which the summation is over the surfaces of density discontinuity,
and the integrals extend throughout the fluid exclusive of the surface
of discontinuity in density, The real and imaginary parts of (20) are

(at u- op)? - It lel? +e"1F 1 -loe6iFl - ) sae lFl =o,

(21)
¥t Hy eed) Zee Bal,
2c,4 plu cote, +k FL) 20. (22)
Writing
G-nlley teri;
we obtain from (22)

(ue = ¢. $a, (23)

then from this and from (21) we obtain

(ut thea tel o + \ gop [F[? + Ded lF

2

(24)

If a and b are respectively the minimum and the maximum of U,

224

Stability of and Waves tn Stratifted Flows

so that a= U=b, we have

o=((u- au-va={u%- (a +») vO +tab\ Q

=[c, te, - (a +b)c, + ab] |0 + geBlFP +) evlFl,

|
after using (23). This means that

[c, - (a +b)/2]’ +07 =[(b - a)/2l’, (25)

that is, the complex wave velocity c for any unstable mode must lie
inside the semi-circle in the upper half-plane, which has the range of
U for diameter. Thus Howard's semi-circle theorem is recovered.

-2 -2
From (19) and noting that |W| Sc, , we deduce that

kc) = max (u'?/4 - gB) (26)

remains valid even if there are surfaces of discontinuity in density.
In (26) we exclude these surfaces in the evaluation of B. It is easy
to see that (26) contains Miles' theorem.

V. SUFFICIENT CONDITIONS FOR STABILITY

Miles' theorem gives a sufficient condition for stability. But
it certainly does not guarantee instability if the local Richardson
number J(y) defined by

Sy) = #6 (27)

ww!
is less than 1/4 in part of the fluid or even all of the fluid. We shall
sharpen Miles' sufficient condition for stability by deriving two
theorems which constitute, more than anything hitherto known, the
natural generalization of Rayleigh's theorem for the stability of a
homogeneous inviscid fluid.

For the discussion in this section it is more convenient to use
the stream function

w= f(y) etk(x-ct) . (28)

Comparison with (6) and (8) shows that

Zio

Vou

f(y) = (c - U)F(y). (29)
In terms of f(y), the governing equation (9) becomes
~ (OU jer det eee
(pf')' + Sly Te t=30., (30)
Equation (30) can be made dimensionless by the use of the new
variables
(31)

a3 iv ipr esp pow c=
p Bi” MA datz U Vv? c

where p, is a reference density and V a reference velocity. Then
(30) becomes, after the circumflexes are dropped,

pf) 9 |i ic eae fo (32)
ay

in which everything is now dimensionless, the accents indicate differ-
entiation with respect to the dimensionless y,

@=kd (33)
is the dimensionless wave number, and
N = ga/v* (34)

is actually the reciprocal of the square of a Froude number. The
appearance of N does not necessarily signify the importance of sur-
face waves, since it appears even if the upper boundary is fixed.

The fact that it is associated by multiplication to p' indicates that
the entire term represents the effect of gravity in a stratified fluid
in shear flow.

Henceforth in this paper we shall consider rigid boundaries
only, for which the boundary conditions are

£(0) = 0 and f(1) = 0, (35a,b)

to be imposed on the function f in (32).

226

Stabitltty of and Waves in Stratified Flows

It is then clear that the system consisting of (32) and (35a,b)
gives, for a non-trivial solution, a relationship

Fi (a7,N,c) = 0. (36)

Since c is complex, (36) has a real part and an imaginary part.
When c; is set to zero and c, eliminated from the two component
equations, a relationship

F,(@,N) = 0, (37)

if one such exists, gives the neutral-stability curve. It is possible,
however, that c is real for all values of @ and N, in which case
c. = 0 inthe entire N-a@ plane, and then of course there is no
neutral-stability curve because one component equation of (36) is
Cir 0, and the other is simply (36) itself, with the c therein real,

In this section, we shall assume rr and U to _be continuous,
analytic, and monotonic. Furthermore, we assume p'< 0 throughout.
We now recall the following known results:

(i) If J(y) is not less than 1/4 for the entire fluid domain,
then the flow is stable [ Miles 1961],

(ii) If c; #0 then c, must be equalto U at some point in
the flow, as a consequence of the semi-circle theorem
of Howard [1961], and

(iii) If an eigenfunction exists for (c,, @,, N,), then near
that point c is a continuous function of @ and N,
[ Miles 1963 and Lin 1945].

Under the assumptions we have made on p and U, and in
view of the known results just cited, we conclude that the non-existence
of any singular neutral mode, which is a mode with a real c equal to
U at some point in the flow, implies the non-existence of unstable
modes. The reason is as follows. In the N-@ plane there is always
a region of stability. For we can imagine g andhence J(y) to in-
crease indefinitely, until J(y) is everywhere greater than 1/4, which
is attainable since B is nowhere zero. Thus there is a region of
large N for which the flow is stable. If unstable modes exist there
must then be a stability boundary dividing the region of stability from
the region of instability, and hence a neutral-stability curve. As we
approach that curve from the region of instability, c, being within
the range of U so long as c; #0 and continuous in @ and N so
long as c is an eigenvalue, according to (iii) above, in the limit,
when c; = 0, c, must be within the range of U, i.e., the limiting
mode must be a singular neutral mode. Hence the non-existence of a
singular neutral mode implies the non-existence of unstable modes.

221

VER

In fact even the existence of special singular neutral modes for which
c equals the maximum or minimum of U does not imply the existence
of contiguous unstable modes, as a consequence of the semi-circle
theorem of Howard. Hence we need not be concerned with these
special border cases. In demonstrating the non-existence of unstable
modes it is sufficient to demonstrate the non-existence of singular
neutral modes with a<c<b, where a is the minimum and b the
maximum of U.

Miles [ 1961, p. 507] has shown that singular neutral modes
are impossible for monotonic U if J(y) > 1/4 everywhere. In his
demonstration he actually showed that a singular neutral mode with
a J(y,) > 1/4 at the place y=y, where U=c is impossible. Hence
we need only consider the case Hy), = 1/4 in our search for the non-
existence of singular neutral modes. For J(y,) = 1/4, one solution
of (32).is

£, = (y - y,)!/?w, (38)
where
w, = 1 +Aly - y,) +e. (39)
with
a= [un GBs yam] , (y=5) (40)
pu' sp c

provided U' does not vanish at y = y,.- [ We shall consider mono-
tonic U only. Hence this restriction on U' does not affect our
results in this paper.] The other solution is found by assuming it
to be of the form f,h, substituting it into (32), and_solving for h.
The result, after division by a constant (which is p, or p at Y)?
is

f= £, In (y-y,) - [2A + (In pi’ (y-y)/ 11 + Bly-y,) teeel, (41)
where B is aconstant. Now the Reynolds stress defined by
eS 1S. Die 9 (42)

where the bar over uv means time or space average, can be ex*
pressed interms of f as

p,V
2ac.t
T = S a(f'f"), e Te (43)

228°

Stabtlity of and Waves in Stratified Flows

in which the asterisk denotes the complex conjugate, and the t, now
interms of d/V, is ee Oo ees as is f, Considering the singu-
lar neutral case, for which c, = 0, it is easy to see from (40) and
(41) that f'f* is real for y > Ye and equal to ~in: for ty <y,., Hence
(£' £*), suffers a jump at y,- Since f' f* is zero at both rigid
boundaries, it cannot afford this jump. [If @ #0, this jump cor-
responds to a jump in the Reynolds stress. But we do not have to
consider the jump in 7, and can consider merely the jump in (f' ft), al
Consequently a eaelae neutral mode with J(y_) equal to 1/4 is im-
possible. And we can henceforth concentrate on the case Jy, )< 1/4.

For J(y,) < 1/4 Miles [1961] gave the solutions of (32):

f(y) =(y - yay ee, (44)

in which
=i tAly —ye)/(1-tv) +2005 (45)

with A given by (40) [but with y = (1 + v)/2 therein] and

S(t 45e), 2 Jo diye le (46)

We can use (44) and (45) with all terms therein considered dimension-
less. Miles [1961, pp. 506-507] showed that for J_<1/4 the solu-
tion, if one exists, must be either f, or f.. We can demonstrate
our point by considering f, as the solution. The demonstration for
the other case is strictly similar.

The study of the eigenvalue problem defined by (32) and (35a,b)
naturally leads to a study of the zeros of f. Since f is given by
(44), it leads to the study of the zeros of w,. This in turn leads us
to consider the differential equation for w (from which the subscripts
are removed for convenience). Denoting w, or w_ by w, we can
easily obtain that equation:

= fas =s ' Pau
(p2z2%w')' — PY) ape rs yp 'z"! 4. fo")? = a*5 = at w = 0, (47)
(ean)
with z=y-y,, and y= (1+ V)72e
We are now in a position to present
Theorem 1. If p and U are continuous and analytic, with p' <0
and U'> 0, and if (pu)! and (In p)" are positive throughout, then

singular neutral modes are impossible.

229

Yth

Proof. We have shown that it is necessary only to consider the case
Jy, )< 1/4. We may consider f, only, since the proof for f_ is
the same, and since the solution ‘is either f_or f. Nowat y=y,
we have f,=0. Near ¥, ve have

Since p' is negative and U' and (pU')' are positive, U" is
positive. Thus U-c is greater than Ujz for z>0O. On the other
hand -p'/p is less than (-p TP). for < > y,» since (In p)" is posi-
tive. We know that for small positive z Q is negative, as can be
seen from (48). Hence for any z>0O the term

Nee
(U - c)”

is less than pJ, /z* and Q is negative. Equation (48) exhibits the
behavior of Q near Yor Let the bracket in (47) be denoted by - G.
Then since Q is negative and U-c is positive for y>y,, and
since p' is negative and (pU')' positive, G must be positive for
y>y,- Multiplying (47) by w and integrating between y, and 1,
we have

|
(p2>%ww'), - if 2°“ pw! + Gw’) dan O, (49)
y

c

where the subscript i indicates that the parenthesis is evaluated at
y = 1. Note that the integral in (49) is convergent in spite of the
simple pole in two terms contained in G-- one of which in Q, as
indicated by (48). Equation (49) clearly shows that w(1) cannot be
zero. .Hence the theorem.

Another theorem is

Theorem 2. If e and _ U_ are continuous and analytic, with p!
negative and U_ positive, and if U" and (In) are negative

throughout, then singular neutral modes are impossible.

The proof for this theorem is similar to that for Theorem 1.
The only modification demanded for clarity is that instead of (44) we
should write

f(y) _ 2h) /2., (2)

with z now definedas y, - ye The equation corresponding to (47)

is now

230

Stability of and Waves in Strattifted Flows

(50)

in which, it must be emphasized, all accents indicate differentiation
with respect to y, not z. The rest is strictly similar to the proof
for Theorem 1, except the range of integration is between z=0 and
z=y, (or between y=y, and y = 0), and we want to show w#0 at
y = 0. Note also that U"<0O now guarantees (pU')'<0.

Since the non-existence of singular neutral modes implies the
non-existence of unstable modes, we have also

Theorem 3. If p and U are continuous and analytic, with p'
negative and (ay positive, and if either (pU")' and (In) " are both
positive throughout, or U and (In P) are negative throughout,
the flow is stable.

This theorem is the natural generalization of Rayleigh's theorem for
inviscid homogeneous fluids in shear flow. Previous attempts at this

generalization [ Synge 1933, Yih 1957, Drazin 1958] have produced
the result that there must be stability if (in dimensional terms)

2Bg(U - c,)

Ju- cl?

_ (su')?

_

does not change sign. This criterion is not useful because it involves
not only c, but also c;.

VI. SUFFICIENT CONDITIONS FOR INSTABILITY

Sufficient conditions for instability have seldom been given in
studies of hydrodynamic stability. In giving some such conditions,
we shall also be able to explain why the @ can be multi-valued for
the same N, at neutral stability.

We assume that p and U are analytic, that p' = 0, and that
at a point where p'=0, U" is also zero. The value of U at that
point will be denoted by U,, for we shall consider the possibility of
having c equalto U at that point. We demand that at any other
point where U=U,, p'=0= U" must be satisfied. If U is mono-
tonic, of course there is only one point at which U = U,.
= Under the assumptions made, p" must be zero at Y,* since
p' is never positive, andnear y,

231

Yth

p' = poly - y,)-

If ps were not zero p' would be positive for y slightly larger than
vo With this realization, it is immediately clear that the bracket
in (32) has no singularity at y_. Let us denote the bracket in (32)
by the sumbol B, which is a function of y, @, and N. Then if m
is the minimum of B/p_ between two points y, and y,, with

2
O0Sy<y,=1, for @ =0, and if

: n = a positive integer, (51)

by the use of Sturm's first comparison theorem we know that there
must be at least n zeros of f between y, and yz, whatever the
value of f(0) and f'(0). (Note that the P in m or in (32) is dimen-
sionless.) We can always choose f(0) =0. If (51) is satisfied then
there must be at least n internal zeros of f. Wecanincrease @
so that, again by Sturm's first comparison theorem

£(1) =

for

a= 2% Os: one eee 39 ays

where
Qi S Op Ais < wae, SO ns

It is evident that for @ =a; there are at least n-i internal zeros.

Hence we have

Theorem 4. Under the assumptions stated in the second paragraph
of this section, if (51) is satisfied there are at least n modes with

c =U, and a=a@, (i=1, 2,..., n), and with a; increasing with i.
For the i-th mode there are at least n-i internal zeros.

It is easy to show, by exactly the same approach used by Lin [1955,
pp. 122-123], which we shall not repeat here, that by varying a
slightly (now not necessarily by decreasing it, as is in Lin's case),
c will become complex. Hence we have

Theorem 5. Near the neutral modes stated in Theorem 4, there
are contiguous unstable modes.

232

Stability of and Waves in Strattfied Flows

Theorem 4 explains why for the same N, given p and U,
there can be many values for @ on the neutral-stability curve (or
curves), which has been observed by Miles [ 1963] for a special p
and a special U.

We can sharpen Theorems 4 and 5 by defining M to be the
maximum of B/p in (y, ; y,) for a=0. Then if (51) holds and

= (h +1)? 1?

M ,
(y,-y,)°

(52)

the words "at least" in Theorem 4 can be replaced by the word
"exactly. "

We note that the analyticity of p and U is needed only near
Y_» and that, as a consequence of Theorem 5, a layer of homogeneous
fluid containing a point of zero U" and adjoining a stratified layer
with uniformly large J(y) is always unstable.

VII. NON-SINGULAR MODES

It remains to study neutral waves with a (real) c outside of
the range of U, whose minimum and maximum will continue to be
denoted by a and b. We assume p and U to be continuous, and
that their derivatives as appear in (32) exist. Thenif m and M
retain their definitions as given by (51) and (52), except that c=a-e,
we have

Theorem 6. Under the assumptions on DP and U_ stated above, if
(51) holds there are at least n modes with c=a-€, a@=q@ (i=

,2,-..,n), and aj increasing with i. For the i-th mode there are
at least n-1 internal zeros. If (52) holds in addition, then there

are exactly n such modes, the i-th of which has exactly n-i in-
ternal zeros. lf (pU')’ is negative, then _n_can only increase as

the arbitrary positive constant ¢€ decreases.

The proof of this theorem is by a straightforward application of the
first comparison theorem of Sturm. Similarly, if m and M are
defined by (51) and (52), except that c=bte, where € is an arbi-
trary positive constant, we have

Theorem 7. Under the assumptions on P and U_ stated above, if
olds there are at least n modes with c= €, @=aqj (l=

»2,-e+e,N), and aj increasing with i. For the i-th mode there
are at least n-i internal zeros. If (52) holds in addition, then there
are exactly n such modes, the i-th of which has exactly n-1i in-
ternal zeros. If (pU')’ is negative, then n can only increase as

€ decreases.

233

Yih

If for c=a-€ or c=Lté€, andany €=0, M is less than
w/(y, “Yi )? for all y, and y, between zero and 1, then there can
be no waves propagating with c equalto a or b, or outside of the
range of U. Onthe other hand, if U'=0 at the point of maximum
or minimum U, and, a fortiori, if there is a region of constant U
where U=aor b, it can be easily shown that waves of any finite
wave length and any finite number of internal zeros n can propagate
with c<a or c>b. All this is in contrast with waves propagating
in a layer of homogeneous fluid with a free surface and in shear flow.
In that case [ Yih 1970], if U is monotonically increasing with y,
waves of all wave lengths can propagate with c greater than b, and
only sufficiently long waves can propagate with c less than b.

ACKNOWLEDGMENT

This work has been supported by the National Science Foundation.

REFERENCES

Drazin, P. G., "On the Dynamics of a Fluid of Variable Density,"
Ph.D. Thesis, Cambridge University, 1958.

Howard, L. N., "Note on a Paper of John W. Miles," J. Fluid
Mech., Vol. 10, pp. 509-512, 1961.

Lin, C. C., "On the Stability of Two-dimensional Parallel Flows,
Part II," Quart. Appl. Math., pp. 218-234, 1945.

Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge
University Press, 1955.

Miles, J. W., "On the Stability of Heterogeneous Shear Flows,"
J. Fluid Mech., Vol. 10, pp. 496-508, 1961.

Miles, J. W., "On the Stability of Heterogeneous Shear Flows,
Part 2aK A Fluid Mech, ? Vol. 16, PPpe 209-227, 1963.

L

Synge, J. I., "The Stability of Heterogeneous Liquids," Trans.

Roy. Soc.’ Can., Vol. 27, pp. 1-18, 1933.

Yih, C.-S., "On Stratified Flows in a Gravitational Field," Tellus,
Vol. 93 lpps 220-227. 1957,

Yih, C.-S., "Surface Waves in Flowing Water," to be published in
J. Fluid Mech. in 1971.

254

Stability of and Waves in Strattfied Flows

DISCUSSION

L. van Wijngaarden
Twente Institute of Technology
Enschede, The Netherlands

The flow with a free surface of a fluid, homogeneous in density,
but with inhomogeneous velocity distribution, is a special case of your
class of stratified fluids. Burns [ 1953] considered this case and I
guess his results are comprised in yours. When viscosity is allowed
for, the problem becomes much more complicated. It may be of
interest to note that Velthuizen and 1[1969a, 1969b] studied this prob-
lem taking viscosity into account. We obtained results essentially
different from Burn's results, which is due to viscous effects.

At large Reynolds number the flow can be divided in an inviscid
region and viscous regions at the critical layer and at the bottom. At
the outer edge of the viscous layer at the wall the Reynolds stress
cannot be put equal to zero a priori because a stress may build up in
the wall layer.

REFERENCES

Burns, J. C., "Long Waves in Running Waters," Proc. Camb. Phil.
Soc. 49, 695, 1953.

Velthuizen, H.G.M. and L. v. Wijngaarden, J. Fluid Mech. 39, 4,
817, 1969a.

Velthuizen, H.G.M. and L. v. Wijngaarden, IUTAM Symposium on
Instability of Continuous Systems, Herrenalb, Sept. 1969.

235

Yth

REPLY TO DISCUSSION

Chia-Shun Yih
Untverstty of Michigan
Ann Arbor, Mtchtgan

It is well known that Rayleigh's sufficient condition for stability

of inviscid fluids flowing between rigid boundaries is satisfied by a
parabolic velocity profile, whereas plane Poiseuille flow, which has
this profile, has been found by Heisenberg and Lin to be unstable at
sufficiently large Reynolds numbers, when viscous effects are taken
into account. Since the present paper is a study of the stability of
inviscid fluids,and, in particular, Rayleigh's criterion for stability

is generalized in it, Professor van Wijngaarden's position that the
consideration of viscosity may force us to modify some of the con-
clusions in the paper is easily acceptable.

In considering viscous effects, however, it is not entirely
self-consistent to assume a horizontal mean flow with a free surface,
as Velthuizen and Professor van Wijngaarden have done [ 1969a,b],
since such a flow obviously cannot be maintained, and must in time
attenuate to a state of rest. This is not to say that any conclusion of
instability reached by them is without significance, for instability of
a transcient nature may well occur, with the disturbances growing
for a short duration of time. In this regard the results of Benjamin
[1957] and Yih [1963] for surface waves ina fluid layer flowing down
an inclined plane are relevant. They found that the speed c, of long
surface waves, be they unstable, neutral, or stable, exceeds the
maximum speed of flow. The absence of long waves propagating up-
stream supports Professor van Wijngaarden's claim in connection
with Burn's result, which is supported by a study [ Yih 1971] of waves
in a flowing inviscid liquid. But the nonexistence of a critical layer
renders rather less cogent the argument given in Professor van
Wijngaarden's discussion. On the other hand, this nonexistence sub-
stantiates the conclusion made in Yih [1971] (and similarly in this
paper) regarding the nonexistence of singular neutral modes, since
the velocity U in laminar flow of a viscous fluid down an inclined
plane is parabolic, with a constant U".

We also recall that Tollmien's sufficient condition for insta-
bility [ 1935] of an inviscid fluid is not much affected by the considera-
tion of viscosity, at least when the Reynolds number is large, and
hope that the same is true with the sufficient conditions for instability
presented in this paper.

236

Stability of and Waves in Strattfied Flows

REFERENCES

Benjamin, T. B., "Wave formation in laminar flow down an inclined
plane," J. Fluid Mech., 2, pp. 554-574, 1957.

Tollmien, W., "Ein allgemeines kriterium der instabilitat laminarer
geschwindigkeitsverteilungen," Nachr. Ges. Wiss. Gottingen,
Math. Phys. Kl., Fachgruppe 1, 1, pp. 79-114, 1935.

Yih, C.-S., "Stability of liquid flow down an inclined plane," Phys.
of Fluids, 6, pp. 321-334, 1963.

(The other references are given either in the paper itself or in
Professor van Wijngaarden's discussion.)

237

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7 rorr }
rae 91 ‘eh

ON THE PREDICTION OF IMPULSIVELY
GENERATED WAVES PROPAGATING INTO
SHALLOW WATER

Paul R. Van Mater, Jr.
United States Naval Academy
Annapolis, Maryland
and
Eddie Neal
Naval Shtp Research and Development Center
Washington, D.C.

ABSTRACT

This report treats the problem of the propagation ofa
dispersive wave system generated impulsively by a
surface explosion in deep water into shoaling and shallow
water regions. A topography consisting of an arbitrary
bottom profile with parallel straight contour lines is
assumed. The linear theory of impulsive wave genera-
tion for water of uniform depth is used as a basis for
evaluating the spectral energy of the wave system ata
point in deep water distant from the explosion. Con-
servation of energy is then invoked to extend the pre-
diction to propagation over a bottom of variable depth.
A cnoidal wave theory is introduced to describe the
changes in form of the individual phase waves and the
wave envelope as the system enters the shallow water
regime. The effect of wave refraction at locations
other than along an axis normal to the bottom contours
is treated. Empirical criteria are incorporated to pre-
dict the occurrence of wave breaking, the decay of wave
height after breaking, and the attainment of stability
in the reforming wave. Attenuation of wave height due
to dissipation of energy at the fluid boundaries is also
considered. All of the above elements have been in-
corporated in a computer program. Details of the
computational procedure are described in an appendix.
Typical predictions made using this program are dis-
played for small, moderate, and large source strengths.
The agreement of the predictions with experimental
observations is discussed qualitatively, but no experi-
mental data are included.

239

Van Mater and Neal

I. INTRODUCTION

The nature of wave systems yenerated impulsively by explo-
sions at or below the water surface is of natural interest to re-
searchers in the naval community because of the ship behavior which
results from such an environment. When water-wave systems enter
shallow water they undergo changes in form which may have an
adverse effect on motions depending on the size of the waves relative
to the ship or small craft. While the motivation of this work from
our point of view is ultimately the prediction of ship behavior ina
shallow-water explosion-generated wave environment, this paper
is confined to the prediction of the forcing function -- the wave
system. In itself this case presents an interesting means of study-
ing the shoaling behavior of a dispersive wave system, an area
which has received surprisingly little attention. Previous efforts
in the direction of predicting impulsively- generated wave systems
entering water of variable depth stem from the work of Dr. William
Van Dorn (cf. Van Dorn and Montgomery [ 1963]) and have been
confined to the prediction of wave envelopes. This paper should be
viewed as a second generation of the Van Dorn model.

As a starting point we shall tabulate some of the rather com-
plex effects which occur in shallow water. Not all of these will be
considered in the present prediction scheme but the exclusions will
be noted.

(a) In deep water, phase and group velocities depend pri-
marily on wave frequency giving rise to the well-known
characteristic of the system known as frequency disper-
sion. As the system moves into water whose depth is
small compared to the lengths of the waves in the system
this frequency dependence weakens and dependence on
water depth and wave height strengthens. Waves which
in deep water moved through the group at phase velocities
up to twice the group velocity now become nearly frozen
in their position in the group.

(b) The nearly sinusoidal form of the waves in deep water
changes to one of sharp crests separated by long flat
troughs. An asymmetry about the horizontal plane
develops in which the crest height above the still water
line is greater than the trough depth. The maximum
slope of the waves increases. This last feature is of
particular importance in ship motion prediction.

(c) As wave height becomes significant with respect to the
water depth and the wave nears the breaking point the
leading face of the wave steepens and a wave slope asym-
metry develops. As the slope of the face of the wave
near the crest approaches the vertical the wave becomes

240

Impulstively Generated Waves Propagating into Shallow Water

(d)

(e)

(£)

(g)

(h)

(i)

irreversibly unstable and breaking follows. In the final
stage before breaking there is an abrupt increase in wave
height known as "wave peak-up." Breaking may fall into
one of three broad categories: plunging, spilling, or
surging,

After breaking the wave continues as a spilling wave until
it either runs up on the beach or reforms as a stable
wave. Energy dissipation accompanying breaking reduces
the wave height.

In the case of dispersive systems entering shallow water
a low-frequency oscillation is superposed on the wave
train. This "wave set-up and set-down" is caused by the
transport of mass with the system, particularly when
breaking or near breaking occurs, and the resulting
counterflow.

When an element of a wave crest passes over a bottom
contour line obliquely it is refracted so as to be more
nearly aligned with the bottom contour line, If the bottom
contours are not parallel straight lines a focusing of
wave energy can occur at "caustic points" and consider-
able wave height enhancement can result.

The height of waves in shallow water is attenuated by
energy losses due to bottom friction, bottom percolation,
internal friction, and surface contamination.

Incoming waves which encounter steep offshore bars or
beaches may be reflected seaward. Under the right con-
ditions standing wave systems of surprising severity may
be produced.

Non-linear instabilities in a shallow-water wave may
cause it to decompose, or split into two or more com-
ponent waves. The hazardous "double rollers" are often-
times an example of this. A different type of decomposi-
tion may occur when a wave passes over an offshore bar
and nearly breaks but then recovers. One or more
smaller waves known as "solitons" may be shed from the
back of the larger wave. Little is known about these waves
at present.

All of these features can affect ship and small craft operations
in shallow water and have been included to emphasize the complexity
of the overall problem. Not all will be attempted in the prediction
scheme presented in this paper. For our purposes we will consider
a wave system resulting from an explosion in deep water of nearly
uniform depth, which propagates into shallow water over a terrain
represented by parallel straight line bottom contours. We will

241

Van Mater and Neal

include change of wave form with wave slope symmetry retained,
wave asymmetry about the horizontal plane, wave peak-up, wave
breaking, wave height attenuation after breaking, stability of the
reforming waves, wave refraction along paths other than normal to
the bottom contours, bottom friction, and surface contamination,
Excluded are: wave slope asymmetry and change of wave form close
to breaking, wave set-up and set-down, the presence of caustics,
bottom percolation and internal friction, wave reflection, non-linear
decomposition, and solotonic shedding. The system will not be
carried all the way into the beach.

Specifically, a linear theory for impulsively-generated waves
in water of uniform depth is invoked to describe the waves in deep
water at a large distance from the source. From this point a
different linear theory based on conservation of energy per unit
frequency is employed to depict the system as it moves into a region
of shoaling topography. The integral expressions in this theory are
evaluated numerically using the conditions at each of a series of
closely-spaced stations as input for evaluating conditions at the next
station. As the system progresses into shallow water its frequency-
dispersive nature gradually disappears and non-linear features
dominate in the wave form and propagation velocities. A non-linear
cnoidal wave theory is matched numerically to the previous solutions
to carry the system inthis region. The cnoidal theory is used to
describe the profiles of the individual waves in the system and the
asymmetry of the system about the horizontal plane. To treat wave
breaking existing experimental evidence has been reexamined and
an improved criterion for wave breaking is incorporated. From the
same experimental source empirical formulations are developed to
account for wave height attenuation after breaking and the attainment
of stability in the reforming wave. The Van Dorn formula for bottom
friction and surface contamination is used to account for these effects.
The system is computed not only along an axis normal to the bottom
contours but also along a series of rays which emanate radially
from the source and change direction continuously due to refraction
as the waves move inshore over the shoaling water.

All these features have been incorporated in a computer pro-
gram. Results of this program for a specified bottom profile and
for several source strengths are presented as figures. For the
researcher working on similar type problems perhaps the most useful
part of the paper will be the computational procedure which is dis-
cussed in some detail in an appendix.

II WAVE GENERATION

Treatment of the subject of water waves produced by a local
disturbance has a long history beginning with Cauchy [1815] and
Poisson [ 1816] each of whom independently solved the classic two-
dimensional wave problem which bears their names. In recent

242

Impulsively Generated Waves Propagating into Shallow Water

years Kranzer and Keller [| 1959] , Kajiura [1963], Whalin[1965a],
and Whalin [1965b] have made significant contributions on the sub-

ject. The last cited is an extension of the Kajiura work and appears
to be the most general treatment on the subject.

The theory as presented by Whalin relates an initial distri-
bution of impulse, surface velocities, and surface deformations to
the waves produced at some distance from the source in water of
finite but uniform depth. To a certain extent the choice of a source
model is arbitrary in that several models may give an adequate fit
to experimental data with each having its own particular advantages
and disadvantages. No physical reality is assigned to the source
model in terms ofthe dynamics of the explosion; fortunately, how-
ever, the dimensions of the source model have been found scalable
in terms of explosive yield so that useful predictions can be made.

A source model that has been found to give good agreement
with experiment is a paraboloidal cavity given by:

nz) = 2d, fe) - 5| a (1)

0 eae
The collapse of this cavity at time t=0O generates the wave system.
In cases of large explosions where the dimensions of the cavity are
not small compared to the water depth, this model yields a poor
prediction and a different source model, perhaps utilizing an initial
time dependency should be employed.

According to Whalin the surface elevation, (r,t), for an
axially symmetric surface deformation is:

00
nr .t). = ae (0) o cos (Qt) J,(or) do (2)
0

where mo) is the Hankel transform of the initial surface deforma-
tion 7(r):

oO
Alo) =| n(z)Jo(or) tr dr. (3)

All unprimed quantities in these equations have been non-dimensional-
ized using the water depth, h. Primed variables indicate the cor-
responding dimensional quantities.

rod, = dimensionless radius and height of initial surface

. deformation = ro'/h, d,'/h

243

Van Mater and Neal

= dimensionless radius to field point = r'/h

r
n = dimensionless elevation of water surface = 7'/h
h = water depth

o = dimensionless wave number = Kh

K = wave number = root of equation: we = gk tanh kh
Q = dimensionless frequency = (w*h /g)!/2

w = frequency, radians /sec

t = dimensionless time = t'Ve/h

t' = time, sec.

The integral in (2) is evaluated using the method of stationary
phase (cf. Stoker [1957]). After doing this and making the substitu-
tions the result is:

n(r,t) = 7982. [- aoe - J,(or,) + cos (or - Mt) (4)

Here, v = group velocity = 3[ (0/x) tanh ale [41+ (20/sinh 20)].

Eq. (4) is valid at distances from the source that are large in com-
parison with the radius of the cavity and in water of uniform depth.
A point is selected which satisfies these conditions and the spectral
energies of the system evaluated as will be discussed in the next
section.

III WAVE PROPAGATION OVER A BOTTOM OF VARIABLE DEPTH

The extension of the solution to regions of variable depth is
based on a conservation of energy approach originally presented by
Van Dorn and Montgomery [ 19631. The equation presented therein
evaluated the spectral energy, that is the energy per unit frequency
of the system, for the special case of propagation along a ray normal
to parallel bottom contour lines. The derivation is presented here in
a slightly different way to permit its extension to include refractive
propagation. The topography considered is represented in Fig. 1
which also shows the coordinate system and a typical refracted
wave ray.

The following assumptions apply:
(a) wave frequency remains constant throughout the region of
wave travel and is unaffected by refraction

(b) energy is transported at group velocity in a direction
normal to the wave crest

(c) energy per unit frequency is conserved between adjacent
wave orthogonals.

244

Impulsively Generated Waves Propagating into Shallow Water

WAVE FRONT
eae — (DEEP WATER)

ORIGIN STATION

| 40 59 64 71 76 100

Fig. 1. Bottom topography and wave refraction

The approach will be to consider the energy patch between
two adjacent wave rays, S, and S,, and between two adjacent
frequencies, w and w + dw, and to establish the dimensions of this
patch as a function of path position. While the total energy of the
patch remains constant the energy density changes with patch size
and this, in turn, determines the local wave amplitude. Establish
a rectilinear coordinate system, (x,y), with x-axis normal to the
parallel bottom contours and the beach. Orient the curvilinear
coordinate system (s,n) shown in Fig. 2 to the median between
rays. Since Kk = K(w,h) only the magnitude of K,, Kz, and Kk will
be the same but due to the curvature of the orthogona s the directions
will be different, Let kx, have components (Z, »m, ) parallel to
(s,n). Since the patch 66 6&, is small £, + fl, = ri m, = m,=m.

It may be shown by applicatinn of Snell's Law that the head of
the vector kK, will shift to the right along line AB in Fig. 3 main-

taining a constant we secke on the beach as the patch moves inshore.
Then, from Fig. 3

m, cos 8, = m cos 8 = mg cos @ . (5)

i]
and also,

m, = Ky59%, (6)

245

Van Mater and Neal

Fig. 2. Propagation of an energy- patch over a variable-depth bottom

b STRAIGHT BOTTOM CONTOURS PARALLEL
TO THE BEACH ARE ASSUMED

Fig. 3. Wave number vector diagram for a shoaling bottom

246

Impulsively Generated Waves Propagating tnto Shallow Water

so that,

_ Cos Q
~ cos 6 60, - (7)

Since dw is small the frequency dependent change in the trajectory
of the phase wave orthogonals may be neglected. The change in the
patch width due to the geometric spreading of the rays as the patch
moves through a distance, ds, will be

m

27 ds.

The patch width will be

sti i
et ath = ds

0)

Sf = 20, - £08 Sods : (8)
cos

Next, consider the patch length, 6§&. If 6t is the interval required
for the patch to pass a given point, i, on s, then

where v, is the group velocity at point i. The time interval, 5t,
may also be represented as follows

_ 6t
dt = =— dw
ata
save! a8 = aw — (1/v) ds
Ow o Vv 0 dw
i i
1 av 1 Ov 8x
=- dol — ~— ds = - dw — = ds.
0 v2 w 0 wt aK Ow
Since v= 8w/8K,
1 90
it = - aw | niet dg
bye OK

247

Van Mater and Neal

i
) 1
&t = ao OK (+=) ds. (9)

The patch length becomes,

i
8& = v, aw J ee (=z) de (10)

If E(w) is the energy per unit frequency of the source dis-
turbance then the energy between orthogonals in a frequency band,
dw, near the source will be

2505

E(w) dw a

Since the energy of the patch has been assumed constant this will also
be the value at the field point, i, and this in turn may be equated
to the local wave energy:

2 1
E(w) dy “B80 Zips n OG65;, (14)

where nj is the local wave amplitude. Substituting the expressions
for 5% and 6€ from Eqs. (8) and (10) gives:

i i
260, _ 1 zi cos 0 2 a7 1
E(w) dw aS Pg Ni 2880 cos ds | [v, aw | OK (=) ds | °

The final result is,

Bled of 222% a6] [of & (ste) a]. ca
a a ee
al B,

The first bracketed factor, @,, represents the effect of geometric
spreading between rays while the second bracketed factor, B;;
represents the effect of the spatial stretching of energy between
adjacent frequencies, or frequency separation.

Computationally, E(w) will be evaluated using the results of
the previous section, Eq. (4), at a point sufficiently removed from
the source and over bottom depths nearly enough uniform to satisfy
the conditions on application of that equation. From that point on
inshore new a's and B's will be computed numerically for each

248

Impulstvely Generated Waves Propagating into Shallow Water

field point desired. Treating E(w) as a constant a new nj = n(w)
will be computed. Details will be discussed in the appendix on com-
putational procedure.

The phase of the waves in water of variable depth requires
attention. The phase function of Eq. (4), cos (or - Qt), for uniform
depth may be rewritten as

Ww 1
cos (K = +) r
Vv

since t'=r'/v'. Now, however, in water of variable depth wave
number and group velocity will depend on location as well as fre-
quency. The argument must now be represented in integral form
with the integration performed along the path, s. The phase term
now becomes

cos | («-5) as'| (12)

where K = k(h,w), v' = v'(h,w), and h = h(s').

The central assumptions involved throughout this development
have been that the system is linear and conservative and that the
energy per unit frequency remains constant. The assumptions are
quite viable as long as the water is of deep to moderate depths, say
one-half wave length or deeper. Inshore of this point the system
becomes progressively more non-linear, non-conservative and the
frequency assumption more vulnerable. An evaluation of the linear
assumption will appear in the section.

IV. NON-LINEAR FEATURES OF THE SHALLOW WATER SYSTEM

The previous section carried the wave system from a region
of uniform depth into a region of variable depth; however, the des-
cription of the system retained its linear character. We have des-
cribed earlier the change in form of shallow-water waves from one
of sinusoidal form to one of sharp crests separated by long flat
troughs with an associated horizontal plane asymmetry. In this
section a particular non-linear theory, the cnoidal wave theory of
Keulegan and Patterson [ 1940], will be incorporated to modify the
form of the waves and the wave envelope.

The first of the cnoidal family of wave theories was presented
by Korteweg and de Vries [1895]. Because the wave elevation was
given in terms of the Jacobian elliptic cn function they coined the
work cnoidal to describe the resulting wave form. Subsequent contri-
butions, in addition to the Keulegan and Patterson paper cited above,

249

Van Mater and Neal

have come from Keller [1948] , Benjamin and Lighthill [ 1954] ,
Laitone [1960] , and Laitone [1962]. Masch[1964] has computed

the shoaling characteristics of cnoidal waves assuming constant power.
Iwagaki [1968] has simplified cnoidal wave equations to a form which
he calls "hyperbolic waves." Masch and Wiegel [ 1961] have pro-
vided extremely useful tables of cnoidal wave functions.

A paper by Le Méhauté, Divoky, and Lin [1968] motivated,
in part, the choice of the Keulegan and Patterson theory for incor-
poration in this prediction scheme. These authors reported shallow-
water wave experiments and compared the results with twelve differ-
ent wave theories. Their finding was that none of the theories was
uniformly satisfactory but that the cnoidal theory of Keulegan and
Patterson was the most generally satisfactory. For the shortest
waves linear theory was the best but failed rapidly as the wave length
was increased. Stokes' second order and Laitone's second order
were consistently worst. Stokes' third and fifth order were better
but not as good as linear theory. In terms of the wave profile the
Keulegan and Patterson cnoidal profile gave the overall best agree-
ment and was accurately placed with respect to the still water line.
This latter finding has also been confirmed by Adeymo [ 1968].

The central equations adapted from Keulegan and Patterson
for use here are:

n'(r',t!) = - ng! + Hon? _— (kr' - ot'), | (14)
\f2
=e b-(2)+) (2) @-2g8)] > a

my! . K(k) - BO), ag y ar (16)

kk” K(k)
“H 16 2
meek K(k)] (17)
2 OW 2 } a
= ie : ea LG} : (2) + (=) O73. 3 BY) | (18)

(primes indicate that the variable is_in the dimensional coordinate
system.) The expression, L?H/h>, may be recognized as the
Ursell parameter, although, in fact, Stokes was the first to identify
it. The parameter gives a measure of the linear or non-linear nature
of the system. Linear theory is generally applicable for values less
than 1 while cnoidal theory is most appropriate for values greater
than 10.

The cn function displays a character that is particularly useful

Impulsively Generated Waves Propagating into Shallow Water

for this application. When the modulus, k, assumes its minimum
value, zero, cn reduces to cosine. When it assumes its maximum
value, unity, it reduces to the hyperbolic secant, sech. Since cos?
reduces to cos by the double angle formula and since the form ofa
solitary wave is given by a sech* function, the cn function by
appropriate choices of k describe the complete transition from
sinusoidal waves to solitary waves. As the value of the modulus
increases from zero toward unity the wave crests become more
sharply peaked and the troughs longer and flatter; the height of the
crest above the still water line increases and the depth of the trough
decreases. The wave form is symmetrical about a vertical through
the wave crest so that no wave slope asymmetry is reflected. Thus
the cnoidal feature may be used to improve the realism of the phase
waves in several ways:

(a) to give the non-breaking waves a more realistic profile

(b) to introduce asymmetry of the waves about the still water
line

(c) to increase the velocity of the waves in very shallow water.
This feature has not been utilized in this application.

Computationally, the frequency, w, and the water depth, h,
are defined in the given frequency and spatial array. The wave
height, H, is obtained from the linear theory of the previous section.
The elliptic modulus, k, is computed from an iterative solution of
Eq. (18). K(k) and E(k) are computed from a series expansion in
terms of k. Further details appear in the appendix on computational
procedure.

The application, then, involves the use of a theory developed
by irrotational, periodic, non-dispersive waves of permanent form
in water of uniform depth to represent a wave system which is dis-
persive and not periodic passing over a bottom of variable depth and
in which vorticity due to bottom friction is present at least to some
extent. The assumptions implicit in this extension are that the phase
waves assume a form appropriate to their local peoquency within the
group and that this frequency content changes only slowly, and further
that rotationality effects are small. The latter assumption appears to

be the most vulnerable.

V. WAVE BREAKING AND ENERGY DISSIPATION

Despite abundant literature on the subject of wave breaking
there does not exist today a fully-adequate mathematical description
of the process, and, in fact, much of the experimental evidence is
contradictory and subject to wide scatter bands. In the case of this
analysis the need is for a criterion for wave breaking which relates
the wave frequency, w, the linearly computed wave height, H, and

254

Van Mater and Neal

the water depth, h. In addition expressions are needed which deter-
mine the wave height decay after breaking and the point at which
spilling waves regain their stability. In the absence of an adequate
theoretical base a purely empirical approach will be used.

Experimental evidence has been plentiful. Iverson [1952]
and Morison and Crooke [ 1953] made classical contributions.
Nakamura, Shiraishi, and Sasaki [1966] presented what is perhaps
the broadest range of data on breaking and decay after breaking. A
fairly complete bibliography of other works on the subject appears
in Van Mater [1970].

No uniformly satisfactory criterion which predicts both the
occurrence and the location of wave breaking has yet been developed.
A commonly used but crude criterion is that a wave breaks when the
wave height-to-water-depth ratio is equal to or greater than 0.78.
The Nakamura et al. paper previously alluded to contains all this
information covering a rather broad range of beach slopes and wave
conditions, and on this basis it was selected to develop the criteria
needed for this application. The paper is essentially a report of
experimental data with no analytical comparisons or proposals. A
comparison of the shoaling coefficients inferred from the Nakamura
data shows lower values for the gentler bottom slopes than would be
obtained by linear theory. This gives rise to suspicion of prominent
frictional effects in the experiments. Although the wave height may
have been attenuated by friction as the waves moved inshore it seems
reasonable to assume that the characteristics, w, H, and h, at the
breaking point would not be severely affected. More vulnerable,
perhaps, is the rate of decay after breaking and the length of the surf
zone reported inthe paper. Nevertheless, the scale of the experi-
ments is approximately the size of those with which we are concerned.
The following formulas are a result of reworking and fitting curves
to the Nakamura data,

(a) Wave breaking occurs if:

5 wh (150)""

bs oth sian ‘
H ~ 2 t+ logi, (wh/g 20.13, S20.01)

zn \/4
p< (10 ae -1.1) +1og ee +0.10 (wh/g<0.13, S= 0.01)
(19)

where S = tangent of the angle of bottom slope. A plot of this criteria
for several bottom slopes is given in Fig. 4.

252

Impulsively Generated Waves Propagating tnto Shallow Water

7s (10 es ai + logie( 92) +10, — <13

s 2.01

Fig. 4. Wave breaking criteria

(b) Decay of wave height after breaking is given by,

yes

h
H, = Hy (72 (20)

where,

H,, h,= wave height and water depth at a point after breaking

H,_, h, = wave height, water depth at breaking

b’ “b

The expression is for two-dimensional waves. In the explosion-
generated wave case to account for geometric spreading the expres-
sion must be multiplied by the ratio a, /a, where @ is obtained from
Eq. (12).

(c) The equation for wave stability after breaking is:

[8 = aa (0.85 - 0.40 _ (21)

The wave heights in Eqs. (17) - (21) reflect experimentally-
measured quantities; however the wave height information we have
at hand is computed from the linear theory of Eq. (12). Linear
theory is known to underpredict wave shoaling as the wave height
becomes a substantial fraction of the water depth. In addition, in
the final stages before breaking the wave front slows more rapidly
than the back. Associated with this developing wave slope asymmetry

253

Van Mater and Neal

is a rapid increase in wave height known as "wave peak-up" which
occurs just before breaking. The effect has been observed by

Le Méhauté, Snow, and Webb [1966]. On the basis of the experiments
reported in that source, Van Dorn, Le Méhauté, and Hwang [ 1968]
state that the increase in wave height due to peak-up is 40% of the
linearly computed value.

Computationally, the wave height computed on the basis of
Eq. (12) is multiplied by a factor of 1.40 to account for peak up and
tested against Eq. (19) for each point in the spatial and frequency
array. If breaking occurs the increased wave height is retained as
H, for use in Eqs. (20) and (21). If no breaking occurs the linear
value of wave height is retained.

The Van Dorn boundary dissipation equation for impermeable
bottoms and modified for a wide basin is:

/2
ang PUD conan aoa

where, v = kinematic viscosity of water and points 1 and 2 are
successive points in the direction or propagation.

VI. RESULTS AND DISCUSSION

In this section the computer program predictions for three
source strengths, corresponding to small, moderate, and large
explosions, will be described and discussed. No experimental data
are included but comments will be made on the agreement between
the predictions and experiments which have been conducted.

The theory and the computational procedure outlined in the
preceding sections are applicable to any arbitrary bottom profile
with parallel straight contours and gentle slopes. However, to
compare the theory with experiment a specific bottom profile, that
of a test basin at the Waterways Experiment Station, Vicksburg,
Miss., was introduced as an input to the computer program. Through
the courtesy of that laboratory access was granted to data from a
series of experiments conducted there. The data has not yet been
formally published by WES at this time, so it cannot be reproduced
here; however general comments on the agreement between predicted
and experimental results will be made.

The computer results are non-dimensionalized on the basis of
the water depth at the explosion, hg Previous notation is used
except that the water depth, h, is taken as the depth at the origin,
ho.

254

Impulstvely Generated Waves Propagating into Shallow Water

Radial distance along axis: r= 2 /ho
Distance along path: s =s'/hg
Wave elevation: y= 7) /B
Offset of path from axis: y=y /o
Source strength: W= yh.

In the last definition Y is the explosive yield in pounds of TNT,
equivalent. For dimensional consistency the exponent should be 1/4;
however experiments have shown that exponents from 0.26 to 0.30
(depending on the submergence of the source) provide better

scaling. The value 0,3 is taken here.

Results of time histories along the axis for three different
source strengths are presented in Figs. 5 - 7:
Figure 5, small explosion, W = 0.139
Figure 6, moderate explosion, W = 0.166
Figure 7, large explosion, W = 0.224.

time —=t=t'V gh,

Fig. 5. Prediction of wave system on axis for small explosion

Van Mater and Neal

—— 7x107 ——

time —= ttV9g7h5

Fig. 6. Prediction of wave system on axis for moderate explosion

—— 7110

1210.67

STA.59
re '/hg= 9.83

20 30 40 50 60 70 80 90
time —= t= Vg/ho

Fig. 7. Prediction of wave system on axis for large explosion

256

Impulstvely Generated Waves Propagating into Shallow Water

Predictions are displayed for the five stations, 59, 64, 71, 76, 100
shown in Fig. 1. The first station, station 59, may be considered

to be in the transition range from deep to shallow water for the larger
waves in these explosions. The remaining stations are all in progres-
sively shallower water. A frequency range was chosen which would
permit the computation of the first four wave groups. The full four
groups are shown at station 59, Fig. 6; however since only the first
two groups are of practical interest these are the only ones shown

in the remaining displays.

Wave breaking occurred for the large explosion but not for
the two smaller explosions. The individual phase waves which have
either just started to break or are continuing to break are indicated
by an asterisk in Fig. 11. The irregular shape of the envelope at
stations 71, 76, and 100 is caused by the fact that breaking and wave
(envelope) height decay after breaking have already occurred for these
frequencies within the envelope. Typically, breaking will start within
a small frequency band then spread to adjacent frequencies as the
envelope moves inshore.

The effect of refraction is shown in Fig. 8 for the moderate
explosion case only. The wave trains correspond to those along the
ray families 0 9* = 0°, 20°, and 40°. Mean offsets of the path from
the axis are indicated. Three stations, 59, 76, and 100 are shown to
give a representative effect.

¥#¥'/ho 78.49
"

ns 10% ——_—_—

60 70 80 90 100
time —=t=t'V9/ho

Fig. 8. Prediction of wave system along refracted rays for
moderate explosion

Z57

Van Mater and Neal

Early runs on the computer used the cavity dimensions sug-
gested in Van Dorn, Le Méhauté, and Hwang [1968] and the wave
peak-up, wave breaking, decay, and stability criteria which have been
outlined previously. To improve agreement between the theory and
observation adjustments were made to some of these empirical coef-
ficients. First, cavity dimensions were adjusted for the best fit
with the results shown below.

; 24
Yo = ae — eel
C,., Cy,
small explosion 7.80 1,53
moderate explosion 7.82 1.33
large explosion 8.14 2.03
recommended by Van Dorn 9.60 2.80

For wave breaking the peak-up ratio was changed from 1.40 to
1.50, breaking and decay criteria were not changed. The criteria
for stability after breaking was changed as follows

2

Ha = Hp (0.90 - 0.40 2he)

hg hp 24
Stable

where H, is now taken as the height before peak-up.

The trajectory of an individual wave ray is dependent on the
initial angle at the origin, 0), bottom profile, and frequency. Con-=
sequently absolute convergence of a family of wave rays representing
an array of frequencies is not possible. To give the best overall
conformity over the spatial domain, 9 is adjusted with frequency,
i.e. 05 = Oo(w). For the conditions assumed a simple linear varia-
tion in @(w) was found to give a variation in path lengths which
generally fell within a 1% band and a variation in offsets from the
axis which fell within a 5% band, The control angle upon which
6,(w) is based is designated gene

With the background now established the following remarks
may be made regarding the agreement between the predicted, or to
put it more accurately, the hindcast wave system and the wave
system observed in experiments.

(a) For the two smaller explosions, W = 0.139 and W =0.166,
agreement of the envelopes at stations 59 and 64 is very good. The
envelopes of the first group are underpredicted at stations 71 and
76 but overpredicted at station 100. This infers that the linear
theory underpredicts shoaling wave height enhancement in very

258

Impulsively Generated Waves Propagating into Shallow Water

shallow water, and also that the boundary dissipation formula used
in the program does not provide sufficient attenuation. The assym-
metry of the envelopes show particularly good agreement and is one
of the strong features of the program. The observed trough levels
at stations 71 and 76 is slightly lower than predicted. This is
attributed to wave set-down due to the presence of a counterflow, or
backwash current.

(b) For these two smaller explosions the theory in general predicts
the correct number of waves in the envelope. Agreement in phase

is poor at station 59 but better at subsequent stations. There is also
some grounds for suspicion of zero-set clock errors in the experi-
mental data so that it is difficult to make definitive statements on this
subject. The same suspicion makes it difficult to comment on the
agreement of the phase velocity of the observed waves.

(c) The agreement in regard to the form of the waves in these
smaller explosions is especially impressive, The change from sinu-
soidal form to cnoidal form quite accurately represents the observed

(d) The observed waves were nearly fully attenuated by the middle
of the second wave group. Stronger attenuation in the higher-
frequency range is required in the boundary dissipation formula.

(e) Time of arrival (based on linear group velocity) of the wave
groups is in very good agreement indicating that transporting energy
at linear group velocity, even in the presence of noticeable viscous
effects, remains valid into quite shallow water.

(f) The fit of the envelope for the large explosion, W = 0.224, at
station 59 is only fair. The observed envelope of the first group
reaches its maximum ata later time. It appears that for this source
strength the explosion may no longer be considered to occur in deep
water and that a different source model, perhaps one yielding a
higher-order Bessel function, is indicated. In addition the observed
troughs of the large waves were much lower than those predicted.
Again the probable cause is the presence of an observed strong back-
wash current which would have the effect of depressing the trough
level. The presence of backwash currents is not reflected by the
theory.

(g) The agreement in phase is quite good at stations 59 and 64, but
not perfect, After wave breaking sets in the phase agreement de-
teriorates. One of the observed waves in the vicinity of the first

node decomposed into two component waves both of which eventually
broke. Otherwise, the theory predicted the correct number of waves.

(h) For the large explosion breaking is predicted for the second wave
at station 64. Actually, the third and fourth waves broke at this

259

Van Mater and Neal

location and the second wave did not break until a subsequent station.
A different source model giving a later peak to the first envelope
would probably also correct this discrepancy. The theory predicts
about the right number of waves breaking at subsequent stations,
although there is disagreement in some cases on which individual
waves break. The decay rate after breaking appears to be about right
for the envelope heights in the surf zone match quite well. As before
there is still distortion of the envelope due to backwash effects. The
stability criteria seems to give a surf zone of about the right length,
but the data are inadequate to say conclusively. Comparison witha
much larger number of experiments is needed to fine tune these
empirical breaking relationships; however impulsively-generated
wave system furnish an ideal vehicle for such studies.

(i) The form of the breaking waves and the near-breaking waves is
very poorly predicted by this program. The wave slope asymmetries
which develop near and at breaking are not reflected at all in the
cnoidal wave form. For future generations of the program the
incorporation of a theory developed by Biesel [1952] is under con-
sideration.

(j) Despite the fact that the "correction factor" approach in account-
ing for wave peak-up is somewhat distasteful it appears useful at this
evolutionary stage. At the minimum the peak-up correction factor
should be refined to reflect the influence of w, H/h, and bottom
slope. Clearly what is needed is a computationally useable non-linear
theory which predicts this phenomena.

(k) No comments can be made on the quality of predictions along the
refracted rays. Such data were taken in the WES experiments but
are not presently available for comparison.

In summary it may be said that for small and moderate size
explosions the theoretical and empirical program presented gives
good predictions of envelope shapes and asymmetry, wave form,
and times of arrival of wave groups in shallow water. Wave height
enhancement in very shallow water and viscous attenuation are some-
what under-predicted. The quality of the prediction of the phase of
individual waves remains to be established. For large explosions
the present source model gives only a fair prediction of envelope
shape. Group times of arrival and form of non-breaking waves con-
tinues to be well predicted. Form of near-breaking waves is poorly
predicted. Counterflow currents have a prominent influence when
waves are very large, but the presence or effect of such currents is
not predicted. The location, size, and extent of the surf zone appears
satisfactory based on a limited comparison.

260

Impulsively Generated Waves Propagating into Shallow Water

ACKNOWLEDGMENTS

The work described in this presentation was performed at the
Naval Ship Research and Development Center, Washington, D. C.
under the joint sponsorship of the Defense Atomic Support Agency
and the Naval Ship Systems Command, Task Area SR 104 0301,
Task 0583.

The authors wish to express their gratitude to the following
persons for assistance and support in this project:

Dr. Ming-Shun Chang Naval Ship Research and Develop-
ment Center

Dr. Hun Chol Kim Korean Institute of Science and
Technology

Professor T. Francis Ogilvie University of Michigan

Mr. John Strange U.S.A.E. Waterways Experiment
Station

Mr. Raymond Wermter Naval Ship Research and Develop-

ment Center

Miss Claire Wright Naval Ship Research and Develop-
ment Center

REFERENCES
Abramowitz, M. and Stegun, I. A., Handbook of Mathematical
Functions, U. S. Dept. of Commerce, Nat. Bureau of Stds.,
Appl. Math. Series 55, 1964.
Adeyemo, M. D., "Effect of Beach Slope and Shoaling on Wave
Asymmetry," Proc. 1ith Conf. Coastal Engineering, ASCE,
1968.

Benjamin, T. B. and Lighthill, M. J., "On Cnoidal Waves and
Bores," Proc, Royal Soc., A.; vs 224; 1954.

Biesel, F., Gravity Waves, .U. S. Dept, .of Commerce, Nat. Bureau
OL otdsee Circulars 21.. 1952.

Cauchy, A. L. de, Mem. de1'Acad. Roy. des Sciences (Memoir
dated 1815), 1827.

Iverson, H. W., Gravity Waves, U. S. Dept. of Commerce, Nat.
Bureau of Stds., Circular 521, 1952.

264

Van Mater and Neal

Iwagaki, Y., "Hyperbolic Waves and their Shoaling," Proc. 11th
Conf. Coastal Engineering, ASCE, 1968.

Keller; JB), Comm. Appl..;Math:.,.v. 1.1948.

Keulegan, G. H. and Patterson, G. W., Jour. Res., Dept. of
Commerce, Nat. Bureau of Stds., v. 24, 1940.

Kajuira, K., Bull. Earthquake Res. Inst., Japan, v. 41, 1963.

Korteweg, D. J. and de Vries, G., "On the Change of Form of Long
Waves Advancing in a Rectangular Canal, and on a New Type
of Long Stationary Waves," Philo. Magazine (Br.), V Series,
V1 29; 1895.

Kranzer, H. C. and Keller, J. B., "Water Waves Produced by
Explosions," Jour. Appl. Physics, v.30, n. 3, 1959.

Laitone, E. V., "The Second Approximation to Cnoidal and Solitary
Waves," J. Fluid Mech., Vol. 9, 1960.

Laitone, E. V., "Limiting Conditions for Cnoidal and Stokes Waves,"
J. Geophysical Res., v. 67, n. 4, 1962.

Le Méhauté, B., Divoky, D. and Lin, A., "Shallow Water Waves:
A Comparison of Theories and Experiments," Proc. 11th
conf, Coastal Engineering, ASCE, 1968.

Le Méhauté, B., Snow, G. F. and Webb, L. M., Nat. Engr.
Science Co., Rpt. S245A, 1966.

Masch, F. D., "Cnoidal Waves in Shallow Water," Proc. 9th Conf.
Coastal Engineering, ASCE, 1964.

Masch, F. D. and Wiegel, R. L., "Cnoidal Waves, Table of
Functions," Council on Wave Research, The Engineering
Foundation, Richmond, Calif., 1961.

Morison, J. R. and Crooke, R. C., U.S. Army Corps of Engineers,
Beach Erosion Board, Tech. Memo. 40, 1953.

Nakamura, M., Shiraishi, H. and Sasaki, Y., "Wave Decaying Due to
Breaking," Proc. 10th Conf. Coastal Engineering, ASCE,
1966.

Poisson, S. D., Mem. de l'Acad. Roy. des Sciences, 1816.

Stoker, J. J., Water Waves, Interscience, Chap. 6, 1957.

Van Dorn, W. G., Le Méhauté, B. and Hwang, L., Tetra-Tech,
Inc. Rpt..TC-130, 1968.

262

Impulsively Generated Waves Propagating tnto Shallow Water

Van Dorn, W. G. and Montgomery, W. S., Scripps Inst. Ocean.
Ref. 63-20, 1963.

Van Mater, P. R., Nav. Ship Res. and Dev. Ctr. Rpt. 3354, 1970.

Whalin, R. W., "Water Waves Produced by Underwater Explosions:
Propagation Theory for Regions Near the Explosion, "
Jour. Geophysical Res.; v. 70, ne. 22, 1965a.

Whalin, R. W., Nat. Engr. Science Co. Rpt. S 256-2, 1965b.

Wwieoel, R. L., Oceanographical Engineerin » Prentice Hall, 1964.

APPENDIX 1

COMPUTATIONAL PROCEDURE

This appendix discusses the details of implementing the theory
outlined in the previous sections in a computer program which will
predict the wave system in shallow water.

Initially, a monotonically decreasing bottom profile is
assumed with parallel straight bottom contours as shown in Fig. 4.
Actually, the specific profile used in this program was chosen to
conform to that of a test basin at the U.S. Army Engineer Waterways
Experiment Station, Vicksburg, Miss. in order to permit comparison
of the analytical predictions with experiments performed there. A
polar coordinate system is established with the origin at the point of
the explosion and the axis taken normal to the bottom contours. The
axis is divided into a number of closely spaced stations, indexed i,
with i=0O atthe origin. The frequency range of interest is also
divided into a number of closely spaced frequencies, indexed j.

A number of rays, or orthogonals, indexed k, are established
emanating from the origin. The local angle of the orthogonal with
the normal to the bottom contours varies with the frequency, w, and
the water depth at a given location is identified as 6jj,. Because of
the frequency dependence absolute congruence of the trajectories of
the orthogonals is not possible. To give the best overall conformity
with respect to both location and path length the initial angle of the
orthogonal at the origin, Oj, is adjusted with frequency. Thus the
index k identifies a family of orthogonals which have approximate
but not precise spatial agreement, except, of course, on the axis.
Throughout indexical notation is for array identification only and
tensor convention is not implied.

A starting station, i=I, is selected in deep water sufficiently
distant from the explosion for Eq. (4) to be valid. That theory strictly

263

Van Mater and Neal

is for water of uniform depth, but for practical purposes as long as
the minimum depth (at i =I) is greater than half the length of the
longest waves with significant energy results will be quite satis-
factory. Accordingly the depth used in Eq. (4) is taken as the depth
at i=I. Equation (4) may be rewritten in indexical notation as

/2
1 qi Hee I
Bie Se [paw | cc. allie) a
ijk ij (ea)
do ij
Nijk = Bijn COS (Kj Sijk - &jtijx) (24)

where,
Bijx = envelope elevation function in dimensional system, (ft)

Nijk = wave elevation in dimensional system, (ft)

ro,dg = radius and depth of cavity in dimensional system, (ft)
Stik = path length to i station along ira ray. Also indexed
j since path varies with frequency
oj, = Kjhi
Vij = group velocity
= /gh, - alah Cj I (1 LNs ) (25)
Bo EN oo sinh 20);
J3 = third-order Bessel function of the first kind
=) . Vgh; . — vil)" ete 2qjj - 20); cosh2ojj toij __ 1 )
ty J sinh 2075; J
(26)

The wave number, Kij » obtained from the equation
2
and may be approximated in closed form by

Ki. = #1 (coth SEL
FoeN cs g

2 2 1/2
Vere: (28)

264

Impulsively Generated Waves Propagating into Shallow Water

The Bessel functions J,(z) and J\(z) are computed from series
expansions given in Aueaniowite and Stegun [1964] (eqs. 9.41-9.46).
J,(z) and J3(z) are then computed from the recursion relation

Jn (2) + Ing (z) = (2n/z)Jp(z).

The time of arrival of each frequency at the starting station
is computed from the relation.

S;;
ijk
It is now possible to compute the wave spectral energy at the
starting station by applying Eq. (12). That equation, rewritten for
numerical integration, is:

a. = ik 2k PTjk (30)
where
: 1
“Tk » Re. canoe aaa (31)
I
Bik = Vy » a (<2) : ee
3
Ae Sy pecs reco
i=O \j (1 ciaece el
As! (32)

cos 0. -l, j,k

The last factor in Eqs. (31) and (32) represents the incremental
distance along the path where As' is the station spacing on the axis.

The energy may now be carried forward from station to
station as:
Beene Uk

or

265

Van Mater and Neal

12 ieee
Tial, i kZiel, j,kPiolik = ijk jn Pijr

Thus
atk Bijk \/2
t -= ! t {
Niet, isk az SP heres | S (33)
Similarly;

Bl. = Blix ijk Pijk qe (3.4)
tote OUR La: ty kPisi, jk

In the computer program it is the wave system envelope, B',
that is carried forward to the next station and the new value at that
station computed from Eq. (34). The phase term is then applied to
obtain the wave elevation.

The envelope function B'(w) is symmetrical about the SWL
by definition. This corresponds well to the observed envelopes at
the starting station in deep water, but as the system moves into
shallow water the envelope and the phase waves develop asymmetries
about the SWL which we seek to describe by the application of the
cnoidal theory as previously discussed.

The first problem is to calculate the elliptic modulus, k,
for once this parameter is known, all other cnoidal properties may
be computed directly. Two difficulties are immediately realized
in the determination of the elliptic modulus. First, Eq. (18) does
not admit an explicit solution in k. Secondly, the form of cnoidal
waves becomes quite sensitive to k as k approaches unity. For
example, there is a noticeable difference between the form of the
wave determined by k* = 0.99990 and that determined by k?=
0.999990. Further, explosion parameters of interest require the
determination of modulus values as large as k”"=1-10°. Thus
the following procedure was employed to efficiently and accurately
determine k from Eq. (18).

Write Eq. (18) as

2 2 2 2
diwih gaSmiy pee ee as H 1 Aes |
oar aie Ee Cad Ge) ta > eae
We then seek the roots of the equation

g(k) = 0 (35)

266

Impulstvely Generated Waves Propagating tnto Shallow Water

Now, smaller roots of g(k), say 0< ages oe 10, are
readily obtained by iteratively searching for zeros of g(k) in
successively finer increments. Larger roots of g(k), say
1-10%<k*=1 - 10°, are then obtained by iteratively searching
for roots of g(k) in half-power increments of 10°". where k=
fae 1 0% ts Nearly exact solutions in terms of n are then obtained
using the approximate interpolation relation

0.575 +

n= a/Q B (36)

where @ and 6 are interpolation constants and Q is the dimension-
less frequency. This approximation is based on a family of curves
(n vs. t) in Wiegel [1964] (Fig. 2.24).

Computer computational difficulties are avoided in solving
Eq. (18) for values of k near unity, since the modulus k and the
complete elliptic integrals K(k), E(k) can be determined from the
value of 10 = 1 - k*, using the approximations given in Abramowitz
and Stegun [1964] (eqs. 17.3.33-17.3.36). Since only the largest
real root of g(k) is of interest, the computer program searches
first for the largest real root. If no real root is found in the pange
1 - 10°4*<k?<1, then the largest real root inthe range 0<k°S
{= 10°* ts computed. The smaller roots or imaginary roots have
no meaning. Whenever no real root is obtained in the range
0<k<1, the modulus is set equal to zero. The computation is
repeated for each frequency at each location. Note that the calcula-
tion is based on the double amplitude of the envelope and not on the
phase wave elevation.

The distortion of the envelope to its asymmetrical form is

achieved by applying Eq. (16). Denoting the elevation of the envelope
above and below the SWL as Hi and H2 those equations become

H1

Huy (Sw - E(kijx) ]

Ki KU;

ijk
ijk)

(37)

_ Hi ijk |
H2ijy = Hijr [1 s ae

The phase term for water of variable depth was given as
Expression (13). Denoting the argument of the function as and
the field point on s as 3g;

v= J (eget,

267

Van Mater and Neal

the expression may be written for numerical integration as follows:

As!
Wijk = > Ki-1,j,k * S00 6. t witii, (38)
ge i-l,j,k
i=O
where
i
1 As!
' — e a eee
‘ike > vi] C08 Onin (39)
i=0

In Eq. (39) tij, is the time of arrival of the jth frequency com-
ponent at location (i,k) and is printed out for each frequency at each
location.

Introducing the cnoidal phase term of Eq. (14) the wave ele-
vation becomes

Nijk = - H2ijx + Hijxon” [ sun) (Hi jx)» ix | ° (40)

The Jacobian elliptic functions can all be expressed in terms
of theta functions, and can be computed from the resulting infinite
series. However, in this program the elliptic function cn in
Eq. (40) is evaluated to any specified degree of accuracy using
Landen's transformations.

Let m=k’, m, = € -m. Then for m sufficiently small
such that m* and higher powers are negligible, we have the follow-
ing approximations for the Jacobian elliptic functions

sn(u,m) = sin u - 0.25 m(u - sin ucos u) cos u (41)
cn(u,m) = cos u t+ 0.25 m(u - sin ucos u) sinu (42)
dn(u,m) = 1-0.50 msin u (43)

For m sufficiently close to unity such that mi and higher powers
are negligible, we have the approximations

tanh u + 0.25 m,(sinh u cosh u - u) sech” u (44)

sn(u,m)
cn(u,m) = sech u - 0.25 m, (sinh u cosh u - u) tanh u sech u (45)

dn(u,m) + sechu + 0.25 m,(sinh u cosh u + u) tanh u sechu (46)

268

Impulsively Generated Waves Propagating into Shallow Water

Using the following transformations, intermediate values of
the parameter m are reduced or increased such that the above
approximations are applicable.

To increase the parameter, let

Arno 1 - me :
Pl tm ee p, = (7)
vie emer
1 + P, =
Then
- \/2, sn(v,p) cn(v,p)
sn(u,m) = (1 +p, ) ante op (47)
dn(u,m) = (1 - (1 - p'”)) sn’(v,p) (48)
: a P| dn(v,p)
2 sn‘ (v p)
cn(u,m) = (1 - (1 - P, UC Rea (49)
To decrease the parameter, let
15: Be u
P= (+=) ’ va— EFT é oy
ie m, ip
Then
an{u;m) = (1 +p") “peek (50)
i +p “an (v>p)
dn(u,m) =r(1 = 2 ___sn(v,p) (51)
> PU TF pent(v ep)
I CERN css BLN (52)

i oh p* sn*(v,p)

Note that in both the descending and ascending Landen trans-
formations, sn and dn are required in order to compute cn. In
the computer program values of m greater than 0.6 are computed
using the ascending transformation. Values of m less than or equal to

269

Van Mater and Neal

0.6 are computed using the descending transformation. The transfor-
mations are reapplied until higher powers of m or m, are deemed
negligible. The currently used cutoff value is m*(m?) = 10 Both
of the Landen transformations converge quite rapidly. Thus the
cutoff parameter value is attained in three of fewer applications of
the pertinent transformations.

Some computational difficulty may be experienced in evalua-
ting the hyperbolic functions used in the ascending Landen's transfor-
mation, for large values of the argument u. This problem can be
alieviated somewhat by reducing the cn argument, u, to its
principal value - 4K(k) Su S 4K(k). Further difficulty may be
resolved by using the descending transformation throughout the
modulus range where applicable.

When k =u, the cn* term in Eq. (40) reduces to cos? (Wijx)/ 2
and Hijj, = H2ijx = (Hijk)/ 2 = B'. Thus, for k=0, Eq. (40) reduces
to

Nijk = B' cos Wijk,

which is the usual wave elevation equation.

The matter of the frequency dependence of the trajectories of
the wave orthogonals has been discussed briefly. In principle it
would be possible to compute an initial angle 0 jx for each frequency
and at each station which give a path length and a path offset from
the axis that would fall within established error limits. Such an
iterative procedure would increase the computation time enormously
and was rejected on this basis. Several schemes were tried in
attempting to find a simple rule for the choice of 99), which would
give reasonable conformity in a given family of trajectories. The
simplest rule turned out to be the best. A linear distribution of
Qik was chosen according to the following relation:

*
Ojk = Sojkl 1 - 0.04(w - 0.2)] (54)

where Ocik is a control angle for the family of trajectories. The
choice of the above relation is quite an arbitrary one and a different
and more complicated bottom topography could necessitate a different
function or the iterative procedure discussed above.

The refraction angle at each station along a given path is com-
puted from Snell's Law:

Bi ik = arcsin Koj_sin Bojk (55)
J Ki

270

Impulstively Generated Waves Propagating into Shallow Water

The path length and offset from the axis are given by

ro As'
=i\ = » cos 011 5 x (56)
t 239
!
1 = °
Yiik =) As' * tan 85 i,k ; (57)

The system is tested for wave breaking by applying Eq. (19) to the
envelope height, Hjj,, increased by a factor of 1.40 to account for
non-linear peak up. If the test succeeds and breaking occurs then

the increased value of envelope height is retained as H, for use in
computing decay after breaking, Eq. (20), and wave stability, Eq.
(21). Indexing the breaking point as i=b and any location in the
surf zone beyond the breaking point as i= a, these equations become:

; haV'25-58a / apy
Hoy = Hy (7) Gr (58)
; 2
ee = pik (0.85 - 0.40 Sita) , (59)
o “Stable bjk 8

Once stability is found the envelope height, or rather the symmetrical
envelope elevation B = H/2 is carried forward in the usual way.

Since the test for breaking is applied to the envelope, strictly,
wave breaking can be considered to occur only if a phase wave crest
occurs within the breaking band of frequencies. As a practical matter
the prediction of the exact arrival of the phase wave is the most
difficult and least reliable part of the whole procedure, so that the
surf zone should be considered to extend over any region where
breaking is predicted for any frequency within the envelope.

The boundary dissipation equation, Eq. (22), is applied to the
envelope height between successive stations outside the breaking zone.

In the present computation the array consists of 100 stations,
120 frequencies, and 3 families of wave rays CH = 0°, 20°, 40°).
The computer program is listed in Van Mater [1970]. The program
requires 23 minutes of running time on the IBM 7094 and provides
about 14,000 lines of output.

271

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HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

Tuesday, August 25, 1970

Afternoon Session

Chairman: J. Hoyt
Naval Undersea Research and
Development Center, Pasadena

Page

Three Dimensional Instabilities and Vo-tices Between
Two Rotating Spheres 215
J. Zierep, O. Sawatzki, Universitat Karlsruhe

On the Transition to Turbulent Convection 289
Ruby Krishnamurti, Florida State University

Turbulent Diffusion of Temperature and Salinity:
-- An Experimental Study S11
A. H. Schooley, U.S. Naval Research Laboratory

Self-Convecting Flows 321
M. P. Tulin, Hydronautics, Inc. and J. Shwartz,
Hydronautics-Israel, Ltd. and Israel Institute of
Technology

273

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i ee t

THREE DIMENSIONAL INSTABILITIES AND
VORTICES BETWEEN TWO ROTATING SPHERES

J. Zierep and O. Sawatzki
Untversttdt Karlsruhe
Karlsruhe, West Germany

We study the motion of a viscous medium between two concen-
tric rotating spheres. Investigating this type of flow is an extension
of the well-known contribution of G. I. Taylor [1], who studied the
motion between two rotating cylinders. Due to the action of the
centrifugal force instabilities are possible. The main difference
between the two flow fields is that for the spheres the centrifugal
force is a function of the latitude. We have here an instability ina
three-dimensional flow and it is possible that there exist different
flow regimes -- stable and unstable ones -- side by side. This prob-
lem is closely related to the cellular convection flow, especially to
those existing over nonuniformly heated surfaces [2, 3, 4]. The
thermal buoyancy corresponds to the centrifugal force, the nonuniform
heating corresponds to the latitudinal dependence of the centrifugal
force,

Now some fundamental things about the used apparatus. * ihe
experiments have been done primarily with the inner sphere (Aluminum)
rotating and the outer one (Plexiglass) fixed. The gap was filled with
silicon oil that contained aluminum powder as flow indicator. Measured
has been mainly the frictional torque that keeps the angular velocity
of the inner sphere constant. The temperature in the gap was con-
trolled by thermocouples and photographs have been taken of the
different flow configurations. The measurements have been done
by Ritter and Wimmer [ 6] as part of their master thesis.

Figures 1 and 2 show the results for two different gap widths.
Plotted is the friction torque coefficient 6, over a Reynolds number
that ranges from 10! to 10§ Covering this wide range has been
accomplished by 1) using silicon oils with viscosities between 3 and
1000 c St and 2) by varying the angular velocity from 0 to 200 revo-
lutions /sec.

In principle we have here independent of the width of the gap
three different domains of fluid motion. For small Reynolds numbers

Sic aang Arm ar RG ga
For detailed information see [5].

275

Zterep and Sawatzkt

R, = 79.95 mm
s

10"
Du: Motion in a spherical gap
Sms & (Re<!) R, = 75,95 mm
>

Fig. 1 CY (Re) for the relative gap width s/R, = 0.0527

10 |
Motion in a spherical gap |
Coe 28.8 Re <1) R, = 67,80mm
R, = 79,95 mm
10° s =12,15 mm
2 “0
Re = Ri i.
Cu es |
M
-} 1 Ce = — |
9, 25. 2
a
aay
Ry -wy-S / Ss
Ta ,, = =: aa R, = 455 |
Pte
2]
10 1 wy 5 6
10' 107 10° 10 10 0

Big. 2 o,,(Re) for the relative gap width s/R, = 0.18

276

Instabilities and Vorttices Between Two Rotating Spheres

the Navier Stokes equations give Cy ~ 1/Re, the law for creeping flow.
Surprisingly this holds up to Re = 3.3 + 103 for the small gap and up
to Re = 600 for the larger one. Next to this regime follows one of
laminar boundary layer type with ¢,~ 1/VRe. Finally the turbulent
flow regime with C,~ 1 Re is Peaches after passing some possible
instable flow configurations in the transition region. In general we
have this behavior in all cases but quantitatively there are important
differences depending on the relative width of the gap. The reason

for this behavior is the multitude of the possible flow configurations.

First we study the case of the small relative width of the gap.
For low Reynolds numbers (for instance Re = 10) the streamlines

are concentric circles around the axis of rotation (Fig. 3). With

Fig. 3 For small Reynolds number the streamlines
are concentric circles around the axis of
rotation. s = 5mm, Re= Rw, /v = 10

increasing Reynolds number the streamlines change to spirals (Fig. 4).
Close to the rotating sphere the spirals are moving from the poles to
the equator but close to the fixed sphere the spirals are moving from
the equator to the poles. The inner and the outer spirals join and

form closed curves. With passing the critical Taylor number

Ta = 41.3 Taylor vortices begin to develop close to the equator. It

is remarkable that the critical Taylor number here has the same

value as for the concentric cylinders. The axes of these vortices

have spiral form and end free in the flow field. (Fig. 5). From

277

Zterep and Sawatzkt

Fig 4 For larger Reynolds number the streamlines
are spirals, s = 5mm, Re = 2350

Fig 5 For large Reynolds number the vortex axes
become spirals and end in the flow field.
s = 1.05 mm, Re = 27,000, Ta = 41.6

278

Instabiltittes and Vortices Between Two Rotating Spheres

the end of the vortices up to the poles the laminar flow remains
stable. With increasing Reynolds number, the axes of the vortices
become wavy (Fig. 6) and the flow turns turbulent after passing some
intermediate states (Figs. 7, 8}. In this case photographs still show
a remarkable distinct structure of the flow.

Fig. 6 The vortex axes become wavy for very large
Reynolds number. s = 2mm, Re = 16,800,
Ta = 68.8

For the larger relative width of the gap this matter is much
more complicated. In the transition from laminar to turbulent flow
we found that altogether five basically different but reproducible
main modes are possible. In the torque diagram all these modes
are noticeable and they are remarkably stable as soon as they have
become existent. For this reason we called them "stable instabilities."
In the experiment these different modes can be established by apply-
ing a suitable acceleration of the angular velocity. In analogy to the
rotating cylinders [7] we have here a case of nonuniqueness. The
mode of instability that is finally realized depends on the initial con-
dition given by the experimentator. Now the main modes I - V shall
be discussed briefly.

I. In spite of having an overcritical state no vortices become

visible. The transition to the turbulent flow occurs by passing through
mode II, that is described below. Mode I is characterized by the fact

219

Zterep and Sawatzkt

Fig 7 The’turbulent)motion. ‘s = 3.5 mm, Re = 53,500,
Taii522

Fig. 8 The turbulent motion. s = 2mm, Re = 158,000,
Ta = 648

280

Instabilities and Vortices Between Two Rotating Spheres

that in the field between the two boundary layers of the rotating and
the fixed sphere -- according to the large gap -- a flow is established
that moves with a constant but smaller angular velocity than the inner
rotating sphere. Obviously this type of motion prevents or at least
delays the development of the Taylor vortices. A similar pattern is
known to exist also in the gap between two discs [ 8] with one of them
rotating.

II. This regime is characterized by a flow with vortices that
begin at the poles (Fig. 9). The axes of these vortices are inclined

Fig. 9 Motion of Mode Il. s = 12.15 mm, Re = 8,300,
Ta. =/630

slightly to the streamlines close to the fixed sphere. With increasing
Reynolds number these vortices advance around the poles to the
equator. The axes become more and more wavy and finally the flow
turns turbulent. The physical explanation of these vortices is by no
means evident. We have the conjecture that we have here a situation
analogous to the occurrences close to a free rotating sphere [9] or
disc [10]. Very often these vortices are called Stuart vortices.
Contrarily to the familiar pattern of vortices, rotating with alternating
direction the Stuart vortices rotate all in the same direction.

III. Two Taylor vertices develop symmetrical to the equator.

Outside the vortex zone we have mode I flow. Surprisingly at the
equator -- where the centrifugal force has its maximum -- the flow

281

Zterep and Sawatzkt

Fig 10 Sketch of Mode III

Fig 1% Motion of Mode III. s = 8 mm, Re = 8,130,
Ta = 318

282

Instabilities and Vortices Between Two Rotating Spheres

is directed inward (Figs. 10, 11). This can be explained by a cellular
motion in the field between the pole and the vortex that forces the
vortex to rotate in the mentioned direction. The result is the sink
flow at the equator.

IV. Two pairs of Taylor vortices develop symmetrical to the
equator but now with an outward motion at the equator (Figs. 12, 13).
Mode III is a limit case of IV reached by increasing angular velocity.
The cell close to the equator becomes smaller and smaller and in the
limit the flow reverses at the equator.

V. This is an unsteady version of mode III. Vortices,
generated at the equator, leave the equator under a small angle of
about 10° (Fig. 14) and move on spiral trajectories to the pole.

It is interesting to see that the critical Taylor number increases
with increasing gap width. Corresponding calculations for rotating
cylinders with arbitrary gap width, done by KirchgaBner [141], agree
very well with our experimental results for spheres having the same
direction (Fig. 15). The explanation for this is that in our case the
instability first begins at the equator and we have there locally a
similar situation as in the case of the two cylinders.

\S

SSS

Fig. 12 Sketch of Mode IV

283

Zterep and Sawatzkt

Fig. 13 Motion of Mode IV. s = 12.15 mm, Re = 2,660,
Ta = 201

Fig. 14 Motion of Mode V. s = 12.145a0m, Re =i ,210,
La =95.6

284

Instabilities and Vortices Between Two Rotating Spheres

35 000 rods

Motion in a

cylindrical gap
if

30 000

Re. Arwyrs
y

25 000

2
Re kr

15 000

10 000

5 000

0
Eno &. 7. 8 IS 17 19 21

Fig.15 The critical Reynolds number for rotating
cylinders [11] and the corresponding
measurements for the spherical gap

As far as theory is concerned we have treated three problems.
Without going into details we give a short summary.

a. In case of fully laminar flow and a small relative gap width
the differential equations can be solved by using an approximation
method like that of v. Karman - Polhausen. The results are simple
expressions for the streamlines. Close to the walls these are loga-
rithmic spirals that fit very well to the experimental results (Fig. 4).

b. Mode 1 -- with larger relative gap -- can also be treated
easily. For the region close to the fixed and the moving sphere esti-
mations can be used for the boundary layer thicknesses. For a first
approximation results for the boundary layer of a rotating disc [ 12]
can be used. Between these two boundary layers we have an already

285

Zterep and Sawatzkt

mentioned full laminar flow. The Navier Stokes equations in spherical
coordinates give a very simple solution here with the velocity linear

in r. The analytical expressions for the three flow regions can be
combined and a short and simple calculation gives a torque coefficient
that fits surprisingly well to our experimental results.

c. The Stuart vortices -- already mentioned in connection
with mode II -- all rotate in the same direction. Experiments done
with a rotating free sphere [9] have confirmed this type of vortices.

A simple cinematic consideration shows how one comes from the
Taylor-Gortler vortices that have an alternating direction of rotation
to the Stuart vortices. For this it is only necessary to superpose a
suitable flow field to the Taylor-Gortler vortices with a flow direction
cross to the vortex axes. With other words: to get Stuart vortices

it is necessary that the Taylor vortices become embedded in a suitable
flow field.

We are dealing here with a real three-dimensional effect that
changes our vortex model. One realizes easily that for a rotating
free sphere or in the gap between two spheres of which one is rotating
the just mentioned situation exists.

REFERENCES

[1] Taylor, G. J., "Stability of a Viscous Liquid Contained Between
Two Rotating Cylinders," Phil, Trans. A 223, 289-293,
19236

[2] Zierep, J., "Thermokonvektiv Zellularstromungen bei inkonstanter
Erwarmung der Grundflache," ZAMM 41, 114-125, 1961.

[3] Koschmieder, E. L., "On Convection of a Nonuniformly Heated
Plane," Beitr. z~ Phys. d. Atmos., 39, 208-216, 1966.

[4] Muller, U., "Uber Zellularstromungen in Horizontalen Flussig-
keitsschichten Mit UngleichmaBig Erwamter Bodenflache, "
Beitr. z. Phys. d. Atmos., 39, 217-234, 1966.

[5] Sawatzki, O., and Zierep, J., "Das Stromfeld Zwischen Zwei
Konzentrischen Kugelflachen, von Denen die Innere Rotiert, "
Acta Mechanica, 9, 13-35, 1970.

[6] Wimmer, M., and Ritter, C. F., "Die Stroémung im Spalt
Zweier Konzentrischer Kugeln," Diplomarbeiten, Univ.
Karlsruhe, Lehrstuhl fur Stromungslehre 1968, 1969.

[7] Coles, D., "Transition in Circular Couette Flow," J. Fluid
Mech. 21, 385-425, 1965.

286

Instabilities and Vortices Between Two Rotating Spheres

[8] Schultz-Grunow, F., "Der Reibungswidstand Rotierender
Scheiben im Gehause," ZAMM 15, 191-204, 1935.

[9] Sawatzki, O., "Das Stromungsfeld um eine Rotierende Kugel,"
Acta Mechanica, 9, 159-214, 1970.

[10] Gregory, N., Stuart, J. T., and Walker, W. S., "On the
Stability of Three-dimensional Boundary Layers with
Application to the Flow Due to a Rotating Disc," Phil.
Trans. Roy. Soc. A248, 155-199, 1955.

[11] Kirchgafner, K., "Die Instabilitat der Stromung zwischen zwei
rotierenden Zylindern gegeniiber Taylor-Wirbeln fur
beliebige Spaltbreiten, ZAMP 12, 14-30, 1961.

[12] Cochran, W. G., "The Flow Due to a Rotating Disc," Proc.
Camb. Phil. Soc. A 140, 365-375, 1934.

DISCUSSION

L. van Wijngaarden
Twente Institute of Technology
Enschede, The Netherlands

The description of mode I reminds me of the result that
Batchelor [ 1956] derived for laminar flow with closed streamlines
of fluids with small viscosity: Thin boundary layers on solid
boundaries separated by a region of constant vorticity. This result
was derived for two-dimensional flow. In your case the flow is three-
dimensional, but it might be that the same conditions which are neces-
sary for Batchelor's result hold in this case of mode I.

REFERENCES

Batchelor, G. K., "On steady laminar flow with closed streamlines
at large Reynolds number," J. Fluid Mech., Vol. 1, p. 177,
1956,

287

Zterep and Sawatzkt

REPLY TO DISCUSSION

J. Zierep
Untversttdt Karlsruhe
Karlsruhe, West Germany

The general condition for the existence of closed streamlines,
given by Batchelor in the cited reference, can be applied to the
present case of a flow with rotational symmetry. We obtained infor-
mation about the velocity distribution that has been confirmed by our
analytical analysis, following a different path.

288

ON THE TRANSITION TO TURBULENT CONVECTION

Ruby Krishnamurti
Flortda State Universtty

Tallahassee, Florida

EINE. RODUCTION

The heat flow out of the sea floor has been observed in close
to 2000 measurements; the mean value for all the oceans is found to
be 1.4 X 10° cal/cm’sec. [Lee and Uyeda 1965] This is three
orders of magnitude smaller than the solar heating at the sea surface
and is surely negligible in any budget of the upper oceans. Yet,
because this heat flux is imposed from below, it may be of some
consequence in the dynamics of the abyssal circulation. If this heat
were to be transferred purely by gonduction through the sea water,

a temperature gradient o of 10.°C°/m would be required. The
largest depth across which such a gradient can exist without con-
vective overturning is determined by the critical value of the Rayleigh
number R, which is defined as follows:

where g is the acceleration of gravity, @ the thermal expansion
coefficient, kK the thermal diffusivity, p the kinematic viscosity,
and d is the depth of the layer in consideration. This largest depth
that can transfer the imposed heat flux by conduction is only around

3 cm. If there are regions or time periods of the abyssal oceans in
which horizontal advection of heat is not the dominant process, then
this vertical convection, with its attendant vertical mixing of nutrients,
can be an important process.

Some understanding of convective processes can be gained
from laboratory studies of a horizontal layer of fluid which is heated
from below and cooled from above. The following is a review of such
laboratory studies and also a report of some recent experiments in
rotating and non-rotating systems.

*
This is contribution No. 33 of the Geophysical Fluid Dynamics
Institute.

289

Krishnamurtt

II. TRANSITION TO TURBULENT CONVECTION IN A NON-
ROTATING LAYER OF FLUID

Unlike the fast transition to turbulence in plane parallel shear
flows, the horizontal convecting layer undergoes a number of discrete
transitions, remaining in each régime for a finite range of Rayleigh
number. The transition to turbulent convection appears to result in
the following manner: at sufficiently low values of the Rayleigh
number the fluid system is stable to all small disturbances. As the
value of R is increased the system becomes unstable to one kind of
disturbance. As R is increased still further the fluid becomes un-
stable to more kinds of disturbances. At sufficiently large Rayleigh
numbers the flow is unstable to so many kinds of disturbances, each
occurring with uncontrolled phase, that the flow may be called tur-
bulent. Before discussing the first three of these transitions, the
experimental apparatus will be described.

Apparatus
One of the possible designs of experimental apparatus is

shown in Fig. 1. The fluid layer occupies a region 51 by 49 cm,
with a depth that can be chosen (usually between 1/2 and 5 cm).

CONSTANT

O76, © 2a. O10 -Ov@

2 8 Se GE

y
j
© y SoS ee YY [recoroer |
g
VOLTAGE. ) elbow ZLEEY O
TRANSFORMER ai g

@LO.O. Oo Ou oe

= =
(ey) ALUMINUM 6061 FLUID
e METHYL STYROFOAM
GU METHACRYLATE INSULATION

Fig. 1. Apparatus

290

On the Transttton to Turbulent Convectton

The plexiglass tank containing the fluid also contains four blocks of
aluminum 6061 T 651. Two of the blocks are 4 in. thick, two are
iin, thick, each is Z0 in. by 20 in. wide. The electrical heater,
which is a fine mesh of resistance material embedded in silicon
rubber, is attached to the bottom of the lowest block, which is 4 in.
thick. The heat input is controlled by a variable transformer backed
by a constant voltage transformer of the line voltage. Above this
lowest aluminum block is a low-conductivity layer of methyl metha-
crylate. A layer of liquid sufficiently thin that it never convects

for the temperature gradients occurring in these experiments effects
constant thermal contact between the layers. Above this low con-
ductivity layer is a block of aluminum 1 in. thick; above this is the
convecting fluid, whose depth is defined by plexiglass spacers. The
arrangement of blocks above the convecting layer is symmetric to
that below except that the cooling is accomplished by cooling fluid
from a constant-temperature circulator flowing in channels in the
uppermost aluminum block. The channels for incoming and outgoing
flows are side by side in order to minimize horizontal temperature
gradients. The channels were cut in a complicated pattern and
spaced so that the separation of channels was not close to an integral
multiple of the expected convection cell size. The maximum flow
rate of the cooling fluid is 2+ 5 gal/min. This apparatus was used
in the studies which will be described with air, water, and silicone
oils. For convection in mercury, the aluminum blocks were replaced
by copper blocks of which two are 2 in. thick, two are 1 in. thick
and each is 20 in. by 20 in. wide.

The thermal conductivity of the aluminum is about three orders
of magnitude larger than that of the oils. The thermal conductivity
of copper is 50 times as large as that of mercury. This is, of course,
an attempt to approach the ideal condition of perfectly conducting
boundaries. With poorly conducting boundaries a horizontal tempera-
ture ripple corresponding to the cellular structure in the convecting
fluid penetrates into the boundaries and may control transitions to
different cellular structures. Also the metal acts as a diffuser of
any horizontal temperature variations arising from the discrete
nature of the cooling channels. The large mass of metal (approxi-
mately 400 1b of aluminum or 700 lb of copper) acts as a large heat
capacity so that temperatures in the blocks are very stable.

The heat transported by the convecting liquid is measured by
concentrating the temperature gradient across the poor conductor in
the manner devised by Malkus [1954]. In the steady state the heat H
transported by the fluid is the average of the heat conducted across
the two poor conductors:

ig ye Tos on Dee
H=k,—7q es :

where Kp and kp are the molecular conductivities of the low con-

ductivity layers, dp is the depth of the layer, and T,, T,, Ts and

29d

Krishnamurtt

T, are the temperatures of the four aluminum blocks. The sub-
scripts are ordered from bottom to top. The conductivities kp and
k' are measured in terms of that of the liquid when it is known that
the liquid layer is in a state of steady conduction. Then the following
relations hold:

Ty AVEL wi) ) Dpeeeignw pa Tail Ty
k, —22—3 = k, 2 =k) —3 4
d Bi) a, pi Bidgig 1°

where k, is the molecular conductivity of the fluid. Thus, once the
conductivity and depth of the poor conductors is determined, a
measurement of the temperatures in the four metal blocks allows
the determination of the Rayleigh number and the heat flux.

Fine aluminum flakes suspended in the liquid were used to
visualize the flow. The aluminum flakes become aligned in a shear
flow, and because they are flakes, reflect light more strongly in
certain directions , depending upon the direction of the shear and of
the illumination. In a uniform shear, the brightness is uniform;
where there is a differential shear, there will be corresponding
bright and dark regions. In the case of water, alumimm flakes
would not stay in suspension sufficiently long, so another tracer
called 'rheoscopic fluid AQW 010' was added to the water. This
tracer displays differential shears, just as do the aluminum flakes,
but remains in suspension about 10 times as long.

Since the fluid layer is bounded above and below by opaque
boundaries, the plan form of convection is obtained by viewing the
flow from the side as shown in Fig. 2. The tracers were illuminated
at mid-depth by narrow overlapping beams of collimated light from
two 2 W zirconium arc lamps. The two beams directed at each other
allow visualization of shear regions at both positive and negative
angles to the line of sight. This line of sight is perpendicular to the
beam. As the light beam is moved horizontally, illuminating differ-
ent regions of the fluid, a camera is moved horizontally on a threaded

Illumination Illumination
at x4 at Xp

Camera Camera
position A position B

Fig. 2. Geometry for photographing plan form of convection

292

On the Transttton to Turbulent Convectton

rod in order to keep the illuminated region in focus. Simultaneously,
the back of the camera rolls on an inclined plane since the camera

is free to rotate about an axis through its lens. Thus, different
regions of the fluid produce images on different parts of the film.

In this way, one obtains a picture of the flow pattern as if one were
viewing from above.

For each steadily maintained external condition, the steadi-
ness or non-steadiness of the resulting flow was to be determined.
This was found to be too difficult by simply observing moving tracers
through the fluid since there were gentle time dependencies with
time scales of the order of several minutes to several hours. In
order to have a record of the flow at an earlier time against which
to compare the flow at a later time, the following photographic
technique was devised. The apparatus used is shown schematically
in Fig. 3. Two narrow overlapping beams of light illuminate
aluminum flake tracers along a line inthe x-direction, say, through
the fluid. The beam remained fixed in space throughout the obser-
vation time. The camera was free to rotate about an axis through
its lens. With the camera aperature open, a synchronous motor
drew a wedge under the back of the camera at a rate determined by
the time scale of the time dependence of the flow. Thus, the photo-
graph displays an (x,t) representation of the flow, where t is the
time coordinate. At t = 0, the camera recorded alternating bright
and dark regions, corresponding to the cellular structure, as a
narrow strip of image across the film. When the flow was steady,
the cell boundaries remained fixed in time, thus producing straight
lines parallel to the t-axis on the photograph. With the beam near
the top (or bottom) of the convecting layer, the tracer particles
have an x-component of velocity which is given by the slope of the
trajectories inthe (x,t) representation.

1Convecting liquid
~N
Light beam

ane
|
p—F

—nn

Fig. 3. The apparatus for photographing the time evolution of flow

293

Krishnamurti

The studies that will be described here were performed as
externally steady, fixed heat flux experiments. Rayleigh number
and heat flux were measured for fluids having Prandtl number from
10° to 104. The Rayleigh number ranged from 10° to 108, Except
in the cases of air and mercury, the "pDlan form" of the convection
was obtained by viewing from the side. The time dependence was
determined by both the (x,t) photographs and thermocouples
internal to the fluid. In the cases of air and mercury time dependence
was determined only by the internal thermocouples.

The First Transition

In the order of increasing R, the first transition occurs at
the well-known critical Rayleigh number R,. This is a transition
from the conduction state to one of steady cellular convection. It
occurs independently of the Prandtl number Pr where Pr = v/Kk.
The nature of the flow and the change in slope of the heat flux curve
have been predicted and experimentally verified. For the vertically
symmetric problem the only stable finite amplitude solution of the
infinite number of possible steady solutions is the two-dimensional
roll [Schluter, Lortz and Busse 1965]. With a vertical asymmetry,
such as that produced by changing mean temperature or by variation
of material properties (v¥,xX,@) with temperature, the conduction
state is subcritically unstable to finite amplitude disturbance, and
the flow near the critical point is hexagonal [ Busse 1962; Segel and
Stuart 1962; Krishnamurti 1968a,b| - Inthis discussion we restrict
our attention to the case in which rolls are the realized flow just
above Re.

As the heat flux, and hence the Rayleigh number, are increased
above Re, steady two-dimensional rolls continue to be the observed
flow up to approximately izZ.R.; ter, 10< Pr < 10%. The size of the
rolls becomes larger in this range, as shown in Fig. 4, where the
wave-number 6 is plotted against Raleigh number. This increased
size of the cell might be rationalized by an argument such as the follow-
ing. By averaging over the entire fluid the non-dimensionalized tem-
perature equation in the Boussinesq approximation one finds

H = Rom + (w9)

where H is the dimensionless heat flux, o, is the vertical tem-
perature gradient averaged over the entire fluid, w is the vertical
velocity, 6 is the departure of the temperature from a horizontal
average, and brackets indicate averaging over the entire fluid.

Thus Ro, is the heat flux due to conduction, (w®8) is the convective
heat flux. As the externally imposed heat flux is increased such that
R exceeds Rg, the fluid transfers this larger flux through the cor-
relation (w®). Consider a fluid parcel near the lower boundary.

Its temperature 9 is limited by the thermal diffusivity of the fluid
material. As H is continually increased, the fluid is forced to

294

On the Transtitton to Turbulent Convectton

The Second Transition

The only theoretical study of stability of two-dimensional con-
vection in the Rayleigh number range of the second transition is that
of Busse [1968]. He shows that for infinite Prandtl number, two-
dimensional rolls having wave-number ff within a finite band (see
Fig. 4) are stable to a restricted class of infinitesimal disturbances
provided that R< 22,600. If R > 22,600 rolls are unstable for all
B. Busse shows further that the roll plan form is then unstable to
a disturbance of rectangular form with one side along the original
roll axis. It is not known from this theory whether the resulting
flow above 22,600 is steady. It is also not known how the selection
of B from this band of possible wave-numbers occurs.

Laboratory studies [ Krishnamurti 1970a] show that two-
dimensional rolls do indeed become unstable near this Rayleigh
number, which will be labelled R,. The "plan forms" (obtained
from the side) are shown in Fig. Ba where that on the left shows
rolls below Ry, that on the right shows the flow pattern above Ry.
The three-dimensional disturbance that forms on the rolls above
R,, is consistent with Busse's instability to a rectangular distur-
bance. Since the method of photography displays regions of strong
shear, the hypotenuse of the rectangle should appear bright. Thus,
the nature of the growing mode (which is found experimentally to
attain a steady state) is in agreement with Busse's result. It may
be noted that the rectangular disturbance of his theory is one with
symmetry in the vertical. The point of transition is also in good
agreement with that computed by Busse, for that wave-number 6
which occurs in the experiment. Figure 5b shows the same transi-
tion when a circular boundary of plexiglass has been inserted into
the rectangular region. Both Davis [1967] and Segel [1969] show
that spatially modulated rolls will line up with their axes parallel
to the short side of a rectangular container. In the almost square
container, there appeared to be little preference of orientation of
the rolls; rolls were seen along the line of sight as well as perpen-
dicular to the line of sight in two different repetitions of the same
experiment. The preference of rolls to line up with their axes
parallel to the short side may be re-expressed as a preference of
the rolls to meet the boundaries rather than lie along the boundaries.
This effect is displayed in Fig. 5b. Presumably circular rolls did
not develop because the plexiglass has thermal conductivity so close
to that of the fluid that there was negligible distortion of the con-
duction temperature field and no fringing of the isotherms since
there was fluid outside of the ring.

Associated with this change from steady two-dimensional to
steady three-dimensional flow, there is observed a discrete change
in slope of the heat flux curve (Fig. 5a). This corresponds to the
second change of slope observed by Malkus [ 1954]. Ry, showed no

295

Krtshnamurtt

move more rapidly to transport this increased heat flux. If the fluid
must move faster, then the cells must be larger in order to allow
the hot rising fluid to be in the vicinity of the cold upper boundary for
a sufficiently long time to lose its heat before sinking and repeating
the process. Although there are many ways in which the fluid could
have transferred the increased heat flux, moving more rapidly with
increased cell size is one ofthem. Of course, if the cells become
very large, the viscous dissipation of energy near the horizontal
boundaries would slow down the flow and defeat its own purpose.
This will be discussed later. It is seen in Fig. 4 that, when the cell
size is allowed to evolve freely (without being forced as in the ex-
periments of Chen and Whitehead [ 1968]), approximately one-half
of Busse's stability diagram is filled with observations, but the
domain B > f, is conspicuously bare.

2x 10°

1x10

Rt
5x10?

Unstable

3x10

2x10

d (cm) R increasing R&R decreasing
Pr iGai 1-2 x ®
Pr 57 2 Oo
Pr 10? 2 + ®
Pr 0:86 x 10° 3 ® @)
Pret x 10* 5 A A
Pr 0-85 x 104 2 *

Fig. 4. The observed cell size plotted on Busse's stability diagram
for two-dimensional rolls

296

On the Transtitton to Turbulent Convection

The Second Transition

The only theoretical study of stability of two-dimensional con-
vection in this Rayleigh number range is that of Busse [1968]. He
shows that for infinite Prandtl number, two-dimensional rolls having
wave-number f within a finite band (see Fig. 4) are stable toa
restricted class of infinitesimal disturbances provided that R < 22,600.
If R> 22,600 rolls are unstable for all B. Busse shows further that
the roll plan form is then unstable to a disturbance of rectangular
form with one side along the original roll axis. It is not known from
this theory whether the resulting flow above 22,600 is steady. It is
also not known how the selection of B from this band of possible
wave-numbers occurs.

Laboratory studies [Krishnamurti 1970a] show that two-
dimensional rolls do indeed become unstable near this Rayleigh
number, which will be labelled R,. The "plan forms" (obtained
from the side) are shown in Fig. 5a where that on the left shows rolls
below Ry, that on the right shows the flow pattern above Ry. The
three-dimensional disturbance that forms on the rolls above Ry,
is consistent with Busse's instability to a rectangular disturbance.
Since the method of photography displays regions of strong shear,
the hypotenuse of the rectangle should appear bright. Thus, the
nature of the growing mode (which is found experimentally to
attain a steady state) is in agreement with Busse's result. It may be
noted that the rectangular disturbance of his theory is one with sym-
metry in the vertical. The point of transition is also in good agree-
ment with that computed by Busse, for that wave-number £8 which
occurs in the experiment, although the selection mechanism of that
B is not understood. Figure 5b shows the same transition when a
circular boundary of plexiglass has been inserted within the rectangu-
lar region. Both Davis [1967] and Segel [1969] show that spatially
modulated rolls will line up with their axes parallel to the short side
of a rectangular container. In the almost square container, there
appeared to be little preference of orientation of the rolls; rolls
were seen along the line of sight as well as perpendicular to the line
of sight in two different repetitions of the same experiment. The
preference of rolls to line up with their axes parallel to the short
side may be re-expressed as a preference of the rolls to meet the
boundaries rather than lie along the boundaries. This effect is dis-
played in Fig. 5b. Presumably circular rolls did not develop
because the plexiglass has thermal conductivity so close to that of
the fluid that there was negligible distortion of the conduction tem-
perature field and no fringing of the isotherms since there was fluid
outside of the ring.

Associated with this change from steady two-dimensional to
steady three-dimensional flow, there is observed a discrete change
in slope of the heat flux curve (Fig. 5). This corresponds to the
second change of slope observed by Malkus [ 1954]. Ry, showed no

(Ae hts

Krtshnamurtt

HEAT FLUX vs RAYLEIGH NUMBER

PRANOTL NUMBER = 860

SHOWING THE SECOND TRANSITION. a

HEAT FLUX « 10

Fig. 5a.

RAYLEIGH NUMBER x10“

Heat flux plotted against Rayleigh number showing the
second transition. Photographs show the corresponding
change in plan form. The Prandtl number is 860.

298

On the Transttion to Turbulent Convectton

Fig. 5b. Photographs showing the plan form within circular side
walls. The transition is the same as in Fig. 5a. The
Prandtl number is 860.

definite Prandtl number dependence in the range 10 < Pr < 107.
There was a marked hysteresis both in the heat flux and plan form
as the Rayleigh number was increased then decreased past Ry .
This transition is shown by the curve labelled II in the régime dia-
gram (Fig. 10).

The Third Transition

The third transition in order of increasing R occurs ata
Rayleigh number which will be labelled R,,,;. It marks a change
from steady three-dimensional to time-dependent flow, and has
associated with it a discrete change in slope of the heat flux curve
(Fig. 6) [Krishnamurti, 1970b]. The change in slope was gmeasured
for each of the fluids shown in Fig. 10, with 10° “@<Pr<104% The
transition point is labelled as curve III. For Rayleigh numbers
above this curve the flow showed two modes of time dependence.

The one is a slow time dependence with time scale of the order of

the thermal diffusion time d?/x. An (x,t) photograph showing

this mode is seen in Fig. 8. The light beam was near the bottom

of the fluid. It is a slow tilting of the cell with height. Below Ryy
there was never a noticeable tilt observed. Above Ry; some cells
would be tilted for times of the order of d?*/k. Fig. 9 shows streak
photographs of tracer particles in a vertical slice through the con-
vecting fluid. Figure 9a shows steady flow in cells of rectangular
cross section at Rayleigh number Re, and Pr = 860. Figures 9b and
9c show tilted cells at Rayleigh number 74 R, and 89 Ry, respectively,
and Pr = 860. The tilted cells often occurred in pairs with the tilt
always such that two rising particles were close together near the
bottom boundary, flaring apart near the top. Two sinking particles
were close together near the top, flaring apart near the bottom.
Untilted cells, as in Fig. 9a are symmetrical about a horizontal

line at mid-depth in the fluid. When integrated over the cell, the

net vertical transport of the x-component of velocity, (uw) is zero.

297

Krtishnamurtt

FLUX X 10°

HEAT

0 2 4 6 Sy SO oe? od 18.20 °22° 24

RAYLEIGH NUMBER X 10

Fig. 6. Heat flux.vs. Rayleigh number showing the third and fourth
transitions. Prandtl number = 102.

300

On the Transitton to Turbulent Convectton

ft
- p

Fig. 7. (x,t) photographs of convective flow. The position x
through the tank is along the abscissa; the total width of the
photograph represents 48 cm through the fluid. The time t
is along the ordinate. The Prandtl number is 57.

(a) R = 28 R,; the total time is 17 minutes

(b)ueR =*200 R,; the total time is 17 minutes
(c) R = 335 R,; the total time is 15 minutes

301

15-0

TIME (HOURS)

Big. 8s

Krtshnamurtt

14:5

14-0

13-5

cas eee POSITION (CM) ©

(x,t) photograph at 45 R,, showing the slow time dependence
corresponding to the tilting of the cells. The light beam is
near the bottom of the fluid layer. The Prandtl number is

5G.

302

On the Transtitton to Turbulent Convectton

(b)

Fig. 9. Streak line photographs of tracers ina vertical slice through
the convecting fluid. The Prandtl number is 860. (a) a ver-
tical slice through steady two-dimensional rolls at R = 6 Reg.
(b) Showing a pair of cells with a tilt relative to the vertical.
R= 74 R,. (c) Showing a tilted cell at R = 89 Rg.

In case of the tilted cells, however, ( uw) over one cell] is non-zero
and is always in the sense of transporting positive x-component of
momentum to regions where the flow is in the positive x-direction,
transporting negative x-component to regions where the flow is in

the negative x-direction. Again one might rationalize the increased
slope of the heat flux curve by the following argument. As the exter-
nally imposed heat flux is increased more and more, the fluid must
move faster and faster, Then, to accomplish the heat exchange at

the boundaries, the cells must become wider and wider. The tilting
of the cells can help to maintain this flow against the increased viscous
dissipation along the boundaries. The tilting of cumulus convection
cells in the earth's atmosphere has often been related with a vertical
wind shear. The tilted cell is believed to be important in transporting
momentum in the vertical direction, thus maintaining the wind aloft.
It is interesting to note that in this laboratory situation convection
cells tilt even in the absence of a mean wind shear.

The second mode of time dependence is an oscillatory mode
with a much shorter time scale. Figure 7 shows (x,t) photographs
taken with the light beam near the bottom of the layer of fluid.
Figure 7b shows a bright region, which is a region of strong shear,
move from one cell boundary to another. This process is repeated
periodically in time. (x,t) photographs synchronized with a tem-

303

Krishnamurtt

perature record at a point within the fluid showed a temperature
anomaly each time a bright region moved pass the point. It is an
oscillation in the sense that the temperature and flow show a time
periodicity at a fixed point in the fluid. This oscillatory mode is
illustrated in the following movie which shows convection ina
Hele-Shaw cell having dimensions 24 in. wide, 2 in. tall, and 1/16
in. thick (that is 1/16 in. in the direction of the line of sight). It
shows hot spots and cold spots (bright regions) forming and being
advected by the mean circulation of the cell.

As the Rayleigh number is increased transition to turbulence
appears to result from the increased number and frequency of these
oscillations.

Ill. TRANSITION TO TURBULENT CONVECTION IN A ROTATING
FLUID LAYER

This topic will be discussed very briefly. A horizontal layer
of fluid heated below and cooled above is rotated about a direction
parallel to the force of gravity. The linear stability theory has been
treated by Chandrasekhar [1961]. The finite amplitude theory with
very clear physical explanations is given by Veronis [ 1959]. Notable
experiments have been performed by Fultz and Nakagawa [1955],
by Rossby [ 1966] , and others.

Recently, Kuppers and Lortz [1969] have shown that for
infinite Prandtl number there exists a critical Taylor number Te.
beyond which there can be no stable steady convection in the vicinity
of R,- The Taylor number T is defined as

_ 4 a*
ee

T

where Q is the rotation rate, dthe layer depth, and v is the kine-
matic viscosity. For T< T, they show that the only stable finite
amplitude solution is the two-dimensional roll solution. For T> T,
there must be a transition from the conduction state to a time de-
pendent flow as the Rayleigh number is increased beyond the critical.

The apparatus consisted of a fluid layer 1 or 2 cm in depth,
18 inches in diameter in the horizontal direction. The fluid was
bounded below by a 2 in. thick aluminum block containing an electri-
cal heater which is a fine mesh of resistance material. Above the
fluid layer was bounded by a glass plate over which the cooling fluid
circulated.

Photographs taken from above by a camera rotating with the

fluid are shown in Fig. 11. Figure 11a shows rolls, Fig. 11b shows
the cross instability forming on the rolls, of the same kind found in

304

On the Transitton to Turbulent Convectton

10° or q
TURBULENT FLOW :
o
TIME 2
DEPENDENT® 6 3-DIMENSIONAL
+105 = —|
g0 2 A pow
e a e es
{ =a Sins =e
3 7 STEADY, 3- °DIMENSIONAG FLOW °
© Sie :
4 ° ae Se sy x
mr y IER ° 8 Pr= ©
> ° ° °
Zi0*+ & :
3 ti ae
i wa Fe STEADY 2-DIMENSIONAL ° FLOW
| 5
10°
10? 10" 1 fe) 10? 10° 10*
PRANDTL NO.

Fig. 10. The régime diagram. The circles represent steady flows,
the circular dots represent time-dependent flows. The
stars represent transition points. The open squares are
Rossby's observations of time-dependent flow, the squares
with a dot in the centre are Willis and Deardorff's obser-
vations [1967b] for turbulent flow. The triangle is
Silveston's [1958] point of transition to time-dependent
flow.

the non-rotating case. Figure iic shows the break-down of rolls
and waves forming on them. The disturbance forms an angle of

58° + 2° to the original roll axis, exactly as predicted by Kuppers
and Lortz. Figure iic is atransient state; iid is the final steady
state. In this state the over-all wavy pattern was not observed to
change with time but the internal striations representing regions of
strong shear were segn to change with a time scale of the order of
one minute. (Here d‘/v = 40 sec, d*/k@1 hr). Figure 12 shows
the régime diagram for the rotating convection. The observed criti-
cal Taylor numbers compare only approximately with those computed
by Kuppers [1970] for finite Prandtl number and rigid boundaries.
The observed transitions occurred at T, =1.5X10° for Pr= 6.7,
R@ R, andat T,=7%10* for Pr=10%, R= R,., The predicted
values are T, =7X10° for Pr=1, Te=1.7X10° for Pr=5,
and T, = 2103 for Pr— oo.

305

Krtshnamurtt

Fig. 11. Rotating Benard Convection, showing cross and wave
instabilities on rolls

(ajlresetee Ry Ts 114 X40") Pe

(Bb) -Reei9ii2 Reset 1 & 102, Pe
instability

(c)i, Ruste UR. § T= 207) X h0> ay Px
developing of waves

(d)) RS 21 Ro) T= 2. 7x40, Pz

6.7 rolls

6.7.07 8S

10°; showing the

I

119% showing

developed waves

306

On the Transition to Turbulent Conveetton

x
STEADY FLOW

Po ss
Wo fe INSTABILITY
STEADY a
= x = 6.
FLOW 3 Pes 60
CROSS 8
INSTABILITY
f 5
13 af 6 WAVE
7} INSTABILITY
> UNSTEADY
12 a4
a fo) i Fs FLOW ee ee @ ® @
Wi 2 STEADY . 13 @ e e
to)

107 TAYLOR NUMBER

RAYLEIGH NUMBER
3

9

8 °

\ UNSTEADY

: STEADY TWO- FLOW
DIMENSIONAL

a7 WAVE INSTABILITY

FLOW

102
TAYLOR NUMBER

2 3 45678910

Fig. 12. The régime diagram for rotating Bénard convection

IV. SUMMARY AND CONCLUSIONS

Series of externally steady, fixed heat flux experiments were
performed to measure Rayleigh number, heat flux and changes in
flow of horizontal, non- rotating convection for 2.5X10°= Pre
0.85 X 10% and 103< Ra<10% The régime diagram summarizing

these experiments is shown in Fig. 10. Each of the curves I, II,

III and IV marks a transition with a change of slope in the heat flux
curve. The first is the transition from the conduction state to one of
steady two-dimensional convection in the form of rolls.

There is a second transition characterized by the following
properties:

(i) There is a discrete change of slope of the heat flux curve
at Rayleigh number R,, near 12 R,, showing no definite
Prandtl number dependence in the range 10 < Pr< 104,

307

Krishnamurtt

(ii) There is a change in the flow pattern from two-dimen-
sional rolls to a three-dimensional flow which is periodic
in space and steady intime. The change occurs ata
Rayleigh number coinciding with R,, to within the error
in determining Rj.

(iii) There is hysteresis in the heat flux as well as in the flow
patternas R is increased from below or decreased from
above, indicating that the transition is caused by a finite
amplitude instability.

The third transition is indicated by curve III in Fig. 10.
Above this curve, the flow is time dependent with a slow tilting of the
cell in the vertical and a faster oscillation which has the nature of
hot or cold spots advected with the mean flow. Transition to disorder
is seen to result from an increased number and frequency of such
oscillations.

Higher transitions observed by Malkus [1954] and confirmed
by Willis and Deardorff [1967a] have not been discussed.

The small amplitude nonlinear theories have been quite suc-
cessful in a small neighborhood of the critical point Rg. The obser-
vation that transition to turbulence occurs near Re for small Prandtl
number in non-rotating convection, and for T> T, for rotating
convection, indicates the possibility of gaining further understanding
of transition to turbulence through the nonlinear theories.

The research reported here was supported by the Office of
Naval Research Contract N-00014-68-A-0159 and by grant number
GK-18136 from the National Science Foundation,

REFERENCES

Busse, F. H., Dissertation, University of Munich. (Translation
from the German by S. H. Davis, the Rand Corporation,
Santa Monica, California, 1966), 1962.

Busse, F. H.,"On Stability of Two-Dimensional Convection in Layer
Heated from Below,"J. Math. and Physics, 46, 140, 1968.

Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability,
Oxford, 1961.

Chen, M. M. and Whitehead, J. A. ,"Evolution of Two-Dimensional
Periodic Rayleigh Convection Cells of Arbitary Wave-
Numbers," J. Fluid Mech., 31, 1, 1968.

Davis, S. H., "Convection in a Box: Linear Theory," J. Fluid
Mech. s 30, 465, 1967.

308

On the Transitton to Turbulent Convection

Fultz, D. and Nakagawa, Y., "Experiments on Oven Stable Thermal
Convection in Mercury," Proc. Roy. Soc. A. 231, 198, 1955,

Krishnamurti, R., "Finite Amplitude Convection with Changing Mean
Temperature, Part 1, Theory," J. Fluid Mech., 33, 445,
1968a. -_

Krishnamurti, R., "Finite Amplitude Convection with Changing Mean
Temperature, Part 2, An Experimental Test of the Theory,"
J. Fluid Mech., 33, 457, 1968b.

Krishnamurti, R., On the Transition to Turbulent Convection,
Part I, The Transition From Two- to Three-Dimensional
Flow," J. Fluid Mech., 42, 295, 1970a.

Krishnamurti, R., "On the Transition to Turbulent Convection,
Part 2, The Transition to Time Dependent Flow," J. Fluid
Mech., 42, 309, 1970b.

Kuppers, G. and Lortz, D., "Transition From Laminar Convection to
Thermal Turbulence in a Rotating Fluid Layer," J. Fluid
Mech., 35, 609, 1969.

Kuppers, G., private communication, 1970.

Lee, W. H. K. and Uyeda, S., Terrestrial heat flow, Washington,
D.C., pp. 87-190. (American Geophysical Union, Geophysi-
cal Monograph series no. 8), 1965.

Malkus, W. V. R., "Discrete Transitions in Turbulent Convection,"
Proc. Roy. Soc. A225, 185, 1954.

Rossby, H. T., Dissertation, M.I.T., 1966.

Schluter, A., Lortz, A. and Busse, F., "On the Stability of Steady
Finite Amplitude Convection," J. Fluid Mech., 23, 129,
1965. ae

Segel, L. A., "Distant Side- Walls Cause Slow Amplitude Modulation
of Cellular Convection," J. Fluid Mech., 38, 203, 1969.

Segel, L. A. and Stuart, J. T., "On the Question of the Preferred
Mode in Cellular Thermal Convection," J. Fluid Mech.,
13, 289, 1962.

Silveston, P. Leis Forch. Ing. Wes. 24, 29-32, 59-69, 1958.

Veronis, G., "Celular Convection with Finite Amplitude in a Rotating
Fluid," J. Fluid Mech., 5, 401, 1959.

309

Krishnamurtt

Willis, G. E. and Deardorff, J. W., "Development of Short-Period
Temperature Fluctuations in Thermal Convection," Phys.
Fluids, 10, 931-937, 1967a.

Willis, G. E, and Deardorff, J. W., "Confirmation and Renumber-
ing of the Discrete Heat Flux Transitions of Malkus," Phys.
Fluids, 10, 1861, 1967b.

310

TURBULENT DIFFUSION OF TEMPERATURE
AND SALINITY: — AN EXPERIMENTAL STUDY

Allen H. Schooley
U.S. Naval Research Laboratory
Washington, D.C.

ABSTRACT

Stratified temperature and salinity conditions in
water have been established in a small laboratory
tank. A method for making measurements and cal-
culating the eddy diffusivities of temperature and
salinity for different controlled levels of turbulence
are described. The ratio of temperature and salinity
molecular diffusivities is on the order of 100. The
ratio of temperature and salinity eddy diffusivities,
for the most turbulent conditions studied, is 14.

The dissipation of turbulent power density (P) due
to viscous friction was found to be on the order of
10’ larger than the power density (P') consumed
in changing the thickness of the pycnocline. The
experiments hint that P/P' may be relatively
constant over a range of turbulence. If this is
assumed to be true, there exists the possibility of
estimating temperature (D) and salinity (D') eddy
diffusivities by knowing the change of density, (Ap),
and (Ap),, with time (At) for a given depth differ-
ence (Ah). Plots of D and D!' in cm/sec vs.
(P'/n)'4 =110(Ap/At)'/2(Ah) in sec”', are shown
where (n) is the dynamic viscosity of water.

344

Schooley

I. INTRODUCTION

The oceans are dominated by several turbulent processes.
For each turbulent situation there are "eddy" diffusivities of tem-
perature and salinity that are much larger than the molecular
diffusivities. In spite of the difficulties in measuring eddy diffusi-
vities at sea, there is considerable, though incomplete, literature
on the subject [ Neumann and Pierson, 1966]. Since turbulent ocean
processes are inherently uncontrollable, several exploratory labora-
tory experiments were conducted in late 1964 and early 1965. This
paper is the first publication of the results of these preliminary
experiments.

Il APPARATUS.

Figure 1 shows the test cell where the experiments were
conducted. It has transparent plastic walls and is 30 cm long,
9 cm high, and 2.5 cmthick. The bottom 4 cm was filled with either
room temperature distilled water or a 0.25% solution of sodium
chloride, depending whether thermal or salinity diffusion was to be
studied. The molecular thermal diffusivity of pure water at at-
mospheric pressure and 20°C is 0.00143 cm*#/sec (4% less than sea
water). The molecular diffusivity of an aqueous NaCl solution is
0.0000141 cm*/sec (9% less than sea water) according to Hill [ 1962].
Convenient distilled water and NaCl solutions were used instead of
sea water because of these relatively small differences.

Fig. 1. Experimental cell. Temperature difference sensor near
center. Salinity difference sensors at right. L shaped wire
at left produces turbulence on demand.

Turbulent Diffuston of Temperature and Salinity

When a thermal experiment was to be conducted, the top 4 cm
of the cell was filled with water having a temperature several degrees
above room temperature. A thin piece of balsa wood was floated on
the top of the bottom layer and warm water introduced through a small
nozzle directed perpendicularly to the top of the balsa wood. This
procedure deflects the downward momentum of the warm water flow
to the horizontal and filling was accomplished with a minimum of
vertical mixing with the cooler more dense water below. Whena
salinity diffusion experiment was to be conducted, a top layer of pure
water was introduced the same way. In this case the bottom water
contained the salt solution.

A repeatable amount of turbulence was introduced by mechani-
cally moving a stiff insulated wire back and forth at the interface
between the two layers at a controlled rate and for a controlled
length of time. The wire is shown in Fig. 1 at the 4 cm level where
the interface was located when the cell was filled. (The cell is
empty in Fig. 1.) The top of the "L" shaped wire is coupled to a
mechanical system outside the picture. It is the bottom part of the
"L" that was rotated back and forth laterally at the interface pro-
ducing turbulence when desired. Table I gives the specifications
for generating the amounts of turbulence that were used.

Table I. Specifications of the Turbulence Generator

Turbulence No. Mixer Dimensions me ee oe nes
0 - 0
4 1.8 mm diam, 7 cm long 10.8° 2.5
2 : : 5.4

When a thermal experiment was being conducted, the center
sensor was used. It consists of a two element copper-constantan
thermopile with junctions 2 cm apart. The upper junction was
placed 1 cm above and the other i cm below the interface of the
upper warm and the lower cooler water. The output of this sensor
was 30 microvolts for each degree difference in temperature. It
was connected to a small commercial micro-voltmeter and recorder
which gave a time record of the temperature difference, 1 cm above
and 1 cm below the interface. The record of vertical temperature
difference decay, with time, gives data related to the effect of mole-
cular diffusion when no turbulence js introduced. The vertical tem-
perature difference decay, with controlled amounts of turbulence,
was recorded by using a succession of carefully timed turbulent
pulses interspaced with short intervals of quiescence.

When a salinity diffusion experiment was being conducted the
two sensors on the right in Fig. 1 were used. They are identical

Schooley

probes consisting of two closely spaced platinum wires set in epoxy.
The conductivity of the solution at the points where each of the probes
were located was read from a meter scale. The upper probe was
placed one half cm above the interface and the other one half cm
below. Standard salt solutions were used to calibrate the conductivity
of the probes in order to measure salinity. This calibration was non-
linear and temperature corrections were necessary.

Figure 2 illustrates the technique that was devised to facilitate
measuring eddy diffusivity by the use of a series of turbulent pulses
of the same amplitude and time. The ordinate of this chart is a
record of the temperature difference (0 to 1.67°C/cm) measured by
the thermopile shown in Fig. 1 vs. time, which progresses from
left to right. The record starts at the upper left corner where the
temperature difference starts to decrease due to molecular diffusion.
At horizontal chart position #5 the mixer shown in Fig. 1 was acti-
vated for 15 seconds and then stopped for about 8 minutes. This
8 minute pause allows the pulse generated internal waves to damp out
so that the temperature difference due to turbulent eddy diffusion
can be measured and separated from the relatively slow molecular
diffusion. Again at chart positions #6.4 and #7.8 similar 15 second
pulsed of turbulence were introduced. The total time of turbulence
was thus only 45 seconds, and three successive temperature differ-
ences due to eddy diffusion were recorded at three 15 second intervals
of turbulence. The effect of eddy diffusion of NaCl was measured
by the same technique using the conductivity sensors.

5 a a a

$f Fy 2&
SS SSS SS = S22 =

Ey SSS
oe eee
——- = =>"

=

= ee eo ee
ee ————— SSS
————

( 4¢ Ses Duke GLI une coca ain NOs ye GUN OLY c: Gioia eM Goo Glad - he Aare

Fig. 2. Temperature difference AT vertically vs. time. The
decay of AT by molecular diffusion is interrupted three
times by turbulent pulses lasting 15 sec each.

Turbulent Diffusion of Temperature and Salinity

III. THERMAL DIFFUSION

The model assumed is shown schematically in Fig. 3. At
zero time ty and depth Zo, a semi-infinite region of water at the
temperature T>, is assumed to be brought together with a semi-
infinite region of water at temperature T,. Further, it is assumed
that z, and T, are essentially constant for t,, to, «+. . Eventually
as t becomes large, the semi-infinite model breaks down in prac-
tice because the effective value of T, decreases.

Fig. 3. Schematic presentation of temperature diffusion as a
function of water depth z andtime t. Semi-infinite depth
is presented vertically with z, a reference. Attime to
a sharp temperature discontinuity is assumed, and T,
defined as (T, - T,)/2. At t, diffusion has started and
To remains constant. For long times t,, t, the semi-
infinite model breaks down because the effective value of
T, does not remain constant in practice.

The heat flow for this model is governed by the one-dimensional
diffusion equation

2
OT ot OT
bvains> Sse (1)

where D is the diffusivity, usually expressed in cm?/sec.

Due to vertical symmetry the problem can be formulated
using the T, part of Fig. 3, where T, =(T, + T,)/2 is the
reference temperature. Taking T, = 0, z,=0, t,=0 the initial
and the boundary conditions for z>0O and t=0 is T(z;0) = Ty =
(T, - T|)/2- For 2=0 and t>0; T(0,t) = T..

Schooley

The analytical solution to this well-defined problem is
T(z,T) = Tel 1 - erf(x/2/Dt)] + T, (2)

where

x/2,/dt
erf(x/2VDt) = 2/Vr iy at ae

OQ

From (1) the gradient of T atthe boundary z, is

oT| _ To
oz Zo wDt
or
oT TT, 4
a 0
E |-3(@) (3)
Zo
and

2
D= (re/m| 4 /(32) (4)
Zz

2
Since I5/2 is constant for any one experiment, a plot of
1/t vs. (AT/Az)* for the data points should give a straight line
through the origin with slope Dr/T>. Figure 4 is such a plot for
the experiment of Fig. 2. A mean square fit gives a slope of 0.66
with a correlation coefficient of 0.99, Since Tp = 2.45°C, in this
case the eddy diffusivity is D = 2.45°(0.66)/m = 1.26 cm®*/sec.

In practice all plots of the experimental data do not yield
perfectly straight lines, particularly for larger values of t (smaller
values of 1/t) than are shown in Fig. 4. Calibration and experimental
errors are always present. In addition, as is illustrated in Fig. 3,
the effective value of T, is not constant for extended lengths of time
(t = ty > tq) because the experiments were necessarily conducted
in a finite size container. However, Fig. 4 does represent con-
sistency of the data with the simple analytical theory under the
assumptions that have been made.

316

Turbulent Diffuston of Temperature and Salinity

-20

| | IA

Natvaz)? 26, c/em)*

(sec")

1/t

Fig. 4. Sample of experimental data showing (AT /dz)? is a linear
function of 1/t. This is in accord with theory that diffusi-

vity D = (T2/n)[(1/t)/(AT/Az)*]

IV. SALINITY DIFFUSION

The substitution of S for T in (4) is all that is necessary.
Thus

2
= (ss/n) [4/38 (5)
O°

where D' is the diffusivity of NaCl dissolved in water in cm®/sec,
S, the mass concentration of salt in gm/cm’, t, the diffusion time in
seconds, z, depth in cm, and Sb, the initial salinity discontinuity
in gm/cm”*,

317

Schooley

0/0!
100 63 40 25 15 10

THERMAL DIFFUSIVITY IN WATER, D (cm/sec)

01 Pg
01
00001 0001 001 O1 A 1

DIFFUSIVITY OF NaC! IN WATER, D! (cm?/sec)

Fig. 5. Thermal diffusivity D vs. diffusivity of NaCl in water D'
for zero turbulence at lower left, and increasing turbulence
toward the upper right.

V. DISCUSSION OF RESULTS

Figure 5 shows thermal diffusivity D vertically, and salt
diffusivity D' horizontally. The point marked (X) at the lower
left represents the handbook values for the two molecular diffusivities.
The nearby circular point was determined experimentally when the
mixer of Fig. 1 was not used. The point nearest the center of Fig. 5
is an average of four experiments using turbulence #1 as listed in
Table I. The maximum deviations from the mean are shown. The
upper right point is for turbulent condition #2. The mean value was
derived from seven experiments. Again maximum deviations from
the mean are shown.

In Fig. 5 a straight line has been drawn connecting the
molecular diffusivity point with the point of maximum eddy diffusivity.
The intermediate point is somewhat below this line but there is
clearly not enough experimental data to determine the shape of the
curve. At the top of Fig. 5 a scale shows how the ratio of D/D'
decrease with increasing turbulence.

Turbulence in a stratified fluid manifests itself in two ways.

318

Turbulent Diffuston of Temperature and Salinity

There is heat energy liberated due to viscous friction. Also, a part
of the turbulent energy is dissipated in changing the potential gravi-
tational energy of the pycnocline by changing its thickness.

The power density associated with a change in potential energy
can be shown for water to be approximately

2
P' = aio peel ergs/cm?® sec (6)

where g is acceleration of gravity in cm/sec”, (Ap) is either (Ap),
or (Ap), symbolizing the change in density due to temperature or
salinity differences across the pycnocline in gm/em®, (Ah) the change in
thickness of the pycnocline in cm, and (At) =(t,- t,) in sec. For tur-
bulence #1, P/=0.0074 ergs/cm?® sec. For turbulence #2, P} = 0.055.

The power density due to viscous friction P was estimated by
measuring the temperature rise in the water due to turbulence #1
and #2 being maintained for measured lengths of time. For turbulence
#1 this was about P, = 0,014(107) ergs/cm* sec. For turbulence #2,
P, = 0.082(107).

The ratio of P,/Pj = 1.9(10’), and P,/P,=1.5(10’). Thus,
it appears that the power density due to viscous friction is on the
order of 10’ greater than the power density associated with a change
in the pycnocline thickness.

However, the ratios for the two conditions of turbulence are
different only by about 25%. This is interesting, for if it should
turn out that P/P' is relatively constant over a practical range of
turbulence, temperature and salinity diffusivities could be estimated
directly from the time rate of change of pycnocline thickness (after
internal waves are filtered out). Possible application to the ocean
is intriguing.

For physical and dimensional reasons, let us divide P' by
the dynamic viscosity of water n = 0.01 gm/cm sec, and take the
square root. Equation (6) then becomes

1/2

(P'/n)"”? = 110(Ap/At)’*(Ah) 1 /sec (7)

This equation contains variables that are relatively easy to measure
and has the dimension of vorticity. It is plotted in Fig. 6 for the
average values of the variables used in the small scale laboratory
experiments.

319

Schooley

(B)be 190 (44)F cam (even

Fig. 6. Tentative extrapolation of experimentally determined diffusion
coefficients D and D' vs. variables that are relatively easy
to measure at sea.

VI. ACKNOWLEDGMENT

I am grateful to Prof. Hakon Mosby, Geofysisk Institutt,
Bergen, Norway, for his interest and early participation in this
exploratory project. I am indebted to Albert Brodzinsky, NRL, for
applying his skill in mathematics and physics to the problems of
data analysis.

REFERENCES

Neumann, Gerhard, and W. J. Pierson, Jr., Principles of physical
oceanography, Prentice-Hall, Englewood Clits, N.J->»

pp. 392-421, 1966.

Hill, M. N. (Editor), The sea, Vol. 1, Wiley and Sons, N.Y.,
Dp. 27; 4962.

320

SELF-CONVECTING FLOWS

Marshall P. Tulin
Hydronauttes, Incorporated
Laurel, Maryland
and
Josef Shwartz
Hydronauttes-Israel, Ltd. and
Israel Instttute of Technology

ABSTRACT

A theory for the motion of two-dimensional turbulent
vortex pairs in homogeneous media has been developed
based on separate velocity scaling of the internal and
external flow fields involved in the motion and taking
into account variations in volume, circulation, momen-
tum, and energy. Based on the results obtained from
this theory (I) a simplified theory (II) is derived to deal
with the rising motion of turbulent vortex pairs in
stratified media. The theoretical results are compared
with systematic experimental observations.

In theory (I) the ratio of internal to external velocity
scales, \, is introduced as an important variable and
the theory is specifically derived for the two limiting
cases of weak (W ~ 1) and strong ( >> 1) circulation.
The weak circulation theory leads to results similar to
those obtained in the past using theory based on com-
plete similarity and momentum conservation; i.e.,
z~t!/3, The strong circulation theory leads to results
which depend very much on the way in which vorticity
from the shear layer is ingested into the vortex pair.
When ingested so as to cause annihilation (cancellation)
of the ingested vorticity, the asymptotic trajectory is
z~t'/2, Under the same conditions the velocity ratio,
Ww, increases toward an asymptotic value, and the
virtual momentum coefficient for the motion tends
to zero. As a result, the asymptotic motion (assuming
vorticity annihilation) corresponds to a motion with
complete similarity and with energy conservation.

3214

Tultin and Shwartz

A comparison of experimental observations of rise
versus time and radius versus height with theory (I)
lend strong support to the strong circulation theory
and suggest that ingested vorticity may be largely
annihilated.

Based on these finding for homogeneous flows, a sim-
plified theory (II) for stratified media was developed
upon the assumptions: (i) the motion is determined
by conservation of volume, mass, and energy (neg-
lecting vorticity and momentum); (ii) complete simi-
larity (dR/dz = 6, a constant). Good agreement was
found between the predictions of this theory and the
results of systematic experiments, and particularly
for the maximum rise of height.

NOTATION
a Density gradient in surrounding fluid, a = (1/pe)(dpe/dz)
A Initial buoyancy parameter (theory), Eq. (44)
A, Vorticity mixing coefficient, Eq. (9)
b Half distance between cores of vortex-pair
B Stratification parameter (theory), Eq. (45)
c Constant
Cy Energy dissipation coefficient
D Energy dissipation parameter, Eq. (46)
E Total energy of convected mass
G Experimental buoyancy parameter, Eq. (44)
j Geometrical parameter, j= 0 for planar geometry,
j = 1 for axial symmetry
k Virtual potential energy coefficient, Eq. (38)
Kin Virtual mass coefficient
K Virtual kinetic energy coefficient, Eq. (37)
M, Vertical component of total momentum
n Parameter defined by Eq. (47)
r Radial distance from center of rising mass, r* = Eg? + ne + co
R Mean radius of rising mass
R Non-dimensional mean radius of rising mass, R= R/R,
S) Experimental density stratification parameter, Eq. (53)

322

Self-Convecting Flows

t Time
me Non-dimensional time, t = Wot/z,
t ax Time at which the maximum height of rise is reached

u,v,w Velocity components, see Fig. 14

W Vertical velocity of rising mass

WwW Non-dimensional vertical velocity of rising mass, W = W/W,
we Vertical velocity of ideal vortex-pair

Z Height of rising mass center above its virtual origin

Zz Non-dimensional height of rising mass center, Z = z/Zo

Zmax Maximum height reached by rising mass

B Modified entrainment coefficient

Y Local vorticity, Eq. (7)

I Total circulation about a single vortex

€,n,»6 Coordinate system, see Fig. 14

p Local density inside convected mass

Pe Density of surrounding fluid

p, Average density of convected mass, Eq. (43)

Ap Density difference, Ap = pi - Pe

Wy Velocity ratio, W,/W,

Le Non-dimensional vertical momentum, (M,/p)/W,,R°
Subscripts

( ), Initial conditions
( ); 7odaternal
{ }, External

I. INTRODUCTION

Ideal Vortex-Pairs. Flow visualization studies carried out
by Scorer [1957] , Woodward [1959] and Richards [1965], indicate
that the shear layer which is formed between a moving isolated mass
of fluid and the stationary surrounding medium tends to roll up and
create a flow field which resembles (in two dimensions) the one
associated with two line vortices of equal strength but opposite sign,
separated by a distance 2b, so-called "vortex-pairs." The possi-
bility of vortex-pair motions in an inviscid fluid was considered and
analyzed over 100 years ago by Sir W. Thomson [1867]. His analysis

Tultn and Shwartz

applies only to an idealized vortex-pair in which each vortex has a
highly concentrated core which is set into motion only by the influence
of the other vortex. Such an ideal vortex-pair moves through the
surrounding fluid in a direction perpendicular to the plane joining

the vortex cores and with a velocity W determined only by the pair
separation, 2b, and the circulation about a single vortex, [,
according to the relation

ee

A unique feature of this idealized vortex-pair motion is the
existence of a closed streamline and a finite captured mass, as
indicated by the oval in Fig. 2a. Thomson [1867] calculated the
semi-axes of the oval-shaped captured mass to be 2.09b and 1.73b
so that the cross-sectional area is approximately 3.62 7b“ and the
ratio of width to thickness is 1.21.

Under certain circumstances it is entirely possible that
carefully balanced vortex-pairs, approximating Thomson's ideali-
zation, can be formed. The motion around the vortex centers must
be affected by viscosity in a real fluid, but as long as the viscous
cores do not extend close to the bounding closed streamline, the
flow within and without this streamline may be so closely matched
that no large shearing motions or accompanying drag are associ-
ated with the motion of the captured mass. In fact, nearly ideal
vortex-pairs are sometimes found in the wakes of lifting surfaces,
Fig. ta, and are known as "contrails," see Scorer [ 1958] and
Spreiter and Sachs [1951]. Of course, the concentrated vorticity
in the vortex cores tends to diffuse, and does so rapidly when the
flow in the core is turbulent.

Turbulent Vortex-Pairs. The probable short lifetime of
ideal vortex-pairs under turbulent conditions gives special impor-
tance to vortex-pairs whose behavior is governed by turbulent
entrainment; indeed, it is these kinds of motions which are most
commonly observed in nature, as in the case of a mass of fluid
forced out rapidly through an aperture, Fig. ib, or in the convection
of isolated masses in nature, Fig. ic, or in the bent-over and rising
chimney plume.

Turbulent vortex-pairs are characterized by the fact that the
interior motion does not match the outer flow at the boundary of the
captured mass, so that a region of high shear exists there, accom-
panied by the production of vorticity and by turbulent entrainment.
In other words, these vgrtex-pairs move with a velocity W not
equal to the velocity W derived from Thomson's model, Eq. (1).

We may, in principle, generalize Thomson's model to con-
sider those cases where the velocity of translation, W, has a more

324

Self-Convecting Flows

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325

Tultn and Shwartz

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3:26

Self-Convecting Flows

general relation to the velocity w* which characterizes the internal,
rotational motions of the vortex-pair. Two distinct cases suggest
themselves, in theory. In one case, W> W , and the convected
mass loses volume to the surrounding fluid and continually shrinks

in size; we denote this vortex-pair as "underdeveloped," see Fig.
2b. In the other case, W < W", and the "overdeveloped" vortex-
pair gains mass through the entrainment of exterior fluid, Fig. 2c.
Of these it is the latter motion which is most commonly observed in
nature and forms the main subject of this work.

Within an overdeveloped motion, the velocity at the boundary
inside the vortex-pair, as seen by an observer moving with it, will
be larger than the velocity of the surrounding fluid just outside the
boundary of the pair. Accompanying the velocity gradients thus
created across this boundary, shear stresses are exerted by the
vortex-pair on the surrounding fluid, resulting in the entrainment of
outer fluid and a general increase in the volume of the convected
mass, see Fig. 3. Within the high shear zones at the boundary on
either side vorticity of sign opposite to that within the respective

SHEAR AND ENTRAINMENT

me Tian

INGESTED VORTICITY
MIXES HERE

Fig. 3. Entraining vortex pair (W, > W,)

327

Tulitn and Shwartz

interior is created, and is pulled down around the bottom of the
rising mass toward the plane of symmetry. To the extent that the
ingested vorticity remains on its own side of this plane, the vorticity
within the interior will be steadily reduced; of course, ingested
vorticity of opposite sign does have a chance to mix and thus to
annul itself, depending on the efficiency of mixing. Should effective
annihilation of ingested vorticity occur, then the initial total vor-
ticity within one side of the pair would be conserved in time.

As for the kinetic energy implicit in the motion of the vortex
pair, it must be continuously reduced with time due to turbulent
dissipation.

Self-Similarity in Vortex-Pair Motions. It is a striking
characteristic of free turbulent flows in homogeneous media at
sufficiently high Reynolds numbers that, under similar circumstances,
the flows at different points in space or time can usually be reduced
from one to another upon normalization by an appropriate length
and velocity scale (self-similarity). This is true, for example, of
the flow at different downstream sections of turbulent jets and wakes.
It is therefore natural to expect that a turbulent vortex-pair exhibits
complete self-similarity during its life time, and this assumption
has been made in all theoretical treatments of the subject, starting
with Morton, Taylor, and Turner [1956]. Two important consequences
of this complete similarity are: (1) cons eryation of the ratio of
internal and external velocity scales, w/w , during the motion;

(2) linearity of the length scale of the convected mass with the dis-
tance traveled from a virtual origin.

This latter result, predicting that the traces of the side
boundaries of the convected mass form a wedge, is independent of
the dynamics of the motion and serves to provide a check on self-
similarity. In fact, a number of previous experiments on self-
convecting masses claim to confirm this behavior to a reasonable
approximation, see, e.g., Scorer [1958] , Woodward [1959] and
Richards [1965].

It is, obvious to ask whether a "natural" value of the velocity
ratio W/W", or the same thing, of the constant B = dR/dz is ob-
served, independent of the original circumstances giving rise to the
convected mass. The answer seems to be no. In the present experi-
ments, two distinctly different ranges of value of dR/dz differing
by a factor 2, have been repeatedly measured; these correspond to
two different stroke lengths in the apparatus used to originate the
vortex motions, Furthermore, although the present data may be
claimed to correspond "in a reasonable approximation" to a constant
value of dR/dz, yet quite consistent deviations from linearity exist
between the traces of pair radius and distance traveled, see Fig. 4.
These deviations are such that dR/dz seems actually to increase
throughout the observed motions.

328

RISE HEIGHT, (z - z,)/R,

Fig. 4.

Self-Convecting Flows

K aK
Eq. [240] Ge = 1/2, = 0.38

RADIUS, R/R,

The variation of vortex-pair radius with height in a homo-
geneous medium, experiment

329

Tultn and Shwartz

In view of these facts, and for other reasons, it seems
desirable to attempt a more general theory of the motion of turbulent
vortex pairs, based on the assumption of separate velocity scaling
of the internal and external motions; i.e, allowing W/W” to vary
continuously. Afterwards, a simplified theory pertaining to motions
in stratified media will be developed and the results compared to
experimental observations,

Il THEORY (HOMOGENEOUS FLOWS)

Separate Similarity of Internal and External Flows. We
visualize the vortex-pair motion to be divided into internal and ex-
ternal flow fields, separated by a thin region of high shear, which

also forms the boundary of the captured mass, see Fig. 3. We
assume that each flow field is itself self-similar.

Internal.
— (x
Wy, (zy): = W, (t) + w, (5:4) (2)
External.

Welxsyst) = Welt) - we(% 5%) (3)

and similarly for the other velocity components.

We let Y= W,/W,, where yw is, in general, not constant in
time as it is in the case of complete similarity.

We choose Wj, as the circumferental velocity averaged over
the inner boundary of one-half of the vortex pair and W, the same
except averaged over the outer boundary. (The inner and outer
boundaries are separated by a thin shear layer.)

Volume Changes. The volume of fluid comprising the vortex
pair increases continuously with time due to entrainment into it.
Because of the similarity assumed, the rate of entrainment of
volume must (in two dimensions) be proportional to a characteristic

velocity and a characteristic length. We take for the former, the
velocity difference W, - W,

a(cR°)

dt

= 2nR(W; - We) > @(¥) (4)

or,

330

Self-Convecting Flows

S = Te (y - 1) + aly) = Qh - tay) (5)

Note that the dependence of the proportionality constant @' on the
velocity ratio ~ has been left unspecified.

Circulation Changes. The circulation I about one half of
the vortex pair is, on account of the inner similarity, proportional
to the length scale and inner velocity scale. On account of the way
in which W, was defined,

T <RW, (6)

The fluid entrained into the vortex pair from the surrounding
high shear layer carries vorticity of opposite sign to that already
within the interior, see Fig. 3 and thus reduces it to the extent it is
not annihilated through mixing with vorticity being entrained in the
opposite lobe. The strength of entrained vorticity must, on account
of similarity, be proportional to the ratio of a pertinent velocity and
length scale. In particular:

y(entrained) ~ He (7)
The flux of entrained vorticity takes the form:
Vorticity Flux “ Volume Flux + y(entrained)
=[2mR(w, - Ww) > ewi[ee“t] a)

Finally, the change in circulation may be related to the flux of
vorticity:

aE <a, + 2maty) + (Ww, =)" 9)
or,
AWLP) 2 aja! + (W, = Wy) + Y= 2 (10)

The value of the parameter A, will depend in part upon the
extent to which entrained vorticity from each side mixes together

335%

Tultn and Shwartz

causing annihilation. In the case of complete annihilation, A,=0.

Momentum Conservation. The momentum, M,, of the vortex

pair is conserved in motion through a homogeneous medium. It
may on account of similarity be expressed in the form,

a2 = Kip) » W,R = const. (11)

where K() is a momentum coefficient. Its form may be deduced
through use of the identity, Lamb [1945], pg. 229,

<3 =f ye dé dt (12)

The latter integral can be taken separately over the interior and
shear layer of the vortex pair, which yields terms upon making use
of similarity,

ey yo d& dt = W; R? (13)
int.
a yt d& dl = (We - Wi)R® (14)

shear layer

As a result, the form of K,{) consistent with our assumptions,
is seen to be,

KA) = K, - Ki (15)
where K, and K, are undetermined constants.

Energy. The total kinetic energy in the vortex pair motion,
taking account of similarity, may be expressed as the sum of two
terms,

KE, _ K, w7R? + K,W.R? (16)

while the dissipation takes the form,

332

Self-Convecting Flows

a KE

a - WR ° Col) (17)

where for large values of WU, Cp must approach some limiting value
Cp', while for small values of | (J+ 1), Cy) C(t) » (We/Wi)>.

Laws of Motion. Limiting Cases; Weak Circulation (W — 1).
In this case the inner and outer flows are almost matched and the
deviation from the ideal vortex motion is small. The laws of motion
in their appropriate form become,

Volume.
& = (p- 1) * a@"(4) (5a)
Vorticity.
anes =- Aja'(1) +» (pb - 1° > W, (10a)
Momentum.
K (1) ° W,R° CONS te (14a)
Energy.
d(WeR) We sse?
K(1) S784 = - (58) WR Colt) (17a)

Combining (5a) and (10a) leads to the result,

|
W, < cep A, #0 (18)
Whereas, (iia) requires
Ww, * = or z=tl/3 (19)

so that the presumed motion can take place only if A, =0 or
A, = (W - 1)-'. Combining (17a) and (10a) leads to the requirement
that the velocity ratio be constant and have the value

3533

Tultn and Shwartz

v=tt[oenytttha | (20)

° @

At the same time, dR/dz is required to be constant and to have the
value,

= dR _/ We Col)
iin PAE Secs age (sp) ; KU) te

We leave till later a discussion of comparison with experiment, but
we may note now that the prediction (19) is similar to that of the
previous theory based on complete similarity.

Strong Circulation (J >>1). In this case the interior circu-
lation is very strong relative to the ideal value and the deviation from
the ideal vortex motion is large. The appropriate laws of motion are,

Volume.
GR Sepa ut
7am a (W) (5b)
Vorticity.
AMAR) = aa'(y) + Wi > (10b)
Momentum.
2
(K, - Kh) + W, Ro = b+ (M,/p) (11)
Energy.
d os 3. gel
KW 3 (W, R) = - W, R* Cy (1 7b)

Combining (10b) and (17b) leads to the requirement that a(ip) be
constant and equal to

Cr
DSK (24)

Since similarity requires that W,/W be constant, we may hereafter
take a' constant in (5b) and (10b). Combining (5b) and (10b) leads
to the result

Self-Conveeting Flows

1+A,
1 Ie
eo RueAy OF W/W, * (#2) 22)
and, substituting (22) and (5b) into (11b) leads to the differential
relation,
Kp dR\w"'A)) — np dR
K, - += = ort (23)
( | (e4 =) a’ dz
which has the solution,
a! _ R* Kos
Ta °Z= x eee + const. (A; # 0) (24)
where
M
u = eee
W, Ro
0
and
a! _ = , Kos
— Kz =4n R + —#R + const. (A, = 0)
al mn
or
1
aK, Bed tn R + S2(R - 1) (24a)

B Ro
Substituting dR/dz derived from (23) into (5b) yields a relation

between | and R,
K
Seta (25)
WR aK
and, finally, it may be shown that,
W te 2 K __ (I+ Aj)
sug ic Bone Seek (26)
K, K,

The type of motions which ensue from this theory in the case
of strong circulation are seen to depend very much on the value of

335

Tulin and Shwartz

A,, the constant appearing in the relation for circulation change,
and which depends in part on the way in which vorticity is ingested
into the vortex pair. In fact, the asymptotic behavior of the vortex
pair changes radically as A; varies around the value unity. This
is demonstrated in the table below.

Asymptotic Behavior ( >> 1)

aK

pt AiKa)
ak. aK,

R(K/a'K,)

R(K,/a'K,)

For values of A, > 1, the velocity ratio is seen to decline,
so that the strong circulation assumption must eventually become
invalid. The case A, =1 yields results qualitatively similar to
the weak circulation case. In the case where A,< 1, however, the
velocity ratio increases to the asymptotic value shown and, most
interesting, the added momentum coefficient (K, - K,¥) vanishes
asymptotically, so that the motion becomes determined by volume,
vorticity, and energy balances alone. Finally, in the case where
A, << 1 (effective annihilation of ingested vorticity), then asympto-
tically the motion ea, tae determined by volume and energy balances
alone, yielding z~t

III COMPARISON WITH EXPERIMENT (HOMOGENEOUS MEDIA)

In Figs. 5 and 6 are shown data from actual experiments on
two-dimensional vortex pair motions in homogeneous fluids. Most
of the data shown were obtained in experiments carried out in our
own laboratory. Suffice it here to show a schematic of the facility
which was used, Fig. 7, and to show Table 1, in which the properties
and characteristics of the experimental vortex pairs are listed.

336

Self-Convecting Flows

LO
Q O TYPICAL

O EXPERIMENT
fo) C)
< 0.5 _o
= O
> O =|
5 O WR
O O
lu
> O
=)
<
V
Zz
lu
a 2

W~R-

0.1
| 5 10

RADIUS, R/Ry

Fig. 5. The vertical velocity vs. radius of a vortex-
pair moving in a homogeneous fluid

Most significant, we found in our experiments and from the
data of Richards [1965] that the measured variation of vertical
velocity and pair fadius (two-dimensions) conformed more closely
to the law W~R™ or z~t'/2 than to the law derived in the past by
others and which is based on complete similarity and momentum
conservation; i.e., W~ R? or z~t'”, A test of the simple con-
servation of momentum, W~ R™, using a typical trajectory is
illustrated in Fig. 5 and, similarly, in Fig. 6 it is shown that the
trajectories, so far as they have been observed experimentally,
conform more closely to the asymptotic law derived earlier for the
case of strong circulation, utilizing a small value of A,

(Oi Ay <0 2).

In the case of strong circulation, the radius grows ina linear
fashion asymptotically, but the theory predicts that during the initial
phases of the motion the quantity dR/dz is less than its asymptotic
value. A similar behavior was observed in our experiments, see
Fig. 4. The matching up of these observed trajectories with the
theory offers an opportunity to determine some of the constants of
the theory. For this purpose we assume to begin with that A, =0,
since the comparison between observed and theoretical trajectories

35

RISE HEIGHT, 2/z,

Fig. 6.

Tulin and Shwartz

THEORY: EXPERIMENTS:
R
ar UN 301

(z/z9) (2+D) (Wot) RUN 302
RUN 304
NOTE: D MAY BE REPLACED BY A
RICHARDS (1965)

D = 1 CORRESPONDS TO CONSERVATION OF MOMENTUM SOLUTION

D = 0 CORRESPONDS TO CONSERVATION OF ENERGY SOLUTION
WITH ZERO DISSIPATION

es) 2 3 4 5 6 7 8 9

TIME, Wt/z,

The rise of vortex-pairs in a homogeneous medium}
comparison of experiment and theory

338

Self-Convecting Flows

HOMOGENEOUS OR LINEARLY
DENSITY-STRATIFIED FLUID
4g"

APERTURE 0.75" WIDE

Fig. 7. Experimental facility for studying vortex-pair motion

suggests a small value. In this case,

eK) EE al = ink + [R - 4] (24a)

The trajectory according to Eq. (24a) with K,/p = 1/2 and

aK /p = 0.38 is shown in Fig. 4. A fair fit with the experiments has
been achieved, and noticeably better than is possible with any linear
trajectory.

The strong circulation theory thus explains the two important
features of vortex pair behavior which cannot be explained by the
usual theory of complete similarity. These features are: (i) the
tendency for forced vortex Peajectorics (homogeneous flow) to more
nearly follow the law z~ t”* rather than z ~ t'/3, and (ii) the ten-
dency for the entrainment coefficient (dR/dz) to grow during the
initial phase of the motion. These results suggest that in vortex

339

Tultin and Shwartz

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340

Self-Convecting Flows

motions in homogeneous flows the internal velocity scale grows
steadily relative to the translational (external) velocity, the ratio
approaching a value considerably larger than unity, while at the
same time, the virtual momentum coefficient associated with the
vortex motion approaches the value zero. The data also suggest
that vorticity ingested from around one half of the vortex pair is
almost annihilated through mixing with vorticity ingested from the
opposite side.

IV. SIMPLIFIED THEORY (VORTEX PAIR MOTION IN STRATIFIED
MEDIA)

Convected masses in nature are often rising or falling ina
medium of varying density, as in the case of a chimney plume pro-
jected upwards into a stable atmosphere. The latter may be
characterized by a characteristic time (the Vaissala period),
1/Jag, where a= - (1/pe)(dp,/dz) and P, is the potential density
of the atmosphere. The same definition can be used to characterize
any density stratified media.

The motion of the convecting mass may also be characterized
at any instant by the time, R/W. It is almost apparent that when
the latter time is long in comparison to the Vaissala period that the
effect of stratification will dominate, and conversely. That is,

Stratification

Effect of
Gira ication Ry ag
Dominates

decreasing Ww increasing
Vanishes _——— ete

Quite clearly, too, as the motion proceeds in time, the ratio R/W
increases continuously, so that stratification must eventually
dominate. When this happens, the vertical motion of vortex-pairs
may become oscillatory, and is accompanied by the collapse and
horizontal spreading of the convected mass, as illustrated in Fig. 8.
This behavior is, of course, not consistent with similarity either
complete or of the kind assumed in the preceding section.

It is sometimes desirable to be able to estimate the tra-
jectory of a vortex pair while it is rising in a stratified media and
particularly to predict the maximum height of rise and the time
required to reach the maximum. For this purpose, we adopt here
a simplified theory based essentially on the assumption of strong
circulation and annihilation of ingested vorticity. In fact, the parti-
cular assumptions adopted would apply if the velocity ratio, w, had
already closely approached its limiting value. These assumptions
are: (i) the motion is determined by conservation of volume, mass,
and energy (neglecting vorticity and momentum); (ii) complete
similarity (dR /dz = B, a constant). For further justification of these

341

Tulin and Shwartz

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342

Self-Convecting Flows

assumptions we shall depend finally upon a comparison between
theoretical predictions and the results of systematic experiments.

The energy balance is expressed as follows,

2 So) [tts + vy? + w*) +(p - pe) 8b] d& dn dt

= - (Rate of Dissipation of Energy) (27)
See Fig. 14 for nomenclature.

For a self-similar, self-convecting flow, the dissipation of
kinetic energy per unit volume which occurs due to the action of
turbulent shear stresses must for dimensional reasons be of the
form,

Dissipation pw (28)
Unit Volume R

As a result, the energy balance, Eq. (27), for a self-con-
vecting mass in a homogeneous medium of the same density takes
the form,

2. 2+} 3
K WRT) 2g W Re (29)
2 dt DR
where
j = 0 (two-dimensional)
j=1 (axisymmetrical)
and K is a constant of virtual energy, defined by the identity,
2 2 2 ;
iY eo dé dn dt = Swr } (30)

and where Cp, is a dissipation coefficient.

Making use of (29), together with the relationship R= Bz,
it may be shown that the height of rise follows the law,

7 (2+ 1/2* Cp/BK) t (31)

343

Tultn and Shwartz

in a medium of uniform density with Ap =0. The dissipation coef-
ficient in nature, D= C)/ BK, may be determined by a comparison
between theoretical trajectories such as given by (31) and obser-
vations of vortex pair rise in homogeneous media. As shown in
Fig. 6, such a comparison leads to the conclusion that D is quite
small (D <"0,. 2).

The trajectory given by (31) may be compared to the law which
would apply if momentum were conserved,

zeta t (32)

which coincides with (31) only if D=1. The variance of observed
trajectories from the momentum law (32) is clearly seen in Figs. 5
and 6.

Volume conservation in a self-similar flow leads to a linear

relation between the nominal radius of the mass and the height of rise
from the virtual origin z = 0:

R = Bz (33)

Conservation of mass takes the form
d 4! 2+) 1+]
= | (3) aR p; | = 2)2nR’ p, WB (34)

where p,, is the density of the surrounding fluid at any given height
z and p,; is the average density within the rising mass, defined by

4\) 24)

(3) mR |p, (z) - p,(z)] = { (Pp - Pe) d& (dn) dé (35)
where the integration is taken over the entire volume of the rising
mass.

The formulation of conservation of energy is based upon (27)

and (28)

K 24) 2+} W>_ 24)
de [2 PiW R™ +(e, - pleeR'” |= - Cori eR oy

where W and R are the observed and measured gross properties
of the rising mass while K and k are the coefficients of the virtual
kinetic and potential energies, respectively, defined in two dimen-
sions, e.g., by

344

Self-Convecting Flows

2_2
Kp, Se -{ £ (we +w*) dé ae (37)
and
k(p; - p,)zR° =f (Pp - Pe)(z +6) d& dg (38)

In most practical instances where one is dealing with a mass
of fluid convected through a homogeneous or stratified medium such
as the ocean or the atmosphere, the difference between the densities
of the convected and surrounding masses is yery small; that is
Ap/p, << 1, being usually of the order of 10~, and therefore pj/pe
can be taken as 1. This assumption, frequently referred to as the
Boussinesq approximation, see Phillips [| 1966], will be used through-
out the analysis presented herein.

Using the Boussinesq approximation and the identity dz = w dt,
the three conservation statements, Eqs. (33), (34) and (36), may be
reduced to the following form in the case of a planar motion:

R = 62, (39)
dpi 4 24p _ 0
dz Zz
or (40)
d (Ap), 27Apy _
dz one alae gts
2
dw Cp 2, 2kg (Ap 4 _
i> +2(1 +e) W eal re az)z = 0 (41)

where a=-(1/p,)(dpp/dz) and Ap = (p; - pe).

Finally, explicit general solutions of (40) and (41) may be
found. They are:

oe = ((40) - Sele * ee 4)
(ye ibe: Be enter ee rile eh a
Wn (+ 2D) T¥D/2° 12D

+ A) 2° - bea] 2” (43)

345

Tultn and Shwartz

where
= (2k) = 298 (Ap
A= (=) Grits anaY ite ane ee) (44)
= (sR an = Wwe (45)
C
= —D
D= BK (46)
n= 2(1.+ D) (47)
and
ZS Z/fZg (48)

The parameter G is a measure of the initial buoyancy of the
convected mean relative to the initial momentum. G is taken as
positive when a net buoyancy force is acting on the mass, i.e.
4p <0. S is a measure of the added buoyancy which would result
from moving the convected mass a vertical distance zo througha
stratified medium. Since fag is the frequency with which a finite
volume of fluid of given density would oscillate in a stratified medium,
often referred to as the Vaisala frequency, the parameter S can
also be considered as the square of the ratio of the characteristic
time of the convected motion, z )/W,, and the reciprocal of the
Vaisala frequency.

The maximum height of the rising mass, reached at the point
where W = 0, is according to (43), given by the solution of the
following:

ee ee) eee)

+B(rep72- T=) =° ee)

It is of interest to consider certain special cases:

1. A mass rising in a homogeneous medium with the
same density as itself; i.e., A=0O and B=0. Then

-n/2
Gy)? Gs) (50)

346

Self-Convecting Flows

or,
(= =1+ (1 +3) wat (51)

This result suggests how to estimate the dimensionless quantity n
(or D) through the analysis of the trajectories of rising masses in
this special case.

2. Amass with initial density difference rising ina
homogeneous medium; i.e., B=0. Then

(ie) = [) = tay]? + Eto)?" ae

If A> 0O, then no maximum height is reached, but if A<0O,

\/(1#2D)
Z max -(- eet (53)

Zo | A |

The predicted rise of the mass as a function of time and the
maximum rise of the mass for a range of values of A (<0) and D,
as obtained from Eqs. (52) and (53), are presented in Figs. 9 and
10. These figures demonstrate clearly the effect of the (negative)
initial buoyancy and energy dissipation parameters on the time
history of an impulsively started rising mass moving througha
uniform surrounding fluid of smaller density.

2a. The same case as above but for W,=0 and A>O.
First of all, (43) may be rewritten:

n
“sw, (22) + eit (5°) ( basco on =

or in this case

es Fctbel GD Sl) | 6

3. A mass with no initial buoyancy rising in a stratified
medium, i.e., A=0O. Then,

347

Tultn and Shwartz

SOLUTION FOR B = 0

= «(ATS 0
A=-0.4
A= -1.0

RISE HEIGHT, z/zo

TIME, Wot/ z,

Fig. 9. The rise of an impulsively started heavy mass of fluid in
uniform surroundings (B= 0), theory

= oa (55)

and the maximum rise of the mass, as a function of B, is obtained
from Eq. (55) by setting W = 0.

Approximate integrated solution for the height of the con-
vected mass, z,; as a function of time can be readily obtained from
Eq. (55) whenever D <<1 and B<<i. When these two require-
ments are satisfied, Eq. (55) can be rewritten as

2
qe aie eid eo (56)

and upon integration we obtain

348

Self-Convecting Flows

0°Ol-

Azoayi ‘(9 =q) ssuypunozans
UlLOFTUN uy pin yo sseur AAeoy pojrejzs AToATSsTNdwy ue so osya unurtxeur oy],

V ‘YOLOVS ADNVAONS IVILINI

ORSs 0°c- Oe ih Sa0s

0 = €@ Y¥Od NOILNIOS

“OF

“BT

XW 7/2) “ IHOIZH ASIY WAWIXVW

349

Tulin and Shwartz

VB ats sin [ sin" VB + 2/BF | (57)

where we have normalized the time t according to

t= Wot (58)

Equation (57) is particularly useful for the approximate deter-
mination of the maximum rise of a mass convected in a stratified
medium and the time at which this maximum height is reached,
tmx: For the maximum height we find

1/4
Zmax = (3) (59)

and the time required to reach this height is given by

ofa eet
ne Ae

This last result is especially interesting. In experiments on
convected masses of fluid moving through a density stratified
medium it was frequently observed that the time it takes the mass to
reach its maximum height is inversely proportional to the Vaisala
frequency of the stratified medium, i.e., tiniV28 = const. Equation (60)
is just a statement of this same fact for small values of B (as usually

exist in nature), since may [cy = (2k BK (trax 2g) «

(60)

ph]

V. COMPARISON OF EXPERIMENTAL AND THEORETICAL
RESULTS (STRATIFIED MEDIA)

In our experimental investigation we have studied the motion
of impulsively started rising masses (or vortex-pairs) in both
uniform and density-stratified surroundings. According to the
theoretical considerations presented in theprevious section, the
time-histories of these motions, expressed in appropriate non-
dimensional terms, are determined by three parameters, A, B
and D, i.e., for given values of these parameters the rise and
growth of the convected mass as a function of time can be predicted.
A and B are determined by G and S and by a third parameter
which is the ratio of the virtual kinetic and potential energy coef-
ficients, K/k, according to Eqs. (44) and (45).

While the parameters G and S are determined in each case
by the initial conditions of the rising vortex-pair, there is no

350

Self-Convecting Flows

practical way for determining a priori the values of D and K/k.
These latter parameters can be determined only by comparing
certain sets of experimental results with corresponding theory.

A series of experiments on the motion of vortex-pairs ina
homogeneous medium of the same density, Series III (see Table 1),
where the parameters G and S (and therefore also A and B) are
identically equal zero, may be used for determining the dissipation
parameter D. The rise of the vortex-pairs in this case is predicted
by Eq. (51) and is graphically depicted in Fig. 9 (with A= 0).

The actual predicted rise of the convected mass depends on the
numerical value of the parameter D (or n).

In Fig. 6 are shown a comparison between experimental and
theoretical results on the rise of impulsively started masses ina
uniform medium of the same density. In a log-log plot, the slope
of the trajectory for large values of (W,t/z,) should be equal,
according to our analysis, to 1/(2+D), and it can be used therefore
for determining the value of the dissipation parameter D associated
with the motion of the rising mass. Included in Fig. 6 are the
experimental results of Richards [1965], on the rise of two-
dimensional puffs in homogeneous surroundings. The best agree-
ment with all experimental results is obtained when we choose
D =,0..2.

The numerical value of K/k enters into the analysis only
when there is an initial difference between the rising and surrounding
fluid densities or when the surrounding fluid is stratified. This
value will be also determined from a comparison of some experi-
mental and theoretical results. For a vortex-pair convected ina
density-stratified medium we found earlier that, for sufficiently
small values of the parameters A, B and D, the time it takes the
mass to reach its maximum height is inversely proportional to the
Vaisala frequency and is given by

(trevas) = (se 5 (61)

This value decreases only very gradually as the value of B (or S)
so that Eq. (61) is very useful for the experimental determination
of K/k.

In Fig. 11 the value of the product (tmav ag » as measured
in the experiments of Series I and II, is presented as a function of
the stratification parameters S; the initial conditions for each
experiment presented in the Figure are included in Table 1. There
are certain inherent inaccuracies in the experimental determination
of tmax which explain the scatter. Also shown in Fig. 11 are the
asymptotic solution for the maximum rise time, Eq. (60), and the
exact solution, according to Eq. (43), with D=0.2 and G/S = - 0.715.

351

Tulitn and Shwartz

Azoey} pue Juoutedxe jo uosjreduios fumypeur
pops}ye1zs-Aj]suap e uy pazJoaAUOD sapzed-xoj,IOA TOF oSTI UMUI]XeU JO OWT “TT “3h7T

S ‘YaLIWVAVd NOILVOISILVULS

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14 2210.0 =" O [09] ba “NOILNIOS DUOLEWASY ——
|-1d l800°0=° 7 SiZ°0- =S/9

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SLINSAY WLNIAWIY3dd x4 2°70 = d *HLIM JONVIVE ADYINA

=
oO

foe)
oO

N
—

0°?

xDwW

4) ‘LHOIZH WAWIXYW HOV34 OL 3WIL

6p

(

35:2

Self-Conveetting Flows

Close agreement between theoretical and experimental results was
obtained when we used 3K/2k = 6 or K/k=4. We have included
in the same figure the experimental results from Series III which
had a markedly different value of B; these too were found to agree
very closely with the theoretical prediction based on D=0.2 and
K/k = 4, indicating that the dependence of these latter two parameters
on £f is probably very weak. We have used these values of D and
K/k for all subsequent comparisons of experimental and theoretical
results. The coefficient of virtual potential energy cannot be much
different from unity, according to its definition in Eq. (38). The
total kinetic energy of a rising vortex-pair was thus found to be
about four times larger than the kinetic energy associated with its
linear convection alone, an indication of the intensity of motion
inside the vortex-pair, which contributes to its total kinetic energy.
This finding lends important support to the strong circulation
assumption.

A comparison between the predicted and actual maximum
heights of rise of a vortex-pair convected in a linearly density-
stratified medium is shown as Fig. 12. The figure includes a
prediction based on the simplified asymptotic solution for small B
(or S), Eq. (59), and a prediction obtained from the exact solution
of Eq. (49) for Zmgx Generally there is good agreement between
the experimentally measured maximum height of vortex-pairs and
the exact theoretical solution.

Finally, in Fig. 13, we compare the measured trajectories
(height versus time) of vortex-pairs with their theoretically pre-
dicted trajectories. The vortex-pairs included in the Figure all had
different starting conditions and they were moving through media
with different density-stratifications. However, their trajectories,
as depicted in the figure are shown to depend only on the values of
the two lumped parameters G and S which combine their starting
conditions with the properties of the surrounding medium.

The fact that trajectories of different vortex-pairs are
grouped according to their G and S values confirms the validity
of the scaling laws and scaling parameters used herein, while the
agreement obtained between the experimental and theoretical tra-
jectories lends further support to the validity of the simplified
theory presented here for the motion of vortex-pairs in stratified
media.

VI. SUMMARY AND CONCLUSIONS

A theory for the motion of two-dimensional turbulent vortex-
pairs in homogeneous media has been developed based on separate
velocity scaling of the internal and external flow fields involved in
the motion. These two flow fields are depicted to be separated by a
thin region of high shear, which also forms the boundary of the

353

Tulin and Shwartz

‘umM]peul peyjji#e138-AR]Suop e UT poyDaAUOD sajed-xXejJIOA FO osTI UNUITXeW oY J,

j-td lezo"0 = © 4a
ji Zio0="° O
,-14 '800°0=° V7
_140z00°0=" O

l

l
SLINSIY WLINIWI8ad x4

Aroay} pue Juewyredxe so uosjzeduios

S ‘YaLaWVYVd NOILVOIdILVals

[6p] “63 “NOUNTOS LOvxX3 —-—-
[6s] “ba ‘NOLNIOS DILOLdWASY
S1Z°0- = $/9
0°9 = %2/X€

Z°0 = CG *HLIM JONVIVE ADYANA

a

Z) ‘LHOI3H 4O 3SIY WAWIXWW

(27

xDwW Oo

354

z/z.,

RISE HEIGHT,

THEORY

WITH

7268)

Fig. 13.

ENERGY BALANCE Eq. [43 |

Self-Convecting Flows

D = 0.2
$/6

= -2.145

@4@ordrpervromgetoO

TIME W,t/z5

EXPERIMENTAL RESULTS

RUN
me ;
0

(SF NS) fe) le} oy fey es fey

355)

Zed

The trajectory of vortex-pairs in stratified media;

comparison of experiment and theory

355

Tulin and Shwartz

captured mass. The theory takes into consideration variations in
volume, circulation, momentum, and energy in the flow field. The
ratio of internal to external velocity scales, wW, is introduced as an
important variable. The virtual momentum coefficient is shown to
be linear inj |, ‘of the form. K, = Koil.

The theory is specifically derived for the two limiting cases
of weak and strong circulation. In the former case; “);— 1} and
the entrainment is weak; the asymptotic behavior of the trajectory
is z~t'® just as predicted by the usual theory based on complete
similarity and momentum conservation.

In the case of strong circulation, w >> 1, the asymptotic
behavior of the trajectory depends very much on the way in which
vorticity from the shear layer is ingested into the vortex pair. In
the case where the shear layer from opposite sides is ingested in
such a way as to cause annihilation of the ingested vorticity, then
the asymptotic trajectory is z~ t¥2, Under the same conditions,
the velocity ratio, Ww, increases toward the asymptotic value
K /Kp so that the virtual momentum coefficient tends to zero. As
a result, the asymptotic motion assuming vorticity annihilation
corresponds to a motion with complete similarity and with energy
conservation. The ratio of growth of the pair radius with height is
shown to increase, approaching a linear relation asymptotically.

Systematic experiments have been carried out, and the results
for rise versus time and radius versus height are compared with the
theory. They lend strong support to the strong circulation theory and
further suggest that ingested vorticity is to a large degree annihilated.

Based on these findings for the case of homogeneous flows, a
simplified theory is derived for the rising motion of vortex pairs in
stratified media. The assumptions of the theory are: (i) the motion
is determined by conservation of volume, mass, and energy (neg-
lecting vorticity and momentum); (ii) complete similarity (dR/dz = By
a constant). General laws of motion in stratified media have been
derived and solutions given; particularly interesting cases are dis-
cussed in detail.

Motions in stratified media were shown to depend on four
non-dimemsional parameters. Two of these depend upon the initial
conditions of the motion and the stratification of the media. The
other two are inherent in the details of the motion and had to be
determined from experiments; one of these, the dissipation param-
eter D = Cp/BK was found to be 0.2 while the other, the ratio of
virtual kinetic and potential energy coefficients K/k was found to be
4. On the basis of these numbers it may be concluded that the dissi-
pation rate is small and that the contribution of internal motions to
the overall kinetic energy is large.

The experiments confirmed the environmental scaling param-

356

Self-Convecting Flows

eters, which were used to collapse data taken under differing con-
ditions. Good agreement was found between predicted and observed
trajectories. Particularly good agreement was found for the maxi-
mum height of rise. The time required to reach maximum height
was found to be inversely proportional to the Vaisala frequency, jag,
and was given approximately by tmexY 2& = 1.8, in good agreement
with the theory. In general the experiments confirmed the utility

of the simplified theory for predictions of the motion of vortex pairs
in stratified media. This theory has been utilized elsewhere for

the prediction of the behavior of chimney plumes rising into a stable
atmosphere, with very good agreement between the theory and full
scale observations, Tulin and Schwartz [1970], and also with
excellent correlation with experiments to the penetration of a density
discontinuity by a turbulent vortex-pair, Birkhead, Shwartz, and
Tulin [ 19691 «

ACKNOWLEDGMENT

This work was supported by the Naval Air Systems Command,
the Air Programs Branch and the Fluid Dynamics Branch of the
Office of Naval Research under Contract No. N00014-70-C-0345,
which support is gratefully acknowledged.

REFERENCES

Birkhead, J. L., Shwartz, J., and Tulin, M. P., "Penetration of a
Density Discontinuity by a Turbulent Vortex-Pair," HYDRO-
NAUTICS, Incorporated Technical Report 231-21, December
1969.

Lamb, H., Hydrodynamics, Dover Publications, N.Y., 1945.
Morton, 5. R., laylor, G. I. and Turner, J. S:, “Turbulent
Gravitational Convection from Maintained and Instantaneous

Sources," Proc. of the Royal Society, A, Vol. 234, p. 1,
1956.

Phillips, O. M., The Dynamics of the Upper Ocean, Cambridge
University Press, Cambridge 1966.

Richards, J. M., "Puff Motion in Unstratified Surroundings ,"
Je of Fluid Mechs, Vole 21, No. 1,.p. 97, 1965.

Scorer, R. S., Natural Aerodynamics, Pergamon Press, 1958.

Scorer, R. S., "Experiments on Convection of Isolated Masses of
Buoyant Fluid," J. of Fluld Méch., ‘Vol, 2, No. 6,.p- 583,
August 1957.

Spreiter, J. R., and Sacks, A. H., "The Rolling Up of the Trailing

Vortex Sheet and Its Effect on the Downwash Behind Wings,"
J. of Aero Sciences, Vol..18,:No..15 p. 21, January 1951.

S5it

Tulin and Shwartz

Thomson, Sir. W., "On Vortex Atoms," Philosophical Magazine,
Series 4, Vol. 34, No. 227, p. 15, July 1867.

Tulin, M. P., and Shwartz, J., "Hydrodynamic Aspects of Waste
Discharge," AIAA Paper No. 70-755, June 1970.

Woodward, B., "The Motion in and Around Isolated Thermals,"

Quart. J. of the Royal Meteorological Society, Vol. 85,
pe 144, 1959.

C, n, & - CARTESIAN COORDINATES

u,v, w - CORRESPONDING VELOCITIES

CENTER OF RISING MASS AT TIME t

ACTUAL ORIGIN OF MOTION

VIRTUAL ORIGIN OF MOTION

Fig. 14. Nomenclature

358

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

Thursday, August 27, 1970

Morning Session

Chairman: G. B. Whitham
California Institute of Technology

Page

Radar Back-Scatter from the Sea Surface . 361
K. Hasselmann,M, Schieler, Universitat Hamburg

Interaction Between Gravity Waves and Finite
Turbulent Flow Fields 389
D. Savitsky, Stevens Institute of Technology

Characteristics of Ship Boundary Layers 449
L. Landweber, University of Iowa

Study of the Response of a Vibrating Plate
Immersed in a Fluid 477
L. Maestrello, T. L. J. Linden,
The Boeing Company

359

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RADAR BACK-SCATTER FROM THE SEA SURFACE

K. Hasselmann”™ and M. Schieler
Instttut fuer Geophystk
Untverstty of Hamburg

ABSTRACT

Doppler spectra of electromagnetic backscatter from
the sea surface are interpreted in terms of general-
ized Bragg models. The observed broadening of the
spectra about the Bragg line is attributed to higher-
order nonlinear processes. At conventional radar
frequencies, good agreement with the measurements
is achieved by an extension of the wave- facet inter-
action model considered by Wright, Bass et al. and
other workers. The correlation of wave slopes and
orbital vélocities in the joint probability distribution
of carrier-wave facets leads to significant differences
between the Doppler spectra for vertical, horizontal and
cross polarization. In the HF band, the Doppler
broadening is interpreted in terms of quadratic wave-
wave interactions. For the usual case that the electro-
magnetic wave lengths are small compared with the
principal wave lengths of the sea, the theoretical Doppler
spectrum consists of the lowest-order Bragg line and
superimposed images of the complete ocean wave fre-
quency spectrum folded on either side of the Bragg line.
Both wave-facet and wave-wave interaction models give
promise of extracting significant information on the
"state of the sea" from electromagnetic Doppler return
at wave lengths short compared with the dominant wave
lengths of the sea,

* :
Presently at Woods Oceanographic Institution.

361

Hasselmann and Schietler

I. INTRODUCTION

The development of numerical wave prediction methods in the
past years [1, 2, 17] has increased the need for wave data ona
synoptic scale, both as a reference for testing and improving the
models and as real-time input for the computations. Synoptic wave
data would also be of value for numerical weather forecasting by
providing indirect information on surface winds in otherwise poorly
covered areas of the oceans. The growing interest in electromag-
netic backscatter from the sea surface stems largely from the
potentiality of the method for furnishing sea-state data of this kind.
Radar scatterometers in satellites could scan most of the world
oceans in a few hours. Alternatively, large areas of the ocean can
be sampled using HF stations on land. Following the pioneering
work of Crombie [6] and others, Ward [22] has recently detected
the backscattered return of ionospheric HF modes from relatively
small, 100 km square patches of the sea surface at distances up to
3000 km.

Unfortunately, both techniques suffer from wave length limi-
tations. Cloud absorption and finite atenna size define an effective
transmission window for satellite scatterometers in the conventional
radar wave length range between a few fractions of a cm and about
50 cms. Backscatter measurements over long horizontal ranges
are similarly restricted to ionospheric modes in the decameter
band. In both cases, the electromagnetic wave lengths are consider-
ably shorter than the principal components of the surface-wave
spectrum, which normally lie in the range between 50 and 500 m.
The bad wave length matching creates difficulties in relating the
backscattered signals obtained by these methods to significant sea-
state parameters.

Scattering experiments in both the centimeter-decimeter and
decameter bands have now clearly established the basic validity of
the first-order (Bragg) wave-wave interaction theory. According to
this model, the backscattered radiation arises from interactions with
two gravity- wave components whose wavenumbers k9 are determined
by the Bragg (resonance interaction) condition for constructive inter-
ference, k?= +2k', where k' represents the horizontal wavenumber
component of the incident radiation. For non-normal incidence,
the wave lengths of the scattering and incident components are then
of the same order, which implies that the scattering surface waves
normally lie in the high-wavenumber, equilibrium range of the
surface-wave spectrum. It appears therefore from first-order
theory that backscatter measurements may yield a useful independent
determination of Phillips' constant [15, 22], but do not contain sig-
nificant information on the more interesting low-wavenumber part
of the wave spectrum which contains most of the wave energy.

Fortunately, the scattering measurements, while supporting

the Bragg theory, also indicate that it should be regarded only as a
first approximation. The Doppler spectra, in particular, exhibit

362

Radar Back-Seatter from the Sea Surface

several features not predicted by the Bragg model. Generally,
there is a marked dependence of the anomalies on sea state, sug-
gesting that useful correlations between backscatter signatures and
significant sea-state parameters may be discovered by extending
the scattering theory to higher order.

Two generalisations have been proposed: the wave- facet
interaction model[ 23, 4, cf. also 3, 9, 10, 20, 24], in which the
Bragg-scattering waves are superposed on longer carrier waves,
and the higher-order, wave-wave interaction model originally inves-
tigated by Rice [18]. The models have been applied hitherto mainly
to the cross sections, which show only weak sea-state signatures. In
the present paper, we consider their extension to the more strongly
sea-state dependent Doppler spectra.

In the cm-dm bands, good agreement with the observed
Doppler spectra is obtained with the wave-facet interaction model,
The Doppler spectra are found to be quasi-Gaussian and can be
characterized to good approximation by the mean frequency and the
frequency bandwidth. Both parameters depend on moments of the
wave spectrum which are governed by the high-energy, low-wave-
number range of the spectrum. They can therefore be used to obtain
independent estimates of, say, the mean waveheight and period.

The model allows only for electromagnetic interactions.
Basically, the hydrodynamical modulation of short gravity waves by
long carrier waves is of considerable interest, not only for the
description of the surface wave field, but also for its energy balance.
The interactions generally lead to an energy loss of the long waves
at a rate which can be estimated from,the observed upwind-downwind
asymmetry of the cross sections [11]!. However, because of the
strong influence of white capping, the interactions cannot yet be
described in sufficient detail to be included realistically in computa-
tions of the Doppler spectra. Their effect on the Doppler bandwidth
is probably negligible, but the mean Doppler frequency may be more
strongly modified.

The wave-facet interaction model is valid for electromagnetic
wave lengths shorter than about 1 m. Thus it applies in the cm-dm
radar band, but not in the dkm band. In the latter case, however,
the Bragg theory can be generalised by straightforward extension of
the wave-wave interaction analysis to higher order. The relevant

TLonguet- Higgins [13] has shown that the momentum loss of short
waves breaking on the crests of longer waves results in an energy
transfer to the long waves. However, the gain in long-wave
kinetic energy due to this process can be shown to be slightly less
than the loss of potential energy arising from the simultaneous
mass transfer between short and long waves. The net result of
both processes is a weak attenuation of the long waves [ 11].

363

Hasselmann and Schteler

perturbation parameter of the expansion is given by the ratio of the
amplitude of the interacting surface wave to the wave length of the
incident radiation. In the first order analysis, the perturbation
parameter is proportional to the slope of the scattering Bragg
wave, which is small for all electromagnetic wave lengths. At
second and higher order, however, the electromagnetic waves
interact with longer surface waves of higher amplitude. In this
case, the perturbation parameter remains small only if the electro-
magnetic wave length is large compared with the amplitude of the
entire wave field. This condition is satisfied by dkm waves, but not
by cm-dm waves.

The requirements for the wave-facet and wave-wave inter-
action models are found to be mutually exclusive, so that the two
expansions cannot be matched in a common region of validity. It is
a fortunate coincidence that the theoretical wave length gap cor-
responds to the gap between the two presently available techniques
for measuring electromagnetic backscatter on a synoptic scale.

The second order wave-wave interaction analysis yields a
continuous Doppler spectrum superimposed on the first-order Bragg
line. The continuum reduces to a particularly simple and useful
form when the Bragg wave length is short compared with the wave
lengths of the dominant surface waves -- the usual situation for
ionospheric modes. In this case, the continuum is identical with
the two-sided image of the surface-wave frequency spectrum,
centered on the Bragg line as virtual frequency origin. The energy
scale of the wave spectrum can be inferred from the observed
energy of the Bragg line, independent of transmission or other cali-
bration factors.

Doppler side-band structures observed by Ward [ 22] and
others are not inconsistent with this interpretation. However, most
Doppler spectra published hitherto have been analysed from rather
short records, so that the continuum is generally not well defined
statistically. Longer records are needed to decide whether the one-
dimensional frequency spectrum of the surface-wave field can indeed
be detected in the Doppler spectrum of backscattered ionospheric
modes above the inherent ionospheric noise.

Il THE LOWEST-ORDER SCATTERING MODELS

For electromagnetic waves short compared with the dominant
waves of the sea, one might attempt to describe the scattered field
by a specular reflexion model, in which the sea surface is repre-
sented as an ensemble of locally plane, infinitesimal facets, each of
which reflects the incident radiation according to the laws of geometric
optics. The cross section o for the backscattered radiation (the
backscattered energy per unit solid angle per unit surface area of the
ocean) is then proportional to the number density of facet normals

364

Radar Back-Seatter from the Sea Surface

pointing towards the source. As the distribution of normals ina
random surface-wave field is approximately Gaussian, the depen-
dence of log o on depression angle 9 is given by a parabola, with
maximum at normal incidence (90° depression angle) and half-width
typically of the order 10° (Fig. 1).

CROSS SECTION
Ow = Onn
(o/ 90° 180°

x x

DOPPLER SPECTRUM
ie) Wa

=W, fo) W, Wy

Fig. 1. Cross sections and Doppler spectra according to the specu-
lar reflection and first-order Bragg scattering models
(qualitative)

Hasselmann and Sehteler

The frequencies of the backscattered waves are shifted
relative to the frequency of the incident radiation by the Doppler
seid OF Nally 2k' * u induced by the facet motion, where
K = (k', Vy is the wavenumber of the incident radiation and u
the local orbital velocity of the waves. For an approximately “linear
wave field, u is a Gaussian variable, and the Doppler spectrum
also has a Gaussian shape.

As the backscattered waves are reflected at normal incidence,
it follows by symmetry that the cross sections and Doppler spectra
are independent of polarisation. Vertical and horizontal polarisation
are denoted in Fig. 1 by V and H, respectively, the first index
referring to the incident field, the second to the backscattered field.
The cross-polarised return VH and HV vanishes.

Although applied successfully by Cox and Munk [5] to the
analysis of sun glitter from the sea surface, the specular reflexion
model fails to describe the observed electromagnetic backscatter
at cm-dm and dkm wave lengths. It appears that for these wave
lengths surface irregularities of length scale comparable with the
radiation wave length cannot be neglected. Accordingly, recent
models have been based on the Bragg scattering theory, in which
these irregularities are regarded as the dominant scatterers.

It is assumed in the Bragg model that the slopes of the
scattering surface waves are small and that their wave lengths are
comparable with those of the radiation field. The backscattered
field can then be expanded in powers of the surface displacement.
The first-order field is linear in the surface displacement and can
therefore be constructed by superposition from the field scattered
by a single gravity-wave component ¢ = Z exp {ik?- x - iugt}. This
corresponds to the classical problem of refraction by a periodic
lattice. The scattered field consists of two waves s = + whose
horizontal wavenumbers and frequencies are given by the Bragg
(resonant interaction) conditions

k' + sk9 = k$
ic ie. ee (1)
W; + sw = W.

(The vertical wavenumber component 8 determining the scattering
angle follows from the dispersion relation | w.| =c lie | » where <
is the velocity of light).

Hae scettcring (k* = k') occurs for the gravity- wave com-
ponents k*= + 2k’. The (rae cattering cross section is accordingly
of the form

CA To T2B ae Tap (2)

366

Radar Back-Secatter from the Sea Surface

where
08 = Tuer g(- 2sk') (2,8 = Vor He or = +)

and F, (k) is the surface-wave spectrum, ngrmalised such that the
mean square surface displacement ((?) = F,(k) dk. The cornered
parentheses denote mean values. (The negative sign of the wave-
number in the definition of ogg has been introduced so that c4g
corresponds to a spectral line with positive Doppler shift, cf. Eq.
(3).) Tag is a scattering coefficient obtained by expanding the electro-
magnetic boundary conditions at the free surface [18]),

T = 20} ee: (1- e)(e[1 +cos* 6] - cos” 8)
VV
(e sin O tye - cos’ 6)
2 2
T,,,= | “2b sin? e Bie
c (sin 8 ie - cos? @)2
Tyy = Tuy = 9

where e€ is the dielectric constant of sea water.

The normalized Doppler spectrum Xg@wg), defined by
Jxag (ea) dwy= ogg where wq= ws - wj, is given according to (1) by
two lines at the gravity-wave frequencies + wg,

Xapled = Xapled) + Xagloa)

with (3)

X gla) = TaB(wWg - S Wg)

Normally, one of the Bragg lines due to scattering from the
surface wave component propagating in the downwind direction is
very much stronger than the other line associated with the wave pro-
pagating in the opposite, upwind direction.

The general properties of the Bragg cross sections and
Doppler spectra are indicated qualitatively in the right-hand panels
of Fig. 1. In contrast to the specular reflexion model, there is a
pronounced dependence on polarisation and appreciable backscatter
at small and intermediate depression angles. The cross-polarised
return again vanishes.

367

Hasselmann and Schteler

L-BAND JULY 29,1965
A SEA STATE "A"
© SEA STATE "B"
x SEA STATE "C"

— THEORETICAL (15 KNOTS)

Ove (dB)

og 10° 20° 30°40° 60° 90°
DEPRESSION ANGLE

Fig. 2. Theoretical and observed Bragg backscatter cross sections
for vertically polarised 24 cm (L band) waves (from Wright
[ 23])

Figure 2 shows a comparison by Wright [23]! of experimental
and theoretical Bragg cross sections for vertically polarised cm-dm
waves. The surface waves were represented by a Phillips’ spectrum
F,(k) = (a/2)k-*5(), with a uniform half-plane angular spreading
function, S(J) = 7m! for 0S || < 1/2, S(W) =0 for - 1/2 < |y|Snm.
The constant @ was chosen to fit the observed cross sections, but
is not inconsistent with other estimates from direct measurements of
gravity-wave spectra (cf. also [15]). Shown in Fig. 3 are theoreti-
cal and experimental cross section ratios oy, /o y Lhe agreement
here is also very good, except for the shortest wave length (3.4 cm,
8910 MHz), where scattering by spray may be beginning to mask the

T

The theoretical cross sections shown in Figs. 2 and 3 were, in fact,
computed for the wave~facet interaction model considered in the
next section. However, the deviations from the first-order Bragg
model are negligible.

368

Radar Baeck-Seatter from the Sea Surface

X 1228 MHz,CROSSWIND 15 KNOTS

© 8910 MHz,CROSSWIND 15 KNOTS

— THEORETICAL DEPENDENCE
FOR RMS TILT ANGLE AT 7.5°

20

/O;°, (dB)

°
vv

5° 10° 20° 30°40° 60° 90°
DEPRESSION ANGLE

Fig. 3. Theoretical and observed ratios of Bragg backscatter cross
sections for vertical and horizontal polarisation at wave
lengths 24 cm (1228 MHz) and 3.4 cm (8910 MHz) (from
Wright [ 24])

weak Bragg return for horizontal polarisation. Not predicted by
first-order Bragg theory is the observed cross-polarised return,
which is generally only slightly smaller than or comparable with the
backscatter for horizontal polarisation; this can be explained by the
wave-facet interaction model [ 23].

Although the observed cross sections oy, and o,, are in
good agreement with theory, the Doppler spectra for these polar-
isations point to limitations of the first-order model. In the cm-
dm bands, the Bragg lines are found to be broadened into Gaussian
shaped distributions with bandwidths of the same order as the Bragg
frequency (cf. Fig. 4., from Valenzuela and Laing [20]). Earlier
measurements by Hicks et al. [13] indicate that the mean frequencies
of the distributions -- which were not measured by Valenzuela and
Laing -- may also be considerably higher, by factors of the order
2to 4, than the theoretical Bragg frequency.

369

Hasselmann and Schteler

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I ‘(uw OL) q spueq uy erzD0ds 1933e9SyDeq TaTddoq paestazejtod Aj{Teo}310A fo sotdurexy

ZH OO} AININO3Z8S 7H OS AININO3ZYS

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($1381930) 3QNLIIdWY 3AILV138

S3AVM Y3L3W
9°0-¢0

(J19NV NOISS38d30 002) RDS (JT9NV NOISS3Yd30 02) AN

"py “Sa

370

Radar Back-Seatter from the Sea Surface

In the decameter bands, the observed broadening and shiit
of the Bragg lines are much weaker. Instead, the Doppler spectra
show pronounced side band structures (cf. Fig. 11, from Ward [22],
and similar spectra in Crombie [6] and elsewhere). The basic
difference in structure of the Doppler spectra observed in the
cm-dm and dkm bands lends support to theoretical considerations
calling for alternative expansion procedures in the two wave length
ranges.

Ill. THE WAVE-FACET INTERACTION MODEL

In order to treat the scattering waves as small perturbations
of a plane surface, it is assumed in the Bragg theory that the wave
amplitudes are small compared with the wave length of the incident
radiation. Ina strict sense, the expansion is valid if this condition
is satisfied not only for the Bragg waves, but for the entire surface
displacement. Thus the theory is not rigorously applicable to short
electromagnetic waves of a few cm wave length, although the long
surface waves of high amplitude which violate the expansion condition
do not enter in the final scattering expressions. Various workers
[e.g. 3, 4, 10, 20, 21, 23] have suggested that this formal short-
coming may be remedied by dividing the surface-wave spectrum into
two parts, a high-wavenumber scattering region, and the energy-
containing region at low wavenumbers which defines the "sea." The
"sea" is then treated as a random carrier wave which modulates
the scattering by the superimposed Bragg waves. If the Bragg wave
length 1/k' is short compared with a typical wave length 21/k°
of the sea, the carrier wave may be represented locally as a plane
facet, and the first-order scattering theory applied in the reference
frame of the moving facet.

The model involves additional conditions besides the two-scale
assumption that it is possible to define a facet diameter D inter-
mediate between the carrier and scattering wave length scales,

(ki)! << D << (k°)| (4)

The finite facet size implies an indeterminacy Ak = O(1/D) of
the scattering wavenumber, which corresponds to an angular spread
Ao = O{(K'Dsin @)"'} of the backscattered beam. The wave-facet
interaction model is meaningful only if A® is small compared with
the change in effective depression angle introduced by the facet
slope 8{/dx = O(k°t) , where € is the carrier-wave amplitude.

a requires k°4DK' sin 9 >> 1, or, since Dk* <<1, on account of
4\,

K'f sin 0 = kyl >>> 1 (5)

cog

Hasselmann and Schieler

Similarly, a wavenumber broadening Ak corresponds to a
frequency broadening of the Bragg line of order Aw= (dwg/dk’) Ak =
(wa/2k9) - Ak (ignoring capillary effects). The model assumes that
this is small compared with the Doppler shift w,=- 2K'+ u induced
by the facet velocity u. For u= O(weS), where weis a typical
carrier-wave frequency, this requires

DK! wk! /w, = DK! (k'k°)!"¢ >> 1

Substituting Dk*° << 1, this is equivalent to

Ki c(i! [oy >>> 4 (6)

Since k'/k° >> 1, the frequency condition (6) is less critical
than the corresponding condition (5) for the angular resolution. The
inequality (5) is normally fairly well satisfied at conventional radar
wave lengths for surface-wave heights of order 1 m and higher (except
for small depression angles, where the model breaks down, in any case
because of shadowing effects). For electromagnetic wave lengths
longer than about 1 m the inequality (5) is normally no longer valid,
even though the two-scale inequality (4) may still apply.

The total backscattered energy is obtained in the wave-facet
interaction model by summing over the contributions from all
scattering facets. Introducing a facet probability distribution p())
with respect to the five basic facet parameters = (A, o Nor Ngo Ngee) »
where

(AX, 2X53) = (u, » U5, U5) = facet velocity ee local
long-wave orbital velocity),

and (h4,X,) = (86 /8x, ,0C /8x,) = (n,,n,) = facet slope, the Doppler
spectrum is given by

Xap = { [ T3g5(wg - SW, - w¢)] p(d) dd (7)

where oy , W, represent, respectively, the Bragg cross section and
gravity-wave frequency in the facet reference frame.

To the modulated Doppler spectrum (7) of the first-order
Bragg field should be added the modulated spectrum of the zero'th
order field reflected from a plane facet, as described by the specular
reflexion model. However, this is important only near vertical
incidence and will be ignored in the following.

Experimentally, the probability distribution p(\) is found to
be approximately Gaussian, in accordance with the theoretical distri-

372

Radar Back-Seatter from the Sea Surface

bution for a random, linear gravity-wave field,

p(M) = (20)? |c[ exp {- 5 Con 2,) (8)

The covariance matrix Cjj can be evaluated from the surface-wave
spectrum and the linear wave solutions,

gk? /k g kk,/k |
—— O
——
g k,k,/k gkj/k !
ie ee ee oie ees ee (9)
gk - WK, = wk,
nn? 2
0 1 7 wk, k Ke
1
where Tykg/k= J F, (k)(kjkg/k) dk, etc.
We note that the facet Doppler shift ws, is correlated not
only with the facet velocity, but through the correlation (u,n,)
also with the facet slope,
(wai) = - 2k; (un) (10)

For small wave slopes, the factor in square parentheses in
Eq. (7) can be expanded in powers of nj. The integration can then
be carried out explicitly for each term of the expansion yielding a
solution of the form

Xqpl@s) = Xqhee) + Xgglma)

with

2
Xgl) = (alo > L- (wg stig) /2¢ 04) } 14 at Hace caer tal)

(2m( ut) v2

where q®, qg®, ... are polynomials in (ws = 60.) of order 1,92; .<<
| 9g
in the facet Slope,

g = damdfacsed | (2a), Aa + Meee (Sa) ca
f

Hasselmann and Schteler

An
25.8 ~ ~s
{5 (SH) says = S290 (2g) + ($e S88)

The subscript 0 refers to values at n= 0.

To lowest order, the Doppler spectra for vertical and hori-
zontal polarisation are identical Gaussian distributions with mean
frequency (w) = Swg and variance

( (w - day) =e) ="2 J Ky kgh Up Yq) + (ky) *(uy)t (14)
(£,m= 1,2)

The distribution represents an ensemble of Bragg lines of equal
energies displaced by their appropriate facet Doppler frequencies
Wee

The higher-order corrections qi Bs qs ye ole represent dis-
tortions of the Gaussian distribution due to the variations in energy
of the Bragg lines associated with variations in the carrier-wave
slope. These affect the shape of the Doppler spectrum through the
correlation between facet slopes and facet Doppler frequency,
Eq. (10). The degree of distortion depends on the depression angle
and polarisation. In the cross-polarised case, the zero 'th and first-
order terms disappear, since (T, wo = (8/8n; 5 Wo = 0, so that the

Vv
Doppler spectrum is non-Gaus sian already to lowest order.

Computations of the Doppler spectrum were made for a
Pierson-Moskowitz [16] spectrum using a half-plane cosine-to-the-
fourth spreading factor,

-4 4
ze ki exp {- B(w,/w)} for k, >0

Fy (k) =

with @ = 0.0081, B= 0.74 and wo= g/U, where U is the wind
velocity, aeeumed parallel to the x, axis. The same spreading
factor was taken for both scattering and carrier waves.

374

Radar Back-Seatter from the Sea Surface

For a Pierson-Moskowitz spectrum, ( wos) Satgeren
(wn;) ~ U. The slope moments (njn;) diverge logarithmically at
high wavenumbers. To obtain finite (njn;) , the "carrier-wave"
spectrum was cut off at an upper wavenumber k /10. The exact
position of the cut-off is not critical for the evaluation of (njnj) ’
and the slope moments themselves enter only rather weakly in the
second-order term q§$ of the expansion (11). However, the
existence of a divergence as such points to a conceptual difficulty of
the wave-facet interaction model. It appears that for an asymptotic
ic? spectrum the carrier-wave region of the spectrum cannot be
rigorously separated from the Bragg-scattering region.

Figure 5 shows the computed half-power bandwidths for the
lowest-order Gaussian spectrum as a function of wave height. The
values compare well with measurements by Valenzuela and Laing
[20] <

Deviations from the Gaussian form due to the higher-order
corrections ay and q& are represented in Figs. 6 - 9 in terms of
the mean frequency (w)/wg and the frequency bandwidth
{ (w - ( w) ie) / (ae) , normalised by their appropriate values for the
zero'th order Gaussian spectrum.

The strongest correction is found for the mean frequency,
particularly for horizontal polarisation. The dependence on depres-
silon-angle and polarisation, shown in Fig. 6 for U = 20 m/s, is found
to be very similar at all wind speeds. The absolute values of the
frequency shifts increase approximately linearly with wind speed,
Fig. 7. Qualitatively, the polarisation and wind-speed dependence of
the mean Doppler frequency are in agreement with measurements
made by Hicks et al. [13] at low depression angles of about 5°, How-
ever, the theory is not strictly applicable in this case on account of
shadowing effects.

The bandwidth corrections (Figs. 8 and 9) remain rather
small for depression angles less than 45° and limited azimuth angles
Ww relative to the wind. Larger deviations in the cross-wind direc-
tions depend strongly on the spreading factors, which are rather un-
certain for these angles. The experimental dependence of the Doppler
bandwidth on radar frequency and polarisation [ 20] tends to be some-
what larger and have a different trend than the corrections shown in
Figs. 8 and 9. Valenzuela and Laing [ 20] suggest that these effects
may be due partly to spray. To a fair approximation, the observed
bandwidths can be represented for small and intermediate angles
and @ by the zero'th order Gaussian bandwidth.

Both the bandwidth and mean frequency vary significantly with
wave height and can therefore be used for estimates of sea state.
For the one-parametrical family of spectra considered in the present
example, the two estimates are not independent. However, in general
the mean square bandwidth ( (w- (w) *) ~ (we) (Eq. 14) and the mean

375

Fig. 5.

Hasselmann and Schietler

P-BAND a

L-BAND 4

C-BAND o

X- BAND

OPEN POINTS (VERTICAL)
CLOSED POINTS (HORIZONTAL)

hp.b [m/s]

1 2 3 4

Comparison of theoretical half-power bandwidths (h.p.b.)
for zero'thorder Gaussian spectrum with measurements by
Valenzuela and Lang 20 . Doppler frequencies are in
units of equivalent velocities Ug=w,/2k'. Theoretically,
a Pierson-Moskowitz spectrum with cos*w spreading
function yields

2
h.p.b. [m/s] = 1.06} £926 (4cos*p +1) +sin?ol'* (H,,[ m] Nes

where the significant wave height Hy * Vena ee 0.209 u*/g.
The computations were made for w = 09, 6 = 20°,

376

Radar Baeck-Seatter from the Sea Surface

u=20m/s

HH
<W> i (4)
Wg
2 2
yz0°
(4) a1)
1 0 1
15 30 45e /60 15 30 45 60
2 2
= (2)
w=20°
1 1
15 30 45 60 15 30 45 _60
(4)

(1)

15 30 45 60 1 30 45 60

5
2 2
(4)
y =60° | SS Wy 0)
(4)
15

1§ 30 45 60 30 45 60

2 1
(1)
(4)
(4)

15 30 45 60
Q -—e
Fig. 6. Ratios of mean Doppler frequency (w) to Bragg frequency

wg for the wave-facet interaction model at windspeed

U = 20 m/s. The indices 1, 2, 3, 4 referto P, L, C and
X bands, respectively. The computations include terms up
to order q§ in the expansion (11). To this approximation,
the cross polarised case yields (w) = wg

S1f

Hasselmann and Schieler

Fig. 7. Dependence of (w)/wg on wind speed U for 06 = 30°,
y= 0°, An approximately linear variation is found for all
depression angles 9 and azimuth angles \W.

378

Radar Back-Seatter from the Sea Surface

u=4m/s
VV HH VH
2 2 2
=0? 4
¥ 0 ‘ (4) , (4) tno RE
1) (1)
16 30 45 60 75 °®»+»15 30 45 60 75 45 "30. 46 160° 95
2 2 2
y=20° a) (4) a
1 1 1 (4)
(1) Be
A
A
Sle. 15 30 45 60 75 15 30 45 60 75 5 30 45 60 75
ae 42 2 3
V
(3) 1 (4)
: ‘ (1)
(1) (1)
y=40
4)
15 30 45 60 75 15 30 45 60 75 15 30 45 60 75

15 30 45 60 75

15 30 45 60 75 16 30 45 60 75 15 30 45 60 75

Fig. 8. Frequency variance of the Doppler spectrym computed to
order q5, normalised by the variance (or) of the zero'th
order Gaussian distribution (U = 4 m/s).

379

Hasselmann and Schtieler

u=20 m/s
1S 30 45 60 75 45 60 75
, = = eS
15 30 45 60 75 15 30 45 60 75 30 45 60
A
a F (1)
Vv i) a.
W =40°
45 60 75 45 60 75 15 30 45 60 75
(4)
w=60° (4)
5 30 45 60 75 15 30 45 60 75 5 30 45 60 75
3 3 3
(1)
2 (1) (1)
w=80°
(4) (4) (4)
15 30 45 60 75 15 30 45 60 75 15 30 45 60 75
e@ —

Fig. 9. Same as Fig. 8 with wind speed U = 20 m/s

380

Radar Back-Seatter from the Sea Surface

product (w,n;) (Eq. 10), which is responsible for most of the mean-
frequency variation, depend on differently weighted moments of the
gravity-wave spectrum. The two Doppler parameters can therefore
be used to obtain independent estimates of two sea-state parameters
--for example, the mean wave height and mean wave period.

IV. HIGHER-ORDER WAVE-WAVE INTERACTIONS

For HF waves longer than about 10 m, the Bragg model can
be generalised by straightforward extension of the wave-wave inter-
action expansion to higher order. In this case, the perturbation
parameter kjf is normally a small quantity even when ¢ is defined
as the surface displacement of the complete wave field, and there is
no need to consider the long waves of high amplitude separately.

In fact, the wave-facet interaction model is not applicable for HF
waves on account of the angular resolution condition (5). The in-
equality kx{ << 1 and condition (5) are mutually exclusive, repre-
senting a wave length gap between the wave-facet and higher-order
wave-wave interaction models extending from a few fractions ofa
meter to about 10 meters.

At second order, the wave-wave interaction analysis yields
scattered waves through interactions with pairs of gravity-wave
somponents a,b satisfying the next-order Bragg conditions

i a b= 4°
i + og? + ok? =
w togw, tow, =, (0,50, = +) (15)

The second-order Doppler spectrum x? (w4) is obtained by
summing over all pairs of surface waves yielding a backscattered
component with the appropriate horizontal wavenumber k‘®= - k'
and frequency w, = 0; + Wg, oy

x?) (wg) = » iY TF, (k°)Fy (ik?) 8(coq- oy 4-004) dk (16)
FD
where k?=-o (2k! + oak") (Eq. 15). pi?) is a scattering function
determfned by thé’ second-order coupling coefficients occurring in
the expansion of the boundary conditions about the undisturbed plane
surface (cf. Ref. (18)). (The polarisation indices are irrelevant for
the following discussion and are ignored.)

Te tenhiicant sea-state signatures are found only for the Doppler

spectra and not the cross sections. On integrating Eq. (11) with
respect to frequency the dependence on the moments ( we) and
(wn; ) disappears, leaving only a weak sea-state dependence
through the slope moments (njnj).

381

Hasselmann and Schteler

The scattering function T'?) includes both electromagnetic

and hydrodynamic interactions at the free surface. For wave lengths
in the HF range and longer, the hydrodynamic interactions can
probably be described to fair approximation by classical hydrody-
namical theory, independent of the effects of wave breaking.

Equation (16) represents the random-field expression of nonlinear
effects such as nonsinusoidal wave forms!, nonlinear phase velocities,
etc., that have been variously suggested as explanation of the observed
side bands of HF Doppler spectra.

In the limit of an incident wave short compared with the
principal waves of the sea, the dominant interactions at finite de-
pression angles are electromagnetic. The largest contributions to
the integral in (16) arise in this case from interactions in which one
of the gravity-wave components, say k°, lies near to the peak of the
spectrum. Since k®° <<k', the second component k? is then approxi-
mately equal to the Bragg component, ok ye - on (ete Eq. (15) and
Fig. 10). The side condition wy = ogwg + cpwp = const (expressed by
the 6-function in the integral) defines an integration curve in the k‘
plane which is given approximately by the circle k° = const. This
follows by noting that, on account of Eq. (15), the variation 6k°
corresponds to an equally large variation + 6k”. But for k°<%k?,
the associated frequency variation 6w, is generally small compared
with the variation 609, since dw,/dk®<< dwg/dk®. Hence the side
condition wg = const reduces to wg= const. It is shown below that
at finite depression angles, T?) ae independent of k° for k* << 1,
and the integration over the directions of k° for fixed k° can then
be readily carried out,. yielding

)+ (2)- (

7 (4) = x? (wy) + x Wg)

where (17)

x2) (44) = 27) F (-s2k!) [Eg(wg- surg) + Ey (stag - w4)]

and = Eg(w) is the gne-dimensional frequency spectrum of the wave
field, with (7) = » ~g(w) dw. (The factor 2 arises through inter-
change of the components a and b in Fig. 10.)

Thus each Bragg line appears as the carrier of a second-
order, two-sided image of the surface-wave frequency spectrum.
Physically, the Doppler continuum arises, as in the case of the wave-
facet interaction model, through the modulation of the first-order

+

These include the often invoked "higher interference orders"
occurring in the Bragg scattering by a lattice. They are generated
only if the periodic scattering field is not a purely sinusoidal dis-
turbance but contains higher harmonics.

382

Radar Back-Seatter from the Sea Surface

Bragg line

Doppler spectrum

Fig. 10. Upper panel. Interacting gravity-wavenumbers k%, ik?
for a spectral-peak wavenumber small compared with the
Bragg wavenumber 2k!. Strong contributions to the
second-order Doppler spectrum y) (w,) arise when ee
is close to the spectral peak. Contour lines indicate
curves of constant spectral energy.
Lower panel. The associated second-order Doppler
spectrum x'*) consists of two images of the wave fre-
quency spectrum Eg reflected on either side of the
Bragg line.

Bragg field by long surface waves of high amplitude. However, in
the present case the Doppler shift is not determined by the frequency
shift w= - 2k'u induced by the long-wave orbital velocities, but
rather by the {ntrinsic long-wave frequencies wg. Each low-
frequency component splits the first-order Bragg line

wi) = w, + op, into two lines w,"' = wi! + wo. Since u® Cw,, the
regions of validity for the wave-facet and wave-wave interaction
models may also be expressed, respectively, as wg <<w¢ and

Wg >> Wee

383

Hasselmann and Schieler

A useful feature of the relation (17) is that it defines the
surface-wave spectrum in absolute energy units independent of
electromagnetic calibration factors, which are difficult to establish
for long-range ionospheric mode propagation. Using Eqs. (3) and
(2) to eliminate the surface-wave spectrum at the Bragg wavenumber,
Eq. (17) becomes

ll) mah (on
Eg(wg - stg) + Eg(swg - wy) = ol =e (s = +) (18)

where ¢'')8= Ix 129. (taa) ae is the energy of the first-order Bragg
line. The ratio T'') /2T(2) can be determined from theory, and
x8 and ef"s may be measured in arbitrary age units.

In the relevant limit k°<<k’, tT!) /at® may be deduced
from the picture of a short scattering wave bb = Ap exp {i(kk>x - wt) }
riding on a long carrier wave fq = Ag exp {i(k® x- wet)} (whf{th is
now, howeyer, assumed to satisfy the wave-wave interaction con-
Aion Aoks << 1, rather than the wave-facet interaction condition
(5)). For small slopes Agk* << 1, the principal effect of the carrier
wave is presumably to alter the phase of the scattered field by raising
and lowering the local mean reference surface of the short scattering
waves!. Thus if the first-order backscattered wave in the absence
of the carrier wave is of the form

gl!) (1)

8
= Ci’ ASA, exp (i(k! + ak? )x - i(w, + opw,)t + ikyx,}

I
where Aj is the amplitude of the incident field, c' is im first-
order coupling coefficient, and it + gpk? & - k', k§~ - kg
the modulated scattered wave in the presence cot the Patiee wave
will be given approximately by

acikste ole (1 + 2ikle, Jo") =o + gi (19)

9 =
Thus
go = Cl) asaya; exp {- ik'x - i(wj; t+ wg)t - ikyxs} (20)

with C® = 2ikic . Expressed in terms of a continuous energy
spectrum, this is readily found to correspond to a scattering function
ratio

T

A more detailed investigation indicates that slope effects can be
ignored if k° << k,=k' sin 0.

384

Radar Back-Secatter from the Sea Surface

= (ky) (21)

For small depression angles (ks << ie), the effect of the
carrier-wave slope becomes comparable with the phase shift induced
by the vertical displacement, and the relations (20), (21) should be
modified to include additional terms dependent on k®. However,
this requires a more detailed investigation of the electromagnetic
and hydrodynamic interactions.

Examples of Doppler spectra obtained by Ward [ 22] from the
sea echo of 21.840 MHz (14 m) waves at ranges near 3000 km are
shown in Fig. 11. The analysis was based on short records of one
minute duration, so that the continuum is poorly resolved statistically
and individual spectra vary strongly. However, there is some indi-
cation of two side-band structures appearing on either side ofa
central Bragg peak. Theoretically, the Bragg line should lie at
0.48 Hz, which agrees well with the central peak of the first spectrum
shown, but is somewhat to the left of the main peaks in the other
cases. The displacement of the side lobes relative to the Bragg peak
is of the order 0.1 Hz expected for typical ocean-wave frequencies.

The ratio of the side-band energy e2) = if x (2) (wy) dwy to the
energy e) of the Bragg line is given according to Eqs. (18) and
(21) by

ive, »2
(2) /e") = 4(kiy(¢ )
Ward estimates a depression angle of 12°, which yields
e) 7" = 0.036 ((6[m] )*)

The observed ratios of order unity correspond to root mean square
wave heights of about 5 m, which appear rather high, but not im-
possible.

More plausible estimates of the wave height may have
resulted from a more accurate determination of the scattering
function ratio T By aia at small depression angles. Contamination
of the observed spectra by ionospheric Doppler shifts may be an
alternative explanation of the high ratios €(2) /é") - A spurious inter-
action between the Bragg line and the low frequency ionospheric
Doppler spectrum could also have been introduced in the present
experiment by the data analysis, since the Doppler spectra appear
to have been computed -- as is often done -- from the time series of
Note added in proof: A detailed analysis has recently been carried

out by D. E, Barrick "Dependence of Second-Order Doppler Side
Bands in HF Sea Echo on Sea State," to appear in 1971 G-AP
Internat. Symp. Digest.

385

Hasselmann and Schieler

4 TITS
1 jlsvecaty
UU rere

easbittH ith:

386

ges of 2700 km (18 ms) and

(14 m) sea echo at ran

Doppler spectra of 21.84 MHz
3000 km (20 ms), from Ward [ 22].

Figo 4.

Radar Back-Secatter from the Sea Surface

the signal phase (or phase cosine), which is nonlinearly related to
the complex signal amplitude. More detailed investigations using
longer time series are needed to decide. whether the ocean wave
spectrum can be extracted from the Doppler spectrum of long range
HF sea echo in the presence of unavoidable fonospheric noise.

ACKNOWLEDGMENT

This work was supported in part by the Office of Naval
Research under Contract No. ONR N00014-69-C-0057.

REFERENCES

14. Barnett, T. P., "Generation, dissipation and prediction of wind
waves," J. Geophys. Res., 73, 513-534, 1968.

2. Barnett, T. P., Holland, C. H. Jr. and Yager, P., “General
technique for wind-wave prediction with application to the
S. China Sea," Westinghouse Res. Lab. Rep., June, 1969.

3. Barrick, D. E. and Peake, W. H., "A review of scattering from
surfaces with different roughness scales," Radio Sci., 3,
865-868, 1968.

4. Bass, F. G., Fuks, I. M., Kalmykov, A. I., Ostrovsky, I. E.,
and Rosenberg, A. D., "Very high frequency radiowave
scattering by a disturbed sea surface," IEEE Trans.,
AP-16, 554-568, 1968.

5. Cox, C. M. and Munk, W. H., "Measurement of the roughness
of the sea surface from photographs of the sun's glitter,"
J. Opt. Soc. Am., 44, 838-850, 1954.

6. Crombie, D. P., "Doppler spectrum of sea echo at 13.56 mc/s,"
Nature, 175, 681-682, 1955.

ai-Daley; J. ‘C., Ransone, J. Ts Jr, Burkett, J. A. and Duncan,
J. R., "Sea-clutter measurements on four frequencies,"
Nav. Res. Lab. Rep. 6806, 1968.

8. Daley, J./C., Ransone; J. T.,Jre, Burkett, J. A. and Duncan,

J. R., “Upwind-downwind-crosswind sea-clutter measure-
ments," Nav. Res. Lab. Rep. 6881, 1969.

9. Ewing, G. C., ed., Oceanography from Space, Woods Hole
Oceanogr. Inst., Ref. No. pe A0, 1965.

10. Guinard, N. W. and Daley, J. C., "An experimental study ofa
sea clutter model," Proc. IEEE, 58, 543-550, 1970,

387

Lis

172

13.

14,

15.

16,

17.

18.

19.

20.

21.

22.

23.

Hasselmann and Sehteler

Hasselmann, K., "On the mass and momentum transfer between
short gravity waves and larger-scale motions," J. Fluid
Mech. , 50, 189, 1971.

Hasselmann, K., "Determination of ocean wave spectra from
Doppler radio return from the sea surface," Nature, 229,
16-17, 1971.

Hicks, B. L., Knable, N., Kavaly, J. J., Newell, Grs-,
Ruina, J. P. and Sherwin, C. W., "The spectrum of X-band
radiation backscattered from the sea surface," J. Geophys.
Res. , 65, 825-837, 1969.

Longuet-Higgins, M. S., "A nonlinear mechanism for the genera-
tion of sea waves," Proc. Roy. Soc. A. 311, 371-389, 1969.

Munk, W. H. and Nierenberg, W. A., "High frequency radar sea
return and the Phillips saturation constant," Nature, 224,
1285, 1969.

Pierson, W. J. and Moskowitz, L., "A proposed spectral form
for fully developed wind seas based on the similarity theory
of S. A. Kitaigorodskii," J. Geophys. Res., 69, 5181-5190,
1964. ears

Pierson, W. J., Tick, L. J. and Baer, L., "Computer based
procedure for preparing global wave forecasts and w nd field
analysis capable of using wave data obtained by a space craft,"
6th Naval Hydrodynamic Symposium, Washington, Office of
Naval Res., Washington, D. C., 1966.

Rice, S. O., "Reflection of electromagnetic waves from slightly
rough surfaces," Comm. Pure Appl. Math., 4, 351-378, 1951.

Semenov, B., "An approximate calculation of scattering on the
perturbed sea surface," IVUZ Radiofizika (USSR), 9, 876-
887, 1966.

Valenzuela, G. R. and Laing, M. B., "Study of Doppler spectra
of radar sea echo," J. Geophys. Res., 75, 551-563, 1970.

Valenzuela, G. R., Laing, M. B. and Daley, J. C., "Ocean
spectra for the high frequency waves from airborne radar
measurements," 1970 (subm. to J. Mar. Res.).

Ward, J. F., "Power spectra from ocean movements measured
remotely by ionospheric radar backscatter," Nature, 223,
1325-1330, 1969.

Wright, J. W., "A new model for sea clutter," IEEE Trans.
AP-16, 217-2235 1968.

388

INTERACTION BETWEEN GRAVITY WAVES
AND FINITE TURBULENT FLOW FIELDS

Daniel Savitsky
Stevens Instttute of Technology
Hoboken, New Jersey

ABSTRACT

A laboratory study of the interaction of deep water
gravity waves progressing into a turbulent flow field
produced by a finite width grid towed in a wide tank
showed wave height attenuation of nearly 90% in the
grid wake and wave height amplifications of nearly 75%
in the still water outside the wake. The transverse
gradient of longitudinal flow in the wake was predom-
inantly responsible for the large wave deformations
and precluded an evaluation of direct turbulence effects.

A simple, analytical solution using wave refraction,
diffraction and superposition concepts is developed
which qualitatively reproduces the measured results.

I. INTRODUCTION

As gravity waves progress from their source of origin, they
encounter a wariety of ocean environments which may interfere with
their ordered motion and, consequently, alter the amplitude and
direction of the wave system. Although an extensive literature exists
on the mechanism of wave generation and their subsequent propaga-
tion through still water or a uniform flow, only recently has some
attention been given to waves moving through a non-uniform flow --
and these have been restricted to relatively weak velocity gradients
normal to the wave direction.

In a realistic ocean environment, gravity waves may encounter
regions of turbulent flow, particularly in the upper layers. These
oceanic turbulent flow fields can be developed by various geophysical
mechanisms. For example, the action of unsteady wind shear stresses
exerted against the surface of the sea; the breaking of wave crests

389

Savittsky

resulting in "splash turbulence" penetrating into the upper layers of
the water; turbulent fields set up in intense currents; turbulence
developed by high velocity, high Reynolds number flows in a tidal
channel; ship wakes; etc. In each case, it is expected that wave
attenuation will result from the interaction between the turbulent flow
fields and wave motion. Such attenuation is of importance in develop-
ing relatively "quiet" local areas in the sea for launching or recovery
of small craft or submarines, or in tracing the progress of, say,

one storm passing through the intensive turbulence of another storm.

Phillips [1959] presents a theoretical study of the properties
of waves on the free surface of a liquid in turbulent motion where the
intensity of the turbulence is sufficiently small to preclude wave
generation in itself and where the mean velocity of the flow is zero.
There are two types of possible interaction, each of which results in
the attenuation of the incident wave. One is an "eddy viscosity inter-
action" in which wave energy is transferred from the wave motion
through a stretching of the vortex filaments in the turbulence which
tends to increase w*, the mean square vorticity associated with the
turbulence itself, This straining process is of second order in wave
height-length ratio and, hence, should be important for steep waves
and when the turbulence scale is much less than that of the waves.
The second type of interaction is a scattering phenomenon where
random velocity fluctuations in the turbulence field will result in the
convective distortion of the wave front, and produce a broad spectrum
of scattered waves. This scattering effect is of first order in wave
height-length ratio and, hence, predominates for waves of small
slope. Phillips shows that, under typical conditions in the open sea,
the attenuation from scattering will be greater than that from direct
viscous dissipation for wave lengths greater than about 10 ft.

An experimental study was undertaken at the Davidson Labora-
tory, Stevens Institute of Technology, to investigate the interaction
between mechanically generated progressive gravity waves and a
controlled turbulence field developed by towing suitable grids ina
towing tank. Since field measurements by Stewart and Grant [ 1962]
supported the applicability of the Kolmogoroff hypothesis (that the
statistical structure of turbulence has a universal form) to turbulence
near the sea surface in the presence of waves, it was believed that
grid-generated turbulence (known to satisfy the Kolmogoroff hypothesis)
would indeed be representative of ocean turbulence on a model scale.
Two experimental studies were undertaken. The first used a grid
which spanned the width of a 12 ft wide towing tank and was towed in
the direction of wave celerity at speeds less than the group velocity
of the regular wave lengths generated by a plunger type wavemaker.

In these studies, the test waves overtook and passed through the
turbulence wake and grid. This so-called one-dimensional grid study
was made in an attempt to develop a turbulent wake with uniform

mean flow across any transverse section aft of the grid. Unfortunately,
as will be subsequently discussed, a uniform flow field was not de-
veloped near the outer edges of the grid wake and this seriously

390

Gravity Waves and Fintte Turbulent Flow Ftelds

influenced the test results. The other series of experimental studies
involved towing a 3-ft wide grid in a 75-ft wide towing tank. The in-
tent of these tests was to allow any scattered wake system to be
defracted outside the turbulence patch. However, the finite width
grid also produced a pronounced longitudinal mean flow velocity
gradient in transverse sections through the wake. Thus, in these
latter tests, the generated waves were simultaneously subjected to
three modification effects: (1) dissipation due to eddy viscosity;

(2) scattering due to turbulent convective distortion of the wave front
and (3) deformation of the wave due to mean flow velocity gradients.

Measurements were made of the wave deformation in the
wakes of both the one- and two-dimensional grids. An analysis of
these results indicated that the velocity gradients in the wakes had a
dominating effect on the wave deformation and thus, unfortunately,
precluded a reliable evaluation of the possible dissipative or scatter-
ing action of the turbulence field upon the incident wave. The studies
are, nevertheless, of importance since they provide unique results,
obtained under controlled laboratory conditions, describing the pro-
nounced distortion of a deep water wave when encountering sharp
current gradients, either naturally existing or artificially produced.
It is shown that the wave distortion can be such as to provide locally
areas of reduced wave motion which can be beneficial in launching
or retrieving small craft or submersibles from a mother ship at sea.

The experimental results are described in some detail and an
elementary analytical model is developed which, using the combined
mechanics of wave refraction, defraction and superposition, at least
qualitatively reproduces the features of the test results and, perhaps
more important, describes a possible physical mechanism respon-
sible for the observed large wave deformations.

These studies were supported by the Fluid Dynamics Branch
of the Office of Naval Research, Department of the Navy, under
Contract NR-062-254, Nonr263(36). They formed the basis for a
dissertation submitted to the Graduate Division of the School of
Engineering and Science in partial fulfillment of the requirements
for the degree of Ph.D. at New York University.

Il EXPERIMENTAL PROCEDURES

Turbulence-generating grids have been used with great suc-
cess in advancing the knowledge of turbulence in air flows, but have
been used only occasionally in hydrodynamics -- particularly in
towing tanks where a grid must be towed in quiet water to generate
a turbulence field. Taylor [1935] has shown that disturbances
generated in the wake of a grid transform rapidly into a quasi-
isotropic turbulent field whether the grid is towed in quiet air or an
airstream passes through the grid.

391

Savttsky

In the present task, vertical turbulence grids of finite draft
and two mesh sizes were towed at various constant speeds in Tank
No. 2 and 3 of the Davidson Laboratory in a direction normal to the
plane of the grid. Regular waves, generated by a plunger type wave-
maker in quiet water, traveled in the same direction as the grid tow
(with initial crest lines parallel to the grid) progressed through the
turbulent wake and grid into quiet water beyond the grid. Waves of
various constant length and height were generated such that the
group velocity of each regular wave was greater than the grid veloc-
ity. The wave lengths and water depth were such that deep water
gravity waves were generated. Wave amplitudes were measured
by resistance type wave wires which penetrated the fluid surface.
Several of the wave wires were towed ahead and behind the grid (at
the grid speed) while others were stationary and located both in and
outside of the grid wake. The outputs of these wave probes were
simultaneously recorded on a "Viscorder" oscillograph tape.

The details of test procedure, grid characteristics, and test
conditions for the one-dimensional and two-dimensional turbulence
grid studies are described separately below. Common to both
studies was the observation that, for the grid sizes and grid veloc-
ities considered, the combination of physical grid and turbulent
wake in smooth water did not produce a measurable wave system of
its own. In fact, soon after passage of the grid and wake relative
to a fixed point in the test tank, the water surface appeared unusually
still. Further, the grid solidity was small enough that, when sta-
tionary, it did not noticeably affect the wave forms which passed
through the stationary grid. Neither was there a measurable wave
reflection from, the grid,

One-Dimensional Grid Studies

The one-dimensional grid studies were conducted in Tank
No. 3 of the Davidson Laboratory. This tank is 300 ft long, 12 ft
wide and has a water depth of 6 ft. A plunger type wavemaker is
located at one end of the tank and a slotted beach of 15° slope is
located at the opposite end to absorb the wave energy with minimum
reflection.

Grid Characteristics: A turbulence grid 11.5 ft wide spanned
the tank width, penetrated the water surface to a depth of 1.6 ft,
was attached to a standard carriage and towed in a direction away
from the wavemaker. Figure 1 shows the test setup. Two mesh
sizes were tested; one had amesh M = 0.36 ft and was made of
crossed square wooden slats 0.80 inches wide; the other hada
mesh M = 0.71 ft and was made of crossed square wooden slats
1.60 inches wide. Thus, in both cases, the grid solidity was
constant and equal to S= 0.40. The grid was towed at speeds of
V = 1.0 and V= 1.7 ft/sec. The hydrodynamic drag and Reynolds
No. of the grid (Re, = VM/w) for these conditions are:

392

Gravity Waves and Finite Turbulent Flow Fields

PlzH Teuojsueurtq euo dn-yeg 3807, JF “81a

393

Savttsky

Mesh Grid Velocity Drag Reynolds No.
0.36 £t 1 ft/sec 13.2 lbs 28,000
0.26 vee ff 37.9 49,000
ay | 4 ie yey 56,800
Oa71 ay 6 37.9 96,500

Measurements were initially made of the mean value of the
longitudinal velocity (in the grid direction) of the grid wake at the
centerline of the grid and at a depth of 0.80 feet below the water
surface. At a distance of 10.0 ft aft of the grid, the wake velocity
was 0.40 V and decreased slowly with distance aft of the grid -- at
a distance 20.0 ft aft of the grid, the mean velocity of the wake was
0.36 V. In these initial tests, a straight line of confetti was sprinkled
across the 12 ft width of the tank parallel to the plane of the grid.
Visual observations of this reference line after grid passage showed
that the confetti moved essentially in one straight line parallel to
the grid, thus indicating a lack of noticeable velocity gradients -- at
least on the free surface. An analysis of the wave distortion data in
this wake yielded anomalous results (these will be discussed ina
subsequent section) that could not be explained by the assumption of
a uniform longitudinal mean flow through transverse sections in the
grid wake. Hence, a detailed survey was then made of the mean
flow at distances of 10 ft and 20 ft aft of the grid. These results are
shown in Fig. 2 which presents a plot of longitudinal mean flow (V,)
versus transverse distance from the grid centerline at a probe depth
of 10 inches below the water surface. The wake velocities (V,) are
normalized on the basis of grid speed (V). It is clearly seen that
the mean flow in the wake is essentially constant for a distance of
approximately 5 ft from the grid centerline but then rapidly decreases
between this point and the tank wall. The significance of this local
velocity gradient will be subsequently discussed.

The wind tunnel results of Dryden [ 1937], who examined the
turbulence aft of a rectangular grid having a mesh size M= 0.41 ft
at a nearly similar Reynolds number, show that the turbulent veloc-
ity fluctuations u' as a function of distance, X, aft of the grid are:

V x
= +
aay) a

Thus, for a distance 10 ft aft of the 0.36' mesh grid x/M = 27.7

and u' = V/37.7 or approximately 3% of the mean flow. At a distance
of 20 mesh lengths aft of the grid, wind tunnel experiments have
shown the establishment of quasi-isotropic turbulence.

394

Gravity Waves and Fintte Turbulent Flow Ftelds

GRID WIDTH =1II.50; DRAFT=20'; MESH SIZE=5.4"
(GRID TOWED IN [2 FT WIDE TANK)
VELOCITY PROBE AT 10° DRAFT
Vw=WAKE VELOCITY ; V=GRID VELOCITY

0.5
TANK
0.4 WALL
|
Vw 23 |
V oe 10 FT AFT OF GRID |
0.1 |

20 FT AFT OF GRID

ie) | 2 3 4 5 6
€ y=LATERAL DISTANCE FROM GRID @ , FEET

Fig. 2 Longitudinal Velocity Distribution in Grid Wake

Wave Height Probes: Wave heights were measured by re-
sistance type wave wires penetrating through the water surface.
The position of the wave wires relative to the grid are shown in
Fig. 1. It is seen that wave wires moving with the grid were
located 12 ft ahead of and along the grid centerline; 13.25 ft and
16.50 ft aft of and along the grid centerline; and one located 13.25 ft
aft and 4 ft transverse to the grid centerline. The wave wires 13.25 ft
and 16.50 ft aft of the grid were used to obtain a measure of the ap-
parent wave length in the turbulence field while the pair of wires
4 ft apart in the transverse plane 13.25 ft aft of the grid were used
to measure any deformation of the wave crest line as it progressed
through the turbulence. A stationary wave wire was located 60 ft
forward of the wavemaker and was used to examine the regularity of
the amplitude and period of the generated incident wave.

395

Savttsky

A range of wave heights and lengths used in these tests were
as follows:

Wave Wave Wave Group Wave
Length Period Celerity Velocity Height
hj dt T, sec V2 ft/sec Vg, ft/sec His ff
20,0 0.025 32:20 1.60 0.05
2.0 0.625 3.20 1.60 0.10
5.0 0.763 a 93 fost 0.10
4.0 0.885 4.52 2.26 0.05
4.0 0.885 4; 52 2.26 0.10
6.0 1.080 5.55 2.18 0305
6.0 1.080 55 2.18 0.10
8.0 1.250 6. 40 52 20 0.04
8.0 1.250 6.40 eA) 0.09

Test Procedure: Several experimental procedures were
used in these studies. In one group of tests, the grid was held
stationary 70 ft forward of the wavemaker until several waves had
passed through the grid. The grid was then towed and wave measure-
ments were made with the moving wave wires. For certain runs,
after approximately 50 - 60 ft of grid tow, the aft moving wave
wire (16.50 ft aft) was released from the tow and remained stationary
in the tank. Thus, wave height measurements were taken both at a
fixed position relative to the moving grid and at a fixed position in
the tank (variable position relative to the grid). The other test pro-
cedure was to first tow the grid for a distance of approximately
50 ft which developed a turbulent wake and then start the waves which
ran through the wake and overtook the moving grid. This technique
avoided the possibility of a secondary wave formation as the incident
wave ran through the moving grid. It was established that the results
obtained with both test procedures were essentially similar.

Two-Dimensional Grid Studies

The two-dimensional grid studies were conducted in Tank
No. 2 of the Davidson Laboratory. This tank is 75 ft square and has
a water depth of 4.5 ft. A plunger mechanical type wavemaker spans
one side of the tank and a sloping beach is installed on the opposite
end to absorb the generated wave energy.

Grid Characteristics: Two turbulence grids, one 3 ft wide
and another 5.5 ft wide, were separately towed in a direction away
from the wavemaker. The grid centerline was 17 ft from one edge
of the tank. Figure 3 shows the test setup. As in the one-dimen-
sional tests, two mesh sizes -- M = 0.36 ft and M= 0.71 ft -- were

396

Gravity Waves and Finite Turbulent Flow Fields

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397 °

Savittsky

tested. The grids were constructed of crossed square wooden slats
0.80 inches wide. The solidity, towing speeds and Reynolds number
of the grids were the same as for the one-dimensional tests pre-
viously described. The hydrodynamic drag of the 0.36 ft mesh grid
was 3.2 1bs and 9.25 lbs at tow speeds of 1.0 and 1.7 ft/sec re-
spectively at a grid draft of 1.67 ft. Ata grid draft of 0.84 ft, the
hydrodynamic drags for this grid were 1.5 1bs and 4. 33 lbs at 1.0
and 1.7 ft/sec. It is to be noticed that the grid drag increased as
the square of the speed for all cases.

Measurements were made of the mean values of the longitu-
dinal velocities at depths of 0.80 and 0.40 ft in the grid wake across
several transverse sections aft of the 3 ft wide, 0.36 ft mesh grid
with a 1.67 and 0.80 ft draft. A plot of the ratio of wake velocity to
grid velocity is given in Fig. 4. These velocity ratios were the
same for both grid drafts and towing speeds.

It is seen that there is a slow attenuation of velocity with
distance aft of grid. Further, there is also a slow lateral spreading
of the wake area. It is interesting to note that the wake velovities
are all in the direction of grid tow. Surveys of the velocity field up
to 5 ft from the grid centerline did not indicate a reverse flow.
Visual observations did indicate a reverse flow along the bottom of
the test tank.

An empirical formulation was established to represent the
wake velocity Vy as a function of distance, x, aft of the grid anda
distance, y, measured from the grid centerline normal to the x-
direction. The wake equation is given by:

ay .
= = [0.45 - 0.00745x] le Pukka oe aaa |

where x and y are in units of feet.

The above formulat was developed for use in the analysis
of wave deformation for reguier waves running into a velocity gradi-
ent. This analysis is presented in a subsequent section of this
report.

Wave Height Probes: The location of the wave amplitude
measuring probes are shown in Fig. 3. It is seen that four probes
were towed with the grid and 7 inches off its centerline; one 3 ft
forward and three others 1 ft, 3 ft and 6 ft aft of the grid. In addi-
tion, seven stationary probes were located in a transverse line
normal to the direction of ,rid tow and at distance 38 ft and then
20 ft ahead of the wavemaker. It will be noted that one of the
stationary wave wires was directly on the grid centerline. This
installation was accomplished by mounting the probe on the floor
of the tank and providing a slot through the grid which passed over

398

Gravity Waves and Finite Turbulent Flow Fields

GRID WIDTH=36', DRAFT=20"; MESH SIZE =2.7"
(GRID TOWED IN 75 FT WIDE TANK)
VELOCITY PROBE AT IO" DRAFT

y 8

V de .062 )
; = -|0.45-0.00745 «| E MOORES
GRID

Vw =WAKE VELOCITY, FT/SEC
V =GRID VELOCITY, FT/SEC

ig X= 0.58 FT

Vw X=20.5 FT

DISTANCE AFT OF GRID, FEET
°
EDGE OF MAXIMUM VELOCITY
052.0% ERO VELOCITY)
EDGE OF GRID WAKE (
<

Vw X=37.5 FT

20
BS
0....2,, 4, §6.);8,, 10

LATERAL DISTANCE FROM GRID ¢ , FEET

Fig. 4 Longitudinal velocity distribution in grid wake

399

Savttsky

the wave probe. Thus, the transverse probes covered an area from
the grid centerline to a distance of nearly 9 ft outboard of the edge
of the grid.

Test Procedure: The range of wave heights and lengths were
essentially Similar to those used in the one-dimensional tests. Also,
the test procedures previously described were followed. The initial
grid location was always 10 ft ahead of the wavemaker. After ap-
proximately 60 ft of tow, the grid was stopped and the wavemaker
continued in operation until the wave amplitudes recorded in the line
of transverse wave probes were again equal to the incident wave
amplitude.

RESULTS OF EXPERIMENTAL INVESTIGATIONS

Selected test results are first described to illustrate the
general behavior of waves ina turbulent flow field. An elementary
analysis of the results is developed in the subsequent section.

One-Dimensional Grid Studies

As previously discussed, the original intention of the one-
dimensional grid study was to provide a turbulent wake with constant
longitudinal mean flow in any transverse section through the wake.
Regular waves would be passed through the wake and measurements
made of the dissipative effects of grid-controlled turbulence on wave
amplitude attenuation. It was expected that the deep water gravity
waves would pass through the turbulence field with the crest lines
always remaining parallel to the grid and that the wave amplitude
would be essentially constant along a given crest line and decrease
as the wave progressed further into the turbulent area. Under these
circumstances, the amplitude attenuation would be due both to
viscous dissipation and to "wave stretching" as it moved into a longi-
tudinal current from an originally quiet area.

This idealized situation did not develop but, rather, it was
found that the wave crest lines were severely deformed; the wave
amplitude was not constant across a given crest line; and, further,
there were pronounced oscillations in the wave amplitude time history
at each wave probe (whether moving or stationary) in the wake. In
all cases, the control wave probe, which was fixed in quiet water aft
of the turbulent wake, indicated a wave of constant amplitude and
period continuously passing into the wake area.

General Behavior: An example of typical wave amplitude
oscillations recorded by both the moving and stationary wave wires
along the grid centerline is given in Fig. 5. The test conditions
represented are for a wave length of 4.0 ft and a wave height of
1.2 inches. The grid velocity was 1.7 ft/sec. The phase speed of
the wave is 5.4 ft/sec while the average wake velocity is approxi-

400

Gravity Waves and Fintte Turbulent Flow Fields

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401

Savttsky

mately 0.60 ft/sec. Trace No. 1 is for the moving wave probe
located 13.25 ft. aft of the grid; trace No. 2 is for a moving wave
wire located 66.5 ft aft of the grid; trace No. 3 represents the
moving wave wire 12 ft ahead of the grid, and trace No. 4 is for

the stationary control wave wire located approximately 20 ft aft of
the start of the turbulent wake. The times of start-up and stop of
the grid motion and the time of entry of the moving wave wires into
the wake are also indicated on this figure. Perhaps the most notable
feature on this typical test record is the pronounced oscillation of
the measured wave amplitude at all but the stationary wave wire.

It is seen that, for the specified test conditions, the measured wave
amplitudes varied from nearly zero to values somewhat larger than
the incident wave. Further, the time between successive minimum
values is approximately 9-10 seconds for the waves in the wake but
considerably longer, although not as clearly defined, for the wave
probe ahead of the grid. For longer wave lengths, the wave ampli-
tude variations were reduced and the apparent period between mini-
mum values increased. A reduction in grid speed reduced the wave
amplitude variation and increased the apparent period between mini-
mum values. There was no discernible effect of grid mesh size on
these general observations.

It is to be noted from Fig. 5 that fluctuations in wave ampli-
tude continued for a long time after the turbulence grid was stopped.
This is, of course, due to the fact that the wake has a mean flow
defined in Fig. 2 and, consequently, moves past the stationary grid.
It is also interesting to note that wave deformation at wave wires 1
and 2 is first evident after approximately 3 seconds or, equivalently,
after a wave crest has traveled nearly 5 ft into the wake.

Specific Behavior: The envelopes of wave height (h) variation
with time, normalized on the basis of incident wave height (h,), are
plotted in Figs. 6 through 11 for a grid speed of approximately 1 ft/sec;
a grid draft of 20 inches; and a mesh size of 2.7 inches. Data are
presented for the 3, 4 and 8 ft wave lengths, each having a height of
approximately 1 inch. The data for the 2 ft wave length are not pre-
sented since the wave heights were most irregular even in the non-
turbulent flow area. The data for the 6 ft long wave were not unlike
those for the 4 and 8 ft test waves and, hence, are not included in
this paper. Two companion plots are presented for each wave length.
For example, the data for the 3 ft long wave are given in Figs. 6 and
7. The envelopes of the ratio h/h,; for the three moving wave wires
are plotted in Fig. 6 along with the phase angle between wave crests
at the centerline and at a point 4 ft outboard of the centerline at a
longitudinal distance of 13.25 ft aft of the grid. A zero phase angle
represents a crest line parallel to the grid. The complementary
data plot for the 3 ft wave is given in Fig. 7 where, in addition to
the envelopes of h/h;,, the apparent wave length is plotted at a
longitudinal centerline position approximately 15 ft aft of the grid.
This wave length is computed from the data obtained at the two
centerline wave wires located at a distance of 13.25 ft and 16.5 ft

402 r

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Gravity Waves and Fintte Turbulent Flow Fields

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408

Gravity Waves and Finite Turbulent Flow Fields

aft of the grid. Similar sets of plots are given in Figs. 8 and 9 for
the 4 ft long wave and in Figs. 10 and 11 for the 8 ft long wave.

The irregular, oscillatory behavior of the wave height envelope
is apparent in all plots and decreases as the wave length increases.
Further, the average wave heights at 13.25 and 16.5 ft aft of the
grid continuously decreases with increasing time of grid travel
(which corresponds to an increasing length of turbulent wake through
which the wave travels).

Repeat runs for otherwise identical test conditions did not
produce identical time histories of wave height envelopes. This can
be seen by comparing the time histories for a point 13.25 ft aft of
the grid as shown in Figs. 6 and 7; 8 and 9; and 10 and 11. The time
histories are much more nearly alike for the 8 ft long wave than for
the 3 ft long wave.

Crest Line Deformation: An examination of the phase relation
(8) between the wave crest at a point 13.5 ft aft of and on the grid
centerline and the wave crest at a point 4 ft transverse to this point
indicates the first clear regularity to these one-dimensional test
results. For the case of the 4 ft long wave (Fig. 6), it is seen that
phase angle is zero, implying a crest line parallel to the grid for
the first 20 seconds (20 ft of wake development) of grid travel. As
time increases, the phase angle increases so that crest at the center-
line precedes the wave crest 4 ft off the centerline. The phase angle
increases nearly linearly with increasing time. For the 4 ft long
wave (Fig. 8), a similar linear phase shift occurs except that the
rate of phase shift is now somewhat slower. The phase shift for the
8 ft long test wave (Fig. 10) has a maximum value of only 45°. For
this long wave, the centerline crest lags the outboard crest. A com-
parison of the wave amplitude shows nearly similar values at both
wave probes when the phase is 0° or 360° and maximum differences
when the phase angle is 180°,

Apparent Wave Length: The apparent wave length along the

wake centerline was determined from an analysis of the time histories
of the wave amplitudes at the two wave probes which were 3.25 ft
apart (probes at distances of 13.25 ft and 16.50 ft aft of the grid).
The apparent wave length generally increases with increasing time

of gridtravel. The 3 ft long wave (Fig. 7) attains a value of approxi-
mately 4 ft after 50 seconds of grid travel and then decreases to a
value of 3.5 ft. The 4 ft test wave (Fig. 9) attains a value of nearly
6.5 ft after 100 seconds of grid travel while the 8 ft wave (Fig. 11)
attains a value of approximately 11 ft after 90 seconds of travel.

The wave lengthening is expected because of the longitudinal mean
wake flow in the direction of wave celerity.

Effect of Grid Velocity: Figures 12 and 13 present the wave
height envelopes for the 3 ft and 4 ft long waves when the grid speed

409

Savitsky

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Gravity Waves and Finite Turbulent Flow Fields

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411

Savttsky

is increased to 1.7 ft/sec. These resuits are to be compared with
those on Figs. 6 and 8 for a grid speed of 1.0 ft/sec. The major
differences are that there are wider variations in wave amplitude at
the higher grid speed and, further, the periodicity of the phase rela-
tion is reduced from 60 seconds to 40 seconds for the 3 ft wave and
from 85 seconds to 50 seconds for the 4 ft wave. Unfortunately,
sufficient data were not collected to determine the effect of grid
velocity on apparent wave length. Similar data exist for the longer
test wave lengths but, for the sake of brevity, these are not included
in the present thesis.

Effect of Grid Mesh Size: Increasing the grid mesh size from
2.7" to 5.4" did not have a discernible effect upon the test results.

Two-Dimensional Grid Studies

The objective of these two-dimensional studies was to investi-
gate the interaction between a turbulent flow field of finite dimensions
and long-crested, deep water, gravity waves. As previously dis-
cussed, the grid wake characteristics were such that the waves were
simultaneously subjected to dissipation effects due to eddy viscosity;
scattering due to turbulent convective distortion of the wave front;
and deformation of the wave due to mean flow velocity gradients. As
in the one-dimensional investigations, it appears that the effect of
the velocity gradients dominated in developing major wave distortions.
Although the experimental program examined wide variations in wave
length, wave height, grid mesh, grid width, grid draft, and grid
speed, the presentation will be limited to a discussion of results for
the following brief range of conditions:

Ace 2. 0546 ONE

H,= 1.0 inches

Grid mesh = 2.7 inches

Grid draft = 0.83; 1.67 ft

Grid width = 3.0 ft

Grid velocity = 1.0, 1.6, 2.6 ft/sec

These limited combinations of parameters serve to illustrate the
major effects of wake-wave interation.

General Behavior: The behavior of the time history of the
wave amplitudes at probe positions, either moving with the grid or
stationary in the wake, were substantially different from the one-
dimensional results previously discussed. The extensive irregularity
in the wave amplitude were not observed -- particularly for those
wave probes which traveled with the grid. There did appear to be
some indication of an irregularity for those stationary wave probes
located approximately two grid widths from centerline -- these were
not, however, well defined. An examination of the crest line defor-
mation as a given wave passed over the transverse line of stationary

412

Gravity Waves and Finite Turbulent Flow Fields

probes indicated a slight concavity to the wave front with the crest
along the grid centerline being in the lead by, at most, nearly 30
degrees for the 2 ft long wave at a grid speed of 1 ft/sec. It will be
recalled that, in the one-dimensional tests, the phase between two
transverse probes 4 ft apart continuously increased with time. In
general, the wave height time histories were characterized by either
a slow attenuation or amplification as the wave passed through the
wake.

Figures 14 and 15 present the envelope of wave height time
histories (normalized on the basis of incident wave height) along the
transverse line of stationary wave probes fixed in the tank for 2 ft
and 6 ft long waves respectively. The wave height was 1.0 inches
and the grid dimensions were: width = 3.0 ft; draft = 1.7 ft;
mesh = 2.7 in; speed = 1 ft/sec. For the probe on the centerline
(No. 7), it is seen that there is a continuous decrease in amplitude
starting from a time when the probe was 13 feet upstream of the
grid. For the 2 ft wave (Fig. 14) the height is attenuated to approxi-
mately 10% of the incident wave height when the probe is 7 ft down-
stream of the grid and retains this reduced height for the entire time
of data collection (80 seconds). It will be noted from Fig. 3 that,
when the grid reaches the transverse wave probes, it has already
developed a wake 28 ft long moving at a mean longitudinal velocity of
approximately 35 per cent of the grid speed.

The probe 2.4 ft from the centerline (No. 1) also shows a con-
tinuous reduction in wave height with increasing time, finally attain-
ing a value approximately 35% of the incident wave. Probe No. 2,
located 4.2 ft from the centerline, indicates only small variation in
wave height with time. The remaining outboard probes (No. 3, 4, 5)
all show increases in wave height for the entire test run. These
probes also indicate the existence of mild "beats" in the envelope
of time histories although not as severe as for the one-dimensional
case previously discussed. The maximum wave height occurs be-
tween probes 3 and 4 attaining a value approximately 75% larger
than the incident wave. It is to be noted that, for all stationary
wave probes, the height modifications are initially noted when the
probes are still 13 ft upstream of the grid. In general, then, the
characteristics of wave deformation in the wake show a significant
reduction in wave height for approximately one grid width on either
side of the centerline and an amplification beyond this region.

Figure 15 represents similar data for a 6 ft long wave -- all
other conditions being equal. The general characteristics of wave
deformation are identical to the 2 ft long wave except that the mag-
nitudes of the changes are reduced. For example, the minimum
wave height along the centerline is now 30 per cent of the incident
wave while the maximum wave height is 35% larger than the incident
wave.

413

Savttsky

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414

Gravity Waves and Fintte Turbulent Flow Ftelds

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416

Gravity Waves and Fintte Turbulent Flow Fields

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417

Savttsky

The envelope of wave height at the moving probes is given in
Figs. 16 and 17 for wave lengths of 2 ft and 6 ft respectively. The
wave probe positions are at distances of 1 ft and 6 ft aft of the grid
and just off its centerline. The grid was 3 ft wide, had a draft of
1.7 ft and a mesh size of 2.7 inches. Figure 16 presents results
for grid speeds of 1.0 and 1.6 ft/sec, while Fig. 17 is for speeds of
1.0 and 2.6 ft/sec. It is seen that there is a continuous reduction in
wave height with time. For the 2 ft long wave and a grid speed of
1.0 ft/sec, the amplitude is reduced to nearly 10 per cent of its
initial value after approximately 20 ft of gridtravel. It remains
essentially at this value for the length of the test record which ex-
tended for 30 seconds after the grid was stopped. The effect of
increasing the speed of the grid from 1.0 to 1.6 ft/sec reduced the
wave height to nearly 8% of its initial value. It is to be noted that
there is a distinct absence of oscillations in these time histories.

The results for the 6 ft long wave (Fig. 17) are essentially
similar to those for the 2 ft long wave. At a grid speed of 1 ft/sec,
the wave height is reduced to approximately 35 per cent of its initial
value. When the grid speed was increased to 2.6 ft/sec, the wave
height was reduced to 12 per cent of its initial value.

An overwater photograph of the wave deformation for a typical
two-dimensional test is shown in Fig. 18. The reduction in wave
height along the centerline wake area and the amplification outside
this area are clearly visible in this photograph.

\*

.

!

Fig. 18 Typical wave deformation for two-dimensional grid

418

Gravtty Waves and Finite Turbulent Flow Fields

Specific Results: To more clearly illustrate the modifications
in wave height along a given crest line, transverse sections through
the wake are plotted in Figs. 19 and 20 for a grid width of 3 ft, draft
of 1.67 ft; mesh of 2.7" and speed of 1 ft/sec. The effect of a grid
draft of 0.83 is given in Fig. 21. The results for the 2 ft wave are
given in Fig. 19 while those for the 6 ft wave are given in Figs. 20
and 21. These data are obtained from simultaneous measurements
of the recorded wave height at times corresponding to the indicated
distances ahead of and behind the grid. A maximum phase shift of
only 30° was discernible in the test records.

It is seen that a substantial reduction in wave amplitude exists
for a distance of nearly one grid width on either side of the center-
line. The maximum wave height amplification occurs at approxi-
mately two grid widths from the centerline and the wave amplitude
appears to be unaffected at distances of approximately 4 grid widths
from the centerline. This pattern exists for distances well aft of
the grid. It is interesting to note that, as the wave passes through
and ahead of the grid, where the wake does not exist, the deformed
crest tends to return to its original uniform height. Again, it is
seen that the 2 ft wave is much more attenuated and amplified com-
pared to the 6 ft wave, all other conditions being equal. (Compare
Figs. 19 and 20.) It is also interesting to note that reducing the
grid draft from 1.67 ft to 0. 83 ft has a negligible effect on wave
deformation for the 6 ft wave. (Compare Figs. 20 and 21.)

TRANSVERSE SECTION
5 AHEAD OF GRID
—— — TRANSVERSE SECTION
0.40 // 6 AFT OF GRID
7

— — — TRANSVERSE SECTION
32 X AFT OF GRID

4 6 8 10 l2
y= DISTANCE NORMAL TO GRID @ ~FEET
Fig. 19 Height of wave oo line in transverse sections normal to
grid’‘@.° X= 2", 1, grid width ="3", mesh ="2.7",
draft = 1.67', V= ne ft/sec.

419

60 2 AFT OF GRID
— —— TRANSVERSE SECTION
7 \ AFT OF GRID
——
De VS
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SESS =
es, 2
h. YY,
' 080 4
m4
eee
040 we
Go 2 4 6 8 10 12
y= DISTANCE NORMAL TO GRID q -FEET
Fig. 20 Height of wave crest line in transverse sections normal to
grid Go. Aw= 16; Hy= i eae grid width = 3', mesh,;= 22 fie
draft = 1.67', V= 1 ft/sec.
2) FORWARD OF GRID
1.60 — —_ 2) AFT OF GRID
——— 4X AFT OF GRID
TG, ea
1.20 a
—
GRID =
LLL LLLLLA
h
hj hie.
0.80
ae
See
040
fe)
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Fig. 21 Height of wave crest line in transverse sections normal to

Savttsky

TRANSVERSE SECTION
2X AHEAD OF GRID

—— — TRANSVERSE SECTION

grid’'G. 4 ="6", H,= 1", grid width =3', mesh =(257"5
draft = 0,83", V = 1 ft/sec.

420

Gravtty Waves and Finite Turbulent Flow Ftelds

Increasing the mesh size from 2.7 to 5.4 inches has a negligi-
ble effect on the wave deformation. Increasing the width of the grid
increased the area of wave attenuation and wave amplification.

III. ANALYSIS AND DISCUSSION
Viscous and Turbulent Effects

The initial analysis of the experimental results was directed
to relating the observed wave deformations to possible physical
mechanisms associated with the turbulent and viscous nature of the
grid-generated wake. It was found that the large changes in wave
height could not be accounted for by these considerations -- parti-
cularly in the two-dimensional grid studies. In this case, the square
of the wave height was integrated along a crest line passing through
the wake to obtain a measure of the energy in the deformed wave.
This was compared with the energy in the incident wave for the same
length of crest line. The results of this comparison are presented
in Table 1 for wave and grid dimensions selected to be typically
representative of the total test program. The length of integration,
Y, along a given crest line was the distance between the grid center-
line and the point where the wave height was again equal to the inci-
dent wave height. The crest length thus includes both the attenuated
and amplified wave height regions. For a given test condition, the
integrations were carried out for several transverse sections through
the wake, both ahead of and aft of the grid. It is seen from Table 1
that the integrated expression representative of wave energy pro-
duces nearly similar results with and without the towed grid. In
fact, for some test conditions, the integrated energy for waves in
the presence of the grid results in values somewhat higher than for
the case of no grid -- but this is attributed to experimental inac-
curies. It thus appears that viscous dissipative effects were quite
small and, although they most certainly existed, their magnitude
could not be accurately detected because the limited number of
transverse wave probes were inadequate to trace the unexpected
large wave height deformation which developed along a crest line.

A measure of the rms of the velocity fluctuations in the turbu-
lent field yielded substantially the same values with or without waves
passing through the wake. This was not surprising since the energy
imparted by the grid to the fluid at a tow speed of 1 ft/sec was
nearly an order of magnitude larger than the wave energy ina
crestline length equal to the grid width.

For the one-dimensional tests, it will be recalled that the
wave height at a given position in the wake exhibited large fluctua-
tions and was characterized by irregularities in the recorded time
histories. These results could certainly not be accounted for by
dissipative mechanisms in the turbulent field. It appeared then
that for both the one and two-dimensional studies the principal

421

Savitsky

TABLE. I
Comparison of Wave Energy With and Without Towed Grid

0 y, ft. Y
y 2 2
E,= J b ay E,g= HyY
(with towed grid) (no grid)

We; Guid" Transverse Section
a *K

oN ght Ts D M V6 Position Eng Ey

6.0 0.09 3.0 1.7 0.22 1.0 ft/sec 5d(F) 0.067 ft? 0.069 £t3
6X(A) 0.063 0.060
32X(A) 0.058 0.064

S200 l09 3.0 1.7 ,.02.22 . tae 2X(F) 0.081 0.082
2X(A) 0.081 0.083
7X(A) 0.081 0.079

2.0 =0:,09 12 340nl4s:7 00529 Akio 5X(A) 0.081 0.084
6X(A) 0.087 0.092
32X(A) 0.081 0.075

60) s0209' 5.5 1.7 0,22 ~ 1.0 2X(F) 0.093 0.100
2X(A) 0.087 0.099
4\(A) 0.087 0.099

22 02%:0) 097 <3. Oret.OS852.01A22)! 2120 5X(F) 0.081 0.098
6X(A) 0.072 0.078
23(A) 0.081 0.091

6.0 0-509, S20" 0. 85, U22- 1.0 2X(F) . 0.081 0.079
Z2XCA), 0, 0,70 0.076
4\(A) 0.070 0.078

*Wave and grid dimension in feet; L = grid width; D = grid
immersion; M = grid mesh

igs is transverse section forward of grid (ft)
(A) is transverse section aft of grid (ft)

422

Gravity Waves and Fintte Turbulent Flow Fields

mechanism of wave height deformation was due to a redistribution of
energy along a crest line rather than to dissipative effects. In this
regard, the results of Phillips [1959] were examined to determine
possible convective distortions of the wave front resulting from
scattering interference between wave and turbulence field. It was
found that the observed results could not be accounted for by the
turbulent scattering.

If, in the present studies then, turbulence is assumed to have
had a minor effect on wave deformation, it remained to examine the
possible interference between the mean flow gradient in the wake
and the incident wave. The velocity profiles for the longitudinal
mean flow aft of the grids are plotted in Figs. 2 and 4 for the one-
and two-dimensional studies, respectively. In both cases there is
a relatively sharp velocity gradient between a region of constant
wake velocity to zero velocity at the tank wall for the one-dimensional
case and to zero velocity in the still water adjacent to the finite grid
wake in the two-dimensional case. By application and superposition
of elemental theories of wave refraction, defraction and interference,
it was found that the observed results could be, at least, qualitatively
reproduced and physical mechanisms described to account for the
large wave deformations observed.

A detailed analysis is first made of the two-dimensional tests
since these results were free of possible wall reflection effects
such as existed in the one-dimensional studies. Further, the 2-D
analysis will provide the foundation for explaining the results of the
1-D studies which proved to be the more complex case.

Wave Interaction with Finite Velocity Field

Two-Dimensional Results. The longitudinal mean flow in the
finite wake area aft of the grid is plotted in Fig. 4 and is quantified
by an empirical formulation, Eq. (2).

——
16740.062x
Vw =[0.45-0.00745x][e. ]
Vi
where
Vw = mean value of longitudinal velocity in wake
V = grid velocity
x = distance aft of grid, ft
y = distance normal to grid centerline, ft

In the present analysis, regular waves in still deep water en-
counter a variable current field Vy (x+y) moving in the same direc-
tion as the waves. The waves are initially refracted by the current
to an extent dependent upon the incident wave length, strength of
current, and the velocity gradients in the wake. The orientation

423

Savttsky

of the wave-wake system is shown in Fig. 22.

For progressive deep water gravity waves in still water, the
phase velocity of the wave, Co, is given by:

Cr= g/k, (3)

where g = acceleration of gravity, \,= wave length in still water,
k, = wave number = 27/X., C, = wave velocity relative to still water.
After the waves have run from still water into a current, the kine-
matical condition that must be satisfied is that the wave period, T,
remains constant while the wave length, i, velocity C, and height
H, change. Given a current velocity V), the constancy of wave
period is expressed as:

27 -k(C + Vy) = k(C, + Vu) (4)
where the subscript, o, refers to the still water conditions.

Thus:

2
ky C+ Vy .S
kia LGastaVead (Cs

For the present case Vy, = 0 so that:

C_S Ny. 9
Co Co Co

and
1
C= 5(Co + VG? + 4V\Co) (5)

which is the wave speed relative to the water for waves progressing
in the same direction as the current.

More generally, for waves whose crest line is at an angle
relative to the x-axis:

1
C= s (Cot Heng + 4V,C, cos 2) (6)

where @ is the angle between a wave ray and the x-axis (Fig. 22).

The wave velocity relative to the bottom C' is the vector sum
of the wave speed relative to the water and the local current.

424

WAVE CRESTS

Gravity Waves and Finite Turbulent Flow Ftelds

DEFORMED WAVE CREST

STILL WATER | FINITE CURRENT FIELD

Vy =LONGITUDINAL CURRENT FIELD IN WAKE
Co = WAVE VELOCITY, STILL WATER

C = WAVE VELOCITY, IN CURRENT
@ = ANGLE BETWEEN WAVE RAY AND X AXIS

Fig. 22 Wave-wake system

425

Savttsky

Grav ae (7)

The wave length for waves progressing in the same direction as the
current is thus:

N -(4 +1 +4(V, om
2

te)

(8)

It can be seen that the effect of a following current is to increase the
wave length relative to the water.

The analytical solution for the refraction of waves traveling
through a finite current field is obtained by application of Fermat's
principle that waves will travel in a path such that the travel time is
aminimum. Applying the method of calculus of variations will lead
to a time history of the path of individual wave rays passing through
the current. For the purposes of this analysis, it will be assumed
that the wake properties do not vary with time. This is a reasonable
assumption since it has been demonstrated that, for a grid-speed of
4 ft/sec, the mean wake-velocity is an order of magnitude less than
the wave speeds. Mathematically, the problem is to determine the
minimum time path of a given wave ray through a current region de-
fined by a position dependent velocity vector. The magnitude and
direction of the current are known as functions of position (Eq. 2).
The magnitude of the wave crest velocity relative to the water is C,
given by Eq. (6). The problem is to determine the path of a wave
ray such as to minimize the time necessary to travel from point A
to a point B. Analogous optimization problems for dynamic systems
are described by Bryson and Ho [1969].

The equations of motion are:

x(t)

- V,(x,y) - C(x,y,) cos @

(9)
- C(x,y,@) sina

y(t)

with initial conditions

x(0) = x,
(10)
y(0) = yo
and at end of computation
x(t,) = xX,

426

Gravity Waves and Fintte Turbulent Flow Fields

where ty is an unspecified terminal time of integration between
points A and B. It is required to find a(t) and t; such that the
above constraints are satisfied and that the performance index J(a)
of elapsed time t, is a minimum expressed mathematically,

"
Ha) = § dt (11)
0)

is a minimum.

From the methods of calculus of variations, the Hamiltonian
of the system is:

H(x,y,%,d,,d) =it A(- bee (o; cos @) = dC sin a (12)

where A, and X
equations are:

y are Lagrange multipliers. The Euler-Lagrange

° 0H
Mme OFS
- _ 86H
Bp
- _ 0H
pee 3
- _ 0H
Vite Ory
ae 0H
~ Ba
The terminal conditions are:
dlt,) =0
(14)
H(t.) = 6

Since the Hamiltonian is not an explicit function of time, Hl=0 and
H is aconstant. Further, since H=0 at the terminal condition,
then it follows that H=0 forall 0StX tye

427

Savttsky

Evaluating 9H/8a from (12) and using the condition H = 0
leads to a determination of the Lagrange multipliers.

C cos @ + SE sina

i, So Cm aS
Vic cos @ ce sin @) +c?

(15)
dC

Ba Cos %- C sina

$s
Vc cos @ +ec sina) + (og

y=

The remaining differential equations are employed and, after
extensive algebraic manipulations (which will not be reproduced
herein), the following expression for a(t), the angular trajectory
for minimum travel time, is obtained.

3 2
att) = [VC ce verge tc Be sina +C S¥w SE sin? a

2
+C B¥ a Sin meos'a HG pe coe. - ac 80 Se cos a
2
- ¢ Bly Se cost a - Ny eS sin a cos @ + C?5™ cosa

6? BE sina +20 902 sina +c°S Gpicos® a
aV, aC dV, acy
- 2
+26 Sw SE sina cosa + Suu( 52) sin e|/

Bop ae | (16)

The partial derivatives of C and Vy contained in Eq. (16)
are obtained from the definition of C and Vy, as follows:

Cc =5 [Co + (C$ + 4V\C, cos a |
V, (x.y) = (0.45 - 0.0074x) exp . (ae meee

428

Gravity Waves and Finite Turbulent Flow Fields

Thus:
c= —— VC, sina (Gc. + 4V,C, cos aye
he = C, cos @ Cos + 4V0, cos oy oNw
ac 2 - OV,
by = C, cos @ (C, + 4V,C, cos a) oe
2 2 “1/2
er = Cy, sin a FV - (C, + 4V_C- cos @)
-3/2
+ 2C, cos (C,+4V,C, cos a) ]
a°C av 2
Byda = C, sina Bye [- (C, + 4V,C, cos a)
-3/2
+ 2C, cos @ (Ge + 4VC, cos @) |
anc _-_ vec a (C+ 4V.C ae
ayes wo <O8 0 wo ©°8
ene, 6 2 -3/2
- 2VC, sin @ (C, + 4VC, cos @)
and
OV = -0.00745V exp | - a
‘i P 67 + 0. 062x
8
+ V(0.45-0.00745x) exp [- (7c) ]
» 3( ' )'[- 0. 062y |
. ° x 1.67 + 0.062x
8Vy = v(0.45-0.00745x) exp - ( )"
y - 7 P e 7 e 2x

429

Savttsky

These equations were programmed and evaluated on the
PDP-10 computer. Refraction diagrams were obtained for several
wave lengths, grid speeds, and initial wake lengths. The present
report presents the results for

= 2, Date
Grid width = 3 ft
Grid submergence = 1.67 ft
Grid mesh = 2.7 in
Grid velocity = 1 ft/sec
Wake length (x,) = 40 ft

Actually, the empirical formulation for the wake velocity is associ-
ated with the above grid geometry. The results for the 2 and 6 ft
waves are plotted in Figs. 23 and 24. These refraction diagrams
are actually constructed by the so-called "orthogonal" method where-
in a wave ray path is obtained from the computer solution. The
crest lines shown on the diagrams are everywhere perpendicular to
the orthogonals and represent the crest position at times correspond-
ing to multiples of the wave period. This time interval, mutliplied
by the local wave speed at each point on the crest, determines the
position of successive wave crests.

It is evident, for both wave lengths, that a large distortion
of the wave front occurs for the length of wave crest initially located
between a point 1 ft from the wake centerline and a point 3.5 ft from
the centerline. In fact, this 2.5 ft length of crest is stretched to
nearly eight times this length after the wave has traveled only 30 ft
into the wake. The local crest line divergence for the 2 ft wave is
larger than that for the 6 ft wave. A similar large stretching is evi-
dent for the length of wave crest between 3.5 and 5.0 ft from the
centerline. Because of this extreme divergence of orthogonals for
localized lengths of wave crest, it is not expected that refraction
techniques alone are sufficient to represent the present wave-current
interference effects. In fact, it is expected that diffraction along
the wave crest must occur to provide for a flow of wave energy along
the crest. This modification will be discussed subsequently.

The qualitative results obtained from the refraction analysis
can be summarized by describing the behavior of adjacent finite
crest lengths as the wave passes through the wake. The incident
wave can be divided into four separate lengths as follows:

1) Crest length between @ and 1 ft.

2) Crest length between 1 ft and 3.5 ft.
3) Crest length between 3.5 ft and 5.0 ft.
4) Crest length beyond 5.0 ft.

430

DISTANCE AFT OF GRID, FEET

Gravity Waves and Fintte Turbulent Flow Ftelds

GRID
=
4.00
4 2.00 2.75
4.50 2.50 3.75
3.50

|

f
/
:

ae. |
16 = = >,
— /
>.
|
20 ——
WAVE RAY LINES
pe AVE RAY LINES —\ |
S |
24 = NS
2
|
28
OTE: (1) SHADED AREA REPRESENTS|

GRID WAKE (FIG. 4) |

(2) Yo =INITIAL RAY POSITION
IN STILL WATER

32

36
WAVE CREST LINES

40
) 4 8 12 16 20 24 28

LATERAL DISTANCE FROM WAKE ¢ , FEET

Fig. 23 Computed wave refraction diagram
X= 2!' V = 1 ft/sec

431

DISTANCE AFT OF GRID, FEET

Savttsky

GRID

Yo =1.00
4.50
75 2.50
te 4.25
2.00 me
~~ af 3.75
= / 3.50
Ss
4 ———~

/ /

8 12

pe
J. /

Na

WAVE RAY unes |

——. NOTE: (1) SHADED AREA REPRESENTS

GRID WAKE (FIG.4)

(2) Yo =INITIAL RAY POSITION
IN STILL WATER

WAVE CREST LINES

16 20 24 28

LATERAL DISTANCE FROM WAKE @ , FEET

Fig. 24 Computed wave refraction diagram

h= 6'

432

V = 1 ft/sec

Gravity Waves and Fintte Turbulent Flow Ftelds

Crest Length Between © and 1 ft. In this region, the current
field is essentially constant across any transverse section and in-
creases slowly in the longitudinal direction. The effect on this finite
wave is to increase the wave length in accordance with Eq. (8). For
a mean flow of 0.30 ft/sec, the 2 ft wave length should increase by
20% while the 6 ft wave should increase by 10%. This is in reason-
able agreement with the results in Figs. 23 and 24. Since the wave
length increases, it is expected that the wave heights will decrease
in order to maintain wave energy balance.

Phillips [1969] accounts for this wave energy balance in
treating the case of long-crested waves running into a current in
which the surface velocity varies only longitudinally. His results,
plotted on Fig. 3.6 of his work, show that for the present conditions
a wave height attenuation of approximately 15% is expected for the
2 ft wave and approximately 8% for the 6 ft wave. This is consider-
ably less than the experimentally attained values of 80% to 90%
attenuation previously discussed. Thus, the refraction procedure
alone does not account for the results observed in the vicinity of
the wake centerline. It will later be shown that diffraction effects
applied to this length of wave crest can indeed result in large local
attenuations of wave height.

Crest rss Soe Between 1 ft and 3.5 ft. The orthogonals for
this crest length diverge rapidly in a direction which causes the
local wave crest to be redirected out of the wake area into the still
water. This finite crest length advances in a constant direction
relative to axis system at a speed and wave length equal to the inci-
dent wave. It then crosses the undeformed incident crest line at a
distance 5 ft from the wake centerline. As a first order effect, it
can be assumed that the wave heights decrease as the square root
of the ratio of the initial wave ray separation to the separation at
any subsequent position of the local crest. For the 2 ft long wave at
a position 12 ft aft of the grid (28 ft into the wake), the wave height
(Fig. 25) indicates that the average wave height for this local crest
is approximately 30% of the incident wave height.

The deflection of this local wave crest length into the area of
the incident wave could account for the irregularity observed in the
wave height time histories at fixed points between 7 and 12 ft from
the wake centerline.

Crest Length Between 3.5 and 5 ft. For this length of wave
crest, it was seen that adjacent orthogonals converge and finally
cross, resulting in a caustic curve [ Pierson 1951]. On the basis of
simple theory, the wave became infinitely high on the caustic which,
of course, is not the case. At present, quantitative analysis of the
wave height at and beyond caustics must still be developed for the
case where variable currents produce wave distortions.

433

Savttsky

Crest Length Beyond 5 ft. This length of wave crest is

always outside of the wake current and thus continuously progresses
through still water with no alteration in wave length or speed.
During its forward progress, it runs into the caustic area and
deflected wave crest originating between 1 ft and 3.5 ft from the
wake centerline.

Application of Analytical Results

Due to the omission of defraction effects for each wave crest
segment, the refraction results discussed in this section are in
themselves insufficient to represent the test results. They are
nonetheless invaluable in forming the basis for providing qualitative
information about the complex processes governing the interaction of
a wave system with a finite current field.

Reflection Effects: The results of the refraction analysis
are first used to compute wave heights along crest lines in transverse
sections through the wake. The detailed results presented herein
are for a transverse section 12 ft aft of the grid and for a current
field extending 40 ft aft of the grid (as shown in Figs. 23 and 24).
Thus, the wave has progressed 28 ft into the wake at the time of com-
putation. The grid was 3 ft wide; had a mesh of 2.7 inches; a draft
of 1.67 ft; and a tow speed of 1 ft sec. Computations were made
for 6 ft and 2 ft long waves. The experimental results for these
conditions have already been presented in Figs. 19 and 20.

Since the purpose of this computation is mainly to compare
qualitatively the measured results with elemental analytical results,
simplifying assumptions were introduced. First, it was assumed
that the local wave height between adjacent orthogonals is inversely
proportional to the square root of the distance between these adjacent
ray s«,..hus:,

Bs = (4) un

where:

separation distance between adjacent rays in still water

se
°
iT]

= separation distance between adjacent rays on the de-
formed crest line

Hy= local wave height in still water

= local wave height along deformed crest line

This enabled wave heights to be constructed along a wave crest from
the centerline to a distance approximately 6 feet from the centerline.
Beyond this point, there is a superposition of the deflected wave seg-
ment with the undisturbed length of the incident wave. In this area,

434

Gravity Waves and Finite Turbulent Flow Ftelds

the two waves are combined in proper phase as indicated by the
crest line plots in Figs. 23 and 24. The section of wave crest that
develops into a caustic has not been included in this elemental con-
struction. The results of this simple refraction analysis are
plotted in Figs. 25 and 26 for the 6 ft and 2 ft wave lengths. It is
seen that this procedure results in essentially unmodified crest
heights just aft of the physical grid; then large reductions in wave
height for areas transverse to the grid and, finally, increases in
wave height in those areas where the deflected segment of the wave
combines with the undeformed segment of the incident wave. The
results of the refraction computations do not entirely agree with the
experimental data -- particularly in the region of the grid wake where
the test results show significant attenuations in wave height while
the computed results show no wave height attenuation.

Considering the variation of computed wave height along the
crest line (Figs. 25 and 26), it is seen that there is a large increase
in wave height for positions less than and greater than approximately
6 ft from the grid centerline. At this 6 ft point, the computed wave
height is aminimum. These transverse gradients cannot remain in
equilibrium and thus represent a source of energy flow along the
wave crest from the regions of large wave height to the point of low
wave height. This is a diffraction phenomenon which exists simul-
taneously with refraction effects. A rigorous theoretical analysis
of this problem appears to be extremely complex and is yet to be
developed. For the purposes of the present study, a simplified
analysis is developed which combines the results of elemental solu-
tions of wave refraction, diffraction and superposition. Although
not completely rigorous, this simplified approach is tenable and
relatively easily applied.

Diffraction Effects: As normally considered, wave diffraction
occurs when part of a wave is "cut off" as it moves past an obstruc-
tion such as a breakwater. The portion of the wave moving past the
tip of the breakwater will be the source of a flow of energy in the
direction essentially along the deformed wave crest and into the
region in the lee of the structure. As explained by Wiegel, the "end"
of the wave will act somewhat as a potential source and the wave in
the lee of the breakwater will spread out with the amplitude decreas-
ing exponentially along the deformed crest line. The mathematical
solution of this phenomenon, which is taken from the theory of
acoustic and light waves, is described by Penny and Price [ 1952] ;
Johnson [ 1952] and Wiegel [1964]. The solutions for two basic
diffraction phenomenon are presented by Wiegel: one is the case of
a semi-infinite breakwater and the other is for the case of waves en-
countering a single gap in a very long breakwater. The solution for
both cases are presented by Wiegel in the form of contour plots of
equal diffraction coefficient, K, defined as the ratio of the wave
height in the area affected by diffraction to the wave height in the
area unaffected by diffraction. For the case of the wave passing
through a single gap, the solutions are presented for various ratios
of wave length to gap width.

435

Savittsky

NOTE: (1) TRANSVERSE SECTION
12 FT AFT OF GRID

(2) WAKE LENGTH =40 FT Po

eke / A EXPERIMENTAL DATA (FIG. 20)

—— FROM REFRACTION COMPUTATIONS (FIG.24)

) 2 4 6 8 10 12
y= DISTANCE NORMAL TO GRID ¢ -FEET

Fig. 25 Results of refraction computation versus measured crest

height... ..= 6';. H,= £"3 . grid width = 3's: meshi=d2eq3
a@ratt =o6rtiecv = Wit/sec:

1.60

NOTE: (1) TRANSVERSE SECTION
(2 FT AFT OF GRID A

(2) WAKE LENGTH =40 FT

/
1.20 /
4 /
4A—_A—
— 080 /
7 ANC,
\
A EXPERIMENTAL DATA (FIG. 20)
0.40 4 \ / = FROM REFRACTION COMPUTATIONS

‘\ (FIG. 24)

te) 72 4 6 8 10 l2
y=DISTANCE NORMAL TO GRID @ -FEET

Fig. 26 Results of refraction computation versus measured crest

height. X= 2'; Hy= 1"; grid width = 3'; mesh = 2.7";
draft =1.67": V= 71 ft/sec.

436

Gravity Waves and Fintte Turbulent Flow Ftelds

In applying these diffraction results to the present study, it
has been assumed that the refraction phenomenon previously dis-
cussed divides the wave crest into several segments which are
separately diffracted as they pass through the grid wake. Specifi-
cally, the segment of the wave crest just aft of the grid is assumed
to behave as though it was a section of the wave which passed through
a breakwater gap equal to the grid width. The justification for this
analogy follows from the refraction results given on Figs. 25 and
26 where it is shown that, for a distance of approximately one-half
the grid width on either side of the grid centerline, the wave height
in the wave cannot be maintained at a constant height since just out-
board of this segment the refraction analysis yields a small wave
height. Thus, it appears reasonable to assume that diffraction
effects will be developed and that this centerline segment of the
wave will reduce in amplitude and spread transversely along the
crest as it proceeds into the grid wake. The diffraction coefficients
will be taken to be those corresponding to a wave at a breakwater
gap as given on pages 188-189 of Wiegel.

One other portion of the incident wave which appears to be
modified by diffraction is that segment of the incident wave which is
located 5 ft outboard of the grid centerline. From the wave refraction
diagrams on Figs. 23 and 24, it is seen that wave rays and crest
lines outboard of 5 ft are not influenced by the grid wake. Simple
refraction considerations then result in a wave of constant amplitude
along this length of the wave front. Again, this constant wave height
cannot be maintained and a defraction process develops which causes
a lateral spreading of the wave crest into the wake area with an
attendant reduction in wave amplitude. This lateral flow of wave
energy can be compared to the case of water passage past a semi-
infinite breakwater, the solution for which is plotted on page 183 of
Wiegel. Typical diffraction diagrams for the case of breakwater
gap and semi-infinite barrier are given in Figs. 27 and 28 of this
report.

The computed results for these two diffraction processes
are plotted in Figs. 29 and 30 for the 6 ft and 2 ft wave lengths
respectively. Again, the computations are made for transverse
section approximately 28 ft into the grid wake. For the 6 ft wave,
the ratio of effective "gap width" to wave length is 3/6 = 0.50; for
the 2 ft wave, the ratio is 3/2 = 1.50. It is seen that the initial
constant height wave segment between the grid centerline and 1.5 ft
outboard is diffracted to approximately 0.30 of this height and is
spread laterally to a distance nearly 12 ft from the grid centerline.
Considering the diffraction of the entire wave segment initially 5 ft
outboard of the grid centerline, it is seen that this section is spread
inboard to the grid centerline with a corresponding reduction in
wave height at approximately 0.30 of its initial height. It is seen
that, for this wave segment, the attenuation of wave height as it
spreads to the centerline is much more rapid for the 2 ft wave than
for the 6 ft wave.

437

Savttsky

x/Xd
10
Oo I 2 4 6 8 10 12 14 16 18 20

Fig. 27 Diffraction of waves at breakwater gap contours of equal
defraction coefficient [ Johnson 1952]

Neclistnlich al eee
very? pbluay pal ae
ay yVAVAN ovale bhepigeey

SAMA tae

\ y\ ATT

z/L

Fig. 28 Diffraction of waves passing semi-infinite breakwater
[ Penny and Price, 1952]

438

Gravity Waves and Finite Turbulent Flow Fields

oes/i yt =

A

ul9*F = 3FeIp

ul°? = ysour

1€ = UIPTM prado, g = “H

yUSIoy JSeIO poanseow snsi9A poynduroy 67 °31T

14 Ob=HLON37 JVM (2)

aiy9 40 13V Lidl
NOLLDAS SSYSASNVUYL (1) -3SLON

14‘ 3 WOUMS JONVISIG ‘2

Vv

LNVLIANS3Y

Vv aguvogino 13S°¢ ONV 14 S'
N33ML39 LNIW9SS JAVM JO NOI LOVYSSY

QuUVOSLNO 143 S°1 OGNV 9D
N3J3ML39 INSW935S 3AVM JO NOILIVYSSIG

dD 4O GuUVOsLNO 14S WOUS
INSW93S 3AVM JYILNS JO NOILIVYSSIG

(O02 91d) VLVO TWLNSWIYN3dX3

<0
M"
RA

Ovo

02"!

439

Savitsky

v!

7

.es/IFT =A

”

él
/

/ a

14 Ov=HLONST JVM (2)

uL9°T = 3Fetp uL°Z = sour ,€=yIpIM pws yp="H ,Z=X
JYWsley 4Sei1d porinseosw snsiz9A payndwioy OF °31q7

Oovo-
~
‘ ¢ ms \
14‘ 4 WOYS JONVLSIO ‘Z 6%
Ol 8 7 9 v A
O
7 x y
<=—_ ey a == ‘\“ & a“
wT” —_— << oo owe es ies ae — «=e ase a=
Fs Ovo
giud 4O Lav 1421 as
NOILOSS JSYSASNVUYL (1) :3LON na :
e g
Vv Ps 080
Vv = VZZZZZZZL2
aiyd
Vv INVLINS3Y 02'I
auvosg ino 14S’€ GNV 14S'
N33M1L38 LNSW93S 3AWM 4O NOILOVYES3IY — —
9 Quvosino 14S) ONV >
N33ML39 LNIW93S JAVM JO NOILIVYSSIG ———
Vv > 40 Guvogino 13S WOYS — 991

LNAW93S SAVM AMILN| JO NOILIVUSSIG
(O02 914) VLVO IWLNSWIY3ad X3 V

440

Gravity Waves and Fintte Turbulent Flow Fields

Superposition of Elemental Results

The results of the refraction and diffraction results have been
superposed in order to provide an analytical estimate of the wave
height distribution along a crest line as it progresses through the
grid wake. The computations were carried out for a transverse
section 12 ft aft of the grid for a grid wake 40 ft long. These are
identical to the refraction calculations previously described. The
following procedure is used in this superposition of elemental results:

a) The initial crest length between the grid centerline and
1.5 ft outboard is diffracted by the breakwater gap technique as
plotted on Figs. 29 and 30.

b) The initial crest length 5 ft outboard of the grid centerline
is diffracted by the technique of wave passage past a semi-infinite
barrier as plotted on Figs. 29 and 30.

c) The segment of wave length initially between 1.5 ft and
3.5 ft is refracted by the orthogonal method and the wave heights
are obtained by Eq. (17). The orientation (or phase) of this wave
crest segment to the transverse section 12 ft aft of the grid is ob-
tained from the computed refraction diagrams such as given in
Figs. 23 and 24. The resultant wave heights for this segment are
plotted on Figs. 29 and 30.

d) The segment of the wave crest between 3.5 ft and 5.0 ft
outboard has been neglected in this simplified procedure since it
develops into a caustic line.

e) The results of (a), (b), (c) and (d) are superposed to ob-
tain the final wave height distribution.

The results of the above procedure are presented in Figs. 29
and 30 for the 6 ft and 2 ft wave respectively and compared with the
experimental data. It is seen that agreement between computed and
measured results is qualitatively acceptable for the 2 ft wave length
and is good for the 6 ft wave length. It appears then that the physical
processes responsible for the observed deformation of waves pro-
gressing into a finite current field have been established. It is
strongly recommended that further studies of this problem be di-
rected towards the development of a unified, rigorous theory which
can be used to quantify this interesting wave-current interaction
phenomenon.

One-Dimensional Results

No similar detailed analysis has been made of the one-dimen-
sional results previously described. It does appear, however, that
the transverse gradient in the longitudinal wake velocity existing
at the outer edges produces a local wave refraction. This refracted
wave segment must then be reflected from the tank walls and prog-
ress across the wake, running into similarly refracted wave seg-
ments from the opposite wall. These continuously crossing wave

441

Savttsky

segments passing over the incident wave may develop distortions in
wave height time histories such as observed in the experiments.
The wave height irregularity at any point in the wake thus precludes
a reliable evaluation of the dissipative effects of the grid-produced
turbulence since only two wave probes were used in this study.

IV. RECOMMENDATIONS FOR FURTHER STUDIES

The original objective of the present study was to investigate
experimentally the interaction between gravity waves and turbulence
fields generated in the wake of a towed grid. Unfortunately, the
longitudinal mean flow velocity gradients in the wake had a dominating
effect on wave deformation and thus precluded a direct evaluation of
turbulence effects alone. Although, ina realistic ocean environment,
turbulence fields can be generated by, and exist simultaneously with,
velocity gradients in ocean currents, it is nevertheless of fundamental
scientific int¢rest to study separately the effects of turbulence fields
with no mean flow interacting with gravity waves. The results of
such an elemental turbulence study can then be combined with velocity
gradients to represent wave passage through realistic ocean currents.
Also, the results can be used alone to study the wave interaction
with isolated turbulence fields such as exist, for example, in regions
of "splash" turbulence developed by breaking waves.

It is thus recommended that the present study be continued
but with an experimental apparatus designated to produce localized
turbulence areas with no mean flow. The experimental procedure
should be capable of generating turbulence fields of controlled eddy
size, turbulence intensities, depth of penetration below the free
surface, and length and width of turbulence patch. It is further
recommended that the turbulence generator be capable of developing
vortices with either a horizontal axis or a vertical axis or a combina-
tion of both.

The control of the vortex direction will be important in the
study of the eddy viscosity interaction in which energy is transferred
from the wave motion to the turbulence. As discussed by Phillips
[ 1959] , the passage of the wave results in straining the elements of
the fluid near the surface in a manner periodic intime. The mean
strain per cycle of the incident wave is of second order, namely
(z7/)® , where a is the amplitude and the wave length of the
incident wave. The wave motion thus provides a mechanism for
stretching the vortex lines that operates in addition to the stretch-
ing inherent in the turbulence itself, and so tends to increase w*,
the mean square vorticity associated with the turbulence. It is ex-
pected that this possible mechanism for transfer of wave energy
will be for waves interacting with vertical vortex fields. In this
case, the vertical velocity gradient in the long-crested wave stretches
the vertical vortices in the turbulence field, but should not effect the

442

Gravity Waves and Fintte Turbulent Flow Ftelds

horizontal vortices. An experimental setup designed to control the
direction of the turbulent vortices can be most instructive in under-
standing this dissipative process.

The experimental procedure proposed to develop controlled
turbulence fields with no mean flow is to sinusoidally oscillate a
series of grids in a physically confined area in still water. The
barrier confining the turbulent field can be constructed of four thin,
vertical plates housing a rectangular box penetrating through the
water surface to a depth below the lower ends of the oscillating grids.
After oscillating the grids, the rectangular barrier can be lifted
above the water surface just as the waves approach so that there is
an interaction between waves and turbulence. The dimensions of
this rectangular container can be varied to represent various sizes
of turbulence areas. The use of an oscillating grid in a confined
area has been investigated by Murray [ 1968] in his laboratory
studies of horizontal turbulent diffusion. In his work, the grids
which filled a 50 cm wide channel were composed of several rods
1 cm in diameter and 5cm apart. The array consisted of 3 grids,
each 30 cm apart, and had a stroke of 40 cm. The following con-
clusions concerning the generated turbulence are described by
Murray.

1. There is no mean flow within the confined area of turbu-
lence generation. This is precisely the objective of the proposed
experimental procedure.

2. The oscillating array of grids produces turbulence fields
which are essentially homogeneous and stationary.

3. The turbulent velocity distributions are Gaussian.

4, Taylor's statistical theory of turbulence effectively de-
scribes the variance, scale time, and scale length of the generated
turbulence field.

In summary then, the proposed turbulence stimulation tech-
nique appears to be adequate for generating controlled and mathe-
matically definable localized turbulence fields.

The experimental procedure will consist in mounting the
turbulence generator over a section of the 75 ft square tank away
from the side walls. The grid array would be oscillated to generate
the turbulence field and mechanically generated gravity waves would
approach this field, Just prior to the waves reaching the area of
turbulence, the rectangular barrier surrounding the oscillating
grids would be lifted clear of the water surface so that the waves
would interact only with the turbulence. It is expected that the
lateral diffusion of the turbulence area will be very slow compared
to group velocity of the gravity waves so that, at least for the pas-
sage of several wave lengths, the turbulence properties may be
assumed to be stationary.

443

Savitsky

Parametric variations in this study will include:
1. length and width dimensions of turbulence area;
2. depth of turbulence area below water surface;

3. spacing of vertical oscillating rods alone to investigate
the interaction between vertically oriented vortices and
gravity waves;

4. spacing of horizontal oscillating rods to investigate the
interaction between horizontally oriented vortices and
gravity waves;

5. combination of (3) and (4) to construct a grid having a
rectangular mesh of varying dimensions to provide for
various scales of two-dimensional turbulence;

6. vary speed of grid oscillation to obtain various levels of
turbulence intensity;

7. vary length and height of gravity waves.

Measurements should be made of the wave-height time
history at various locations both inside and outside of the turbulence
patch. A spectral analysis of these wave height time histories
should be carried out to determine the extent of wave scattering due
to the presence of turbulence. Further, a hot film probe should be
slowly towed through the turbulence area to characterize its statisti-
cal properties with and without the presence of passing waves.

It is believed that the suggested experimental procedure is
practical and can provide data necessary for basic studies of gravity
waves interacting with local turbulence areas.

V. CONCLUSIONS

An experimental study was undertaken to investigate the
interaction between deep water gravity waves progressing into a
turbulent flow field generated by a finite width grid moving in the
wave direction in a large towing tank. It was found that the lateral
gradient of the mean longitudinal flow in the wake had predominant
influence on wave deformation and precluded an evaluation of the
direct effect of turbulence.

The presence of the velocity gradients resulted in combined
refraction, diffraction and interference between finite and adjacent
segments of the incident crest line. Their combined effects were to
reduce the wave heights in the wake area to approximately 10% of
their original value. The wave heights outside of the wake were
increased to values 75% larger than their original value.

444

Gravity Waves and Finite Turbulent Flow Fields

An elementary analysis was performed of the refraction of
waves entering a finite current field. A combination of these results
with simple diffraction considerations qualitatively reproduced the
measured crest line deformations. A unified theoretical study of
this complex problem is required to provide quantitative results.

Recommendations for further investigation of wave interaction
with turbulence field with no mean flow are made.

It appears that the present results may be useful in develop-
ing full-scale procedures for local "quieting" of the deep water waves
behind support ship for retrieving or launching submersibles or
landing craft in a following sea.

ACKNOWLEDGMENTS

The author wishes to express his appreciation to Dr. R. Hires
of Stevens Institute of Technology for valuable discussions and tech-
nical advice rendered during the course of this study. He is also
indebted to Professors W. J. Pierson, Jr. and G. Neumann of
New York University for their continued encouragement and helpful
suggestions throughout the study. Professor Eric S. Posmentier of
New York University is thanked for his thorough review of the
dissertation.

REFERENCES
Bryson, A. E., Jr. and Ho, Hu-Chi, Applied Optimal Control
Theory, Blaisdell Publishing Co., 1969.
Johnson, J. W., "Generalized wave refraction diagrams," Proc.

Second Conf. Coastal Eng., Berkeley, Calif., 1952.

Johnson, J. W., "Generalized wave diffraction diagrams," Proc.
Second Conf. Coastal Eng., Berkeley, Calif., the
Engineering Foundation on Wave Research, pp. 6-23, 1952.

Murray, S. P., "Simulation of horizontal turbulent diffusion of
particles under waves," Coastal Engineering Proceedings
of Eleventh Conference, London, England, Vol. 1, pp. 446-
466, Sept. 1968.

Penny, W. G. and Price, A. T., "The diffraction theory of sea
waves by breakwaters and the shelter afforded by break-
waters," Phil. Trans. Roy. Soc. (London) Sec. A, 244,
pp. 236-53, March 1952.

445

Savttsky

Phillips, O. M., "The scattering of gravity waves by turbulence,"
J. Fluid Mech., Vol. 5, part 2, pp. 177-192, 1959.

Phillips, O. M., The Dynamics of the Upper Ocean, Cambridge
University Press.

Pierson, W. J., Jr., "The interpretation of crossed orthogonals in
wave refraction problems," U.S. Army, Corps of Engineers,
Beach Erosion Board, Tech. Report No. 21, January 1951.

Stewart, R. W. and Grant, H. L., "Determination of the rate of
dissipation of turbulent energy near the sea surface in the
presence of waves," J. Geophysical Res., Vol. 67, No. 8,
pp. 3177-3180.

Taylor, G. I., Statistical Theory of Turbulence) /Partst(=.IW,
Proc. Royal Society A; CST Ppp. 421-428.

Wiegel, R. L., Oceanographical Engineering, Prentice Hall, Inc.,
1964.

x ok 3 oi *

DISCUSSION

Dr. N. Hogben ;
National Physical Laboratory, Shtp Divtston
Feltham, Middlesex, England

Dr. Savitsky has undertaken a very interesting investigation
of the effect of turbulence on waves in an oceanographic context.
His main finding is that waves can be dramatically attenuated by
turbulence from travelling grids. He explains this in terms of
refraction and diffraction and comments on the potential use for
quieting sea waves.

Whilst listening to his presentation it occurred to me that his
findings may also have an important bearing on the understanding of
wavemaking by ships. It is a common experience that the wave
system originating from the stern region of a ship tends to have
much smaller amplitudes than would be predicted from the usual
theories. I would be glad if Dr. Savitsky could comment on whether
this suppression of wavemaking by ship sterns may be at least partly
explained in terms of a refraction and diffraction analysis such as he
has described in the paper, applied to the interaction between the
vorticity and turbulence in the boundary layer and wake and the stern
wave system.

* * * % *

446

Gravity Waves and Finite Turbulent Flow Fields

REPLY TO DISCUSSION

Daniel Savitsky
Stevens Institute of Technology
Hoboken, New Jersey

It may be possible for ship wakes to locally attenuate the wave
generated by the afterbody. If the mechanism described in the paper
is applicable, it would necessarily require that wave amplitudes be
larger at some distance transverse to the ship wake. This, of
course, follows from considerations of preserving wave energy.
Much further study of afterbody generated waves would be necessary
to determine the association of ship-wave attenuation with the
mechanism described in the present paper.

447

CHARACTERISTICS OF SHIP BOUNDARY LAYERS

L. Landweber
Untverstty of Iowa
lowa, City, JTowa

I. INTRODUCTION

When I accepted the invitation to lecture on ship boundary
layers, my original plan was threefold: a) to review three-dimen-
sional boundary-layer theory, b) to discuss the few available appli-
cations of the theory to ship forms, and c) to present certain un-
published results on ship boundary layers that have been reported
in several theses at the University of lowa. In the course of
attempting to "catch-up" on the literature on three-dimensional
boundary layers, so that I could pretend to be an authority on the
subject, I encountered so many excellent review articles, that it
became apparent that a review-of-reviews was hardly likely to
match the immortality achieved in its category by the "song-of-
songs." Rather it seemed to be more useful and interesting to
examine the validity and applicability to ship forms of the assump-
tions of existing methods for computing three-dimensional boundary
layers, and to suggest and partly to implement certain approaches
which appear to be better suited to the ship problem,

Some of the common assumptions of three-dimensional
boundary-layer theory are the following:

1. Assumption of small cross-flow -- that the direction of
flow within the boundary layer deviates by only a small angle from
the direction of the streamline at the outer edge of the boundary
layer.

2. Assumption of methods of calculating two-dimensional
boundary layers for determining the velocity component parallel to
the outer streamline, even when the small cross-flow assumption is
avoided.

3. The assumption of monotonic cross flow -- that as the
wall is approached from the outer streamline, and angle of deviation
of the boundary-layer streamlines increases monotonically up toa
certain value at a small distance from the wall, beyond which it

449

Landweber

remains nearly constant.

4. The assumption that three-dimensional boundary-layer
problems are best treated with equations in streamline coordinates.

A stimulating article by Lighthill [1] on the fundamental
significance of vorticity in a boundary layer initiated the development
of a proposed method for treating ship boundary-layer problems.
This will be presented in two sections; a first in which the vortex
sheet on the ship hull, which generates the irrotational flow about it,
is determined; a second in which the vorticity equations for a three-
dimensional boundary layer, in terms of a triply orthogonal coordi-
nate system, are derived. The significance of the first part is that
it furnishes the initial values for the second.

So there will be no "review"; but it still seems desirable to
touch upon the ship boundary-layer treatments of Lin and Hall [2],
Webster and Huang [3], and Uberoi [4], and the contributions in the
theses of Pavamani[ 5], Chow [6], and Tzou[7].

II NATURE OF THE SHIP PROBLEM

In comparison with other three-dimensional boundary-layer
problems, that for the ship is much more complex because of the
presence of a free surface at which the body is moving partly im-
mersed. Some ship boundary-layer problems will now be described.

1. The first step in a boundary-layer calculation, the deter-
mination of the irrotational flow outside the boundary layer (the outer
flow) is a difficult problem. Solutions employing linearized free-
surface boundary conditions and thin-ship theory furnish inadequate
approximations. The development of more accurate methods of cal-
culating the irrotational flow about ship forms is a current research
problem [ 8].

2. At Froude numbers sufficiently low so that the free sur-
face may be treated as a rigid plane, (zero-Froude-number case),
the three-dimensional flow about the double model, obtained by
reflecting the immersed portion in this plane, is of considerable
interest. Methods of computing the irrotational flow for this case
are available [8, 9]. Calculation of the viscous drag for this case,
and its ratio to the frictional resistance of a flat plate of the same
length, wetted area and Reynolds number, would yield the so-called
form factor of the hull form which is required in one method of
predicting ship resistance on the basis of model tests Pat.

3, The three-dimensional boundary layer is very sensitive
to the shape of the bow. The nature of the boundary layer near the
forefoot, which determines whether or not bilge vortices will be
generated, can also be studied at zero Froude number. Bows

450

Charactertsttes of Ship Boundary Layers

frequently are designed with zones of reversing curvature, at which
boundary-layer profiles with S-shaped cross-flows may occur,

4, At higher Froude numbers the boundary layer will lie
over a hull surface area which depends upon the equilibrium trim
and draft and the surface-wave profile along the hull at that Froude
number. The curvature of the outer streamlines at the free surface
strongly affects the cross-flow components of the boundary-layer
velocity profiles [6,7], an effect which is completely ignored in
boundary-layer studies at zero Froude number.

5. Near the stern the boundary layer thickness becomes of
the same order of magnitude as the radii of curvature, and the
methods of thin boundary-layer theory cannot be used without modi-
fication. A detailed study of boundary-layer characteristics in this
region is desirable in connection with the development of improved
rational methods of computing the viscous drag, and the design of
stern appendages from the point of view of strength and cavitation.
Of course, if such a calculation could be extended into the near wake,
it would be a great boon to the propeller designer,

6. The draft and trim of a ship may vary greatly, depending
upon its cargo. It operates at various speeds or Froude numbers,
and if model tests are involved, the effect of the scale or Reynolds
number would be of interest. Since the flow pattern would vary
with each of these four parameters, one may wish to calculate the
boundary layer for many combinations of parametric values,

III. SHIP BOUNDARY-LAYER CALCULATIONS

Ship boundary layers at zero Froude number have been calcu-
lated by Uberoi[4]. To determine the outer irrotational flow he
introduced a distribution of n discrete sources lying within but
close to the hull and determined their strengths by solving simul-
taneously n linear equations, obtained by satisfying the boundary
condition on the hull at n points. This source distribution was then
used to calculate the streamlines.

For calculating the boundary layer, the flow was treated as
two-dimensional along each streamline, and the momentum thickness
and shape parameter determined by an available two-dimensional
semi-empirical procedure [10]. A better approximation could have
been obtained with little additional effort had one of the available
three-dimensional boundary-layer procedures assuming small cross-
flow been used [11], since these would have taken into account the
important three-dimensional property of the spreading of streamlines.
Nevertheless, since the spreading of the streamlines is small except
near the bow and stern, the results should furnish a useful approxi-
mation.

451

Landweber

Finally, to determine the viscous drag, an empirical formula
relating the shape parameter H, with the outer velocity U, at the
tail (designated by the subscript t) and the velocity of the uniform
stream at infinity, U,,

Geode ttent
Won

is assumed, as well as that the equations of thin boundary-layer theory
may be integrated to the very tail, a dubious assumption. Since

this empirical relation is unlikely to be universally valid, the fore-
going procedure, which is that usually employed to compute viscous
drag, emphasizes the need for additional research on the character-
istics of the thick boundary layer near the stern.

An approximate method for computing the boundary layer on
a ship form at a nonzero Froude number has been developed and
applied by Webster and Huang [3]. Guilloton's theory of ship wave
resistance [12] as presented by Korvin-Kroukovsky [ 13] furnishes
tables from which the outer flow can be determined along three
streamlines on the hull. The boundary layer along these streamlines
is then computed by a small cross-flow method employing streamline
coordinates, due to Cooke [14]. This method has been applied to
two Series-60 forms of 0.60 and 0.80 block coefficients, over a range
of Froude and Reynolds numbers. Although the assumption of small
cross-flow is basic to the method, it was nevertheless applied to
estimate the locations of separation points on these streamlines on
the basis of Cooke's criterion that separation occurs when the cross-
flow is 90°.

Smith's comparative study of five different methods of com-
puting a turbulent three-dimensional boundary [11] indicates that a
method which does not assume small cross-flow, and which employs
a three-dimensional extension of Head's entrainment hypothesis [ 15]
for the variation of the streamwise shape parameter gives better
predictions of the cross-flow than methods which assume small cross-
flow, and a constant value of the shape parameter. All five methods,
however, yielded values of the momentum thickness in poor agree-
ment with experimental results. Smith conjectures that this failure
is probably due to the adoption of empirical relations for the shear
stress from two-dimensional theory.

These results of Smith indicate that the Webster-Huang pro-
cedure for calculating separation points could be improved considerably
by the adoption of the best of the five methods. None of the methods,
however, can be used reliably to calculate the viscous drag.

Lin and Hall [ 2] also employ streamline coordinates and the

small cross-flow assumption in computing the boundary layer on a
ship form. As in the method of Cooke [14], the momentum integral

452

Characteristics of Ship Boundary Layers

equation in the streamwise direction becomes a differential equation
for momentum thickness after assuming a power law of variation for
the streamwise velocity profile and a semi-empirical relation from
two-dimensional boundary-layer theory between the shear-stress
coefficient and the momentum-thickness Reynolds number. An
additional assumption, that the cross-flow angle varies as the square
of the distance from the outer border of the boundary layer,:is intro-
duced to determine the cross-flow, Finally a new auxiliary relation
between the shape parameter and the momentum thickness is derived
by combining the streamwise momentum and energy integral equations
and introducing one more assumption, another semi-empirical rela-
tion between the dissipation coefficient and the momentum thickness
Reynolds number, also borrowed from two-dimensional theory.

Each of the five assumptions of the method used by Lin and
Hall is of doubtful validity for a ship boundary layer. Boundary-
layer data on ship forms, which are discussed in subsequent sections,
indicate that the cross-flow is not everywhere small, that the two-
dimensional relations are not generally valid in a three-dimensional
boundary layer, that a power law is not a good approximation for the
streamwise velocity profiles, and that the cross-flow angle cannot
obey a quadratic relation.

Finally, a paper due to Gadd [16], which the author has not
yet seen, should be mentioned. He determines the outer potential
flow, taking wavemaking into account, and applies this to calculate
the boundary layer on an equivalent body of revolution, neglecting
cross-flow. In referring to this paper, Shearer and Steel [17] remark
that "Gadd has recently applied a three-dimensional boundary-layer
theory to the pressure distributions obtained using the Hess and Smith
method, taking account of the free surface, to give friction distribu-
tions which agree very well with measured values. Comparison of
this theory with some of the experimental values detailed herein (in
[17]) are given in... " (in[16]).

IV. BOUNDARY-LAYER DATA FOR SHIP FORMS

It has been indicated that the relations for the shear stress
used in calculating two-dimensional boundary layers may not be valid
for a three-dimensional boundary layer. In order to investigate the
applicability to ship forms of these and other empirical relations that
have been proposed, it would be desirable to have a set of data, in-
cluding pressure distributions, mean velocity profiles for both the
streamwise and cross-flow directions, and shear stresses, for some
shiplike forms.

Full scale boundary-layer measurements ona 210-foot ship,
the USS Timmerman, have been reported by Sayre and Duerr [18].
Mean velocity profiles are given for four points along the hull, at
speeds of 5, 10, 15 and 20 knots. The measured boundary-layer

453

Landweber

thicknesses are in poor agreement with values computed from a for-
mula for two-dimensional flow on a smooth flat plate. Although no
other analysis was attempted, these data offer an opportunity to test
procedures for computing a three-dimensional boundary layer, e.g.,
by the suggested modification of the method of Webster and Huang.

Some boundary-layer measurements on a 70-meter research
vessel "Meteor" [19] and a 1:30-scale double model in a wind tunnel
[ 20] have been reported by Wieghardt. The full-scale measurements,
taken at a point 40 per cent of the draft from the free surface and
40 meters from the bow, yielded a value of the shear stress approxi-
mately equal to that for a flat plate, but a definitely lower value of
the momentum thickness. The results for the boundary layer at the
corresponding point on the model were consistent with the full scale
measurements in spite of the neglect of free surface effects in the
wind-tunnel tests. Several phenomena peculiar to ship boundary
layers were displayed by the model study. One of these is the unusual
shape of the boundary layer (vorticity-containing region) around the
girth of a fore-ship section, showing bumps at the sides and a great
increase in thickness at the keel, attributed by Wieghardt to secondary
flow (i.e., large cross flow) initiated near the bow. The shear-stress
coefficient at midship section was nearly constant at about Cy =,0'.,0885,
but decreased to 0.0025 as the keel was approached, and then increased
rapidly to 0.0039 at the keel. The momentum thickness 6 varied
even more, from a mean value of @/x = 0.0013 down the sides, in-
creasing to a maximum of 0.0028 as the keel is approached, then
falling to 0.0018 at the keel. These results indicate that, at least
near the keel, the two-dimensional shear stress formulas frequently
assumed in computing three-dimensional boundary layers, are very
inaccurate. Wieghardt concludes that "much more experimental
knowledge about the flow in ship boundary layers, including secondary
flows and trailing vortices is needed for semi-empirical calculation
methods for such three-dimensional boundary layers ..."

A project to obtain full-scale measurement of ship boundary
layers is under way in Japan, and some resuits of this work were
reported at the 12th International Towing Tank Conference in Rome
[21]. The unusual shapes of certain velocity profiles astern of the
parallel middle body were attributed to the presence of vortices
separated from the hull. Clearly these profiles could not be repre-
sented by a power law. On one ship an array of five longitudinal
vortices was observed in the wake, of which one pair originated at
the bow, another pair was shed astern of amidships, and the fifth
was due to the propeller.

A recent paper by Shearer and Steel [17] is noteworthy in
that it presents the results of shear-stress and pressure surveys on
two ship models at a particular Froude number. The effect of the
Froude number on the shear-stress coefficient C; was found to be
small except at the uppermost measurement locations along a water-
line at 25 per cent of the draft from the free surface, for which the

454

Characteristtes of Shtp Boundary Layers

curve of C, against longitudinal distance along the waterline undulated
180° out of phase with the wave profile. The most interesting feature
of the C, curves for various waterlines is their large variation along
the waterline even at depths where the free-surface effect should be
negligible, in agreement with Wieghardt's results. Furthermore, the
variation was found to be sensitive to the shape of the bow. This again
indicates that one is not free to assume a simple formula for the shear
stress in calculating a three-dimensional boundary layer.

The boundary layer on an ellipsoid with axis ratios 20:4:1 and
the incident flow in the direction of the longest axis was investigated
in a wind tunnel by Pavamani [5]. He measured the distribution of
both pressure and shear stress, the velocity profiles, as well as the
flow directions in the boundary layer. With the equipment used it was
not possible to probe the boundary layer in the regions of largest
curvature. It was found however that the shear stress in a transverse
section increased in the direction of increasing curvature.

Two shear-stress formulas that are used in computing three-
dimensional turbulent boundary layers are one due to Young,

-0.2
c,= 4% = 0.0176 (Y)
2 74
pU
and another due to Ludwieg-Tillmann,
0.678H tae
C,=0.246x10° (S2)

Here 7 is the shear stress at the wall, p is the mass density of
the fluid, U is the velocity at the outer edge of the boundary layer,
@ is the boundary-layer momentum thickness computed for the
streamwise component of the velocity, v is the kinematic viscosity,
and H is the shape parameter of the boundary-layer velocity profile.
Although not done by Pavamani, his data can be used to compare the
predictions by these formulas with his shear-stress measurements
by Preston's method. Comparisons at two points in the midsection,
one on the centerline and the other in the vicinity of the edge, and
given in the following table.

COMPARISON OF MEASURED AND COMPUTED SHEAR STRESS
AT MIDSECTION OF 20:4:1 ELLIPSOID (R, = 109)

Measured Young Ludwieg-Tillmann
0.00330 0.00380 0.00283
0.00466 0.00444 0.00318

at centerline

near the edge

455

Landweber

These results for only two points already indicate that neither of the
above formulas gives good agreement, although Young's seems to be
preferable. It is planned to continue the analysis of Pavamani's data
with the aims of representing his shear-stress data by an alternative
formula, and to compare his measurements with computed values of
the boundary-layer characteristics.

V. SHIP BOUNDARY-LAYER PHENOMENA

At a ship's bow certain streamlines of the outer flow pass
downwards along a side, turn around the bilge, and continue along
the underside of the hull. Because of the large curvature at the
bilges, the cross-flow angle in the boundary layer may become large
and the resulting secondary flow has been observed to roll-up into
a pair of so-called bilge vortices [22]. Clearly the small cross-flow
assumption is not suitable for treating this phenomenon.

It has been observed that these bilge vortices can be eliminated
by attaching a large bulb to the bow [ 23]. A possible explanation of
this effect is that the curvature reversals as an outer streamline
passes from the bulb to the bow, and then around a bilge result in an
S-shaped velocity profile, i.e., one in which the sign of the cross-
flow angle changes in passing from the outer limit of the boundary
layer to the wall. In any case, since bows are frequently designed
so that streamlines would undergo changes in the sign of the curva-
ture, S-shaped velocity profiles would occur, so that the assumption
of monotonically varying cross-flow angle, and in particular its fre-
quently assumed quadratic variation, would be improper.

The surface wave profile along the hull affects the boundary
layer in two ways, as has been shown by Chow[6]. Climbing from
a wave trough to a crest is equivalent to passing through a region of
adverse pressure gradient. If the free-surface slope is large enough
and continues long enough, separation will ensue near the free sur-
face. Secondly, the curvature at a surface-wave crest along the hull
tends to generate a secondary flow. Chow[6] has attributed a second
zone of separation at some distance beneath the free surface to this
phenomenon.

The conjectured mechanism of the effect of a surface wave
on a boundary layer was confirmed by Tzou[7]. He simulated the
free surface by a sinusoidal ceiling in a wind tunnel, and observed
and photographed the flow directions in the boundary layer ofa
vertical ogival strut, as indicated by an array of fine threads sup-
ported at various distances from the wall. He also verified the effect
by solving the Navier-Stokes equations and the equation of continuity
numerically, by a combination of a finite-difference method together
with the Blasius solution for a flat plate, for a simplified model of
his experiment. These results indicate once again the unsuitability
of the small cross-flow assumption for ship boundary layers.

456

Characteristics of Shtp Boundary Layers

VI. THE COORDINATE SYSTEM

A set of mutually orthogonal lines on a surface S can be
selected in infinitely many ways. Such a net, together with the
distance along the normal to S form a system of space coordinates
which, in general, are triply orthogonal only on S. Although a non-
orthogonal system of space coordinates is usually an awkward choice
in formulating the Navier-Stokes equations, when these equations
are simplified in accordance with the usual assumptions of thin
boundary-layer theory, Squire [| 24] has shown that the boundary-layer
equations are identical in form with that for a fully orthogonal system.

When the third coordinate is the distance ¢ along the normal
to S, the surfaces © = const. are, for obvious reasons, said to be
parallel to S. It is shown in texts on differential geometry that the
lines of principal curvature, and only these lines, have the property
that the surface normals along them generate developable surfaces
C = const. and = const., and that these, together with the parallel
surfaces (€ = const., form a mutually orthogonal family. For this
reason Howarth [ 25] and Landweber [ 26] employed the lines of
principal curvature as surface coordinates in formulating the equations
of motion. Nevertheless, according to Crabtree, et al., [27], "this
is an undesirable restriction," a feeling that seems to be shared by
most of the contributors to the subject of three-dimensional boundary
layers. Preferred is the streamline-coordinate system, although
geodesics and rectangular coordinates have also been used. Only
Howarth [ 28] has adopted the lines of principal curvature for the
coordinate system in his treatment of the three-dimensional boundary
layer near a stagnation point.

There are two good reasons for using streamline coordinates.
One is that, in the cases to which they have been applied, the inviscid-
flow streamlines could be readily obtained; the other is that practical,
approximate methods of solving the boundary-layer equations, em-
ploying techniques developed for two-dimensional boundary layers,
are available for the equations in streamline coordinates. The simplest
of these methods are based on the assumption of small cross-flow in
the boundary layer. According to Smith[ 11], however, who applied
five of these methods to compute the boundary layer on a yawed wing,
none of these was found to be completely satisfactory, as has already
been indicated.

For the case of present interest, the boundary layer ona
ship form, the first of the aforementioned reasons does not apply.
Calculation of the velocity distribution and the streamlines on a ship
form at the particular Froude number is a task of the same order
of difficulty as that of solving the three-dimensional boundary-layer
equations. For the zero-Froude-number case, methods are available
for computing the potential flow [8,9]; at nonzero Froude numbers an
approximate method due to Guilloton [12,13] furnishes tables for
the calculation of three streamlines along a ship hull.

457

Landweber

Another consideration is that the streamline pattern on a ship
form is a function of four parameters, the Froude number, the
Reynolds number, the trim angle and the draft-length ratio. Thus,
if streamline coordinate were to be used, it would be necessary to
calculate a great many coordinate systems. It appears to be more
practical to select a unique coordinate system which depends only upon
the geometry of hull and is independent of the above four parameters.

If it sufficed to study thin boundary layers, there would be a
free choice of orthogonal surface coordinates on the hull surface.
But the boundary layer near the stern cannot be considered thin, and
a continuation of the boundary-layer calculations into this region
could not be undertaken with the equations for an orthogonal coordi-
nate system unless the surface coordinates had been selected to be
lines of principal curvature.

VII. DETERMINATION OF LINES OF PRINCIPAL CURVATURE

First suppose that the equation of the surface S is given by
F(x,y5z) = 0 (1)

where (x,y,Z) _a: are the rectangular Cartesian coordinates of a point

P on S Let ds=idx +tjdy +kdz denote a vector element of arc
along one of the lines of principal curvature, where i, j, k are unit
vectors along the x,y,z axes. Then

grad F=VF=iF, +jF, tkF, (2)
is a vector along the normal at P and
dVF = ds - VVF (3)

is the change in this vector along the normal in moving an increment
‘ds from P to P' along a line of principal curvature. It can be
shown [ 29] that the normals to S at P and P' intersect if and only
if ds is an element of arc of a line of principal curvature. This
implies that the vectors

ds, VF and ds*« VVF
are coplanar, and hence that

ds-« VE X(ds'* VVF) =0. (4)

458

Charactertstics of Ship Boundary Layers

Also the condition that ds be normalto VF is
ds > VF =0 (5)

Equations (4) and (5) are the differential equations of the lines of
principal curvature.

In terms of their components, (5) becomes
F dx + F dy + F dz =0 (6)
and from (4) we obtain
(FF, - FF ,,)(dx) HER, - FF )(dy) H(E FY, - FF )(dz)’
t Care aa - ee a EK. - FF) dy dz
1 (EE - BP yy ze FF y - FF 2) dz dx

+(FF -FF +FF_ -FF_ )dxdy=0. (7)
z Xx z yy y yz x = XZ

Because of the quadratic nature of (7), the simultaneous solution of
(6) and (7) yields a pair of solutions for (dx, dy, dz), which can be
shown to be orthogonal. Thus, from an initial point P, one can

calculate the lines of principal curvature in step-by-step fashion.

If the equation of the surface is given in the form
y = f(x,z) (8)

where x is directed from bow to stern, z is positive upwards,
and the plane y = 0 is the vertical plane of symmetry, then (6) and
(7) can be combined into the differential equation of the projection of
the lines of principal curvature on the plane of symmetry,

2
[ pqt - s(1 +q?)] (<2) +[ (1 +p%t - (1 +¢2)r] @ +[(1+p?)s - pqr] = 0

where
t=f (10)

and the principal radii of curvature p are given by

459

Landweber

(rt - s*)p2 + K[t(1 +p) + r(1 +q2) - 2pqs]p + K* = 0 (11)
where
ali
K =[ t+ p® #ge]

Other relations between the geometric parameters of the
orthogonal coordinate system based on the lines of principal curva-
ture are given in [ 26].

VIII. EQUATIONS OF VORTICITY IN A BOUNDARY LAYER

Lighthill [1] makes a convincing case for the primary im-
portance of vorticity in a boundary layer. If the vorticity is known,
the velocity field can be calculated by the Biot-Savart law. Secondly,
vorticity is diffused and convected more gradually than other fluid
properties and hence is more readily determinable numerically.
From the mathematical point of view, Lighthill implies that it is
easier to solve the diffusion equation for vorticity than the boundary-
layer momentum equations governed by an outer irrotational flow.

Sherman [ 30] has also been impressed by Lighthill's views,
and has contributed a more mathematical discussion of "sources of
vorticity." Neither he nor Lighthill, however, have formulated the
vorticity equations for a three-dimensional boundary layer. This
will now be undertaken.

The Navier-Stokes equations for an incompressible fluid may
be written in the vector form
Wy xGtgrad(4¥-v+2 + gz) =-v cud (12)
at 2 p

where Vv is the velocity at a point ot the fluid, w = curl v is the
vorticity, t denotes time, p is the pressure, p the mass density,
g the acceleration of gravity, z is a vertical coordinate, positive
upwards, and v is the kinematic viscosity. An immediate conse-
quence of (12), obtained by applying the nonslip condition at the wall
surface S, is

grad ‘S + gz ) =-vecurlw on § (13)

which relates the vorticity at the wall to the pressure gradients of
the flow outside the boundary layer. By taking the curl of the mem-
bers of Eq. (12) we obtain the Helmholtz vorticity diffusion equation

460

Characteristics of Ship Boundary Layers
dw - - —
3 7 curl (v Xw) - v curl curl w (14)

a form from which the pressure gradient has been eliminated. The
velocity, however, still appears.

In rectangular coordinates we would have
curl curl w= VX(VX0)=VV° o- Voz - Vo
since V°* w = 0, and (14) could be written in the form

af _¢ Qu,,9u,, du_ a ab _ ae at

— —— + == SSe i == ——

at ox | dy dz ox dy oz Ox

an _¢ 4,247 a¥_ yan yan _ any on

Bee ha by ie oe ay (eye a)
2

06 _~ Iw Ow Ow | OO eo OG ag

Be ae | by Set bs Oy Nios. oe

We wish to obtain the equivalent set of equations for a three-dimen-
sional boundary layer, employing a triply orthogonal coordinate
system (2,8, y), where hda and h,dB are elements of arc along the
lines of principal curvature on S, and y is distance along the nor-
mal, with y=0 on S.

TetG.; Cos €, denote unit vectors in the directions of in-
creasing @, 6, y. Put

veeutevtew, w=eé ten tet.

| 2 3 |

From w= curl v we have in this system of coordinates, with h, =a,

1 fow _ O(hev) 1 Ow 8v_ v dhe

6 h, Lop Oy ] h, 068 Oy h, dy
178 dw]_ du 1 dw, u Oh
=e —— = — SS — ieee peel eset! IF
1= t Lay 1) - Gal By” h 8a” h, By

|

Mat ihe p dv 1 du
= aloe th) 3p |= 5 oa” he OB

461

Landweber

Put h (2, 6,.0)%= H, on AC 6,0) H,; let K; K_ be the curvatures
in the plane tangent to °S of the arcs @= const. and 6 = const.; let

K,)

h,= H(i +Ky), h,= H(t + Ky)

whence

8h __Ky 4 Bhp _ Ka

1
h, oy ey he oy fy

and, also from [ 26]

Hence the expressions for &,n,G become

potest (sOieroiy Ove Kav
h, 0p dy ft Ky

we see that

This indicates that the vorticity lines and the skin-friction lines

S form an orthogonal net, as is well known.

462

K, be the principal curvatures of the surface S corresponding
to the directions of increasing @ and 8. Then we have [ 26]

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

on

Charactertsttces of Ship Boundary Layers

Since in a boundary layer w is small in comparison with u
and v, and derivatives with respect to @ and B are small in com-
parison with derivatives with respect to y, we are justified in omitting
the derivatives with respect to @ and f in (19), (20) and (21). Ina
thick boundary layer it may be necessary to retain the terms K,y
and Ky in the denominators of (19), (20), and (21), but we shall
neglect these terms in the present treatment. Thus the expressions
for the vorticity components in a boundary layer become

_ Ov
E =- By = Kv (25)
du
=- — +
n ay K, (26)
¢=Kv-Kwu. (27)

Near the wallthe y derivatives are dominant so that the expression
for the vorticity remains that given by (22). Farther into the boundary
layer, however, the terms Kv, K,u, K.u, and K,v may become
appreciable when the curvatures are large, as at the bilges of a ship
form.

When € and n are known, the corresponding values of u

and v, obtained by integrating the differential equations (25) and (26),
are given by

“Kix?” K.y
wee ol ne > dy (28)

0
EY" se
ve-e 2 te” ay (29)
ie)

somewhat more simply than by the Biot-Savart law.

We can now obtain the components of curl w in the boundary
layer by replacing u, v, w by §,7, © inthe right members of (25),
(26), and (27). Thus we obtain

curlw-+ é€, = - ri K,n (30)
ae ot9

curl we, 5 = + K§ (35)

Gurl wsttege uk 1.5 K§ (32)

Landweber

and similarly, from (19), (20), and (21),

curl (v X w) = ele op (un > WE) - wy (w& - ub) - K,(w6 - ut) |
2

+e, e (vo, 0) cee Z (un - v§) + K(v6 - om
ae [= aa (w§ - uf) ee (vG - wn)

+ K\(w6 - ul) - Klvg - wn) | (33)

For the components of curl curl w inthe boundary layer we obtain,
neglecting small terms,

2
Pe ee 0g
curl curl w « e, =e By2 = (K, F K,) dy (34)
curl curl w- e -.2'n_ K ie (35)
2 dy2 3 4° Oy
aan ese ag Sioa
curl curl w ae = K By aie Ko By a K,K,§ ar K,K,n ° (36)

Substituting these results into (14) yields the vorticity equations

2
ee =5 a (un-vé) _ Fy (we-ut) a K,(w§-ub) + [S54 (KtK)) = | (37)

2
an _ 9 _ ee ee) = _ an an
at zo OY (vo wn) h, oa (un vé) a K{vS wn) Ee + (K, +K,) 5 | (38)

Be = a (wE-Ue) ~ He gp (ve-wn) + K(wE ub) - Kvo-wn). (39)

Here u and v are given in terms of the vorticity by (28) and (29);
w can then be obtained from the continuity equation.

In order to start the calculation, conditions at time t = 0
are required. This may be taken to be the vortex sheet for irro-
tational flow about the hull in a uniform stream, since this gives the
initial vorticity distribution when the body is impulsively accelerated
from rest to its constant speed. A procedure for determining this
vortex sheet is developed in the following section.

464

Charactertsttes of Ship Boundary Layers

IX. INTEGRAL EQUATION FOR A VORTEX SHEET FOR IRROTA-
TIONAL FLOW ABOUT A THREE-DIMENSIONAL FORM

A three-dimensional form bounded by a surface S is im-
mersed ina uniform stream of velocity U inthe positive x-direc-
tion, of unit vector i. We shall suppose that the fluid is inviscid
and incompressible. Let us assume that the disturbance of the flow
due to the body may be represented by a vortex sheet of strength
‘Y= yo where o isa unit vector tangent to the surface S such that

the fluid within the body is at rest.

In crossing S in the direction of its outward normal,
designated by the unit vector n, there is a discontinuity in the tan-
gential component of the velocity of the fluid, of magnitude y, in
the direction with unit vector

B= ox ne (40)

By continuity, since the fluid on_the interior side of S is at rest,

the velocity components inthe o and n directions at the exterior
side of S must also vanish, and hence the velocity at the exterior
side of S is given by

_

u=yoXn=y xu. (41)

Since, a priori, the mutually orthogonal directions of the
streamlines, %S, and of the vortex lines, o, are unknown, it is
necessary to introduce a set of orthogonal, curvilinear coordinate
lines,on S, € = const. and 1 = const. _Denote unit vectors in the
directions of increasing § and » by e, and e,, with sense such

that e Xe,=n. Put
Vee +e u=ze +e
VS Ghee. We Ute ve (42)
Then, by (41), we have
w= Yor v=-y¥,- (43)

An integral equation for the vorticity vector ‘y can be derived
from the condition that the contributions to the velocity on the interior
side of a point P of S must sumto zero. This gives

2
2

—_—

Yp% mp = U (44)

— 1
( YaX Vp (s+) a8, + s

1
4m Js PQ

in which the integral, obtained from the Biot-Savart law, represents

465

Landweber

the velocity at P induced by vortex elements at points Q of S,

and, by (41), the negative of the second term is the contribution

from the local vortex element y,. Here r,, is the length of the
chord joining the points P and Q of S and Vp denotes the gradient
with respect to the coordinates of P.

The integral in (44) is not suitable for numerical evaluation
in the given form because rpg which goes to zero as Q approaches
P, occurs in the denominator of the integrand. This singularity can
be eliminated, however, in the following manner.

First take the cross-product of (44) by iis to obtain

1 - 1 = fo ie eee
RA cae) Xn, dS, +5 (y,Xn,) Xn, = Ui Xn, (45)

Since, in the neighborhood of P, both ap and Volt /tpq = Tep/ Tay lie
very nearly in the tangent plane at Q, their cross-product is very
nearly parallel to no, and hence the integrand of (45) is proportional
to the angle between n, and n, or r,o/R, where R is the radius
of curvature of the"arc,of 75 subtended by the chord PQ.” Thus the
order of the singularity of the integrand of (45) has been reduced to
that of 1/r,..

In order to eliminate this singularity, consider the relation

be 1 nasty 1 are {
[ 3% Vo (s+) ] xB, = Ye + HMa(st) - We * Yo(=*)

Q PQ
he 1
SS yen Re a7 (4) (46)
Bel! Nea
since Ae ° a. = 0. Also we may write
= 1 — 1 - 4
np * Vo(=—) dSg= \ | p+ Yo(=— a era) dS, + 2m
5 TPQ S PO PQ

(47)

since f, Hg+ Vg (t/rpQ dS, is the flux through S due to a sink of
th at? i. Applying (46) and (47), and noting that

unit streng
1 4
eine vag)
Pt Q lpg

we obtain from (45),

466

Characteristics of Ship Boundary Layers

The singularity has been removed from the first integral in (48)
because a factor proportional to tpg is contained in

—

Yo '= Ve Zep (Ver

The second integrand is also singularity-free at P since

= { no° © |
° PS Pe 2 —_—
oe Vp ( ) am) REA

and

n ° V. (=—) a é
Q P Teg 2Rreo

Thus we see that (n, + n)) ° V50eyr..) is regular at P.

A procedure for obtaining a numerical solution of the integral
equation (48) consists of replacing the integrals by quadrature for-
mulas to obtain sets of linear equations. Expressing y in terms of
e, and e, as in (42), for each of n points P the quadrature for-
mula yields a linear equation in the unknown values of u and v at
n points Q. This gives n vector equations or, resolving in the
directions e, and €, at P, 2n scalar equations in 2n unknowns.
When @Q coincides with P, the integrand is set equal to zero.
hs Taking the scalar products of the members of (48) by e,, and

IP
€op, we obtain the pair of scalar equations

er as rl _ es AF
{ CYo'> Vp) x Vp (=) - e,, dS, t a | (np tng) °V, (=) dSg

PQ “PQ

=4nUi +e, (49)

ie cs he Nc, ae {
Wo- We XV (z-) *® ds tye) & +h “Veli -) ds
a Gaur P=. ara Sea ke ee oe Peo Q

= 4nUi + eop- (50)

In applying these equations, one needs to express Yg and ng in

467

Landweber

terms of the unit vectors aes: Cre and Tips This requires that the
direction cosines of C19? Cogs and No relative to C\p» Cap, and np
be calculated for each combination of P and Q; i.e., $n(nt1)
tables of direction cosines. Furthermore, if (x,, y,, z,) and
(Xg, y Yq: 2 Q) are the coordinates of P and Q ina rectangular
Cartecian coordinate system with unit vectors ie Vs k, then

Ean = I(x, rica as ay, Set Ke eee)

and the expression of Vp(1/rpqd = Tpo/Tpg in terms of €,p, €pp and
np requires that n tables of direction cosines of the latter set of
vectors with, respect;to.the .i,;j, k system also be obtained. These
direction cosines and the components of rpg can be readily deter-
mined if the equations of the surface are given in the form

<= (Ee .), y= GE sn); z = H(€,n). (51)

A procedure for solving (49) and (50) by iteration is suggested
by the following modifications:

= = { = _ Hl
if (Yo ~ Yo) x Vp (+) ip dSq . Up nel i (np x Ny) T Vp (=) dSg
s PQ ) PQ

= 4nrUi e ip (52)

{ (a - Yen X Yo( s+) * Sep 45g Pein a, toa: Ye (=e ) a8,
S PQ PQ

= 4nUi + €5p. (53)

For ship forms the foregoing procedure can be used to deter-
mine the velocity and vorticity distributions and the streamlines and
vortex lines on a double ship model at zero Froude number, At non-
zero Froude numbers, a similar pair of integral equations can be
derived, but these would be considerably more complicated because
of the contributions of the wave potential to the velocity on the body
surface S.

X. CONCLUSIONS

It has been indicated, on the basis of the limited available
boundary-layer data on actual ships and ship models, that the various
integral methods, with or without the small cross-flow assumption,
and employing streamline coordinates, are of dubious applicability to
ship forms because three additional assumptions concerning the

468

Characteristics of Ship Boundary Layers

velocity profile, the cross-flow angle, and the shear-stress coefficient
are not in accord with these data. If the energy integral equation is
also used to obtain an auxiliary equation, then an additional assump-
tion concerning the dissipation coefficient comes into question.

Two significant ship boundary-layer phenomena, the generation
of secondary flows and possibly of vortices at the bilges near the bow
and at a wave crest along the hull, indicate that cross-flow angles
may become large, so that the small cross-flow assumption would be
inappropriate. The possibility that the cross-flow may change in
sense and that the velocity profiles may become S-shaped both at the
bow and along the wave profile on the hull must also be taken into
account.

Lines of principal curvature are recommended as the basis of
the orthogonal coordinate system for treating ship boundary layers
because, in contrast with alternative choices, this system remains
orthogonal even in the thick boundary layer at the stern, and because,
unlike the streamline coordinates, the former system does not change
as the draft, trim, and the Froude and Reynolds numbers are varied.
For this reason, equations for determining the lines of principal
curvature have been included.

Since integral methods seem to be wedded to the use of stream-
line coordinates, the recommendation that these be replaced by the
lines of principal curvature implies that a differential method must
be adopted. One such method, based on the work of Bradshaw,
Ferriss and Atwell [31] for a two-dimensional boundary layer, has
been extended to the case of a three-dimensional surface by Nash [ 32].
An alternative approach based on determining the vorticity in the
boundary layer, strongly promoted by Lighthill [1] , motivated the
derivations of the vorticity equations in principal-curvature coordi-
nates and the integral equations of a vortex sheet for irrotational
flow about a three-dimensional form. Considerable further develop-
ment is required for application of these vorticity equations to a tur-
bulent three-dimensional boundary layer.

Lastly it should be remarked that presently we cannot deter-
mine the outer flow about a ship form with sufficient accuracy for
reliable boundary-layer calculations due to a combination of errors
due to linearization of the free-surface boundary conditions, approxi-
mate satisfaction of the hull boundary condition, and the effects of
viscosity on the wave making. In comparison with the outer-flow
approximation for the flow about a body without a free surface, the
effects of viscosity are experienced much farther upstream along the
body because of the phenomenon of interference between waves
generated near the bow and stern. Because of the strong interaction
between the outer flow and that in the boundary layer and wake, it
appears to be necessary to develop an iteration procedure, alter-
nating between these regions, which hopefully would converge toa
solution for the flow about a ship form.

469

Landweber

ACKNOWLEDGMENT

This study was supported by the Office of Naval Research,
under contract Nonr i611-(07).

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470

Characteristics of Shtp Boundary Layers

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General Curved Surface," Phil. Mag., Ser. 7, Vol. 42,
1951.

Landweber, L., "Appendix A. Equations in Curvilinear Ortho-
gonal Coordinates," Advanced Mechanics of Fluids,
Edited by H. Rouse, John Wiley & Sons, New York, 1959.

Crabtree, L. F., Kiichemann, D., and Sowerby, L., "Three
Dimensional Boundary Layers," Chapter VIII, p. 415 of

Laminar Boundary Layers, Edited by L. Rosenhead,
Oxford University Press, 1963.
Howarth, L., "The Boundary Layer in Three Dimensional Flow

-- Part IL, The Flow Near a Stagnation Point," Phil. Mag.
Ser. 7, Vol. 42, 1951.

Smith, C., An Elementary Treatise on Solid Geometry,
Macmillan & Company, London, 1891.

Sherman, F,. S., "Introduction to Three-Dimensional Boundary
Layers," Rand Corporation Memorandum RM-4843-PR,
April 1968.

Bradshaw, P., Ferriss, D. H., and Atwell, N. P., "Calcula-
tion of Boundary Layer Development Using the Turbulent
Energy Equation," J. Fluid Mech., Vol. 28, 1967.

Nash, J. F., "The Calculation of Three-Dimensional Turbulent

Boundary Layers in Incompressible Flow," J. Fluid Mech.
Vol. 37, Part 4, 1969.

472

Charactertstics of Ship Boundary Layers

DISCUSSION

Dr..N. Hogben
National Physical Laboratory, Shtp Diviston
Feltham, Middlesex, England

This paper performs a valuable service in laying the founda-
tions for a new method of calculating ship boundary layer properties
in terms of the vorticity field. A verdict on the merit of this ap-
proach must await the results of actual computations and comparison
with experiment. The purpose of this contribution is to supplement
the review of experimental data given in the paper by drawing atten-
tion to work which I reported in 1964 (Ref. (*)). It comprised
boundary layer explorations covering the surface of a model with
mathematical form as defined by Wigley (Ref. (**)) having parabolic
waterlines and section shapes. Measurements were made at 5 speeds
in the range 0.16 < F,< 0.32 and it is of interest that features
noted by Wieghardt as cited in the present paper were also observed.
In particular considerable stretching of the boundary layer below the
keel, likewise attributed to secondary flow effects, and bumps on the
velocity profiles attributed to trailing vortices, were found.

REFERENCES

(*) Hogben, N., "Record of a boundary layer exploration ona
mathematical ship model," Ship Division NPL Report No. 52,
July 1964.

(**) Wigley, W. C. S., "Calculated and measured wave resistance
of a series of forms defined algebraically," Trans. R.I.N.A.,
1942.

473

Landweber

DISCUSSION

H. Lackenby
The Brittsh Ship Research Association
Northumberland, England

I certainly agree with Professor Landweber's plea for a more
detailed study of the relatively thick boundary layers in way of ship
sterns especially in very full tanker forms.

In Ref. 17 (Shearer and Steele) it is apparent that some of
the models of the full tanker forms considered were suffering from
gross separation at the stern. There is little doubt that this accounts
for the viscous pressure resistance being as high as 30% of the total.
Incidentally, in this reference the corresponding wave making re-
sistance or gravitational component was stated to be only 3 to 5% of
the total.

Reference is made in Professor Landweber's paper to con-
ditions which bring about separation but not, as far as I can ascer-
tain, to methods of calculating the shear stress after separation and
the added pressure resistance which this brings about. I would be
glad if the Author would care to comment on the development of
calculation methods for these conditions. I would also mention that
in some of the latest full tanker forms there is little doubt that
separation is taking place on these ships at sea and the situation is
having to be accepted.

The viscous component of ship resistance has always been
an important one but owing to the developments in tanker forms it
is becoming even more important than it was in this class of ship.
Professor Landweber's paper is, therefore, of particular interest
and importance at this time and I look forward to further develop-
ments in his approach to the problem.

474

Characteristics of Ship Boundary Layers

REPLY TO DISCUSSION

L. Landweber
Untverstty of Iowa
Ttowa City, Lowa

Mr. Lackenby emphasizes the importance of boundary-layer
studies on tanker forms. Since the resistance of a tanker is mainly
viscous, and the power-wasting phenomena of bilge-vortex formation
and stern separation are viscous in origin, it is clear that such
studies are needed, if only to develop designs which avoid their
occurrence.

For the particular tanker form to which Mr. Lackenby refers,
the wavemaking resistance was stated to be only 3 to 5 per cent of
the total. Our experience has been that the wave resistance of a
tanker model may be about 10 per cent of the total, and that ifa
proper bow bulb is fitted, it may be reduced to about 6 per cent.
Since others have found that bulbs on tanker models reduce viscous,
rather than wave resistance, this indicates that the problem remains
to be resolved.

I would like to thank Dr. Hogben for reminding us of his
important 1964 boundary-layer paper, one of the very few available
studies of ship boundary layers. Explorations of this kind on other
ship forms are urgently needed to guide the development of methods
of computing ship boundary layers.

475

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STUDY OF THE RESPONSE OF A VIBRATING
PLATE IMMERSED IN A FLUID

L. Maestrello
WASA Langley Research Center
Hampton, Virgtnta
and
Peele wo inden
European SpaceOperattons Center
Darmstadt, Germany

I. INTRODUCTION

A large aircraft in supersonic flight undergoes large variations
in flow field over its surface. This paper is concerned with studying
the response of a structure excited by convected turbulence at nearly
zero pressure gradient and by shock-boundary layer interaction,
with the inclusion of the coupling due to the acoustic field on each
side of a panel. Shock waves on thin-walled structures can impose
severe loading problems, the most common of which is the self-
induced oscillation which is generated by an oscillating shock. The
shock wave can easily couple with the forcing frequency present in
the environment, including panel resonances.

From interior noise point of view, the upper region of the
airplane fuselage is considered the principal noise radiator. The
aerodynamics in this region are known from the Prandtl- Mayer
relation, and further downstream by shock-boundary interaction. In
addition, the fuselage skin experiences traveling shock waves which
run up and down the skin during the acceleration period, which might
last twenty minutes for a Mach 3 airplane.

In supersonic flight, the vibration of the surface is influenced
by the back pressure resulting from the radiation of sound on both
sides of the surface, so that, the surface motion and radiation are
coupled phenomena. The interior noise level is determined by skin
panel vibrations. For radiation below the critical frequency, the
major source of sound arises from the interaction of the bending wave
with the discontinuity of the boundary. Above the critical frequency,

477

Maestrello and Ltnden

the action of discontinuities like tear stoppers, etc., have little
effect on altering sound pressure level, since the sound radiation by
the panel is in the form of Mach wave radiation.

The experiment described in this paper indicates that some
simplifications in the model can be made, viz. (1) that there is no
significant interaction between the plate and the aerodynamic forces
on the plate; and (2) that the panel displacement is small in compari-
son to its thickness so that thin plate theory may be used. The plate
is, however, acoustically coupled to the external flow field and the
internal cavity.

Lyamshev [1968] has solved a similar problem for a com-
plex structure. Dowell [1969] computed the transient, non-linear
response of a simply supported plate coupled to an external flow
field and a cavity. Dzygadlo [1967] presented a linear analysis
allowing mutual interaction between the plate and the external flow.
Fahy and Pretlove [1967] have computed a first order approximation
to the acoustic coupling of a flexible duct wall to the flow field through
the duct. Maidanek [ 1966] considers an infinite, orthotropic plate
coupled acoustically to an external flow field. Numerous other in-
vestigations have been reported on acoustically coupled structures
with varying degrees of approximation, Irgens and Brand [ 1968],
White and Cottis [| 1968] , Strawderman [1967], Creighton [ 1970],
Ffowcs- Williams [1966], Crighton and Ffowcs- Williams [1969],
Peek [1969], Feit [1966], Lapin [1967], Pal'tov and Pupyrev

1967].

II. MEASUREMENTS
a) The Experimental Arrangement

The flow investigated was the sidewall boundary layer of the
Jet Propulsion Laboratory 20-inch supersonic wind tunnel; the shock
was induced by a 30° wedge mounted outside the boundary layer,
off-center and on the same side that the measurements were made.
This was done to offset the position of the reflected shock from the
opposite wall. The position of the shock was determined by observing
the displacement of a line of tufts, and by a static pressure survey.
For zero pressure gradient, detail of flow field and panel response
has been previously reported by Maestrello [ 1968].

The experiment was arranged to perform three basic measure-
ments: mean velocity profile ahead of the shock with static pressure
distribution across the shock, wall pressure fluctuations and measure-
ment of displacement response of a simple panel structure. The
titanium test panel measure 12 X 6 X 0.062 inches and was brazed
on all four sides of a 3/4 inch X 3/4 inch titanium frame. The brazing

Uap isuportt less %
TI-6AL-4V Titanium alloy containing 6% aluminum, 4% vanadium,
90% titanium.

478

Response of a Vibrating Plate in a Fluid

was intended to simulate the clamped edge condition. The panel
formed most of one wall of a rigid cavity measuring 14 X 8 X 6.6
inches. The other surface of the panel was exposed to the flow field.
The pressure differential across the panel was variable. The experi-
ment was conducted at two pressure differentials, viz. 0.06 and

14 psi; the latter corresponds to the actual differential between wind
tunnel pressure and local ambient.

The side wall of the tunnel was modified to accommodate
two identical, rigid, steel plates, which supported the necessary
instrumentation. One plate contained an array of holes in which
pressure transducers were mounted. The pressure transducers
were mounted on the center-line of the tunnel in the streamwise
direction at the same locations where the mean static pressure
measurements were made. Two types of pressure transducers
were used; one, the conventional lead zirconate titanate type made
by Atlantic Research, the other a capacitance type made by Photocon
Corporation with sensitive diameters of 0.06 inch and 0.09 inch
respectively. Correction due to finite size transducers was made
adopting the Corcos [1963] approach. The panel displacement was
measured with Photocon capacitance, displacement transducers
mounted on brackets which could slide along a bar and could be set
precisely by means of a screw mechanism.

The output of both pressure transducers and displacement
transducer were recorded on Ampex FR-1800H 14-channel tape,
recorded in the FM mode. Four channels were used for simultane-
ously recording data for correlation measurements. The maximum
dynamic range was obtained by splitting each data channel into two
tape tracks through phase matched filters to separate the lower and
higher frequencies.

b) The Wall Pressure Field

Measurements indicated that the flow field in front of the
shock closely approximated the properties of equilibrium of an adi-
abatic flat-plate boundary layer | Maestrello 1968]. The flow in
front of the shock has the following characteristics: Mach number
Me, = 3.03, free stream velocity Ue = 2,100 ft/sec, total tempera-
ture T, = 5679R, boundary layer thickness 6 = 1.37 inch, bound-
ary layer displacement thickness 5* = 0.445 inch, momentym thick-
ness = 0.083 inch, Reynolds number R = Ue5/U = 4,87 X 10", skin
friction coefficient Cr = 1.27 X10~, and C; Rg =39.8 Coles param-
eter [ Coles 1964].

The pressure ratio across the shock is a well defined function
of Mach number, for a 15° half-cone angle, the pressure ratio is
approximately 8.5. Experimental results show, however, that this
ratio is considerably smaller (Ap = 2.3). It is postulated that inter-
action with an expansion wave originating at the base of the wedge is
responsible for lowering the pressure differential and producing an

479

Maestrello and Linden

Psd
Ps

0 1,0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
x/§

Fig. 1. Static Pressure Fluctuations and Mean Pressure
Distribution Downstream of the Shock

effective decay downstream, Fig. 1. In the present case, the wedge
angle induces a shock in the boundary layer large enough to cause a
separation: farther downstream, the flow becomes reattached and
goes back to the flat plate condition. This transition takes place
within a few boundary layer thicknesses.

Downstream of the shock, the ratio of the mean pressure
distribution p,g/p, and the ratio of the rms pressure fluctuation
Piq/P, vary with a consistent relationship and both reach a maximum
at x/5= 2.3, where subscripts s and sd, mean upstream and
downstream of the shock, respectively, Fig. 1. Beyond x/6= 6
the effect of the shock on the static pressure vanishes. Kistler
[1963] indicates a similar behavior between mean and fluctuating
pressure in the spearated region ahead of a forward-facing step at
the same Mach number and upstream Reynolds number. The differ-
ences in the flow geometry only alter the magnitude of the pressure,
in that the ratio of the mean pressure to the fluctuating pressure
Pq/, Pa = 14 in the present experiment while Kistler found that

Py/ Pig * 32.

The normalized power spectral density measured upstream
and downstream of the shock are shown in Fig. 2. The spectra are

normalized by requiring Jno) d=i1 in order to demonstrate the

deviation from the zero pressure gradient ease. For the spectra
just downstream of the shock more energy is concentrated ina
narrow low frequency band while further downstream at x/6= 4,
the energy is distributed over a much broader bandwidth and
approaches the shape and level of the spectrum taken upstream of

480

Response of a Vtbrating Plate in a Fluid

x
1.0 OP. \

1 (w)U,/5

7
<-— Zero Pressure
Gradient Spectrum

0.01 0.10 1.0 10 100

Fig. 2.. Power Spectral Density of the Wall Pressure
Fluctuation

the shock. The normalized power spectral density found upstream
of the shock corresponds to the zero pressure gradient, and peaks at
w5/Ue™= 2 while downstream the spectral density is modified in the
region below the peak. It is significant that by altering the local
flow conditions, only the low frequency ends of the spectra are
appreciably affected. It is noticed that the pressure flucutation
measurements at x/5= 0 where the shock impinges show a notice-
able deviation from the general pattern in the higher frequencies.
This is attributed to an intermittant signal superimposed on the
regular pressure signal as seen on the oscilloscope. It is possibly
due to the characteristic fanning of the shock as it goes through the
boundary layer.

Measurements of the cross-correlation are shown in Fig. 3.
The cross-correlation characteristics are a function of position
downstream of the shock. The cross-correlation between positions
x/§ = 0.33 and x/6 = 3.80, the farthest apart, has characteristics
similar to those found at zero pressure gradient boundary layer in
that the ratio between the convection velocity and the freestream
velocity Uc/Ue = 0.72 and that the correlation between those two
points is still significant. The cross-correlation of the shortest
distance between x/6 = 0.33 and x/&=0.75, shows that the con-

481

Maestrello and Linden

Fig. 3. Longitudinal Cross-Correlation of the Wall
Pressure

vection velocity is very low Uc/Ue = 0.13 and the correlation is very
weak. The correlation between x /6 = 0555 and x/5 = 0:25, where
x/5 = 2.25 corresponds to the maximum static pressure ratio is
negative. The shock induces the boundary layer to separate and the
recirculation within the separation region permits the sign of the
pressure to change. Kistler argued that the fluctuating pressure

in the separated region arises from the combined action of the turbu-
lent shear layer and the recirculating flow. The picture, however,

is not yet clear enough to develop a model for time dependent loading,
since the geometry of the separated region is the primary variable

in estimating the pressure amplitude and resulting phase.

No measurement of the lateral cross-correlation was made
during the test; however, for the purpose of computing the response
of the panel, it is assumed that the pressure decays similarly to
that in the case of zero pressure gradient e 7/92 where a = 0.26
and 7 is the spatial separation [ Maestrello 1968]. This choice
overestimates the lateral cross-correlation, since the flow field is
far from being homogeneous. However, the overestimation may not
bé exceéded by a factor of Z.

482 |

Response of a Vtbrating Plate in a Fluid

c) The Panel Response Field

Measurements were made of the power spectral density and
cross-correlation of the displacement. Typical results are shown
in Figs. 4 and 5 for a pressure differential of 14 psi. The static
deflection of the panel was 0.06 inches at the center, and the dynamic
deflection was small in comparison with its thickness.

The displacement spectral density at the center of the panel
show pronounced spikes, the lowest frequency of which corresponds
to the lowest mode of the panel. The accuracy beyond a frequency of
3100 Hz was poor due to the spatial resolution of the capacitance
transducer, and therefore the spectrum beyond 3100 Hz was ignored.

Space-time correlation measurements were made along the
panel centerline from x =x'=3 in. y =y'=3 in. at one-inch inter-
vals up to a maximum separation of 6 in. The correlogram indicates
a convected feature with a phase velocity + Ucp = 770 ft/sec. This
convection velocity corresponds to that found in the previous experi-
ment using the same arrangements, except that no shock was present
[ Maestrello 1968].

f = 556

N
=
a> 996

is] =

3

750 x=x'=0.5 ft.

= =y'=0,25 ft.

z ss yay

a 10 M_ = 3,03

FB e

i Press, Diff. Ap = 14 psi

pe 1300

ey {

a * 1664

FA 10 1930

4

A 2720

rat ey 2340
10 3150
10724
10° eae 1 N L 1 ! N ! oer ame wee ee

700 900 1100 1300 1500 1700 1900 2100 2300 2500 2700 2900 3100 3300
FREQUENCY Hz

Fig. 4. Displacement Spectral Density

483

Maestrello and Linden

In comparing the results of the present and previous experi-
ments, it is concluded that the sign change of the convection velocity
is attributed to the presence of the shock. Furthermore, the cross-
correlation of the wall pressure also reflects a phase change fora
separation of 2.5 inches, which is in the same location as the phase
change which occurs for the displacement correlation in Fig. 5.

III ANALYSIS OF ACOUSTICALLY COUPLED PANELS
a) Two-dimensional Finite Panel

The vibration of the panel is induced by an arbitrary, external
pressure field F. It is assumed that the panel motion does not
interact with the turbulent boundary layer, i.e., the forcing field is
not altered by the plate motion. However, the panel is acoustically
coupled to the fluid on both sides of the panel.

The equation of motion for an harmonic component of the dis-
placement, W, of a thin panel with a force, F, and a pressure
differential, pp. - pj t 6p acting upon it, obeys the equation

BAW - ppw W =F tp, -p, + 6p (1)

where the bending stiffness, B, may include hysteretic damping,
and where pp is the mass per unit area of the panel, w is the
angular frequency, py is the acoustic pressure on the streamside of
of the panel, p, is the acoustic pressure below the panel and dp

is the static pressure differential.

The perturbation pressures, p, and pp, are related to the
velocity potentials, which satisfy time-independent wave equations in
the appropriate regions. In solving these equations one uses a
boundary condition which relates the potentials to the panel displace-
ment. These relationships may be made more obvious through the
use of Green's theorem. Thus, it is required to solve a system of
three coupled partial differential equations, the first of which is not
separable for the clamped edge boundary condition.

Pp; and py may be found directly as function of W. Thus,

consider first the cavity. The acoustic velocity potential, gy, satis-
fies the Helmholtz equation

Ag +k’ =0 (2)

with boundary condition 89/8n = 0 on all walls except on the plate
where 09/8n = - iwW.

The Greens function, g, for a cavity with hard walls satisfies

484

Response of a Vtbrating Plate in a Fluid

pa ie Tg |
o AER ies - 0
ee ee a ee
| See ae

%= xX’ = 0.25 FT

Y = Y’ = 0.25 FT

PRESS. DIFF. Ap = 14 psi
400 Hz HI PASS FILTER
M,= 3.03

UPSTREAM
CORRELATION

|

POSITION OF SHOCK
IMPINGE MEN T

!

DOWNSTREAM
CORRELATION

R(E 9,7)

Fig. 5. Broad Band Space Time Correlation of the Panel Displace-
ment Along the Center from x = x' = 0.25 ft, y=y' = 0.25 ft.

485

Maestrello and Linden

the equation
Ag(r|r) + keg (3 | x") = 4n6(r - r') (3)
and is given by Morse and Feshbach, Vol. II [1953]

'
mx m1rx n n
Cc Oslo leone

re cos ac os ae Be De
g(r/r') = aabe » » CS) =" =
c Kinn Sin. (Kmnd)

m=0 n=0
cos kyj,z cos k,,(z'+d) a ae Ae

Xx (4)
cos k,j2' cos kantZ +4) z<z!

where
2 2 mr : nw 7
nn Bega. acne)

and kg = w/cg where cy, is the speed of sound in the cavity of dimen-
sions a,, by, d.

By applying Green's theorem, the integral equation for ¢g is
obtained,

—_ —_> Q
o@)= 2 T eet at (5a)

Now using the boundary conditions, this becomes
— iw we — wea sea =,
g(r) =- = | a(n 71) W(r))'d x (5b)
plate

The pressure p, is related to g by

Pp, = - iwp.¢e

where pc, is the mass density of the fluid in the cavity.

To compute p,, it will be more convenient to operate with the
differential equation. Let the acoustic velocity potential in the flow

486

Response of a Vibrating Plate in a Flutd

field be denoted by WW By applying the Fourier transform on the
(x,y) coordinates, one gets the ordinary differential equation

2 aA
POP 2) + Pila,p,2) = 0 (6)

dz
where

2 2 2 2 2
GC =k +(M - 1)@ - 2kMa - £B
@ Ao oxy)
+ Aa
W(x, yz) -{ ( da dp Suen w(a,B,z)
“0 “-09

k=w/c, M is the flow Mach number and c the speed of sound in
the region above the plate. Only the positive exponential solution to

Eq. (6) is chosen, since it is the solution representing outgoing
waves. Thus,

b(e,B,z) = A(a,p)e™ (7a)

The boundary condition, arising from the continuity of normal dis-
placement is

du(a,B,z) - . ear
dz 520 =O

where the differential operator

L=k+t+FiM —
Ox
Thus,
Aa ~~ .
H(a,B,z)=- 5M eM (7)

Now, since
nN a 1) Es ' '
LW -{ { dx' dy'e lide LW! sy)

then

487

Maestrello and Linden

a b
W(x,y,2) = - a hs dx' dy' G(x,y,z|x',y',0)L W(x',y') (8)

Ther 23 © 00 ila(x-x') +B(y-y')+C(z-z')]
a(F | =f if SO ee, a (9)
=00 ““00

which is found in Appendix A to be for supersonic flow,

where

ie(Muty/ ue -R*)
e

271i
me M’-1 an - R°
G(r | x") =
0 outside the Mach cone

and for subsonic flow,

ix (Mu+yue+R® )
Gi bel ot (10)
Vvi-M2 vut+ R?

YN
dieept inithis case, K = efyl- Mo hand aasi(x-at) (=e = aha

had been evaluated as

a

(k - aM)W

then Eq. (8) would read

a b
xe
U(x,y,z) = - =|) y dx' dy' W(x',y")L G(x, y,z|x',y',0)(11)
TT

*
This equation is formally correct if L G is interpreted as a distri-
bution, which is to say that one partially integrates to obtain Eq. (8).

Now using Eq. (8)

Pp(x,y +2) at ip cLi(x,y,z)
OY aa 2
= leae’ ( dx' dy' G(x,y,z|x',y',0){[L| Wlx',y’)
4% Yo Yo
(12)

488

Response of a Vibrating Plate in a Fluid

where fp is the density of the fluid above the plate, and where partial
integration has been utilized. Had Eq. (11) been used instead,
Eq. (12) would read

e 2

1P9C
Po(x sy» Z) = wee

40 re)

a

b
2
{ dx' dy' W(x,y)|L | G(x,y,z|x',y',0) (13)
ié)

which is reducible to Eq. (12) by partial integration. Thus, super-
sonic flow does not present any especial difficulty aside from the
fact that G is singular all along the Mach cone, and this is an
integrable singularity.

Inserting the expressions for p, and p, into Eq. (1) results
in a single partial integro-differential equation to solve, viz.,

a b
2 2 Es ipoc? ( a
BAW - ppw W = F + 6p es A , G(x,y,|x',y")|L| wW(x',y") dx'dy'

2
+ reas if B(XosVo|xXbrve) W(x, y) dx dy (14)
plate

where the subscript c refers to the cavity, thus

x, =x + 7?

u
<
+

Ye

Equation (14) presents a formidable computational problem. The
Green's function g is known as an infinite series which is slow to
converge (1/n) thus compounding the difficulty by an increasing num-
ber of necessary operations to maintain a given accuracy.

An alternative to solving Eq. (14) is to convert it to an integral
equation for its Fourier amplitudes and to solve the resulting equation.
The advantage is that this equation is simpler (though it is a singular
integral equation). The following notation shall be employed:

489

Maestrello and Linden

The result of applying the Fourier transform to Eq. (1) is
2+ Qa a a ve
BA W- pw W=f + Po - py (15)
where
f=F + 6)
The first term in (15) may be evaluated using Green's theorem;
thus,

—> <=>
-iK-r

2a 4a — rs] 2
AB WEIOWIF OF Grete (Aw +K'w)| (16a)

Now, for a plate clamped on its edge to a rigid, plane support, the
following boundary conditions hold,

on edge

where s is in the direction of the edge, i.e., the tangent. Thus,
22 4°
Aw=KWtdAl Ww] (16b)

where

If more general boundary conditions are to be considered (e.g.
elastic foundation) the above expression must be replaced by the
right side of 16a. From the Eq. (7b) and the equation prior to Eq.
(12) it is found that

E 2 eel
p, = Pec en MY Wik) (17)

From Eq. (5b) it is found that

490

Response of a Vibrating Plate in a Fluid

2
~ W —~ —_ —_ —
P, = - So { dr, g(K | r,) W( rg)
plate
e
- “a[w]
where
g(k| r') = = { dre g(r |r') (18)

plate

Substituting these results into Eq. (15) gives

BT(K)W(k) + BA[W] - Q[w] =£(K) (19)
where
oe 2 ear a
1(K) = K*- be _ oc eM) (19a)

Let Wp denote the finite Fourier transform of a beam eigenfunction,
9,2 It follows from the Fourier representation that the Wn, form an
orthogonal set on the infinite interval. Thus, expanding W as

w(K) -) Winn'im(%2) Up (Bb) (20)

m,n

or alternatively, W as

W(r) = » Wmn9m(= ) Pn (Z) (21)

and introducing these expressions into (19) and subsequently utilizing
the orthogonality, gives

Win * > Dace as enn (22)

491

Maestrello and Linden

and

Pas ={ § ak YadodbolP) (pal, (2) o,(Z)|

om Oo t(K)

- a[¢, (2) « ()])

The computation of the integral Iimnr, may be simplified by
deforming the contour on the @-plane. Due to the manner in which
the Fourier transform was chosen, the integrand, except the term
T(k), is single-valued and analytic in the lower half-plane. The
contour will, thus, be deformed in this half-plane. This deformation
is determined by the analytic properties of the function T(k),

Eq. (19a).

2
T(a,B) = (a2 + p2)? - Se ee
Bo V2 + (M2 - 1)a? - 2kMa - &

This function is two-sheeted with square-root type branch points at

kM + Vk* + (M? - 1)p?
pba ESI Ee

M* - 1

The sheet associated with the positive value of the square root will
be termed the physical sheet, since it corresponds to outgoing radi-
ation.

The function has ten zeros on the two sheets, four zeros on
each sheet with the same values, corresponding to resonances of the
plate and the other two zeros are located near the branch points on
one of the two sheets, independent of each other.

It is convenient to make the following substitutions

2
= Po and 4 = Pp©
oy Milsias

The Eq. (19a) may be written

492

Response of a Vibrating Plate ina Flutd

aM

i
T(a,B) = (a + B*)* Bec a
k2 + (m2 - 1)a2 - 2kMa - Bp?

In the present case,p is a small number (= 0.0015) so that approxi-
mate values of the zeros may be found, expressed as a power series
in p. To the second order these zeros are

[xe + (M2 - 1)B7(Az* + 2A,°B +B - y*)”
where

+ kM+ yk? +(M?- 1)p°
ay = eee ee

M2 - 1

(£), (4), > A(4), (4),

(ee (“en (22) J

i 4A (p2+A2 )[x° +(M?-1) A> -2kMA eras
(+), (+). (+), (+) (+) (+). (+) (+)

where
= / 2 2
Avs), (£)g 7 i aE (*),¥

The last four zeros exist on both sheets. The location of the first
two zeros may be distinguished into three possibilities: when

ve— A, then a, and a; are respectively on the unphysical and
physical sheets; as y is increased such that A,<y< As, then a;
moves off the unphysical sheet and crosses over to the physical
sheet and as remains unchanged; as y is further increased such
that AZ<y then a; crosses over to the unphysical sheet and aj
remains unchanged. A typical configuration for the poles and branch
points is shown in Fig. 6.

493

Maestrello and Linden

a Plane

Fig. 6. Integration Contour for Typ.

The above contour, Fig. 6, is deformed to circulations about poles
and branch cut in the appropriate half-planes of analyticity as indi-
cated below, Fig. 7, for the upper half-plane,

Fig. 7. Deformed Integration Contours for I,

The function T has been analytically continued into the k-plane by
giving k a small negative imaginary part; so that, the branch points
and the poles are displaced off the real axis as indicated in the
previous figures.

The branch-cut integral is given by

494

Response of a Vibrating Plate in a Fluid

wo Ym(ae) [BA(a, B)-2(2,B)] C oe

y k?+(M?2-1)a?- 2kMa-f2
1(B) -f dt [Ba +P - pur? + poce(k-aM
Z Vie+(M2-1)02- ae)

where @ = A_- it. So that

oo
ie - 2ti s residues dB + { Ya (Bb)I(B) dB

The branch-cut integral is exponentially damped rather oscillatory,
so it may be readily performed numerically using Laguerre-Gauss
quadrature. The second integral is more difficult, it oscillates with
a period 21/b.

In solving Eq. (22) maximum values of the indices are polstu-
lated. This is justified, since the index is inversely proportional to
some length on the panel. Now there certainly exists, from the
experimental point of view, a smallest length to which a disturbance
may be localized. After having solved Eq. (22) the plate displace-
ment is simply the Fourier transform of (21), thus,

WOE) =) Wanem(2) on (Z) (23)

where 9m is a beam eigenfunction.

To find the sound pressure level in the cavity the expression
for W from Eq. (23) is inserted into Eq. (5b) to give

cos 8cos DWE cos Kmn(z + d)
a b
y Wrs linrJns

— 2
(Z)- oe S
Ey Kmnsin Kn

agb,k Ly
m,n 1,8
where I, and Jj, are given in the appendix B.

Similarly, the radiation may be computed from Eq. (12).

The force is not a deterministic function as has been im-
plicitly assumed from the outset, but a stochastic variable whose
correlation properties are known, either via a model or directly
from experimental data. Thus, it would only be meaningful to com-
pute statistical averages of the response based on statistical averages
of the force (i.e. cross-correlation). The procedure to be used is
an amendment of a procedure due to Rosenblatt [1962], the notation
is that of Rosenblatt.

495

Maestrello and Linden

Consider a homogeneous, stationary, random process >
(Rosenblatt writes this as X»,(w) to explicitly indicate that it is a
function defined on a sample space) with a cross-correlation defined
as

R( 2 , tr’; t") = (Xo XTh > (24)

where ( ) is the expectation operator, i.e., R is defined through
the ensemble average. Because the process is homogeneous and
stationary,

R(r ,t;r',t') = R(t - r', t-t')
or
R(r - r',t-t!) dr dt = (dM (xz*,) dM (Xm 4, ))

where dM (x> +) is the Stieljes measure of the process. The pro-
cedure may be simply stated as the problem of finding a Fredholm
expansion of R and subsequently representing X,, by such as
expansion. Such an expansion is provided by the eigenfunctions and
eigenvalues of the integral equation

(e8)
w(t ,t) = NY R(r-r',t-t')y(r',t") dr! dt (25)
-©

The spectrum, is of course, continuous. The eigenfunctions are
plane waves and the eigenvalues the inverse of the power spectral
density as can be seen by applying the Fourier transform. Thus the
desired expansion for R is

R(T-T",t- ay ay fe eee Pt Rik aids

Now let

@

ir - wt)

re

- = e dM (X=, ) (26)
° (2m)P VR(K ,w) “© fs

It follows from (24) and (26) that

(Zgu2ZFw) = 6K - K')5(w- a")

496

Response of a Vtbrating Plate tn a Fluid

since (dM(X—,)dM (X~,)) = R(r-r',t-t') dr dt.

Thus the Zr, are independent random variables with unit
variance and with zéro mean if EX, ,=0. The transform of (26) is,

00 —> —>
—> -i(Ker-wt) >
Xo, - Z JR(K,w)e dK dw (27)
’ Kw

oO

A simple calculation reveals that (27) satisfies (24).
These results will now be applied to the plate displacement.

Thus, the cross-power spectral density (CPSD) of the plate response
is given by (asterisk denotes complex conjugate)

(ww) =D om (S) en) (EZ) eC) (Wan Wie") (28)

m,n,r,s

Now if the solution to Eq. (22) is represented as

Winn o » Ymnij Pj
i,j

then

( Winn Wes ) ~ » Ynniy Yren! § 2ij Pkt >
ijkl
Barther,
oo ,0 ve os
(5 ©) -( dK dK' Seeyeeeeee abbr (BY) (5) ery
-00 “-00

The force F is now identified with Kooy so that

(FER) = [RE wa Kol} (Zu Z 0)

* —_ A —_. VA > _.
[R (K a) RK, )] * 5K - KY)

In summary then

= — * x
(Winn Wes.) = y Vinny Yrek ) - am Vea (bby Bie) Meee

ie
nk [T(K) | (29)

a77

Maestrello and Linden

Analogous to (28), the expression for the CPSD of P, can be written
in terms of E Wmn Ws from (29). The same can ailize be done for the
radiation.

b) One Dimensional Model

To simplify the computations we assume that the transverse
plate dimension is very large and that no flexural waves propagate
in the transverse direction. With these simplifications Eq. (22)
may be written

w, + > Paw ue, (30)
where
00 ur (Ka)A Pm =
r 2 ( dK [ (=) | (31)
am -© T(K)
where
4 K NAP
ipy (1 -
T(K) = K* = “i aN et ae ae
Vk? + (M2 - 1)K? - 2kKKM
and
00 1
a -\ ak Yol(aK)F(K) (32)
Bon T(K)

The circumvention of the branch points in the above expressions will
be described shortly.

If Gmn denotes the inverse matrix to 6mn +t Im then the

solution to (30) is
Wa = » Gam?m

m

Now performing the ensemble averages as before gives

(WnWa) =) Gurl rs) Gr (33)

r,s

but

498

Response of a Vtbrattng Plate in a Fluid

(a Hel2K) (yc Yala) Ket
(O45) -{ ak Sea io dK Fieey Waele (K'))

To make the discussion concrete let

*
(F(K)F (K')) = P(a)6(K! - (K - 3))
which corresponds to a spatially uncorrelated pressure field with
convection velocity U, and power spectrum P(w).
Thus

aw

U;,
TK (Ke >)
Cc

UF (aK) uglaK -

m 00
(4,0) = Pt) | ax (34)
-00

The major contribution to the integral for In, Eq. (31), comes about
when the peak of Up is close to the peaks of 1/f(K); since Wn is a
highly oscillatory function (period = 27/a) with a peak at x, /a and de-
caying with the distance from this point and 1/Y(K) is a non-oscilla-
tory function with peaks whenever K equals the real part of the poles
which are roughly located at y times the four roots of unit and the
trapped wave poles near the branch points (k/(1+M) )(k/(M-1)). But
for the frequencies we are considering (up to 3000 Hz) only the pole
near k/(1+M) lies on the physical sheet. For this frequency range

the trapped wave pole is bounded by 0 and 0.15 and the pole near y

by 0 and 0.6. Now, the peak of py is given by y,,/a which is numer-
ically (see Appendix B) 0.155, 0.26, 0.36, 0.465, 0.57, 0.67, ...

and the period is 0.206. The height of the peaks decays roughly as

(1 /Ka)> is 0.05 of the first. The function 1 /T(K) also has peaks that
decay in the same manner. Thus, the infinite matrix Ip, has appreci-
ably non-zero entries only in the upper left-hand corner. Consequently,
we need only compute Imn for the first 4 or 5 modes, say, and then
invert this matrix + I to obtain the upper left square matrix of order
4or 5 of Gmp and thus for higher modes

Grn = 5mn
So that
* X
(Wn Wn) = (bmPn >

for other than the first few modes.

The contour for the integral in (34), Fig. 8, is similar to one

499

Maestrello and Ltnden

K-Plane

K - Plane

|

e S|\7r

eee

wer wmr neem me ew KH

Fig. 9. Deformation of Contour in Fig. 8

described earlier but the term Y*(K - (k/m)) introduces further
poles and branch cuts. The contour and its deformation are shown
in the Figs. 8 and 9. In Fig. 9 only the contours for that part of the
integrand which is analytic in the upper half-plane are shown.

Wm(K) may be written

Um(K) = Sp (K) + T,(K)e 8°

where
H 3 3
4 4
(aK) - Xm
and

500

Response of a Vibrating Plate in a Fluid

5 Tm(K) = Ney [bn + ia, aK) ( cos Xm

Eg ae)
2 2 2
(ak)* - x2 (ak)? +x,

a

- (amXm - iaK) (8 Eis ee :)|
(aK) - Xm (aK) + xq

with @, and Xm defined in Appendix B.

Now

dm(K)Yy(K - +) = LOK) + U(K)
c

where

-i(K-—)a
ae Ww (a) Uc
EK) = sm (K) (s,(K - U,) + T,(K - U,/° )

and
i(aw/Uc)

i(k-7)e
U(K) =e T,, (K) (s,(K -T) TAK = Tw, Uc )

Thus, we have

[ee]
(mon) = P(u) f dK ee
c

The above expression is evaluated as a sum of residue contributions
plus branch--cut integrals.

5

1 * L(z})
aoe (o,¢,) = 2mi Res | —_Ha)__| +I +I
P(w) 2 T(2, )r* (2, - ) | 2

U,
l2
ap Anil dq nes |e) | tit,
os 252) (r(z)T (z - we)
ET U,
j=9

where the z, are identified in the above figure.

5014

Maestrello and Linden

Fig. 10. Location of Poles and Branch Cuts

As indicated previously, zg and Zg are not on the physical
sheet for the range of frequencies considered (up to 14,000 Hz).

k
a geremale acme a
“0 (aoa Get iy) | "(ar i ty) ter)

where T*((k/M +1) 5 iy) is the expression following Eq. (31),the
plus being included only to explicate that it is the positive side of the
square root branchcut. Y7((k/M+1) + iy) is obtained from T* by
replacing pp by - wp.

She UGa—a tt ty) 1 1

c(i L (477 - iy) 1 4

dy oe le

os (Wo ai) txt: iy) TGS + iy)
ae Met) | ot
0 ol eee aed, "(art - iy) "(art - iy)

It is felt, that the effect of the pole at z3 cancels the contribution of
I; and similarly the residue at z,, cancels the contribution of I,.

502

Response of a Vibrating Plate in a Fluid

Thus only I, and , will contribute. Therefore

5
4 * : U(z)
Pay 6 omen = 2ni_) res |__| +1
2 ) er z= 2; T(z)t*(z - wT) °
j43 :
aN: L(z) |
+ 2ni Res) ee a
i: 2, Reel ate ys)
j79 S

By the same argument which eliminated I, and Iz we may replace

I, and I, by pretending that z, and zg are on the physical sheet
and thus

; U(z)
I,=- 2ri Res 4 ————
5 2726 are = ‘
Cc

and

: L(z)
Ig=- 2mi Res 4,——
4 ‘i 2=2q Sere = =}
4

To evaluate the residues it is necessary to determine

T(z) = 423 + ipy(32M7k* + 2°M* - 2°M® - 3kM°2* - kz - k°M)

vk* + (M® - 1)z* - 2kMz°

The following relationships among the poles are valid

* *

Za = yt 27 Zs= a + Ze z= at Zs
Uc Ue Ue
7) * W *

Z10 aie or Zy = tie ce

*4 4 ipy*(1 ri Sa)

Zz iy +
Vice + (M2 - Hee z 2kMz

503

Maestrello and Linden

So
T(z y= Tite)
Thus
bipil pee aa |e? te:
U
Similarly

'
eee ee

Lim | - i | Ag (z,)

cei, T(z -

zz, L7"(z, - a)
4 * U(z,) U(ze)
: bd 1k SUNS ES EN e 2 EAE ne
ZaTP (a) ‘ Smt) T'(z,)T (z, - a T'(2,)T"(z» - Tu
(a) * Ww *K
2 U(5- + z2) . U(—- + 2g)
-s * SAT ae
li +z 7)T (z,) si +2,)T (zg)
‘ Uae +P Zg)
Tg #zJt7 (zg)
P L(z7) P L (za) :
Tz)T (z_- u,) T'(z,)T (zy - wr, )

504

Response of a Vtbrating Plate in a Fluid

EE e3)) V2 425)
ete eo Oe ee Tee 2
xs *
Ta + 2))T (z.,) Ta + Z2)T (zo)

____ Lz)
T'(zg)T "(2g - a)

In a similar, though simpler fashion, we may evaluate the integral
in Eq. (3.2).

Al m(=)] = Am(K) + Byik)e

where
2
ZN X: [ COS, Xm CO. Xm fe ]
==, (| AM = AM
Am(K) a es Xm sin Xm + sh Xm Boag
and

2 °
Bn(K) = - SSm (Xm) = sh x, + sinx,chx,,) - iKa sin x,sh |

sin Xm + sh Xm

*
Thus , Al $m] may be decomposed into factors analytic in upper,
and lower half-planes, respectively. Denoting these terms by G
and G we have,

” GtiK) + G"(K)
1 ae - an aa dK
-00
where
G"(K) = Am(K) (Sn(K) + Tr (K)e"")
and
G"(K) = Bm(K)(Tn(K) + Sp(K)e)

The contour of integration is indicated below.

505

Maestrello and Linden

Fig. 11. Contour-for I, Eq. (31)

The contour is deformed in the appropriate half-planes and
branch cuts are replaced by poles as discussed previously. Thus,

Tat Tmo = Res {Sc} + nae {S0P}

ae Moma vaeicnd

We now invert the matrix 6mn+Imn- Let us denote this inverse by
Gij - Then

* ee
Co “,) = Ginp < %%q) Gon

where the tilde denotes Hermitian conjugate and sum of p and q is
implied. The final step, then, becomes the diagonalization of the
CPSD matrix ( Wm Wa) thus giving the PSD for the actual degrees
of freedom of the system.

c) Acoustic Power, Power Radiated
In computing the power radiated, we make use of the unitary

matrix Ujj which diagonalized the CPSD matrix. The average
power radiated, to the far field, P, is given by

P~ (Im jy" Set)

506

Response of a Vtbrating Plate tn a Fluid

With the use of the asymptotic form of the Hankel function, the
asymptotic form of § may be computed. This form is

; ke 1- M@sin®@ + Mcos @)
A(8)e lig

w(r, scrap
oe Jonikr V1 Evie sin* @

where

it G a —— -M)
A(@) -\ dx'e W(x!)
0

The acoustic velocity in the radial direction u, is given by

a, = S¥ (r, 6)
_ ik(yi - M® sin® @+ M cos 6) sin* 0+ M cos Oy ie r)

Mo 4

The pressure, p, a microphone in the far-field would measure is
given by

ipo oa oe

BP ox

cL
pc

ikM?
EET Rs
Mo=4

iT]

_ ikp(r, ®)
Me ol
The average radiated intensity is given by the expectation value

1 *
= Re(pu )

mM
iT]

2
- ; ik(¥i- M sin 6+ aM coe 5 Re(wy')

2 2 2
> pck (y1 ait arn + M cos 8) Re (yy*)

Now,

Maestrello and Linden

* 4 *
Cb) Se (ALBA (8)
(2mkr)) 1 - M* sin* 0
where
— (xx) cos 8 7
| |-M@ sin se

(.A(@)A7(0)) - ax dx'e ( W(x!) W*(x))
0 )

and where
(W(x!) W" (x) = > (Win War) Pun 21) q (20)
m on

*.
Let U;; denote the unitary matrix which diagonalizes (W,,W,);
LCs 5 the transformation to the actual degrees of freedom. Further,
let \m denote the power in the m!" degree of freedom, then we have

( w(x") W(x) = yi 9; (x') Uni (5); Uaj _(x)

i,j,m,n

The integrals have already been encountered, they are simply the
finite Fourier transform of the g(x) so that,

(A(0)A"(8)) = », Wi (2) Usa i 8ij Uny Yj(-2)

i, j,mn
where
ik cos 6 _M
yi - M2 sin@ 6
Z = T.f 2 feo. Gop 2 ee
Ms = 1

508

Response of a Vibrating Plate in a Fluid

REFERENCES

Corcos, G. M., "Resolution of Pressure in Turbulence," J. Acoust.
Soc'.'Am:'; ‘Vol. '3'5, 'N2;°1963.

Crighton, D. G., "Radiation from Turbulence Near a Composite
Flexible Boundary," PRDC. Rog. Soc. London A314, 153-
173, 1970.

Crighton, D. G. and Flowcs-Williams, J. E., "Real Space-Time
Green's Functions Applied to Plate Vibration Induced by
Turbulent Flow," J. Fluid Mech., Vol. 38, Part 2, pp. 305-
343, 1969.

Dolgova, I. I., "Sound Radiation from a Boundary Layer," Soviet
Physics Acoustics, Vol. 15, No. 1, July-Sept. 1969.

Dowell, E. H., “Transmission of Noise from a Turbulent Boundary
Layer Through a Flexible Plate Into a Closed Cavity," J.
Acoust. Soc. Am., Vol. 46, 1969.

Dzygadlo, Z., "Forced Vibration of Plate of Finite Length in Plane
Supersonic Flow," Proc. Vibration Problem, Warsaw, 1.8,
1967.

Fahy, A. J. Petlove, "Acoustic Forces on a Flexible Panel Which is
Part of Duct Carrying Airflow, " J. Sound and Vibration,
Vol. 5, 1967.

Feit, D., "Pressure Radiated by a Point-Excited Elastic Plate,"
J. Acoust. Soc., Vol. 40, No. 6, pp. 1489-1494, 1966.

Flowcs-Williams, J. E., "The Influence of Simple Supports on the
Radiation From Turbulent Flow Near a Plane Compliant
Surface," J. Fluid Mech., Vol. 26, Part 4, pp. 641-649,
1966.

Lapin, A. D., "Radiation of Sound by a Vibrating Non- Uniform Wall,"
Soviet Physics - Acoustics, Vol. 13, No. 1, pp. 55-58,
July-Sept. 1967.

Lyamshev, L. M., "On the Sound Radiation Theory From Turbulent
Flow Near an Elastic Inhomogeneous Plate," The 6th Inter-
national Congress of Acoustics, Tokyo, 1968.

Maestrello, L., "Radiation From and Panel Response to Supersonic
Turbulent Boundary Layer," J. Sound Vibration, Vol. 10,
Zant I O90

509

Maestrello and Linden

Magnus, W. and Oberhettinger, F., Formulas and Theorems for the
Special Functions of Mathematical Physics, Chelsea, 1961.

Morse, P. and Feshback, H., Methods of Theoretical Physics,
Chap. 11, McGraw Hill, New York, 1953.

Pol'tov, V. A. and Pupyzev, V. A., "Vibration and Sound Radiation
of a Plate Under Random Loading, Soviet Physics - Acoustics,
Vol. 13, No. 2, pp. 210-214, Oct.-Dec. 1967.

Rosenblatt, M., Random Processes, Chapter VIII, Oxford University
Press, New York, 1962.

Strawderman, W. A., "The Acoustic Field in a Closed Space Behind
a Rectangular Simply Supported Plate Excited by Boundary
Layer Turbulence, USL Report No. 827, 1967.

White, P. D. and Cottis, M. G., "Acoustic Radiation From a Plate

-- Rib System Excited by Boundary Layer Turbulence,
Measurement Analysis Corporation 702-06, 1968.

APPENDIX A

If we make the following changes in variables

= ¥(M* - 1) a@- KM

&

K=
M2 - 4

See
M* - 4

then expression for the Green's function in Eq. (9) becomes

| Qixmu : aieueB- y')+z,/ €2 -(x? + B?)]
G(x,y,z|x',y',0) ari ie d dp =
: VM? - 14 40 “00 Je? - (x? +p?

@iMu fl, or age i[Bly-y' +2] (€2+x2)- B2)

2 os K*) = B?

510

Response of a Vtbrating Plate in a Fluid

(see M+F, Vol. 1, Page 823)

ieMu

00
el dé e*" rt”) (RYE - K*)
s -@

R? = (y - y')* +22

The contour of integration for the above integral is shown below.

& - Plane
Branch Cut

i) ;
I(u) -{ dé Ht” (nye? — Ky

-0O

Define

By considering the asymptotic form of the Hankel function, the above
integral is seen to be convergent in the upper half-plane for values
of u and R _ such that

u>R

that is, the region inside the Mach cone.

The contour may be deformed to be a contour along the branch
cut as shown below.

€- plane

Thus

Maestrello and Linden

He) «| ( (eles “Hl (VE? - ) a€

K foe)
below cut above cut

It

oo .
J 2{ ely (RVER - K*) ab

We thus get (see Magnus and Oberhettinger, p. 179)

in/ju?-R?

I(u) = 2i—
u2 - R2

so that
in(Mut,/ u2-R2
2Tri e! ( : )

GCx:| 2) eee Be
a Meee fue Bo

elk MR (cos O+sin @)

Me = 1 R sin 0

APPENDIX B

Lr = ‘a cos — or (= ) dx

fe)

(oe NeeC deer E

9, (=) See = she te (cos “= -
if a a r a

412

[o @) ise) 4
\e ee Rye one “i - be alr, (RYE - IC) aE

Response: Of a Vibrating Plate in a Fluid

Set = aoa fo (cn [Ce B2)»-BECGS)] +anf22C54)])
+ coofZE (%8)] ~coo[(A-B2) aE (858)]}
+ Spm coe ESEC)] -coel te BD ECT
+o (ain [(EeoBt)a +22 (28) tn [BE@2))}
eel Sie cos [22 (25%)| (ch x, tash x '

+ ayParmp oie RECE] -otoREEFA)] (ane, ta.cnah
as

where

= Gos K= ch ik
r sin K, t sh K,

and the normalization N, is given by [ Dzygadlo 1967]

-2 L 1
N, = aK, (sh 2K, - sin 2K,) 2 ee K, COS K, - sin Kr ch K,)

Ar °
2K (ch 2K, - cos 2K, - 4 sin kK, sh K,)
+ ar ag +3 (sin 2x, + sh 2x,)

= = (sin K, ch K, + sh kK, cos K,)|

and the eigenvalues xk, are the roots of the equation

cos K, cosh K, = 1

Maestrello and Linden

and are approximately given by
ul
K, = 4.730 K, = 7.853 K, = 7) (2n “+ 1)

Jns is the same as Ings with a and a, replaced by b and kh,
respectively.

APPENDIX C

3 2 na aaa
Ow > —- dw -iK-r
Atw] =§ sues BS ae e |

Now
A[en(2) ea(E)] = Otn.0.2,8,a,0
- Q(n,m,f,@,b,a)
where g is defined in Appendix A and

O(m,n,@,8,a,b)

2 . : i
= 208) [B- tnp+ oPaen us iP ate] J, oe enC§)

The integral may be found by appropriately combining the following
four integrals

i(x_ + @a) - i(x_- aa)
e - |

i(aa-«,) i(aa+x,)
ale oA Ke A |
2 aa - K, aat K,

ea
n
he
=}
xs
—]
|x
O_
Q
x
Qu
*
I

a i(aa+x«,) i(aa-«,)
x iax _ ee -i +2£ -i
- cos Kas e dx= “Stl aa hk. = Ee K,

Response of a Vibrating Plate in a Fluid

(aati x,) i(aa-ix,)
‘ che Zeta ai | s =i Ue =

aa PEK, aa - LK,

; i(aatix,) (aa-ik,)
‘ Ohi a | Serie Shee hearst -1]

na Zi aa tik, aa - ik,

and where a, is given in Appendix B.

where

and J+

APPENDIX D

(a —-> svar aa a
Ql w] -aiPs ( d r,/g(k/r,) W( r,)

00
QL o(x/algsly/b)] = wp, » Emén amiaae)in(Bbn)Emeins

Kn Sin Kinae

is given in Appendix B.

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

Thursday, August 27, 1970

Afternoon Session

Ghairman: Ts Inui
University of Tokyo, Tokyo, Japan

Page

Recent Research on Ship Waves 519
J. N. Newman, Massachusetts Institute of Technology

Variational Approaches to Steady Ship Wave Problems 547
M. Bessho, The Defense Academy, Yokosuka, Japan

Wavemaking Resistance of Ships with Transom Stern oye:

B. Yim, Naval Ship Research and Development Center

Bow Waves Before Blunt Ships and Other Non-Linear
Ship Wave Problems 607
G. Dagan, Technion-Israel Institute of Technology,
Haifa, Israel, and M. P. Tulin, Hydronautics, Inc.

Shallow Water Problems in Ship Hydrodynamics 627

BE. O, Tuck and P. J. Taylor, University of Adelaide,
Adelaide, South Australia

BAT

RECENT RESEARCH ON SHIP WAVES

ee rm she)

de Nig Newman 2 - 3 -\ ° ai had
Massachusetts Instttute of Technology
Cambrtdge, Massachusetts

ABSTRACT

This paper is concerned with various aspects of the
far-field wave pattern generated by a ship or other
moving body in steady translation. Section II contains
a brief derivation of the classical Kelvin wave pattern,
based upon linear inviscid wave theory. In Section III
full-scale aerial photographs are presented for the
waves generated by the Ferry Boat UNCATENA, and
compared both with the theory and with photographic
observations of a small scale model of the same vessel.
In Section IV we discuss a towing tank experiment,
designed to compare the waves generated by a "wavy
wall" with the nonlinear theory for this hull form.
Finally, in Section V, a third-order solution is out-
lined for the Kelvin wave system, which indicates the
occurrence of a nonlinear instability on the cusp line.

I. INTRODUCTION

Ship waves are intriguing from several viewpoints. To the
observer of a moving vessel, either layman or professional, they
are a fascinating pattern on the free surface -- in otherwise calm
water one might, in fact, regard them as a thing of beauty. To the
naval architect they are of primary interest as a source of energy
radiation, and hence of wave drag on the vessel. More generally,
to ship hydrodynamicists of all disciplines, they are a source of
complication, interacting with, and affecting the boundary conditions
of, such fields as viscous drag, propeller-hull interactions, sea-
keeping, and maneuverability. To anaval vessel ship waves are a
possible source of detection, visible for hundreds of wavelengths or
ship lengths downstream and to each side of the vessel's track.
Similarly, to the operators of all types of vessels, they are a source
of damage to property and of personal injury to persons, in or be-
neath the water surface, or on the shoreline. And, to the theoretical
hydrodynamicist, they are the source of seemingly endless challenges,
both analytical and computational.

5.9

Newman

The classical analysis of ship waves using linearized inviscid
wave theory, originated in the nineteenth century by Kelvin and
Mitchell, has enabled us to understand and predict the qualitative
features of ship waves. The period since 1900 has seen a wide vari-
ety of refinements and applications of this basic model, and the
present unsatisfactory state-of-the-art in no way detracts from the
dedicated contributions of Havelock and others, some of whom are
in this audience, who labored with ship-wave theory before it was
facilitated by digital computers and a more widespread understand-
ing of the methods of mathematical physics.

As in most other branches of ship hydrodynamics, our present
knowledge of ship waves is sufficient only for a qualitative under-
standing of the phenomena, and does not permit quantitative predic-
tions with the accuracy required by most engineering situations. In
the past decade many ambitious scientists and engineers have, there-
fore, abandoned the assumptions of linearization, or of an inviscid
fluid. Some of these attempts at fundamental improvements of the
Kelvin-Michell approach were summarized by the author in a Panel
Report (Newman [ 1968]) to the Seventh Symposium on Naval Hydro-
dynamics. The present paper is not intended to cover such a broad
range of contributions, but only to report on my own recent and very
limited activities in this area, both experimental and theoretical. In
the experimental domain, photographic observations have been made
of the Ferry Boat UNCATENA and of a small scale model of the same
vessel, in order to verify the Kelvin wave pattern prediction and to
search for variations in the wave system resulting from scale effects.
In addition, a series of experiments have been made in a towing tank
with the objective of confirming the striking nonlinear phase-jumps
predicted by Howe [1967, 1968]. Finally, in an-extensive theoretical
investigation, which is reported upon in more detail elsewhere
(Newman [ 1971]), we consider the possibility of third-order nonlinear
resonant interactions in ship waves, motivated by the importance of
these interactions in the field of ocean waves (Phillips [1966]). For
the sake of completeness, we shall first give a brief outline of the
classical Kelvin ship-wave system.

Il.’ THE KELVIN SHIP=“WAVE PATTERN

Disregarding the local effects close to a ship hull, we can
assume that ship waves are a distribution, in wavenumber space, of
individual plane water waves. Generally speaking, these are ob-
served to be of small amplitude relative to their wavelength, and the
relevant Reynolds numbers are of order 10®to 10°, so that we are
led to a linear potential-flow model. The individual plane wave
system can then be described by the free-surface elevation

ik(x,cos O+y, sin@) -iwt

C (xosy,) = Ae (1)

520

Recent Research on Shtp Waves

where A is the wave amplitude, k is the wavenumber (20/d, if

X is the wavelength), 9@ the direction of propagation of the wave with
respect to variations of time t. The kinematic and dynamic properties
of the wave motion can be readily determined from the velocity poten-
tial, which differs from the above only by a factor

(sip jue ©

if the fluid depth is large, the vertical z,-axis is positive upwards
with z,=0 the plane of the undisturbed free surface, and the

(X55 Vo» Zo) coordinates are fixed with respect to the bulk of the fluid
volume. Finally, with the above restrictions, the frequency w and
wavenumber k obey the dispersion relation

k= w*/g.

The most general distribution of these elementary plane waves
is obtained by integrating over all scalar wavenumbers k (or fre-
quencies w) and wave directions 90, so that

Ik(x, cos @+y, sin @)- iwt
e

(00) 27
E(x 5 Vo) -{ ax f dé A(k,8) ° (2)
(@) 0

However, steady-state ship waves are independent of time, when
viewed from a moving coordinate system which translates with the
ship, say with velocity V inthe +x direction, and this condition
restricts the frequency, or wavenumber, of the contributions to the
integrand in (2). If (x,y) denote moving coordinates with

Xg= xt Vt
Yo Y?
then by direct substitution

ik(x cos O+y sin@)+it(kV cos O-w)

00 27
t(x,y) = l ax { d® A(k, O)e (3)
fe) 0

This integral will depend on time, for arbitrary amplitude functions
A(k,9), unless

w = kV cos 8 (4)

521

Newman

or from the dispersion relation,

kee (g/V?) sec’ 0, (5)

Finally, if the ship's velocity V is positive, it follows from (4) (or
from an obvious physical argument) that the wave direction ® must
lie in the interval - 1/2 S05 1n/2. Thus we arrive at the "free-
wave" description of the ship-wave system

i(g/V°) sec” @(x cos @+y sin@)

11/2
Heme -{ de A(o)e (6)

-7/2
which is the starting point for many analyses of wave resistance.
Kelvin's ship-wave pattern may be obtained from (6) by noting
that if the polar radius R = (x? + y?)!/2 is large compared to the
typical wavelength 271V?/g, then from the method of stationary phase

the dominant contributions to (6) will arise from those angles @
where the phase function

(g/V’) sec’ 9 (x cos 8 + y sin 8) (7)
is stationary, or

& sec’ Q (x cos 8 ty sin 8) = 0. (8)

Carrying out the indicated differentiation, it follows that
x tan @ + y(2 sec*@ - 1) =0,

or that the significant ship waves will be situated at points such that

sin 8 cos 9
- |y/x| Se eT a ma (9)
The behavior of this function is indicated in Fig. 1, and the essential

features of a Kelvin system are immediately clear:

-/2
1. The waves are confined to a sector ly/x|< 8 *

tan 19°28".

2. On the boundaries of this sector, or cusp line, the waves
are oriented at an angle |0| = cot! 2V@ > 35016",

5iZz

Recent Research on Shtp Waves

In the interior of the sector, two distinct wave systems

Sis
will occur, the diverging (|@| > 35°) and transverse
(|@| < 35°) systems.
| x
<= --+4y= tan/2°28"
~ — Nbcos® aN O= 35%"
Vx 2-cos*@ ae O=-35°le' 0 0

Yne~tan 13°28

Fig. 1 Plot of y x, from Eq. (9), as a function of the
wave direction 0.

Finally, if the loci of a given wave crest are plotted, by
requiring that the wave phase (7) be constant, while (9) is satisfied,
the familiar Kelvin wave pattern shown in Fig. 2 is obtained. (The
phase difference of 1/2 between the two wave systems along the cusp
line is not explainable from the above simplified argument, but is a
natural consequence of the method of stationary phase. Fig. 2 is

reproduced from Lunde [ 1951].)

Transverse
Wave Crest

Fig. 2 Wave crests of the Kelvin wave system (from
Lunde [ 1951] )

The amplitude of the individual elements in the Kelvin wave
system will vary, in accordance with the function A(®@) or the "free-
wave spectrum" of the vessel. Moreover, the wave amplitude will
vary as R for large R, in consequence of the radial spreading

523

Newman

of wave energy. This result follows, too, from the stationary phase
approximation, wien in addition, tells us that the attenuation rate
is changed to R"’3 on the cusp line. A uniformly valid expansion
near the cusp line has been obtained by Peters [1949] and by Ursell
[1960]. The latter work includes numerical computations.

Ill PHOTOGRAPHIC OBSERVATIONS OF SHIP WAVES

Kelvin's ship-wave pattern, as developed in the preceding
section or as originally developed by Lord Kelvin using an initial-
value approach, is well known and widely accepted, since the final
results are consistent with our observations of ship waves. Never-
theless, and perhaps to the surprise of many, the author knows of no
definitive experimental confirmation of the Kelvin pattern. Aerial
photographs are generally of the near-field (e.g., Guilloton [ 1960];
Inui [ 1962]) or from oblique angles (Wehausen and Laitone [ 1960],
Fig. 23). Stoker's "Water Waves" contains striking high-altitude
photographs which are from directly above the vessels, but during
turning maneuvers or while in convoys. One exception to the above
may be a special volume™ on aerial reconnaissance prepared during
World War II, but this is not generally available, nor has it been
used at all for a comparison with theory.

Two years ago I had the opportunity to obtain aerial photo-
graphs of the Ferry Boat M. V. UNCATENA. This vessel is 147 feet
long by 28 feet waterline beam and 9 feet draft, displaces 400 tons,
and operates at a speed of 153 knots between Woods Hole and Martha's
Vineyard, Massachusetts. Propulsion is from three propellers,
turning at 1,200;,1,000, and 1,200 rpm. The water depth injgiie
area where photographs were made ranges from 50 feet to 80 feet,
with depths of 70 feet predominant. Originally, this vessel was
chosen for observation because of the severe wave systems which
it generated, in consequence of its high (0.38) Froude number. Pre-
liminary observations from a surface vessel indicated the most severe
waves to be substantially shorter than those which are predicted on
the cusp line of the Kelvin wave system, but it is apparent from the
subsequent photographic observations that these waves, in fact, are
located inside the 19°28' boundary of the cusp line.

Three of the photographs obtained are shown here as Figs.
3-5. These were made with a Hasselblad 2-1/4 X 2-1/4 inch camera
and a wide-angle lens of 38 mm focal length. Figures 3 and 4 were
taken consecutively, looking directly downward from an altitude of
1,000 feet. Figure 5 is an oblique shot from 1,600 feet. As is clear
oe these photographs, the predominant waves generated are a
portion of the diverging wave system, lying inside of the 195° cusp
line. In Figs. 3 and 4 we have drawn in boundary lines + 19°28'

*
"Speed of Shipping," revised edition, Central Interpretation Unit,
revised November 1943,

524

Recent Research on Shtp Waves

*

7

eas ORM

4

oe

Fig. 3 Aerial view of the UNCATENA wave system with
19°28' boundary lines and typical 35916' cusp-
crest tangent lines superimposed

Newman

2
;

”
>

3

further down-

boundary lines and

tangent lines superimposed

>

stream, with the 19°28'

Fig. 4 Aerial view of the UNCATENA
typical 35°16"

526

Recent Research on Shtp Waves

Fig. 5 Aerial oblique view of the UNCATENA from
ahead of the bow

from the center of the wake, which, in principle, should lie on the
cusp line or boundary of the wave system. The apex of this center

is to some extent arbitrary, since, without further knowledge of the
amplitude function A(6), as it appears in Eq. (6), for this particular
vessel, the location of the ship's bow relative to the origin of the
(x,y) coordinates is arbitrary. It is clear, in fact, from Fig. 3 that
the apex of the UNCATENA cusp lines is somewhat ahead of its bow,
by a distance of about one ship length. This conclusion is consistent
with other observations of ship waves (e.g., Gadd [1969]), and we
emphasize here that, in principle, there is no contradiction between
this observation and the linear Kelvin prediction.

Also shown in Figs. 3 and 4 is a typical pair of 35°16' angles,
which should be tangent to the wave crests on the cusp lines, Both
the 19°28' boundary angle and the 35°16' wave crest angle are sub-
stantially confirmed by these observations, within the accuracy
obtainable from these photographs. In fact, we have not observed
any phenomena in these tests which are inconsistent with the linear
Kelvin description of the far-field waves, in spite of the obviously
nonlinear near-field disturbance associated with this vessel, espe-
cially at its bow. There is also no noticeable effect on the waves
from the ship's viscous wake and propeller wake region, in spite
of the persistence of this wake far-downstream. (However, the
latter wake effects may be expected to affect the transverse waves
near the centerline; this effect could not be detected here because of

Newman

the relatively weak transverse wave system of this vessel, and the
fact that aerial photographs from directly above tend to emphasize
the shorter, and hence steeper, diverging waves.)

As can be seen in all three photographs, but most noticeably
in the downstream portion of Fig. 5, the diverging wave system
includes three discrete groups of waves, separated by relatively
calm regions or "nodes." The observation of three wave groups is
also noted by Wehausen in the text adjoining Fig. 23 of Wehausen
and Laitone [1960]. The explanation of this phenomenon is somewhat
controversial. One possibility is in terms of the conventional "bow"
and "stern" waves, and possibly others associated with the shoulder
or knuckle of the vessel. My own view is that, while this synthesis
is applicable near the ship, it is inappropriate in the far-field where
the stationary phase approximation developed in the previous section
is valid. Indeed, if two discrete "bow" and "stern" wave systems
are superposed, in the Kelvin stationary phase approximation, the
result is only one wave system, provided that the observation distance
downstream is large compared to the separation distance between the
two disturbances. There will, of course, be interference effects,
with regions of reinforcement and other regions of cancellation, just
as in the simpler radiation and diffraction patterns which we associ-
ate with nondispersive wave problems. In the present context these
will be introduced via the amplitude function A(@), and my own view
is that the nodal regions between the three observed wave systems
correspond to zeros of the function A(9) for this particular vessel.
Indeed, it is not difficult to perceive, in some parts of these photo-
graphs, a phase difference of 180° across the nodal regions. In
principle, there should be additional nodal lines and discrete diverg-
ing waves in the interior portion of the wave system, but these will
correspond to relatively short waves which are not so strongly
generated by this vessel, and more quickly attenuated by viscous
and other effects.

As one possible measure of the validity of Froude's hypothesis,
that ship-wave effects are dependent only on the Froude number and
the hull form, a series of photographs have also been made with a
scale model of the UNCATENA. For this purpose a six-foot (scale
ratio of 24) fiberglass model was constructed, and equipped with a
single battery-powered electric motor and screw propeller, anda
radio-controlled rudder system. Figures 6 and 7 show the model,
as fitted with its single propeller and rudder, with an antenna mast
for the radio control receiver. Test runs were made with this model
on the Charles River, with photographic observations from the Boston
University Bridge at a height of approximately 50 feet. These photo-
graphs were made with a Minolta 35 mm camera and 28 mm focal-
length lens. Figures 8 - 10 show the model wave system from various
oblique angles. Unfortunately, no observations could be made from
directly above, so that measurements of the wave angles are not
obtainable for the model scale. Figures 8 - 11 show indeed that
close to the model discrete bow and stern wave systems are

528

Recent Research on Ship Waves

Fig. 6 UNCATENA model Fig. 7 UNCATENA model
bow view side view

distinguishable, whereas further downstream these blend into a single
diverging wave pattern with several nodal regions. It is apparent
that, on the model scale, more than three wave groups can be
distinguished far downstream; in Fig. 10 four or possibly five dis-
crete groups can be noted. Since viscous attenuation is stronger at
the smaller Reynolds numbers corresponding to the model scale,

it must be presumed that the attenuation of the short diverging waves
in the full-scale tests is due to other effects, such as the higher level
of ambient waves and turbulence in the full-scale flow. Another
noticeable difference is the obvious presence of transverse waves

in the model tests, especially in Fig. 9 which is from approximately
the same viewing angle as the full-scale photograph shown in Fig. 5.

Fig. 8 UNCATENA model and wave system
(compare with Fig. 5)

529

Newman

Figs 9 UNCATENA model and wave system

aerial view

SONG

Fig. 10 UNCATENA model and wave system

530

Recent Research on Shtp Waves

This infers a significant difference between the transverse wave
amplitudes for the model and full-scale, which could have important
ramifications on the predictions of wave resistance from conventional
model testing, but this tentative conclusion may be biased by minor
differences in camera angles or lighting™, and it is felt that a quanti-
tative measurement of the transverse wave amplitudes for the model
and full-scale vessel should be made, with wave buoys or stereo
photographs.

IV. TANK TESTS OF A WAVY WALL

In perhaps the only truly nonlinear analysis of ship waves
carried out to date, Howe [ 1967, 1968] has considered the waves
generated by a "wavy wall" or ship hull form consisting of a slowly
damped sine wave. This geometrical form generates preferentially
only one wave system. By suitable choice of the hull wavelength
and velocity, a diverging wave system can be generated which,
according to Howe's theory and based on the analysis of slowly vary-
ing finite amplitude waves as originally developed by Whitham [1965],
will become unstable. The most striking feature of Howe's compu-
tations, resulting from this instability, is the occurrence of a shock
or "phase-jump" across which there is an abrupt change in phase
and wavenumber. Figure 11 is reproduced from Howe [ 1968], and
shows the calculated wave system and the region where a phase-jump
is predicted. Also shown, on the abscissa and with an exaggerated
scale, are the waterlines of the hull form.

Fig. 11 Cross sections of the free surface perpendicular to the
phase-jump. The broken line segments indicate a possible
form for the free surface in the neighborhood of the phase-
jump (from Howe [ 1968]).

*In color slides shown during the oral presentation of this paper,
some weak transverse waves can be noted in the full-scale tests.

531

Newman

It should be noted that Howe's choice of a specific problem to
which to apply the Whitham technique was based largely on the rela-
tive ease of veryifying the results with a suitable experiment. We
therefore set out to conduct such an experiment in the MIT Ship Model
Towing Tank. For this purpose a model was constructed of Formica
plastic laminate, bent to conform to Howe's damped sine wave, with
fiberglass and polyester resin reinforcement and fairing of the back
side of the Formica. The model was 10.4 feet long, by 1.5 feet
vertical depth, and was immersed to a wetted draft of 1.0 feet. This
"model" was fitted to the towing carriage in an off-center position
to maximize the effective width of the tank and minimize reflections
from the tank walls. The tank width is 8.4 feet, and the model was
set up to give a separation of 5.5 feet between the wavy side and the
facing tank wall. Tests were carried out at a speed of 4 feet per
second, and the wave system was observed visually, photographically,
and with a pair of wave probes which were placed at varying distances
from the model to obtain a total of sixteen longitudinal wave records.

Figures 12 and 13 show the model in operation, and the re-
sulting wave system. In no case was a phase-jump observed, in the
region where it was anticipated. One can discern a somewhat irregu-
lar local effect along a longitudinal line about one foot from the tank
wall, but this phenomena extends to the front of the wave group, is
parallel to the longitudinal axis, and, moreover, originates further
away from the model than the predicted phase-jumps. This discrep-
ancy is unexplained, although D. J. Benney (private communication)
has pointed out that the existence of phase-jumps can be questioned,

Fig. 12 Photograph of the wavy wall and wave pattern looking down-
stream

532

Recent Research on Shtp Waves

Fig. 13 Photograph of the wavy wall and wave pattern looking
upstream

533

Newman

in principle, on the grounds that its existence violates the preassumed
condition of a slowly varying wave system which is the basis for
Howe's work.

V. THIRD-ORDER INTERACTIONS IN KELVIN WAVE SYSTEMS

One of the fundamental properties of a linear boundary-value
problem is the principle of superposition; thus, for example, Kelvin's
ship-wave pattern, although originally derived for a single "pressure
point,” is valid for any distribution of singularities and hence for
arbitrary ship hulls. But as soon as the assumption of linearity is
discarded, the possibilities for nonlinear interactions, among the
previously independent components of the solution, must all be
examined. In water-wave theory it was shown ten years ago by
Phillips (cf. Phillips [1966]) that for deep water gravity waves the
second-order interactions are relatively uninteresting, but when
third-order effects are included it is possible for "resonant" inter-
actions to occur. Thus two or three primary waves can interact,
over large scales of time and distance, so as to transfer a substantial
portion of their energy into a completely new wave system of a differ-
ing wavenumber. This striking result has been confirmed by others,
both theoretically and experimentally, and can be regarded as well
established.

Motivated by the occurrence of third-order interactions in
ocean wave systems, and by the striking nonlinear effects obtained
for a special case of the ship-wave problem by Howe [ 1967, 1968]
as noted in the previous section, I have studied the third-order per-
turbation solution of the Kelvin wave problem. The details of this
investigation are "messy," to say the least, and will be presented in
a separate paper (Newman [1971]|), but I shall briefly describe the
technique employed and the form of the results. First, as a pre-
liminary approach to this problem, we may examine the possibility
that, at any point in the Kelvin wave field, the transverse and diverg-
ing waves are such as to satisfy the criteria developed by Phillips
for resonance between two primary waves. It is not difficult to show,
in fact, that the wavenumbers of the diverging and transverse waves
are not resonant, except possibly on or near the cusp line, where the
simple stationary phase results are invalid.

To develop a complete solution of the ship-wave problem valid
to third-order would be a formidable task; local effects near the hull,
and nonlinearities associated with the boundary condition on the hull
would have to be included, and the possibility of a breaking wave near
the bow would raise fundamental questions of validity of the solution.
Instead, we focus on nonlinearities associated only with wave propa-
gation on the free surface, and taking place slowly over scales of
many wavelengths, so that local effects and hull nonlinearities can
both be neglected. The first-order linearized velocity potential must
satisfy the familiar free-surface condition

534

Recent Research on Shtp Waves

2
8, ae P isk = 0 a z= 0 (10)

where subscripts denote partial derivatives. The notation and co-
ordinate system are as defined in Section II. By suitably non-
dimensionalizing the coordinates, we may replace (10) by the con-
dition

>, + $),,=0 on 7a Ole (11)

The general solution of this free-surface condition and of Laplace's
equation, not including local effects near the disturbance, is (cf.
Eq. (6))

2
is | do £(0) ef? *k'x (12)
[@)

where k= (g/V*) sec* 8, k=(kcos 0, k sin 0), and x= (x,y). (In
Section II the wave eae were restricted to the sector - 1/2<@<
1/2. Here we allow all values of @ in the integrand of (12), in
order to avoid taking the real part of the complex exponential;

Eq. (12) will be real if £(6) = f (w- 0), and to avoid difficulties with
the radiation condition we shall assume that (12) holds only if x is
large and negative.)

The second-order free-surface condition, analogous to (11),
is

az Poxx = 2V >, V Ox - Pix Pi 27 a Pixxz)e (13)

By inserting the first-order solution (12) for 9, in (13), and replac-
ing products by repeated integrals, it follows that a particular solu-
tion of (13) will be

27 21 k z+ik *x

bo= J) do i do, £8) £(0,) W(0,,O,)e Ge enle (14)

where

hae

ig = k(8,) + k(®,).

The weight function W is an algebraic function, determined by the
various derivatives in (13), and it can be shown that this function
contains only removable singularities. Thus by repeated application
of the method of stationary phase, $)= O(R"!) for large distances

R from the disturbance, and this second-order potential will be

535

Newman

masked by the first-order potential 9, = te

)e
Extending these results to third-order involves straightfor-
ward but tedious analysis. The third-order free-surface condition

is analogous to (13), but involves more terms on the right-hand
side:

1

3 if P34 ate 2V6, ° vive,)° : 26 (Vo, ; hate

Te y Vo) + 2(V >, , V bo. 7. ee ° Vo)

l
- b(doz7+ doxxz) (15)

A particular solution, analogous to (14), is given by the triple integral

2a 2r 2a eT
os 123% 123) =
d3 = i do, if de, \, de, £(0, ) £(0,) £(05) W(0, »9,,6,)e (16)

where

Ky = RO) TKO), F k(0,).

The weight function W(0, 8., 9.) is singular at points where its
denominator vanishes or where

2
Kio, — (sec 9, t+ sec 6, + sec 8,) = 0 (17)
and it is necessary, therefore, to study the roots of this equation.
It can be shown that the strongest singularities occur along the cusp
line; for example, at the point

The integral (16) is improper at these points and we, therefore,
conclude, as in many linear wave problems, that a steady-state
solution cannot be assumed _a priori, but must be derived as the
appropriate limit of an initial value problem.

An expedient initial value problem is obtained by regarding
the right-hand side of (15) as a pseudo-pressure distribution, im-
posed on the free surface from an initial state of rest, and then
looking for the steady-state limit which results. To avoid unneces-
sary algebra we rewrite (15) in the unsteady form

536

Recent Research on Shtp Waves

$5, us % xy Pe 2b sys t 344 =e! P(x,y) (18)

where P denotes the right-hand side of (15). A solution correspond-
ing to (16) is readily obtained, and in the limit € ~ 0 we find that
the only modification is to replace Eq. (17) for the roots of the
denominator of W by the new equation

3
Kio + (€ 70) sec @))" = 0. (19)
jr

Thus the singularities in (16) become slightly complex, and for e > 0
the integrals in (16) are proper.

Finally, the behavior of (16) as € ~ 0 must be examined.
Here the algebraic details are critical, since cancellation occurs
between many of the leading-order terms. In view of the numerous
possibilities for error, the following surprising result must be re-
garded as tentative, and I would hope that it will be verified inde-
pendently by others who are willing to tackle the algebra involved.

As €—0, Eq. (16) predicts waves on the cusp line of the
same form (0 = 35°) as the first-order solution, but with an ampli-
tude which tends to infinity logarithmically in €. Thus there can
be no steady-state solution of the third-order initial value problem,
as posed in Eq. (18) and, presumably, in the more general case of
a "steady" moving disturbance initiated from a state of rest. Ulti-
mately, as in the analogous case for ocean waves, the logarithmic
growth rate will be modified by further nonlinearities, but, never-
theless, we must conclude that significant amounts of energy can be
exchanged, through nonlinear processes, in the region of the cusp
line, among adjacent wavenumbers.

VI. CONCLUSIONS AND RECOMMENDATIONS

The caption above is the standard one for theses and reports.
In this paper we have obviously raised more questions than we have
answered. The observations of the UNCATENA show that Kelvin's
wave patterns are confirmed, even for a highly nonlinear near-field,
provided the observation point is sufficiently far downstream (much
further than is possible in a conventional towing tank). But are the
differences noted in the photographs of the full-scale vessel and the
1/24th-scale model due to differences in photographic conditions
and experimental errors, or are the transverse waves (and very short
diverging waves) of substantially larger amplitude on the model scale?
Here we would emphatically recommend further experiments in which
the wave heights can be measured quantitatively, both for the full-
scale vessel and for its model. This task can be simplified if only

537

Newman

the transverse waves are examined, but the difficulties of carrying
out full-scale measurements in an ambient wave system are well
known. (In spite of their appearance to the contrary, Figs. 3 - 5
were made early in the morning of a relatively calm day to ensure
little ambient wave motion. The low rising sun in the model photo-
graphs is explained similarly !)

Turning to the wavy wall tests described in Section LV, we
have attempted, without success, to verify a theoretical anomaly --
Howe's phase-jumps. It is an open question whether this failure is
due to experimental error (an obvious possibility is the effective
modification of the wavy wall shape due to viscous effects), or if,
in fact, the phase discontinuity obtained by Howe is a consequence of
pushing Whitham's slowly varying finite amplitude technique too far,
with a solution which is not always slowly varying but contains local
"singularities" or "shocks." It is not likely that this question can be
answered by further experimental work, unless possibly a viscous
correction can be incorporated in the model's shape (to allow for a
turbulent boundary layer in the presence of a slowly varying sinu-
soidal pressure gradient !).

Finally, in Section V, we have outlined a nonlinear analysis
of the Kelvin wave system which predicts an instability along the cusp
line, but for which (unlike Howe's instability) the conclusion depends
critically on a delicate avoidance of algebraic errors. Having thus
gone out on a limb, I can only express the hope that independent
verification is soon forthcoming.

VII. ACKNOWLEDGMENTS

The three experiments described here were carried out with
the generous assistance of many persons. The full-scale photographs
of the UNCATENA were made by Mr. F. Claude Ronnie of the Woods
Hole Oceanographic Institution; the Woods Hole Oceanographic Insti-
tution also furnished the aircraft for these tests and the services of
Mr. Robert Weeks as pilot. The subsequent experiments were per-
formed by several MIT graduate students: David MacPherson and
William McCreight built the fiberglass mold and model of the
UNCATENA; Albert Bradley kindly loaned his radio control system;
and the model completion and testing was carried out by Charles Flagg
and Nan King, with photographs made by Ronald Walrod. Messrs.
Flagg and King also built and tested the wavy wall model, and photo-
graphs of this test were again provided by Mr. Walrod. The propeller
for the UNCATENA model was kindly loaned by Professor Daniel
Savitsky of the Davidson Laboratory, Stevens Institute of Technology,
and the wave probes used for the wavy wall test by Professor Jerome
Milgram of MIT.

538

Recent Research on Ship Waves

REFERENCES
Gadd, G., "Ship Wavemaking in Theory and Practice," Trans. Royal
Inst. Nav. Archs., Vol. 111, 4, pp. 487-506, 1969.

Guilloton, R. S., "The Waves Generated by a Moving Body," Trans.
Inst. Nav. Archs., Vol. 102, 2, pp. 157-174, 1960.

Howe, M. S., "Non-Linear Theory of Open-Channel Steady Flow past
a Solid Surface of Finite-Wave-Group Shape," J. Fluid Mech.,
Vol. 30, 3, pp. 497-512, 1967.

Howe, M. S., "Phase Jumps," J. Fluid Mech., Vol. 32, 4, pp.
779-790, 1968.

Inui, T., "Wave-Making Resistance of Ships," Trans. Soc. Nav.
Archs. and Mar. Engs., Vol. 70, pp. 283-353, 1962.

Lunde, J. K., "On the Linearized Theory of Wave Resistance for
Displacement Ships in Steady and Accelerated Motion,"
Trans. Soc. Nav. Archs. and Mar. Engs., Vol. 59, pp.
25-685, 1951.

Newman, J. N., "Panel Report -- Nonlinear and Viscous Effects
in Wave Resistance," Seventh Symposium on Naval Hydro-
dynamics, Rome, 1968.

Newman, J. N., "Third-Order Interactions in Kelvin Ship- Wave
Systems," J. of Ship Research, Vol. 15, 1, pp. 1-10, 1971.

Peters, A. S., "A New Treatment of the Ship Wave Problem,"
Communs. Pure and Appl. Math., Vol. 2, pp. 123-148, 1949.

Phillips, O. M., "The Dynamics of the Upper Ocean," Cambridge
University Press, 1966.

Ursell, F., "On Kelvin's Ship-Wave Pattern," J. Fluid Mech.,
Vol. 8, 3, pp. 418-431, 1960.

Wehausen, J. V., and Laitone, E. V., "Surface Waves," Handbuch
der Physik, Vol. IX, Springer-Verlag, 1960.

Whitham, G. B., "A General Approach to Linear and Non-Linear

Dispersive Waves Using a Lagrangian," J. Fluid Mech.,
Vol. 22, pp. 273-284, 1965.

559

Newman

DISCUSSION

SOME DEVELOPMENTS IN SHIP WAVE PATTERN RESEARCH

N. Hogben

National Phystcal Laboratory, Shtp Diviston
Feltham, Middlesex, England

I. INTRODUCTION

This note briefly reports two developments in ship wave
pattern research. The first concerns progress in application of the
'Equivalent Source Array' concept described in Ref. 1; the second is
the development of a fully automated system of recording and
analyzing the waves, a more detailed account of which is being
prepared asi /Refs<2.

II. EQUIVALENT SOURCE ARRAYS

An 'Equivalent Source Array' means for the present purpose,
a source distribution which according to linear theory would generate
a given wave pattern. It can be used for evaluating and interpreting
the correlation between wave theory and experiment and also for
predicting the effects of wavemaking on changing tank width and depth.

In Ref. 1, this concept was invoked to interpret wave pattern
measurements behind 2 series of 3 models with parabolic hull forms
and systematically varied beam. More recently, a similar investi-
gation has been made of the wave patterns behind 3 trawler type hull
forms (aparentand 2 derivatives) tested by Everest (Ref. 3). In the
case of these more realistic models, the 'equivalent source arrays’
were found to vary significantly with speed as indicated by the sample
results for one of the models shown in Fig. 1. It may be seen that
sources and sinks appear at a distance ahead of the bow which in-
creased with increasing Froude number. This effective lengthening
of the array may be explained in terms of 2nd order increases of
wave phase velocity due to nonlinearity of the waves generated in the
bow regions.

III. AUTOMATED RECORDING AND ANALYSIS
A prototype for a fully automated recording and analysis

system has now been developed and operated. It comprises a station-
ary array of 4 capacitance probes with paper tape output and a com-

540

Recent Research on Shtp Waves

O68% 130OW

sA@izre ad.1no0s JusTeAINby

ey

BIg

541

Newman

MODEL 4890

—— SOURCES

— — —)— — EXPERIMENT

a
|

0-25 030 035 0-40 0-45 0sO0 0:55

Fig. 1b Wave pattern resistance

puter program which analyses the tapes as punched by the recording
digitizer. The probes themselves are as described in Ref. 4. The
computer program uses the Matrix method of analysis developed
with this application in view and described in Refs. 4 and 5. It works
by a least square fitting of an appropriate function to the wave sur-
face defined over a suitable grid of positions.

Some sample results for a 20 foot model of one of the trawler
type models tested by Everest (Ref. 3), are shown in Figs. 2 and 3.
Fig. 2 is a copy of part of a computer output. At the top a tabulation
defining the wave ordinates Z(X,,Y,) 'as measured! by listing R,
Xp Yr and Z (R isaserial number, Xp, Yp are longitudinal and
transverse coordinates respectively in feet, and Z is the wave

542

Recent Research on Shtp Waves

R X Y Zz

1 25.218) + (8.0005 st 1643

2 + 25.436 + 6.000 + 0.244

B “+ 2'5.653 + 4.000 + 0.722

4, 4°25.871 + 1.333 + 0.893

5 + 26.307 + 6.000 + 0.7

6 + 26.524 + 4.000

tt 26.042. 95 1.2

PARE WAC
22 0.739

146 + 5%. + 1.333 - 0.623
147 + 60.058 + 8.000 - 0.538
148 + 60.275 + 6.000 - 0.804
149 + 60.493 + 4.000 + 0.353
150° + 60.711) + 1.333 = fit

CASE 535.100 SPEED 8.710

N THETA AN By 100DC AL F. Fo
0 00.00 0.695 -1,112 0.0608 08.481 -0.0371 0.0232
1 32.09 -0.460 -O.111 0.0051 10.019 -0.0030 -0.0124
2 45.87 -0. 636 0. 683 0.0235 12.191 0.0319 -0.0298
3 53.12 -0.127 0.218 0.0019 14.144 0.0166 -0.0097
4 57.70 0.005 -0.277 0.0024 15.887 -0.0334 0.0006
5 60.92 -0.028 -0.397 0.0050 17.468 -0.0748 -0.0052
6 63.35 -0.048 -0.217 0.0016 18.923 -0.0635 -0.0139
7 65.25 -0.137 -0.021 0.0006 20.276 -0.0093 -0.0620
8 66.80 0.056 0.145 0.0008 21.546 0.1011 0.0387
9 68.09 0.076 0.068 0.0003 22.747 0.0728 0.0812

10 69.18 0.185 0.038 0.0012 23.888 0.0621 0.3033

11 70.13 -0.068 0.047 0.0002 24.977 0.1174 -0.1708

12 70.96 0.003 -0.082 0.0002 26.021 -0.3172 0.0100

13 71.68 -0.074 0.063 0.0003 Zils O25 0.3709 -0.4387

44 72.35 -0.021 0.062 0.0001 27.993 0.5631 -0.1856

[ This is to be compared with 100C W=0.09375 obtained by

100CW= 0.1041 Everest with 15 foot model of the same form (Ref. 4).]

R x Y CZ DZ
1 + 25.218 + 8.000 + 1.661 + 0.018
2 + 25.436 + 6.000 + 0.194 - 0.051
3 + 25.453 + 4.000 + 0.764 + 0.042
1,333 + 0.852 - 0.041

+
pyar

59.840 + 1.553 Uli 478
147 + 60.058 + 8.000 - 0.502 + 0.036
148 + 50.275 + 6.000 + 0.766 - 0.038
149 + 60.493 + 4.000 + 0.607 + 0.254
150) #_ 60.7145 F 15333 = OF7693 + 0.418

RMS RESID 00.163 RMS Z 01.080

Fig. 2 Sample computer output

543

Newman

elevation in inches). In the middle is a tabulation of wave spectrum
parameters accompanied by the resulting wave resistance coefficient
100 Cw. The first 4 columns define the amplitudes and resistance
contributions of the various wave modes in notation which corresponds
to that used for example in Ref. 4. The last 3 columns define functions
used for computing ‘equivalent source arrays' in notation explained

in Ref. 1. It may be seen that the resistance coefficient 100 Cy checks
reasonably well with a result obtained by Everest for a smaller model
of the same form, using manual pointers on transverse cuts analyzed
by the method of Eggers (Ref. 6).

At the bottom is a tabulation defining the wave ordinates CZ
(Xp-° Yr) 'as fitted' in the same format as the 'as measured’ results
but with an extra column listing the difference between measurement
and fit. Fig. 3 shows a sample profile plotted from the computer
output.

=o AS MEASURED:

—_§\f--——_ AS. FITTED’
3

PROFILE AT y=|-333 FEET FROM TANK CENTRELINE

Fig! 3 “Profile at» y ="1.533 feet from tank centerline

544

Recent Research on Ship Waves

ACKNOWLEDGMENTS

Appreciation is expressed for the contributions of Mr. B.

Garner and Mr. H. G. Loe in developing the automated wave recording
system and of Mr. E. J. Neville and Mr. M. Wilsdon in conducting
the experiments.

REFERENCES

Everest, J. T. and Hogben, N., "An experimental study of the
effect of beam variation and shallow water on 'thin ship’ wave
predictions ," Trans. RINA paper W11 (1969) issued for written
discussion.

Hogben, N., "Automated analysis of wave patterns behind towed
models," in preparation as Ship Division Report No. 143,

Everest, J. T., "Some comments on the performance in calm
water of a single hull trawler form and corresponding
catamaran ships made up from symmetrical and asymmetrical
hulls," NPL Ship Division Report No. 129, February 1969.

Gadd, G. E. and Hogben, N., "The determination of wave re-
sistance from measurements of the wave pattern," NPL
Ship Division Report No. 70, November 1965.

Hogben, N., "The computing of wave resistance from a wave
pattern by a matrix method," NPL Ship Division Report No.
56, October 1964.

Eggers, K. W. H., "Uber die ermittlung des wellenwiderstandes
eines schiffsmodells durch analyse seines wellensystems,"
Schiffstechnik Vol. 9, part 46, p. 79, 1962.

ssh 54 £ BLANK
S47] FOLLOWS

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i awamien POU eee: ah is ; i
: s ee Ay ety a ue eee i me B i ‘ 5 Fendi Mi
f fed 4
pol a) re t
aae ie v age ij F Oe i Ah eee | Oy

Wh ont Bach al heat, 8 oh SS Baa

ae rie AG) C1 oe ; j

APR AG m i he ba, r ¥, ,

| a » : +“, v hey 7 m bes i, oe ; “ _ a : y y" t 1” ssl p29)
dh ee eh ath vehiteih Nad NIA Ae oigod ta Be a ;
a" By a ae Wy (i? ny i H Lay ey) iy wet i renee ! tJ 5 iF) ay |
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etiegg Te CMM ERIN Se) watts! ree Siberadt ho) OW: gf big Ps
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MVA Set LN o
RWOLIOA ANG

VARIATIONAL APPROACHES TO STEADY SHIP
WAVE PROBLEMS

Masatoshi Bessho
The Defense Academy
Yokosuka, Japan

INTRODUCTION

Although there have been many fruitful. engineering applica-
tions of the theory of the wave-making resistance of ships, it is still
not possible to completely explain the wave resistance of the usual
surface-piercing ships. The so-called order theory gives us insight
into the structure and composition of our approximate theory; however,
we do not yet have a consistent and practical theory which is univer-
sally acceptable.

The author has speculated on what would be the best approxi-
mation to our boundary value problem. In this connection, is there
a useful principle which corresponds to the Rayleigh- Ritz principle
in the theory of elasticity? The present paper will provide a partial
answer.

Our first aim is to introduce a variational principle which
corresponds to the linearized boundary value problem. This is
accomplished by introducing Flax's expression from wing theory. [ 6]

Our second aim is to find an alternate expression which will
enable us to treat blunt bodies, since Flax's method is useful only
for thin wings. Gauss' variational expression [ 24,25] for the
boundary problem of a harmonic function is introduced for this pur-
pose. This is shown to be equivalent to extremizing the Lagrangian
or kinetic potential. The resulting dynamical interpretation of the
boundary value problem is similar to the approaches of many other
authors who have studied free surface problems by using the
Lagrangian [| 3,12,13,14].

I. FLAX'S VARIATIONAL PRINCIPLE
The variational principle introduced by A. H. Flax in wing

theory [6] may be directly applied to our problem. Those unfamiliar
with this principle are directed to Appendix A.

547

Bessho

If the Kutta-Joukowski condition |6,7,8] is satisfied at the
trailing edge, we have the reciprocity relation

‘i pw dx ay= | { pw dx dy (f.1)
S S

by (A. 8) and (A.24), where p is the pressure, w is the vertical
velocity component, and tildas denote reverse flow quantities. The
integration is over the wetted portion of the ship hull S.

Let C(x,y) be the free surface elevation. The variation of
the integral

1299 [tw - Be, - Bul ax ay (1.2)
S
due to variations of p and p takes the form

51: = iy [ 6p(G, -w) - 6p(%, + w)] dx dy. (1.3)
S

Since the variations 6p and 6p are arbitrary, the pressure which
extremizes the integral I is equivalent to the solution of the boundary
value problem (A.25) and (A.26); that is, the problem for the pertur-
bation potential $ with the conditions

6, = -we ¢,
(1. 4)
t= W= -4,

x

on the free surface. The stationary value of I is the drag; namely,

[1] = SS pb, dx dy, (1.5)

where p, denotes the correct solution. [6,24,26] Thus, the bound-
ary value problem is converted to a variational problem, the solution
of which is suggested by various methods of approximation. |[ 6]

If we introduce the error integral,
* ms
E -{f (p - p,)(® - &,) dx dy, (1.6)
S

we see from (1.1), (1.4), and (1.5) that

548

Variattonal Approaches to Steady Shtp Wave Problems

Deer ee (1.7)

Therefore, Flax's principle produces an approximate solution which
makes the error integral (1.6) stationary. [23]

This method suggests powerful means for obtaining approxi-
mate solutions, but unfortunately it has been applied only to thin
hydroplanes and wings. [ 7]

Il. GAUSS' VARIATIONAL PRINCIPLE
In this section, we assume there is no free surface. Then

the velocity potential has the following representations for the source-
sink and doublet distributions:

o(P)= 355 J) aE 45(2), $2 04eteee Veet)

and

oP) = a5) Hi (Q) & AES 4512), PeO; 422,600 ee)

Here, quantities with the suffix zero stand for the correct solutions
while those with other suffices are not necessarily correct. For these
potentials we have the following reciprocity relations:

\{ oo, dS -\{ To, as; (223)
S S

i Poo) dS = pF b,O,,dS, (2. 4)
Ss Ss

ia $15, dS -{f $6), dS. (2.5)
Ss Ss

Gaus s's variational principle for the Dirichlet problem states
that if we consider the functional

and

G =3/ (b - 2f)o0 dS, (2. 6)
S

where

549

Bessho

f=%, is givenon S, (Zea)

then the function which gives the maximum value to G is the solu-
tion of the Dirichlet problem. [9,10] This is easily verified by
making use of the reciprocity (2.3).

In the same way, we may construct a variational principle for
the Newmann problem as follows: Let us consider the extremum
problem for the functional

= ant (6, - 2f,)y dS, (2.8)
$
where
f= Oo, 18° givenron (Si (2.9)

This problem is seen to be equivalent to the present boundary value
problem by making use of (2.4).

Alternately, we may construct a variational problem by making
use of (2.5); namely, by introducing the functional

J= an $(2f, - ,) ds, (2, 10)

and taking the variation, we have

6J = a S4(f, - $,) dS. (2.44)
S

From this we see the equivalence to the boundary value problem.
[ 24,25]

Now, since

G0, eraas= $06 vaive, ar ety

where D is the entire water domain and d7 is a volume element,
a natural measure of the error of an approximate solution 96 is

ey [Vib - 9] ar, (2.13)

E

550

Vartattonal Approaches to Steady Shtp Wave Problems

which becomes

E= Ane (o i bo) (>, - bo, dS = J a J; (2. 14)

by Green's theorem. Here,

Jo -3(( Mobo, dS (2, 15)
S
is the correct value. We see clearly that
SE = - 6J. (2.16)
Since E is non-negative, we have the inequality [ 10]

t,o. (2347)

It is well-known that among all functions $ having a finite

energy integral,
2
r=3((( [Vo] drt, (2.18)
D

and a given normal derivative on S, the one which minimizes T is
a harmonic function [1,4]. Accordingly, if we solve this minimiza-
tion problem, say by the relaxation method, we have the inequality

ea en eles (2.19)

fe)
This is the dual of (2.17) and we now have the variational problem
(2.7) as an involutory transformation of the latter minimization prob-
lem. (See, for example, the textbook on variational calculus [11] .)

Ill. A VARIATIONAL PROBLEM FOR THE LAGRANGIAN

The preceding principle can be easily extended to flow ina
gravitational field. Let us consider the functional

L=T-V, (3.4)

where

Bessho

A annie Ive] dt (3. 2)

and
v= 8 O° dxidy (3.3)
F

are the total kinetic and potential energies, respectively. L is just
the kinetic potential or Lagrangian. [5] Assume that the function
@ has a given normal derivative

dy=-xXy on S and F. (3. 4)

Taking the variation of L, we have

6L= any $V 8h dT al $56, dS +

si [ 656, + {(374)* - gt}6v] ds.

Making use of (3.4), which is also true for the new deflected free
surface, we have

ai=-\|( $v so dt + an pév dS, [3,14] (3.5)
D

where

p/p = - 6, - 3(Vd) - gh. (3. 6)

Hence, if the pressure at the free surface vanishes, the
stationary value of L will be attained when 6¢ is harmonic. This
is just an extension of Kelvin's minimum energy principle. [1,4]

On the other hand, if 6¢ is harmonic, then the stationary value
of L is attained when the free surface pressure is constant and
zero. The latter is an extension of Riabouchinsky's principle of
minimum added mass. [3,14]

The variational problem can be transformed so that the con-

straint condition is converted to a natural condition. Let us adda
term which is zero at the stationary point. Consider the functional

baz

Variattonal Approaches to Steady Ship Wave Problems

P=T- v-\ (xv + o,) dS. (3. 7)

S+F

Assume that $ is harmonic and, for simplicity, assume that the
integral over an inspection surface at infinity vanishes. Making use
of Green's theorem we have

p= - SVG Cont uvorl ar - 89 0? ax ay
= ait p dt + Const, (3. 8)
p D
where
Const = §'( H? ax dy -3{\ z* dx dy, (3.9)
S

and H is the depth to the bottom.

Taking the variation, we have

4
6P = SS. pév dS - SS. 5o($, + x,) dS. (3. 10)

Therefore, when
p=0 on F and $,+x,=0 on S and F, (3.41)

P is stationary. This result was first given by J. C. Luke [12,13],
who pointed out that the volume integral of the pressure is equivalent
to the Lagrangian.

Furthermore, we may write (3.8) as
P=M-H, (3.12)
whe re
H= Tt V (3.13)

and

553

Bessho

M = - (Ah oy dT. (3.14)

M is the total momentum of the system in the x-direction and becomes
equal to twice T,

M = 2T = ae éx, dS -{{ oo, dS, (3.15)
StF S+F

when 96 satisfies the boundary condition.

Hence, 6P=0 means that
6H = 6M. (3.16)

That is, when the variation of the total energy equals that of the total
momentum in the x-direction, the potential satisfies the boundary
conditions (3.11).

For purposes of application, it may be convenient to write P
as

P=- ({ (x, + 46,) dS - g(( OF se doy. (3.47)
S+F F

This principle is applied to a regular, two-dimensional wave-
train in Appendix B. In general, there is some difficulty in the appli-
cation of this theory since the integrals P and L may not be finite.
This is because the kinetic energy exceeds the potential energy for a
finite amplitude wave. [1,2,4]

One way to bypass this difficulty may be to assume a flow
model like the Riabouchinsky model [3] in cavitation theory (see
Fig. 2); however, this may be impossible in the three-dimensional
case. Another way may be to introduce Rayleigh's friction coef-
ficient so that the waves far downstream will die out. In any case,
there are still some problems which make us hesitant to begin the
actual numerical computations.

Finally, let us consider the linearization of the free surface
condition. Neglecting higher order terms in the integral over the
water's surface and assuming that

go(x,y) = - 6,(x,y,0), (3.18)

we have

554

Vartattonal Approaches to Steady Shtp Wave Problems

(a)

(b)

L: SUFFICIENTLY

LARGE DISTANCE

Fig. 2. Riabouchinsky Models

P=P-.+P,,
where
Poe Sf. (x, + $4,) dS,
and

Pe = - 4). (xx + goz) dS.

Accordingly, if we set

xx + go, = 0 on F,

which is just the dynamic boundary condition, then

and we are left with a variational calculus problem for Py °

555

(3.19)

(3. 20)

(3521)

(322)

(3523)

Bessho

IV. THE LINEARIZED PROBLEM
The variational problem for Pg (3.20) is not satisfactory
since there is no reciprocity relation for this form. We must intro-

duce the reversed flow potential as was done for Flax's principle.

Let us consider the integral

Tete e) = Ll" Go.4) = - Sy Vo,Vo, dT - aS C05 dx dy.44o8)

Assuming that 9, and $5 are harmonic and satisfy the free sur-
face condition, we have, by Green's theorem,

L*($,,) =-2 \h, $140, 4S =-% Mi ox, ds, (4. 2)

where S is the surface of a submerged body. This is the recipro-
city theorem for a submerged body. [8]

If $,= - $,, then

1*(,) = +\ i bb, dS = L(, 4), (4. 3)

where

L($,9) = 3 aa (74) dr - g(( 1 dx dy. (4. 4)

*
L_ is called the modified Lagrangian integral [5]. Note that L(¢, 9)
has a finite value in the linearized case but not in the finite amplitude
case.

If S is the wetted part of a surface-piercing body which is
under the waterline before the free surface is disturbed, there is an

additional term from the surface integral. [15,16,19,20,21] The
reciprocity theorem, in this case, is

oa ‘f bib dy - a\t $192, dS

rf $6, dy -2 i $,6,, dS. (4.5)

L*(4,,6,)

556

Vartattonal Approaches to Steady Shtp Wave Problems

When 4, = 9, $52 @ and ¢$,=-+x,, Tae becomes

1 (6, 8) Se \ oLx, dS + aft ox, dS, (4. 6)

where n is the inner normal to the waterline curve L inthe hori-
zontal plane. Thus, the first term in the right-hand side of (4.6) is
the correction for the change of the wetted surface S. [16] This is
justified, on the one hand, by the dynamical meaning of the Lagrangian
and, on the other hand, by the linearization procedure of the pre-
ceding section.

For the case of a pressure distribution over the water surface,

we may integrate (4.5) by parts and make use of the formulas in
Appendix A. This results in the expression

-2 ae 15 dx dy = af. $.,6, dx dy

i : 1 x
a4). [P, + pgt|c, dx dy = 75, [P, + pgt.|t, dx dy.
(4. 7)

1*($,, 90)

Thus, the reciprocity becomes [ 8]
S (pice) = taxdy= tll 52 dx dy (4.8)
P, »Po 2p s P, 2 yy. 2p S 2 |

where

£*(p,.B,) = 116.4) - all tt, dx dy. (4.9)

Making use of these reciprocities, we may easily show the
equivalence of the boundary value problem to the variational problem
for the functional I”, where

i a [ $6, - (@ - $)x,] as, (4. 10)

for a submerged body, and

*

I =- a [pe "(b= pit de dy, (4,11)
Puvs

for a pressure distribution. [ 24,26]

557

Bessho

Alternate representations for these integrals are

To sd (bordel eD (bos Ooh): (4, 12)

2K 2k ~ * ~ ~
I = (po,Po) - £ (p - pos Pp - Po), (4.13)

where the suffix zero stands for the correct solution. These for-
mulas show that the variational principle extremizes the Lagrangian
of the error and that the stationary values are just given by the
Lagrangian.

The difficulty arises in the case of a surface-piercing body.
From (4.12), the functional to be extremized is

*
I

= - 16,4) +a ($£0- $f) dy + =e) ($ - $)x,dS. (4.14)

Taking the variation, we have the boundary conditions equivalent to
this variational problem,

>y= - glo $,= 8% on L, (4.15)
by= - Ove -Xy,y on Se. (4.16)

But we have no knowledge of the surface elevation on L, a priori,

as this problem may be indeterminant. [17,23] We must remember
here that the solution is unique only when the detachment points are
fixed by the theory of cavitation. [3,14]

This difficulty may be avoided by introducing a homogeneous
solution for the two-dimensional, linearized case (see Appendix C).

For the present case, we might proceed as follows: Let us
consider the difference between a surface piercing body and the
limiting case of a submerged body moving very close to the free
surface as in Fig. 3. [23] The boundary condition on the water

surface above the submerged body must be 9, = 0, but since the top
is also the free surface, this is equivalent to

>, = &x(x,y) =O on aoe (4,17)

or integrating, we have

o,(x,y,0) = - g€(x,y) = Const = func (y) on FF. (4.18)

558

Variattonal Approaches to Steady Shtp Wave Problems

(a) SUBMERGED

OT ae
\ YU)

(b) VERYSLIGHTLY SUBMERGED

(c) SURFACE PIERCING

Fig. 3. Slightly Submerged Ship

This formula shows that there may be a thin layer of uniform flow
over the top of the submerged body.

When this layer moves with the body,

$x(x,y,0) = - go(x,y)=-1 on F, (4.19)

and we clearly have the case of a surface-piercing body.

On the other hand, the boundary value problem of a submerged
body is equivalent to the variational problem (4.10). After solving
this problem, we may calculate the surface elevation over the top
water plane by (4.18), but it will differ from (4.19), in general.

In this case, it might be necessary to introduce another potential
which satisfies condition (4.19), in addition to the above potential.
This procedure may not be practical because the treatment of the top
water plane is difficult.

In this case, it would be more convenient to consider the follow-
ing two boundary value problems: Let us split the velocity potential
into two parts,

d= 6, +, (4, 20)

with boundary conditions

559

Bessho

>, = 0 on LL
(4. 21)

>, =- Xy on S

to = Lo = Leo, given on L

(4. 22)
$9,=9 on S

The corresponding functionals are of the form (4.10) for 9),
and of the form (4.14), without the third term on the right-hand side,
for Oo.

For the present case, 4) must be equal to 1/g by (4.19);
however, in general, it will be arbitrary and, perhaps, a constant
oa form (4.18). ¢$. is called the homogeneous solution. {18,22,
26

Finally, it should be noticed that the Lagrangian is closely

related to the far-field potential. For a submerged body, we have,
from the boundary conditions, (A.9), (A.41), and (4.3),

B

: i xxy dS + 21($, 9)

if)

2L(>,9) + V, (4. 23)

where V is the displaced volume. For a surface-piercing body,
interpreting condition (A.10) as a correction for the real wetted
surface, we have

iv xg,as +i x, dy =V, (4, 24)
s EYL

where V is the displacement volume under the still waterline.
Therefore, we can write (A.1i1) as

B=V +21 (9,6, + $), (4, 25)

where 9, and $9 are defined by (4.21) and (4.22), with C,5= Wierd

For a pressure distribution, we have, from (A.18) and (4.8),

B = 2p "(p,P,)s (4. 26)

560

Vartational Approaches to Steady Shtp Wave Problems

where pz, is a homogeneous solution, as is 5, and to= tg

Since B is also a measure of the total lift, this formula shows that
the homogeneous solution for the constant surface elevation influences
the lift, as we have easily verified by the reciprocity (4.8). [ 26]

It should be noticed that, in this case, the condition A=0 in (A.18)
insures the continuity of the planing hull.

Kotchin's function (A.17) is also given in the form

a)

wo-- $f

where ‘by has the boundary values

6.x, dS =| o.6 dy + 2L.°(6,%), (4. 27)
L

Pay = ~ Per ede Ss

and (4. 28)

~

gt, = ee mS Pay on L

@qis called the diffraction potential. [23,26] Here, the second term
of (4.27) may be omitted as in (4.25).

For a submerged body, there is no integration along L and
H may be written as

H(6) = - ae (be + dy)x, dS. (4. 29)

Finally, for a pressure distribution,

H(0) = 2p£"(p, py) (4. 30)
where
a oe (4.31)
d g ex

V. CONCLUSION

We have presented two variational principles for the boundary
value problem associated with the waves of a ship advancing at a
constant speed: The first is Flax's principle, which makes use of the
stationary character of the drag. This principle is useful only for

561

Bessho

planing boats or for submerged thin wings. [6,24] The second is
based on Gausz's principle, which converts the boundary value prob-
lem to a variational problem. This method is shown to be an exten-
sion of Riabouchinsky's principle of minimum virtual mass. [3,24]

The latter principle is based on the stationary character of
the Lagrangian and has recently been used by Luke, in a more general
form, to study water wave dispersion problems. [3,12,13] We
also have analogous principles for light and sound wave diffraction
and for the radiation of energy due to the heaving, swaying, and
rolling oscillations of ships. [25,27,28,29,30]

The variational principles emphasize the dynamical meaning of
the boundary value problems and permit us to solve them approxi-
mately by the Rayleigh- Ritz-Galerkin procedure. [6,28,29] How-
ever, when we try to apply these principles to our problem, there
are two difficulties:

The first is that our system is not conservative because of
the trailing wave. This may be bypassed by introducing an artificial
model, as in Fig. 2, or by introducing a reversed flow for the
linearized case.

The second difficulty is for the surface-piercing body, in which
case the wave profile is not known, a priori, even in the linearized
case. This difficulty may be avoided by introducing homogeneous
solutions [27] which appear in the case of a surface pressure distri-
bution. [ 26]

Finally, although a variational method does not necessarily
represent a new method of analysis, it does suggest new methods of

approximation. For this reason, it may be useful, especially for
engineering purposes.

Rk FERENCES

i. Lamb, H., Hydrodynamics, 6th Ed., Cambridge University
Press, 1932.

2. Wehausen, J. V. and Laitone, E. V.,"Surface Waves,’ Handbuch
der Physik Bd. 9, Springer and Co., 196U.

3. Gilbarg, D., "Jets and Cavities," Handbuch der Physik Bd. 9,
Springer and Co., 1960.

4, Milne-Thomson, L. M., Theoretical Hydrodynamics, 4th ed.,
Macmillan and Co., 1962.

562

i

5 i

13.

14,

15s

16.

re

18.

19.

Vartattonal Approaches to Steady Ship Wave Problems

Morse, P. M., and Feshbach, H., Methods of Theoretical
Physics, in 2 volumes, student ed., McGraw Hill and Co.,
1953.

Flax, A. H., "General Reverse Flow and Variational Theorems
in Lifting-Surface Theory,"J. Aeronaut. Sci., vol. 19, 1952,

Ursell, F., and Ward, G. N.; "On Some General Theorems in
the Linearized Theory of Compressible Flow," Q. J. Math. and
Mechs, vol. 3,.1950.

Hanaoka, T., "On the Reverse Flow Theorem Concerning Wave-
Making Theory," Proc. 9th Japan Nat. Congress for Appl.
Mech. ? 1959.

Frostman, O., Potential d'Equilibre et Capacité des Ensembles,
Lund Univ., 1935.

Inoue, M., Theory of Potential, Kyoritsu, Tokyo, 1952
(Japanese).

Hayashi, T., and Mura, T., Variational Calculus, Corona,
Tokyo, 1958 (Japanese).

Luke, J. C., "A Variational Principle for a Fluid with a Free
Surface," J. Fluid Mech., vol. 27, 1967.

Lighthill, M. J.,"Application of Variational Methods in the
Non-Linear Theory of Dispersive Wave Propagation," Proc.
IUTAMSymposia, Vienna, 1966.

Garabedian, P. R. and Spencer, D. C., "Extremal Methods in
Cavitational Flow," J. Ratl. Mech. Anal., vol. 1, 1952.

Wehausen, J. V., "An Approach to Thin Ship Theory," Proc.
Int. Semi. on Theor. Wave-Resistance, Michigan, 1963.

Yim, B., "Higher Order Wave Theory of Ships," J. Ship Res.,
September, 1968,

Kotik, J., and Morgan, R., "The Uniqueness Problem for Wave
Resistance Calculated from Singularity Distributions Which are
Exact at Zero Froude Number," J.S.R., March 1969.

Van Dyke, M., Perturbation Methods in Fluid Mechanics,
Academic Press, 1964,

Bessho, M., "On the Theory of the Wave-Resistance,"
Je.Zosen Kyokai, vol. 105, 1959,

20.

21.

226

23.

24.

25s

26.

27.

28.

29.

30.

Bessho

Bessho, M., "On the Theory of the Wave-Resistance," (2nd
Rep.), AA ee vol. 106, 1960,

Bessho, M., "On the Formula of Wave-Making Force Acting on
BWOnips | de LieKke Vole 1101960,

Bessho, M. and Mizuno, T., "On Wave-Making Resistance of
Half Immersed Circular Cylinder and Vertical Plate," Rept.
of Defense Academy (Japanese), vol. 1, 1963.

Bessho, M., "On the Boundary Value Problem in the Theory
of Wave- Making Resistance," Memo. Defense Academy,
vol. 6, 1967.

Bessho, M., "Gauss' Variational Principle in Boundary Value
Problems," Read at Sea-Keeping Sub Com. of Japan tow.
Tank Comm., October 1967.

Bessho, M., "On Boundary Value Problems of an Oscillating
Body Floating on Water," M.D.A., vol. 8, 1968.

Bessho, M. and Nomura, K., "A Contributiion to the Theory of
Two-Dimensional Hydroplaning," M.D.A., vol. 10 (in print).

Miles, J. and Gilbert, F., "Scattering of Gravity Waves by a
Circular Dock," J.F.M., vol. 34, 1968.

Mizuno, T., "On Swaying Motion of Some Surface-Piercing
Bodies," M.D.A., vol. 9, 1969.

Mizuno, T., "On Sway and Roll Motion of Some Surface-
Piercing Bodies," read at Spring Meeting of Jap. Soc. Nav.
Arch. -°1970%

Isshiki, H., "Variational Principles Associated with Surface

Ship Motions ," read at Korea-Japan Seminar on Ship Hydro-
dynamics, Seoul, 1970.

564

Vartattonal Approaches to Steady Shtp Wave Problems

APPENDIX A
The Linearized Velocity Potential [2,23]

Let us consider the flow of water around a ship S, taking the
coordinate system as in Fig. 1 and the velocity of the stream at up-
stream infinity to be unity.

x
We Vee tir amtevs

(UNDISTURBED)

) ow, ss ae a —_
yf

Fig. 1. Coordinate System

The pressure p(x,y) on the water surface is given by

= (x,y) = - $,(x,y,0) - g&(x,y), (A. 1)

in the linearized theory, where p is the water density; g, the
gravity constant; (€, the surface elevation, and 9, the perturbation
potential (d@= -udx-vdy-wdz). The suffix stands for differ-
entiation.

The kinematic condition on the water surface is
$(x,y,0) = € (x,y). (A. 2)

Since the pressure on the free surface is constant, the potential
must satisfy the condition

$,,(x»y 29) + go(x,y,0) = 0. (A. 3)

A solution which has a source singularity at a point Q and

Bessho

satisfies the above water surface condition can be expressed as

t tl - & lim

(P,Q) ” (pO) My +0

T fo) k(z4z') +ik(@+ @')
, e dk dé
i fre ne co
“Tr

k cos*6 - g tyul cos 8
(A. 4)

where P= (x,y,z), Q= (x!,y'sz'), C= (Ov laee )s r(P,Q) = PQ
and w=xcos@+ysin®@, w'=x'cos@+y'sin 0. Hereafter, we
will call this the fundamental singularity. This solution approaches
the following values. asymptotically:

4 1 1
(P.O) eae eo) tea FY (A. 5)

x>>x!

2
f x sec O{(z+2') +i(@+")} 2

S(P,Q) ame E tm ec“0 dé. (A.6)

x<<x! om /2

By considering the integral
VV tetarse ay - (Q)SAP,Q)] dS(Q)
about a point P in the interior of the fluid, we have the expression
$(P) -{( [$,S - $S,] ds.
“S+F
Since $ and S satisfy condition (A.3) on F, we have, finally,
$(P) -{f [ ¢(Q)S(P,Q) - HQ)S,(P,Q)] d4s(Q)
- +, [ (Q)S,(P,Q) - $(Q)S(P,Q)] dy', (A. 7)
where L is the curve on which F cuts S.

When the water motion is due to a pressure distribution over
the water surface, we have

#(P) = 254. p(Q)Sy(P,Q) dx! dy", (A. 8)

566

Vartattonal Approaches to Steady Shtp Wave Problems

where we have used (A.1) and integrated (A.7) by parts. We have
also assumed that the potential and surface elevation are continuous
over S and F, including L.

Making use of asymptotic characters (A.5) and (A.6) of S,
we obtain the asymptotic expansions of ¢$ as follows:

He) Sst ter At EP» (A. 9)
2ur
where r=FPO and

a=\I s,as +4 6, dy, (A. 10)

s BL
1

B=(( Cax- oe) as-4) [o-xe) ay. 9 (att)

s BIL

The expression for A may also be written as

= hae a
A -\\'4, ds aa ,, dx dy ye $,, dS ane 6, dx dy, (A.10')

by using (A.3). Thus A is the total outward flux from the water
domain. This must be zero; otherwise, we would have a large source
of the resistance other than from the wave and splash.

We also have the kinematic condition on the surface of the ship,
$,=-'‘x, on S-. (A.12)

Therefore,

SS. o, dS = - Mee x, dS = 0, (A. 13)

where S is the wetted surface of the ship below the undisturbed
water surface. From (A.10) and (A,13), we have

1 = = =
+f o, dy = i‘ GC dy =0. (A. 14)

But this condition is not adequate in practical cases. One way to
avoid this difficulty may be to take the real wetted surface as S.
On the other hand, for the consistency of the theory, it may be

5a?

Bessho

preferable to take

x
P) S34 eS B,

instead of (A.9).

Far downstream, we have

27/2
op) —co—t Im te(P,0)H(8) sec6 dO,
-17/2
where
$(P,0) = exp[g sec’O(z + ia)]
and

1
H(0) = 15) [ oo at be, dS - if ($,, a eq) dy.
For a pressure distribution, we have simply

A=0

iy
B =--—_—_ dx d s
ped J p(x,y) y

where

4
H(6) ane p(x,y),, dx dy.

(A.9')

(A.15)

(A. 16)

(A. 17)

(A. 18)

(A. 19)

If the flow direction is reversed, the conditions corresponding

to (A.1), (A.2), and (A.3) are as follows:
1w~ ~ ~
ppixsy) = (x,y ,0) s gC(x,y),

$,(x,y,0) = - t, (sy).

and

568

(A. 20)

(A. 21)

Vartattonal Approaches to Steady Shtp Wave Problems

$,,(%,y +0) t go,(x,y,0) = 0, (A. 22)

so that the fundamental singularity is the same as that for the direct
flow, except that the wave follows on the downstream side. This may
be expressed as

S(P,Q) = S(Q,P), (A. 23)
we also have
S\(P;Q).= 5,(@,P). (A. 24)
The boundary conditions for this case are

y=-o)=xXy,y on S, (A. 25)

$,=-w=-6=w=-f, on S. (A. 26)

APPENDIX B
The Progressing Wave

Let us obtain the solution for a periodic progressing wave,
moving at constant unit speed, by the variational method of §3.

We take the form of the complex potential to be
o tiv = - ia exp (kz - ikx), (B. 1)

where the origin is on the undisturbed water level.

The integrals to be evaluated are

‘ w/k fe)
P=M-T-V, tye. ax | o, dz,
e -w/k -0
(B. 2)
1 = 4 2 1 a 2
ee |e (V 4) dx dz, ries ar ax;

569

Bessho

where 21n/k is the wavelength.

Assuming a surface disturbance of the form,
{ =b + ¢ cos kx +d cos 2kx, (B. 3)

and integrating the expressions for M, T, and V, we have

2
1 d k 2 2 2
5M = neal 1 +k(b +5) +--(c* + 4b° + 2a" + Abad],
1 ae 2.2 2 3 2
T= [1 + 2kb tki(c + 2b +d) +k°c*(2b + d)], (B.4)

trp me ee 2
-V= 3 lc +a + 2b).

Differentiating P with respect to a,b, c and d, and equating
the derivatives to zero (by the principle (3.16)), we obtain the follow-
ing stationary values, neglecting higher order terms:

22
= a(1 arr ee

a
i}

8
Be Ol).
(B. 5)
dy 5ka’,
g/k=1 - k*a’,
k p22 a ee
a7 5 ea (4,-+ kre pi;
evs BE M1 +5 Kec?) (B. 6)
oT D 4 9 e
k . a Pg ess
Sarah oe ote

These expressions agree with other well-known results. [1,2]

570

Vartattonal Approaches to Steady Shtp Wave Problems

APPENDIX C
A Variational Principle for the Stream Function

In the two-dimensional case, we may use a stream function
instead of the velocity potential. Let us introduce the stream
function as follows:

O(x»Z) = U,(x.2), ,(x,z) = - p(x,z). (c.4)

Then, the boundary conditions for become
(x0) - g(x,0)=0 and Y,(x,0) - gib(x,0)=0, — (C.2)
Wlx,z)'=- b(x,z)=-2 “on S, (Cs)
C(x) = - W(x,0) and &(x) = U(x,0). (C. 4)

Introducing a modified Lagrangian integral,

* - « a ~
L (Wy >) = -4 Sf. Vp Vb, dx dy - ral CS, dx, (C. 5)
F
we have, directly, the reciprocity

L(y, 09) = sc ,)
(C. 6)

i Yi bou ds - Yet, ds
s s

In particular, from (C.3),

L(y.) = - 4 \. Uso, AS = - 4 \ zp, dS = aN Yolo, IS

=4 I Tue) dx dy - ral to dx = L(bo, Ho). (C. 7)
J dp F

The variational problem with the function

byt

Bessho

I” = L"(Yorto) - L(h - Gord - Ye)
1 rw ~~ ~~
= a) (Web, - Yoby - Yoh) dS. (C. 8)
Ss

is equivalent to the boundary value problem for yw Here, the
boundary values, Wo and wo, are given by (C.3). Since a stream
function has an arbitrary constant, we should also consider the
modified problem with boundary conditions

Wo=- Wo = C: constant on S, (C.9)

which is the homogeneous problem. [ 22]

If condition (C.9) holds, the surface elevation at the fore and
aft ends is C (instead of zero for the condition (C.3)), but the x-
component of the velocity at the same points is-(1 +gc), by (C.2).
Hence, the water flows in and out the body unless C=-1/g. Thus
an adequate condition for a surface piercing body is

w=-z-— on S&S. (C. 10)

Throughout this section, we have treated a class of functions
yw and { which are finite and continuous everywhere. As long as
the integrals considered exist, the method may be applied with some
minor changes to other classes of functions.

The question of the uniqueness of solutions will be left to the
future.

572

WAVEMAKING RESISTANCE OF SHIPS
WITH TRANSOM STERN

Be. Yim
Naval Shtp Research and Development Center
Washington, D.C.

ABSTRACT

The wave resistance of ships, having transom sterns
and bow bulbs, is analyzed by an indirect method.
The total wave resistance of a combination of singu-
larity distributions for such ships is minimized with
various parameters of the bulb and the transom stern
at each Froude number. The effect of free surface on
the body streamlines near the stern is also analyzed.

INTRODUCTION

There has been an increasing interest in ships having transom
sterns ,not only for high-speed ships like destroyers but also for
cargo vessels, because of the advantages of more cargo room as
well as modern improvements in techniques of loading and unloading
cargoes over the stern. Recently a theoretical study was made by
this author [1], and a mathematical model was suggested in view of
applying a higher-order ship-wave theory [2]. In this paper, we shall
look at the problem again from a different angle, i.e. , approaching
the problem in a more practical manner.

There are usually two approaches to the ship hydrodynamics
problems, direct and indirect. In the direct approach one starts
with a given ship and finds a corresponding mathematical representa-
tion for it, then, calculates physical quantities, and verifies the
results by experiments. On the other hand, one could start with a
mathematical model such as a singularity distribution, and calculate
the physical quantities of the model, find the corresponding ship form,
check the behavior with the experimental results, and utilize the
knowledge in designing practical ships. This is an indirect approach.
Both approaches are useful in the development of ship science. In
Reference 1 by a direct approach, it was found that the transom stern
might be represented by a sink line along the stern, having a strength

519

Yim

proportional to the stern draft. Now, we would like to use the in-
direct approach to the transom stern theory. Noting that a sink line
on the free surface is tantamount to the constant pressure distribution
ahead of the line and that it produces a negative cosine regular wave
and a depression in the free surface immediately behind the sink line,
we expect to get streamlines similar to the transom-stern ship from
a combination of a normal ship-singularity distribution and the tran-
som sink.

First the free-surface streamline due to a two-dimensional
sink line is plotted to establish the validity of this model, which will
be used later for plotting the streamline near the transom. Then,
several simple original ship singularities are considered so that
basic ship models can be modified to those having transom sterns.
Since the transom sink is supposed to behave like a stern bulb [ 1]
to cancel stern waves, a bow bulb [ 3,4] made of a source is also
considered together with the stern sink to cancel bow waves as well
as to supply source strength which helps form a closed body. Thus,
optimum strengths for the bow bulb source together with the transom
stern sink are calculated to minimize the total wave resistance.

The wave resistances with and without the bow bulb and the
transom stern are calculated. The Sretensky formula for wave
resistance is used since it is much simpler to program in the high-
speed computing machine than the Havelock formula. Finally,
approximate waveforms near the stern are investigated, which will
help in designing a good afterbody near the transom stern.

This is part of a project in which the ultimate goal is to under-
stand more the physical meaning of a transom stern in the wavemaking
resistance of a ship; to obtain better design criteria for ships with
transom sterns; to find out the possibility of an improved ship design
with the gained knowledge, and, hopefully, to design a good ship with
a transom stern, making full use of high-speed computers as well as
testing the model in a towing tank.

Although this is a small part of ship-designing problems, it
is not easy to complete ina short time. At this stage, it is merely
hoped that this paper will achieve several objectives: (1) to validate
the mathematical model of a ship having a transom stern as a stepping
stone to analytical investigation of transom sterns, (2) to determine
the practical ranges of parameters within which the application of
bulbous bows and transom sterns would be beneficial, and (3) to
initiate a computational procedure which would be used for an overall
design program using a high-speed digital computer.

574

Wavemaking Reststance of Ships wtth Transom Stern

Il. A SINK ON THE TWO-DIMENSIONAL FREE SURFACE

Lamb [5] showed a formula for the free-surface shape due to a
point sink with the strength M located at x =0, y = 0, moving with
constant speed U to -x direction on the free surface, where +x
is on the mean free surface pointing right, and ty points vertically
upward. The wave height is

mx

Z

where

Gee

720 -mx
Ky | fe Cin {3 - Si kx) cos kpx + Ci k,x sin kox}

3 m* +k-e zZ
in sd>"0 (2)
oe es oat (3)
6 Ge geen eo (4)
y = 0.577215665 (5)

and g is the acceleration of gravity. If this is compared with the
wave height due to the distribution of constant pressure p, on x<0O,
it can be easily seen that

(6)

holds. Thus, from the Bernoulli equation the form of the free surface
in x<0O can be given by

00 mx

ee, k :

n=-Bo(i+ kal —z—z dm) , in x<0O (7)
Pg 0 EK,

The wave height for x= 0 is plotted in Fig. 1 and it clearly shows

575

Yim

Fig. 1. Stem Waveform by a Sink Line Transform Stem

the appropriateness of this flow model in the vicinity of the transom
stern. The integral term is the local disturbance which dies down
rapidly with aie and the expression of n in x <0 can be inter-
preted as the body streamline of the half body which is formed by a
sink on the free surface with a total flux equal to ut.»= upo/(pg) = 27M.
This kind of two-dimensional half body was treated by Afremov[ 6] in
investigating the flow near a transom stern, thus obtaining pressure
distribution of the two-dimensional half body,

y=-me, in’ x=0 (8)

with the parameter a=0O. He obtained a sharp rise of pressure near
the edge of the transom stern where the pressure is zero, the value of
atmospheric pressure. This sudden rise of pressure at the stern can
also be observed from experiments of planning ships [7].

However, the application of the two-dimensional analyses is
generally valid only near the transom stern. In addition, in designing
the afterbody near the transom stern, the investigation of hydrodyna-
mic interactions between other parts of ship hull and transom stern
is important, especially the superposition of ship wave systems from
bow, stern, and any discontinuities of the hull; this will be discussed
later.

Ill SHIPS WITH TRANSOM STERNS AND BOW BULBS

It seems reasonable to say that the general representation of a
ship with a transom stern and a bow bulb may be achieved by com-
bining singularity distributions: a source distribution along a given
base surface with either point or line doublets or sources for the bow
bulb, and a line sink distribution for the transom stern along a line

576

Wavemaking Reststance of Ships with Transom Stern

that is on the free surface, at the stern, and perpendicular to the

ship centerplane. In the previous section it is known that the transom
stern can be represented by a sink line at the stern, it is therefore
natural to deduce that the sink could contribute in cancelling stern
waves [1]. Since a moving point sink produces negative cosine regular
waves behind, the proper main hull should have a hull shape that
produces positive cosine stern waves. To investigate the best shape
of transom sterns, in a simple way, we have chosen two simple bare-
hull forms, represented in the Michell sense by the following equa-
tions:

B -1<x<0
—~ |
y, = 1 - cos (27x)+, in (9)

oe 8 - <2 <0
and

-i1<x<0
2Be 2 ;
y iwi Santas +x‘), in (10)
s -<2<0

where L, B,, and H are the ship length, the beam, and the draft,
respectively; (x,y,z) is the right-hand rectangular cartesian co-
ordinate system with the origin 0 onthe free surface; the x axis is
in the direction of the uniform flow velocity at infinity; and the z
axis is vertically upward. The hull in Eq. (9) produces the favorable
stern waves for the transom stern but has cusps in both ends and is
called a cusped cosine ship here. The hull in Eq. (10) is rather a
common parabolic hull form, not particularly favorable for the tran-
som stern. The corresponding source distributions by the Michell
approximation are

o, = ge! sin (2mx) (14)
in
D({y=0,-1<x<0, - H/L<z <0) (12)
and
op = -syt(i +2x), in D (13)

The theory of superposition can be allowed in the sense of the Michell
ships. For convenience, we set

it

Yim

c=, +0, (14)
with

B=B, +B, (15)

where B is the beam of a superposed ship.

For the transom stern, a triangular shape of stern draft is
considered as

g=toy + HL, in 65 eA se (16)

where @ denotes the deadrise which takes on a negative value for
y <0, and a positive value for y>0O,

Hs =. a7 (17)

is the maximum stern draft; and @ is a parameter to be determined
for the total optimum wave resistance. Following Ref. 1, the source
distribution for the transom stern is taken as

He =
o, ron) ~ $+ ay ) (18)

although actual shape of the transom stern corresponding to this
source distribution will be found later by streamline plotting. In
addition, a point source located at bow stern, x=-1, y=0,2=42,
with the strength o, is considered as a source type of bulb that con-
tributes to form a closed body with the sink distribution at the stern
as well as to reduce bow waves.

IV. WAVE RESISTANCE

The wave resistance due to the source distributions mentioned
previously can be written according to Sretensky [8]

@

2
{ont 2b 2p ie
pesca 35 —= » En eo (P +) (19)
SoU L mio 7A oon |
where
gL 1
k= 2S
Ue Fe

578

Wavemaking Reststance of Ships with Transom Stern

1 when sagt — (0,
a (20)
ts 2 when m2 i
_ {i 1 47™
be /5ts f1+(5=) (24)

Lw = tank width where the ship is tested,

Pee) -{ o(x,y »z) exp} kb(zb tix) t+i2ty =| dS(x,y,z) (22)
s

‘Substituting in Eq. (22) all the source distributions for the
bare hulls, the transom stern, and the bow bulb, and performing the
integration, we obtain

TT B,
= _2 0 __2Be_\ 4 sin (kb)Ba] ,
P= E -cos (xb) ace 7: mer + SL E (23)
| m
P,= ——, {1 -cos(B,7— (24)
Ww
P, = cos (kb) exp (kz, b’) (25)
_ wB, /(2L) 2Be2 Bo Bo
Q, = [p= (kb) Pee = | + nmkbL +cos (kb) oa E
(26)
Q, — (0) (27)
Q, = - sin (kb) exp (kz,b’) (28)

where subscripts 1, 2, and 3 correspond to the bare hull, transom
sink, and bow bulb, respectively,

E= |t- exp (KO TH Se (29)

'U
1

= P, +eP, +*)P, (30)

oat ed

Yim

and
Q=Q, taQ, +2, (31)

with the functions P and Q evaluated and substituted into Eq. (19),
the wave resistance may be minimized with respect to such parameters
as @ and o,-

V. OPTIMUM TRANSOM STERN AND BOW BULB

A usual technique is employed to obtain the optimum values
of @ and oc, for given k, B,/L, B /L,2,, and H/L. Namely, we
solve two linear BGG! eancees pean he in @ and o,

8 =e ae ee Sat a
ZS = 2a(P; + Ob) + 20,(P,P, + Q,Q,) + 2(P,P, + O,Q,) = 0
(32)
aR a
Bey = 2UlPP, + Q,0)) + 20,(P; Pp? +04 +2(P P, +QQ)=0
for @ and Op? where
= 2
_ 16n°k 2b

m=O
etc.

from the formula of wave resistance. Cases of B, = 0, B, #0;
B, #0, B,=0 and B,#0, B,# O were calculated for each Froude
number F,, which will be shown later.

VI. SLENDER BODY THEORY

For the case of B,=0, the cross sectional area curve is a
cusp at both ends. Thus a slender body theory can easily be applied
here. The result will be only the’ change of

Diced | (34)
and
al = (35)

in P, and Q, in Eqs. (23) and (26) of the wave resistance of the
previously described Michell ship, where A is the area of the mid-

580

Wavemaking Reststance of Ships with Transom Stern

ship section.

The slender ship approximation is useful for the ship with a
transom stern because this may give better chances to represent the
ship shape near the transom than Michell approximation.

Yet for the case of low Froude numbers it is well known that the usual
slender ship theory is also very poor. To improve this situation, the
slender ship theory can be modified further from that developed by
Maruo [11].

In the equations of wave resistance Eqs. (19) through (22), by
consecutive applications of integration by part to Eq. (22)

P +iQ -{ o exp | kb(zb + ix) + i2my = ds
Ss
1

= 15 Se o(0,y,z) exp (eebe + i2ny =) dc

Ae o(-1,y,z) exp | kb(zb = 4) erp i2ny —| dc

fax eli a a o exp (kzb* Tarai Bde
c(x) ~

S m
tis |S, o(0,y,z) exp (kzb + i2ny ail dc

“i o(-1,y,z) exp | kb(zb - i) + i2ny mt dc
c(-1) be

2 . x=0
=m 2.6 o exp (kzb + i2ry —) ac{ ik
kb = c(x) A Soe 1

22 ikxh ae (" 2 m
- \ dx e'** a, \ o exp (kzb + i2ty ee) ad (36)
c(x)

where
dS = dx de (37)

The last integral can be approximated by

5814

Yim

1 ie ikxb n
3 dx e M*"(x)
kb ¥¢,,

where

M(x) -/ o(x,y,z) de (38)
c(x)

This reduces the influences of the line singularity approximation of
ship hull singularities in two ways, (1) by the factor 1 /b*, which
corresponds to cos°@ inthe Havelock wave resistance formula,
(2) by the factor 1/k® which is the smaller if the Froude number is
the smaller. Of course, the number of terms have been increased
in the singularities along c(0), which is the intersection of the free
surface and the transom stern, and c(-1), which is the straight bow
stem line.

VII. STREAMLINES

To establish the validity of the mathematical model of the ship
with the transom stern, the double model[9,10] scheme is inadequate
because free-surface waves play a vital role in the flow field of a
transom stern.

Fortunately a slender ship model [ 11 ,12] gives an easy repre-
sentation of the wave height along the train of ship. The wave height is

Q
and
k Bt
g(x) +y,>0) pet 4 o(x,y,zZ) ds
Ss
w © eitw
x Re | secto | —__¢ 7 __at dé (40)
_W o t-ksec*@-ipsec®
where

= (x, = x) cos 0+ (y, - y) sin 0 (41)

and S is the ship surface.
The inner double integral was treated by Havelock and was

represented by both a Bessel and a Struve functions. [13,14] We
will consider, for simplicity, a source distribution which becomes

582

Wavemaking Resistance of Ships with Transom Stern

zero at both ends of a ship such as we considered in Eq. (11), although
this is not a basic necessity for evaluating physical quantities, namely,

o(0,y,z) = o(-1,y,z) = 0

By making use of this assumption we integrate the wave-height Eq. (39)
by parts

x
C(x, sy)) ae x (x) sy; 0)
4 ti -imxcos@ qd preeie pectpe | | dade doe
= - = e dx a> odc rel eee
wVe-| o(x) 0 t-ksec’O - ip secé
’ Gis
. Re ("f° ey 8) eee ee aed (42)
0 t-ksec *o - in sec
where
w, = x, cos 8 + (y,; - y) sin 0 (43)
0 itx cos 8 d 2. tliw) +z)
I(x,,y,3t,9) = - e dx aS o sec Oe dc (44)
= * JYo(x)

Integrating I by parts with respect to x, we obtain

1 d tiiw) + 2)
I(x, ,y,3t» 9) = ee o(0,y,z) gece ble : dc

A ¢(0)
: & A Cites exp |ti(x, FH cos 0 +y,-y sin@)ttzbdc
(8)
-it 8 2
J axe ay {x,y ,z)sec® 0 el(l'*2)| ac (45)
= dx “¢(x)

For simplicity o,(0,y,z) and o x(-1,y,z) are assumed to be uni-
formly distributed , respectively, along the stern on the free surface
and along the bow stern vertically. For investigation of the flow
field near the transom stern, the contribution of o,(0 Vee) tO tue
wave height may be approximated by two-dimens tonal values; the
contribution of o,(-1,y,z) may be approximated by the stationary
phase; and the contribution from the last integral of I(%, yit, 9). may
be approximated by a slender body theory, and will be investigated
first, here.

583

Yim

Now let us investigate the last (third) term of the above inte-
grallI. From the slender body theory

2
ee a o(x,y,z) exp {ti(x, cos © + y, ~y sin 6) ttz} dc

5 d°M(x)
= awe

z exp {ti(x, cos © + y, sin 6)} (46)
x

With this approximation, we go back to the wave-height Eq. (42) and
consider the values only on y, = 0

ti(x)-x) cos @
tabs,.0) = - B60" axe mttay (" f sectee ST at ae (an
ai O it(t-ksec a= ip sec 8)

Changing the contour of integration with respect to t in a complex
plane such as Havelock [13] used, we have

i

sels ac mtey seco ae | GEN sesame) oo

£5(2, ,0) ae

Ky T
+f dx M(x) { 2m sec 0 cos (kx, -x sec 0) dO
-| -7

(0)
T Z
a) dx M"(x) [{ 2 log |x,-x| - Yo(k|x,-x|)}
X sign (x,-x) + Hy(lex, =) |
cee.

T ;
+ dx M"(x}Y,(k|x,-x|) (48)
where Y is the Neumann function, H is the Struve function, and

sign (x,-x) is +1 for x,-x>0 and -1 for x,-x<0.

When only the integral of the first term of I in Eq. (45) is
taken, we may approximate the wave height as follows, for x, <0:

584

Wavemaking Resistance of Ships wtth Transom Stern

B/2 it(x,cos @- y sin @)
Re 39 J t y
_ Reo Ace ( ay J do a ie BO ea se

C
B/2 it(t - ksec“O - in sec 6)

' é @ ave ee it(x cos @- y sin @)
= dx R x { d ‘ ao { CSS SS SSS
f. * aU -B/2 _ = Oo : t-ksec“@ -ipsec 6 =

x

\,

dx Re oo "ay of at | ce ee
-B/2 t-ksec@ - ip sec ®

eit(x cos 9- y sin 4), J

x, B/2 x 1/2
my ax ot | dy ~ Eu ox lim ao { dt
fe) B/2 Tt ¥,; > 0-1/2 0)

x genes oir (ty, sin 6) +J
sin @(t - ksec*6 - ip sec 6)

ao] e* oO itx
Vx? +B2/4 +B/2 o» e'* dt
- a\ dx log UX tB/4+B/2 4 4 dx +J
» XRD eS, > eet

|x|

(49)

Note that t= 0 is not a singularity because of the zero of
sin {t(x, cos ® - y sin 0)} in the numerator of the real integrand.
The integrand J is defined as follows:

8/2 =
ee a (a Z ne of eee iwy sin®

al 7?

o it(t - ksec’® - ip sec 6)

B/2
= Re ay [ a"? at sec of =

1 \ -ity sin @ ity sin @
e te

ee )
t-ksec*@ - in sec 0

+

o wv ik ae
Re =a lim an( d@ cos 8 |! $ sec 8 sin@
TY B--0 o ik sin 8

20,7 /(k°U) (50)

585

Ytm

Letting
x .
_ al dt
ies pen er Se
re) 0 a
we have
00 ™ 1
I -\ = dm in ‘sx< 0
| 0 me +k? 1
ee sis con koe Cie a iioc
% Zak ii ee! | 1 1 igs *
and
_ 20 30S us
I, = = cos kx, - EF - % U( 5 - Silex,) cos kx,
+ Cikx, sin kx,} in x, >0

Therefore, for x, < 0
x
|
Oy ,
op ru | 4x1 (log [x,| - 1) - a) dx log (jx? + B2/4 + B/2)
0

+2 (( 2 - sikx,) cos kx, + Cikx, sin kx}] (52a)

and for x, > 0

x

I
C a [4x,(log x, - 1) + sf dx log (¥x* + B“/4 + B/2)

8T 4n 4 T
Lis cos kx, - erie es - Sikx,) cos kx,
+ Cikx, sin kx,}] (52b)

The wave height related to the second term of I can be approxi-
mated by the method of stationary phase, [5] neglecting the local
disturbance:

/ -1,0,0 u
Co=- 4 a ie een a} cos (kx +7) (53)

586

Wavemaking Reststance of Shtps wtth Transom Stern

The wave height near the stern due to a point source at the
bow can be approximated also by the method of stationary phase:

tp = 4k exp (kz, ) / a cos (kx, + z) (54)

The wave height near the stern due to the transom stern sink,
approximated by the two-dimensional one, was given earlier, say ¢..

As a result, the total sum of ¢,, Cor Sp, and C€, would repre-
sent an approximate wave height in the wake of the ship near the
transom stern. The streamlines on the body near the stern can be
obtained by considering the pressure which is given by the singularity
representation as was done in Eqs. (6) and (7). The streamlines near
the bow may be obtained by a double model approximation. The inte-
grated scheme to produce approximate body streamlines will be pro-
grammed for a high-speed computer in the near future.

VIII. NUMERICAL RESULTS AND DISCUSSIONS

The optimal strength of singularities for the transom stern
and for the bow bulb are shown in Figs. 2, 4, and 6. The former is
shown in terms of the deadrise angle a of the afterbody near the
stern. The latter is shown in terms of the radius of the correspond-
ing half body produced by the point source located in the infinite
medium. These are all functions of Froude numbers for the given
hull shapes. The wave resistance at each Froude number is computed
for the given hull with the transom stern and the bow bulb optimal at
a given Froude number} see Figs. 3, 5, 7, and 8.

587

=z
2
=
a
-|4
eee
1
0
0.2
Fig. 2.

0.3

Yim

oli

Fn

0.8

0.6

=
=
Ct rad

0.2

0.4 0.5

Optimal Values of Bulb Size and Transom Dead Rise Angle

for a Cusped Cosine Ship Using the Michell Ship Theory

588

Wavemaking Reststance of Ships wtth Transom Stern

2.5
By
T 70.12, B, =0
0.05, 2 = 0.035
2.0 1
oom
Ys
15 WITHOUT BULB
a> WITHOUT TRANSOM
NN
Qin
—
De
[a4
x
~o
=
1.0 ;
OPTIMUM AT Fn =0.4 WITH BULB
es WITHOUT TRANSOM
0.25
0.3
(4
0.5 “Pp 0.35
0.2 0.3 0.4 0.5

Fn

Fig. 3. Wave Resistance of Cusped Cosine Ships with Optimum Bows
and Transom Stems, using the Michell Ship Theory

589

(r2/A) x 100

Fig. 4.

Yim

Optimal Values of Bulb Size and Transom Deadrise Angle
for a Cusped Cosine Ship, using the Slender Body Theory

590

Qt rad

Wavemaking Resistance of Ships wtth Transom Stern

2.5
= 0.006
L
2, =-0.035
WITHOUT BULB
2.0 WITHOUT TRANSOM OPTIMUM AT Fn=0.4
— 1.5
NS
wr
>
Qin
~~”
=
[= 4
=x
oO
2
1.0
0.5 y,
0.2 0.3 0.4 0.5

Fn

Fig. 5. Wave Resistance of Cusped Cosine Ships with Optimum

Bulbous Bows and Transom Sterns, using the Slender Ship
Theory

Yim

(r/L)" x 10°
NR

Fig. 6. Optimal Values of Bulb Size and Transom Deadrise Angle
for Cusped Cosine-Parabolic Ships B, = Bo, B,/L = 0.06,
H/L = 0.05, Z, = - 0.035

592

Wavemaking Reststance of Ships wtth Transom Stern

2.0

PARABOLIC CUSPED COSINE OPTIMUM BOW
BULB +TRANSOM STERN
AT Fn =.325,AT Fn =375

1.0

PARABOLIC
Bas 0, Bo= 0.12

10° x r/( 2v'.’)

0.5

PARABOLIC+SINE, B, = By = 0.06

0.2 0.3 0.4 0.5
Fn

Fig. 7. Wave Resistance of Ships of Cusped Cosine and Parabolic
Waterlines with and without Bulb and Transom Sterns
(x, = 0)

593

Yim

2.0
B, By
— = — = 0.06
L L
bile p= 0.035
r70.05 71

WITHOUT BULB AND TRANSOM STERN

OPTIMUM AT Fn=0.375

Fn=0.325

0.5

0.2 0.3 0.4 0.5
Fn

Fig. 8 Wave Resistance of Ships of Cusped Cosine and Parabolic
Waterlines with and without Bulbous Transom Stern
(x, = 0.05)

594

Wavemaking Reststance of Ships wtth Transom Stern

For a slender body model, Eqs. (11) and (35) are used for
the bare-hull source distribution, which is called a cusped cosine
ship here. For a Michell thin ship model, computations are per-
formed for Eq. (11) for cusped cosine and parabolic ships. For the
combined bare-hull source distribution, the influence of the location
of transom stern xg, is shown in Figs. 6 through 8. It can be under-
stood that there is an optimal location for the minimum wave resis-
tance as in the case of a bulbous bow; however, it is not computed
here.

It is interesting and reasonable to see that the optimum size
of the bulb becomes the smaller for the large Froude numbers over
0.4, and eventually the strength becomes negative at F,>0.5. In
other words, for a large Froude number, a ship behaves like a single
point doublet far behind the ship so that the only way to reduce the
wave height is to reduce the ship volume.

Indeed it is possible to take advantage of the transom stern
as well as the bulbous bow to reduce wave resistance in the Froude
number range F,< 0.5 by a proper combination of the ship hull
shape and the transom stern and the bow bulb. For the case ofa
high-speed ship such as a planing boat, there is no alternative to
evade the detrimental cavitation without having the full separation
occur at the transom stern, whether it is beneficial to the wave
resistance or not.

The numerical results of streamlines are not given in the
present paper because of their complexity. The approximate
method of computation of the streamlines near the stern is shown in
the previous section. When the ship draft is fairly large, compared
with the wavelength, the ship shape from the singularities can be
approximately computed from the double model. However, fora
transom stern, the free surface follows immediately behind the
usually shallow drafted afterbody. Thus, the modified slender body
theory used in the previous sections, combined with a double model
approach to the forward part of ship hull seems to be promising.
Some imaginative approximate configurations from the concerned
source distributions are shown in Fig. 9.

Last, but not least, the importance of experiments on the
design of ships with transom sterns should be emphasized. There
are very few experimental results available [15]. However, this
has to be done in close coordination with the theory so as not to grope
in the dark. The theory is now on a solid foundation. More time and
effort are needed to achieve experimentally usable and complete
results on ships with transom stern. In the future, the author hopes
to finish a systematic computer program for designing ships with
bulbous bow and transom stern that includes information about the
wave resistance, the bulb size, the transom-stern draft, and the
main hull shape.

595

Yim

ae

A Cusped Cosine Ship with Bulb and Transom Stern 1

=

A Cusped Cosine Ship with Bulb and Transom Stern 2

eee er

A Cusped Cosine - Parabolic with Bow Bulb and
Transom Stern (x # 0)

Fig. 9. Imaginative Diagram for Ships with Bulb and Transom Stern

ACKNOW LEDGMENT

This work was carried out under the General Hydrodynamic
Research Program of the Naval Ship Research and Development
Center, The author expresses his thanks to Mr. J. B. Hadler, Head,
Ship Powering Division, NSRDC, for his encouragement in numerous
discussions. Thanks are also due to Dr. P. C. Pien for his valuable
advice, Mr. H. M. Cheng for reviewing the manuscript and his help
in editing, and Mrs. L. Greenbaum for her patient effort in the
preparation of the manuscript.

REFERENCES

1. Yim, B., "Analyses of Waves and the Wave Resistance due to
Transom-Stern Ships," Journal of Ship Research, Vol. 13,
No. 2, June 1969,

2. Yim, B., "Higher Order Wave Theory of Ships," Journal of
Ship Research, Vol. 12, No. 3, Sept. 1968.

3. Maruo, H., "Problems Relating to the Ship Form of Minimum

Wave Resistance," Proceedings of Fifth Symposium on Naval
Hydrodynamics, ONR, Department of the Navy, 1964.

596

Te

Wavemaking Reststance of Ships wtth Transom Stern

Yim, B., "Some Recent Developments in Theory of Bulbous
Ships," Proceedings of Fifth Symposium on Naval Hydrody~
namics, ONR, Department of the Navy, 1964.

Lamb, H., "Hydrodynamics," Cambridge University Press,

Cambridge, England, 1932, Sixth Edition.

Afremov, A. Sh., "Sbornik Statey po Gidromekhanika 1 Dinamike
Sudna," USSR, L967, PPe 130-146,

Sottorf, W., "Experiments with Planing Surfaces," Tech. Memo.
No. 739, NACA, 1934,

8. Sretensky, L. N., "On the Wave-Making Resistance of a Ship
Moving Along in a Canal," Philosophical Magazine, Vol. 22,
Seventh Series, 1936.

9. Inui, T., "60th Anniversary Series, Vol. 2," The Society of
Naval Architecture of Japan, 1957.

10. Pien, P. C., and Strom-Tejsen, J., "A Hull Form Design Pro-
cedure for High Speed Displacement Ships," Transaction of
The Society of Naval Architects and Marine Engineers, 1968.

11. Maruo, H., "Calculation of the Wave Resistance of Ships, the
Draught of Which is as Small as the Beam," Journal of the
Society of Naval Architects of Japan, 1962.

12. Tuck, E. O., "The Steady Motion of a Slender Ship," Ph.D.
Thesis, Cambridge, 1963.

13. Havelock, T. H., "Ship Waves: the Calculation of Wave Pro-
files," Proc. of the Royal Society, A, Vol. 135.

14. Wigley, W. C. S., "Ship Wave Resistance," Trans. N.E.
Coast Inst. Engineers and Shipbuilders, Vol. 47, pp. 153-
196, 1931.

15. Michelsen, F. C., Moss, J. L., Young, B. J., "Some Aspects
of Hydrodynamic Design of High Speed Merchant Ships,"
Trans. of SNAME, Vol. 76, 1968.

LIST OF SYMBOLS

A Midship section area

b Defined by Eq. (21)

B Ship beam

BB, Ship beams associated with two hull forms given by

Eqs. (9) and (10), respectively

597

oe Pars Se ober ee eG

“amet OD DY

6,620,503

by

Yim

Cosine function defined by Eq. (4)
Domain defined by Eq. (12)
Defined by Eq. (29)

U /(gL)
Acceleration of gravity
Draft of ship

Stern draft

Froude number, Ve

Expression defined by Eq. (44)

Expression given by Eq. (51)

Expression defined by Eq. (50)

Lg/u*

Ship length

Source strength

Pressure

Expression defined by Eq. (22)

Expressions defined by Eqs. (23), (24) and(25), respectively
Expression defined by Eq. (22)

Expressions defined by Eqs. (26), (27) and (28), respectively
Bulb radius

Wave resistance

Ship surface

Sine function defined by Eq. (3)

Velocity at x - @

Ship speed

Tank width

Rectangular coordinates

The x coordinate of the location of transom stern sink
Ship hull forms given by Eqs. (9) and (10), respectively
The z coordinate of the location of the point source for bulb
Dead rise angle of transom stern defined by Eq. (16)
Wave height

Two-dimensional wave height

Wave heights due to the first, second and third term of I
in Eq. (45)

Wave height due to bulb source

598

Wavemaking Reststance of Ships wtth Transom Stern

Gs Wave height due to transom stern sink

o Source strength for ship hull

|S Source strength given by Eqs. (11) and (12), respectively
oT, Source strength for bow bulb

0; Source strength for transom stern

p Density of water

Y Quantity given by Eq. (5)

DISCUSSION

Georg P. Weinblum
Institut fur Sehtffbau
Hamburg, Germany

Some general remarks may be permitted, especially from
the point of view of application.

So far investigations of more practical character deal pre-
ferably with the bow wave formations, while linearized wave re-
sistance theory treats with equal love the forebody and the afterbody
of displacement ships. Few experiments only have been conducted
to check, ceteris paribus, the advantage of form symmetry with
respect to the midship section in real fluid. Such tests have been
performed by the present writer with bulbous forms by comparing
simplified ship hulls:

a) without a bulb (naked hull),

b) a bulbous bow only,

c) a stern bow only,

d) bulbs symmetrically arranged at bow and stern.

These experiments are useful in the present context, starting
from the author's and my personal viewpoint, that in ideal fluid a
similarity can be reached in wave effects due to a transom stern and
a bulbous bow (because of a similarity in form representation by
dipole arrangements). The sketch annexed shows an impressive
improvement by the symmetrical bulbous ship design (d), and this
indicates, that the combination bulbous bow + transom stern should
be useful as shown theoretically by the author.

599

Yim

The application of linearized wave resistance theory to slow
full ships following our present state of knowledge overstrains this
theory heavily. This theory should not be discarded, however,
completely as long as it is used as a heuristic principle only, i.e.
as means to look for solutions which must be checked experimen-
tally.

It is recommended to use in this sense several earlier inter-
esting papers published by the author. Considering the present
critical attitude towards linearized wave resistance theory in general,
I wish to state that its use (including perhaps some correcting "im-
provements" for practical purpose still can be highly recommended
in case of medium or especially high Froude numbers. It should be
remembered that in the range of the large wave resistance hump
values computed by Michell's theory may differ by an amount only
from experimental results which corresponds to the scatter of the
latter derived from models in different scale. Therefore I welcome
the author's present second approach on the subject of transom sterns
although the correlation between form and generating singularities
(so far rather indicated than carried out) may still require further
studies.

With regard to the author's statement about lack of systematic
experimental evidence it is suggested to look into report No. 167
Institut fuer Schiffbau Hamburg and to check if something useful can
be found there.

600

Wavemaking Resistance of Ships wtth Transom Stern

DISCUSSION

S. D. Sharma and L, J. Doctors
Untverstty of Michtgan
Ann Arbor, Michigan

In an oral discussion at the Symposium Professor Maruo and
the first-named discusser challenged the validity of the author's
Fig. 1 because they felt that the wave profile should have been dis-
continuous at the location of the line sink or the pressure step,
x=0,. Dr. Yim insisted that his figure was correct, arguing that a
similar curve is shown in Lamb's "Hydrodynamics," p. 405, In
the meantime, we have examined the problem more closely and
arrived at the following conclusions.

Let us examine the case of the pressure step first. Consider
a two-dimensional pressure distribution,

p(x) = po{1 + sgn(x)} /2, (D1)

on the mean free surface, z=0, moving steadily with speed U
along the direction of Ox. The resulting motion can be described by
a velocity potential ¢(x,z) subject to the conditions

,,(x»2z) + ,,(x,z) = 0, (D2)
p(x) - pU¢,(x,0) + pgo(x) + ppUd(x,0) = 0, (D3)
US (x) + o,(x,0) = 0, (D4)
$, 2(*»-00) = 0, (D5)

where z = ((x) describes the free surface elevation and the limit
. — +0 is understood as usual. It is easy to verify that the solution

is
” exp (ikx +kz) 2
(x,y) = - (eo/now) | ae aa dk, ky = g/U ’ (D6)
@
G(x} = - (p,/pg) {1 tsgn(x)}/2 - p,/no8) | ae dk. (D7)

601

Yim

The limit » — +0 then leads to the following real expression

00
pgt(x)/p, = -{1 +sgn(x)} /2 - Senta) { exp (-wlkgx|)
i ) 1 + w*

- {1 - sgn (x)} cos (k,x), (D8)

which is indeed continuous at x =0 although the wave slope E.
becomes infinite at that point. This is evident in the accompanying
Figure (see Curve 1), but does not show up on the scale used by the
author in his Fig. 1.

___. _ _ Curve2: Wave profile for a
line sink, sk
2TIm

|
~~ ~ Curve 1: Wave profile for a

pressure step, “p~
°

gx/U? >

Fig. Di. Comparison of Wave Profiles for a Line Sink and
a Pressure Step

On the other hand, one can also approach the problem as the
limiting case of a submerged line source as the submergence tends
to zero, The velocity potential for a line source of strength m
(that is, output 2mm per unit length of line) on the line x =0, 25 - f
is found to be

2 2 i
- x + (ztf) exp {k(z-f + ix)}

602

Wavemaking Reststance of Ships with Transom Stern

and the wave profile

C(x) = Ud,(x,0)/g, (D10)

now becomes

© -Wikox! yds
_ sgn (x) e {cos (wkof) + w sin (wkof) }
- US (x) /2mm =. sane) | See dw

-k.f
- {1 - sgn (x)} e Ka cos (kof). (D1i1)

It is obvious that for f= 0 the wave profile of the line source be-
comes discontinuous at x=0. For any nonzero value of f the
profile remains theoretically continuous at x = 0. However, for all
practical purposes it is discontinuous in the limit f - 0 as shown in
the accompanying figure (see Curve 2) for kof = 0.00001.

If we assume p,/pg = - 2mm/U, then the potentials of the
pressure step (D6) and the line source (D9) become identical in the
limit f—- 0. But the wave profiles differ by the first term of (D8).
Incidentally, the author's Eq. (1) differs from our (Di1) by a factor
of 2. But his relation (6) seems to have a compensating error of
factor 1/2 so that his wave profile (7) does agree with our (D8).

We have not investigated what effect this discrepancy has on the
author's further calculations of wave resistance. But we did notice
an obvious slip in Eq. (18) for the strength of the line sink repre-
senting the transom stern. If o, is regarded as a line density,
apparently a factor L is missing on the R.H.S. On the other hand,
if o, is interpreted as a surface density, then the R.H.S. should
contain the Delta function 6(0) as a factor.

We also find the idea of using a line sink to represent the
transom stern rather unconvincing. The line sink would tend to
force the flow around the corner of the transom, which in practice
occurs only at low speeds, but in a highly viscous manner not
tractable by ideal fluid theory. The case of real interest is the
one at high speeds where the flow separates smoothly from the
transom. In this regime, we feel that the line sink should not be
used so that the excess sources in the hull can produce a semi-
infinite half-body. We would appreciate the author's comments on
this point.

Notwithstanding minor differences of opinion, we wish to

congratulate the author on his imaginative approach to a very inter-
esting problem.

603

Yim

REPLY TO DISCUSSION

B. Yim
Naval Ship Research and Development Center
Washington, D.C.

The author would like to acknowledge Prof. Weinblum's
encouragement. The author fully agrees with him on everything he
mentioned. As is indicated in the text, the model of the transom
stern assumes the linear free-surface condition although, in
practice, very often nonlinear phenomena, e.g., a rooster-tail or
cavity collapse, do occur. Therefore, this point also needs care,
in addition to the error in Michell's thin ship theory or the slender
ship theory.

REPLY TO DISCUSSION

B. Yim
Naval Ship Research and Development Center
Washington, D.C.

The author sincerely appreciates the deep interest shown by
Drs. Sharma and Doctors regarding his paper.

About the validity of Fig. 1, the author will attempt to make
a detailed explanation. First, the author would like to point out that
the discussers agree by their Eqs. (D6) and (D9) that, with P, /Pg =
- 2mm/U, the potential due to a point sink located on the free surface
at x =0 and the potential due to the corresponding uniform pressure
distribution along the free surface from x=-ooto x=0 are
identical everywhere in the flow field and on the boundary. This fact
has long been known. Thus, velocities of the two cases are identical
everywhere, and the wave heights of the two cases are identical from
the relation (Di0). Namely, one problem with the given pressure
distribution is in fact the same problem with the properly given source
distribution as in many fluid mechanics problems. Admitting this fact,
it is impossible to claim that the representation by pressure gives

604

Wavemaking Resistance of Ships wtth Transom Stern

the smooth boundary and that the representation by source gives

the discontinuous boundary. To elaborate a little more, the discussers
did not notice that the boundary ahead of the location of sink or at
x<0, z=0 is no longer a free surface but has become a part of

body boundary formed by the sink flow field, where the pressure is

a constant different from zero. This may be understood better if we
consider another identity of potentials due to a point sink on x = 0,
z=0 and a uniform distribution of doublet in - x direction on
-o<x<0, z=0. The body boundary created by the two-dimensional
sink is considered to form a part of a transom stern heuteristically

in the second section to justify the three-dimensional mathematical
model in the following sections. It will be easily noticed with a real
scale how readily acceptable the boundary would be and how proper
and simple this model is.

Here the author also appreciates being advised that the factor
of 2 is missing on M in his equations in the second section of the
author's paper. This factor has nothing to do with any of the numeri-
cal results. In fact, the second section has no numerical relation
with the results derived in other sections.

605

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rr

BOW WAVES BEFORE BLUNT SHIPS AND
OTHER NON-LINEAR SHIP WAVE PROBLEMS

Gedeon Dagan
Technton-Israel Institute of Technology
Hatfa, Israel
and
Marshall P. Tulin
Hydronauttiecs, Incorporated
Laurel, Maryland

NOTATION
a Draft at bow of a completely blunt shape
(dimensionless)
b, b Outer and inner coordinates of point B in the
t and @ planes
al aah a ah
He ee Arbitrary constants
e.. »€,..d), rd,.
| | t I
Cp Drag coefficient
D'! Drag force; D = D'/opu*t!
f' = »' + iy' Complex potential; F = f'/g'/2T'3/2, £ = f'g/u'>;

t = £'/u'T!
Fr, = U'/(gL')'”? Length Froude number
Fr,= u'/(gT')”2 Draft Froude number

h'(x') Function describing the body shape; H = h'/T';
b= he

A,' Forebody length; £, = £,' g/u'®

ti Characteristic length; £ = £'g/u'?

Nea / 1 Dimensionless free-surface elevation

P = p'/pu'? Dimensionless pressure

t' Jet thickness; t = t'g/u'*; ee th/T!

Tr Draft

wisu! + iy’ Complex velocity; w= w=w!'/U'; W= w'/(gt')'#

607

Dagan and Tultn

U' Velocity at infinity
Ze x! tay’ Complex variable; z = z'g/U"; 2 =2'/T';
22 T!
8 Angle at bow
5,2 55, A, Gauge functions
e= t'c/u'* Small parameter
to=i6 + i Auxiliary variable; @ = ¢/e
n' Free-surface elevation; 7 = n'g/U"; n= n'/T';

Q = £n(1/w)

N= 7/7
Logarithm of complex velocity = T + i@

I. INTRODUCTION

The conventional linearized theory of ship waves is based on
a first-order perturbation expansion in which the length Froude num-
ber is of order one, while the beam Froude number (thin ships)
and/or the draft Froude number (slender of flat ships) tend to infinity.
While the theory is in fair agreement with laboratory results in the
case of schematical fine shapes (e.g. Weinblum et al. [1952]), it is
of a qualitative value at best in the case of actual hulls. To improve
the accuracy of the linearized solutions, second order nonlinear
effects have been considered, either in the free-surface condition
or in the body condition (e.g. Tuck [1965] , Eggers [1966]).

A different nonlinear effect, overlooked until recently for the
case of displacement ships, is that associated with the bow bluntness.
It is well known from the theory of inviscid flow past airfoils or
slender bodies (Van Dyke [ 1957]) that the linearized solution is
singular near a blunt nose in the stagnation region. The singularity
may be removed by an inner expansion in which the length scale is
a local one associated with the nose bluntness.

In the case of a free-surface flow with gravity the phenomenon
is more complex. The pressure rise in the stagnation region is
associated with the free-surface rise and the formation of a breaking
wave or spray and the existence of a genuine bow drag. The inner
expansion of the Bernoulli equation shows that the inertial nonlinear
terms become more important than the free gravity term, for
sufficiently high local Froude numbers.

The bow nonlinear effects have been recognized a long time
ago in the case of planing plates (Wagner [1932]), but they have been
always associated with a relatively high Fr,, such that the lift /buoy -
ancy ratio is of order one. Here we are primarily interested in the
case of displacement ships which move at a small Fr, and the hull

608

Bow Waves and other Non-Linear Shtp Wave Problems

position beneath the unperturbed level is practically independent of
Fr,. Nonlinear inertial effects may be important nevertheless
near a blunt bow.

A systematical experimental confirmation of the role played
by the bow bluntness has been provided recently by Baba [1969].
From towing-tank tests with three geosims of a tanker (Cp = 0.77)
it was found that in ballast conditions ata Fr,* 1.2 a breaking wave
appears before the bow. At the maximum Fr, tested (Fr, = 0.24,
Fry= 1.7, Fig. 1a) the energy dissipated in the breaking wave con-
tributed 18 per cent of the total resistance, while the energy radiated
by waves gave only 6 per cent. Baba has suggested a two-dimensional
representation of the breaking wave of this experiment, as if it were
uniform and normal to the bow (Fig. 1b), and has estimated
equivalent length as half the beam. The drag coefficient per unit
length, corresponding to a two-dimensional flow across the breaking
wave is Cp=D'/0.5 pU"T' = 0.08 for Fr,=1.7. Sharma [1969]
has indicated a larger breaking wave resistance for a higher block
coefficient tanker and has suggested that the bow bulbs main effect
is to reduce the breaking wave resistance. With the development of
large tankers, as well as large and rapid cargo ships, the study of
the bow free-surface nonlinear effect becomes particularly important.

We present here some of the results of our last year's
studies, which are reported in detail in two reports (Dagan and
Tulin [1969, 1970]).

In this first stage we have attacked the two-dimensional prob-
lem of free-surface flow past a blunt body of semi-infinite length.
The two-dimensional study is a necessary step in the development
of a theory for three-dimensional bows since it provides a valuable
gain in insight at the expense of relatively simple computations.
Moreover, it gives an estimate of the bow drag of flat ships and opens
the way to more realistic computations by further approximations.

Taking the length as semi-infinite is very useful from a
mathematical point of view and it is equivalent to the limit Fr, ~ 0.
This assumption is entirely justified for the small Fr, considered
here and for determining the bow flow, which is not sensibly in-
fluenced by the trailing edge condition.

Il. THE FREE SURFACE STABILITY (SMALL Fr, EXPANSION)

We consider the two-dimensional gravity flow past the body of
Fig. 2. The box-like shape has been adopted for the sake of com-
putational simplicity, but the method can be easily extended to any
other shape.

When Fry is small the free-surface is smooth. We assume

that breaking wave inception is related to the instability of the free-
surface. According to Taylor's criterion (Taylor [1950]), the

609

Dagan and Tultin

(a)
(VIEW FROM FRONT)

“KT

bh WAVE
(UNIFORM)

Fig. 1. Baba's [1969] experimental results. (a) Breaking wave
before a tanker; (b) Baba's two-dimensional representa-
tion of the breaking wave.

>

SHVUEUI

(0) ‘b)

“
A 01) B(+1)

ae me

(c)

Fig. 2. Small Fr, flow past a box-like shape body. (a) The
physical plane; (b) the linearized dimensionless physical
variables; (c) the auxiliary ¢ plane.

610

Bow Waves and other Non-Linear Shtp Wave Problems

free-surface becomes unstable when the normal acceleration
vanishes. In our case this occurs when the centrifugal effect related
to the free-surface curvature offsets the gravity acceleration. Since
we expect the free-surface to become steep as Fry increases,
there must be a critical Fr, characterizing instability.

The gravity free-surface problem is, however, nonlinear,
To linearize it we consider a small Fr, perturbation expansion,
i.e. and expansion for a state near rest. Referring the variables
tor (Pig: 2)? and (gT ')'/2 and expanding as follows:

F(Z)= @ +iW = Fr,F(Z) + Fr °F,(Z) +... (1)
W(Z)= U - iv = Fr,W (Z) + Fr 3w,(Z) +... (2)
N(X) = Fr/°N (X) + Fr,4N,(X) +... (3)

we obtain from the exact free surface and body boundary conditions
the following equations:

at first order (Fig. 2b)
Vv, =0 (ASBA) (4)
wW,=1 (X — - o) (5)

i.e. a flow beneath a rigid wall replacing the free-surface at its
unperturbed elevation. In addition

Napii-u,). (<0, ¥ =6)

second order
W, = - UN, (AS, X<0, Y= 0) (7)
U, = 0 (SBA, X>0, Y=H(x)) (8)

i.e. a flow generated by a source distribution along the degenerated
free-surface, and

N,= 4 0,0, Der Onna p) (9)

It is easy to ascertain that Wo is zero at infinity such that

6i1

Dagan and Tulin

the total source flux is zero. Similarly N, is zero at both the origin
and infinity. In fact the first order solution gives the exact values

of N at infinity and at the stagnation point, the higher order approxi-
mations correcting only the free-surface shape between these two
anchor points. The above expansion is consistent and hopefully
uniformly convergent. It differs from that suggested by Ogilvie
[1968] who has kept terms of different order in the same equation

in order to obtain waves far behind a submerged body.

The solution of the first order approximation for the box-like

shape body (Fig. 2a) is obtained in terms of the auxiliary variable
C as

72
W, = ($3) F,=t/n (10)

where the mapping of the linearized Z plane (Fig. 2b) onto ¢
(Fig. 2c) is given by

z= (e? - 1% +2 tn [(e? - 1)? - 2) (14)

T

Hence, by Eq. (6) we have

N, = tes
For the second order approximation (Eq. (7)) we get

/2
y, = 64H

one a7 (SiS Sad rip. =O) (12)

W, given along the ¢ real axis (Eqs. (8) and (12)), leads by a
Cauchy integral to

{ mr dd
&, (6) a T Re \; We () (2p ©

1 ‘ork (ee eee ay/2
-i\2; +(g44) BEE aS

U2 and No as functions of €, are easily found from Eqs.
(143) and (9) (for details see Dagan and Tulin [1969]). The shape of
the free-surface at second order is given in Fig. 3. As expected,
the profile becomes steep as Fr, increases.

612

Bow Waves and other Non-Linear Ship Wave Problems

VV
l

0.25
X=x'/T
N=n'/T'

0.20
N (Xx) = Fr? N, X)+ Fr# No (x) +...

” 0.15
ae een (xy :
ie =
Z Ti Fr? 0.10
T=9.5
~~ 1.0
== Gp
: 0.
Pas) S
3.0)

2.0

Fig. 3. The free-surface shape in front of a rectangular body

The dimensionless pressure gradient component normal to
the free-surface is proportional to

2 2
a{ tard, \ © * <M)

2 (2
+ Fr So dN _ y 2a Ne Aug OL)

dX dX ! qx? "12 ax eS

where the first two terms of the expansion contribute to order Fr_®
in the pressure gradient. Taylor's marginal stability is reached for
the value of Fr, which renders Eq. (14) equal to zero. This value
has been found to be Fr,# 1.5 and the point of instability at X= 0.3.

Although the expression for the pressure gradient can hardly
be expected to converge rapidly at such a high Fr., the’ result iis ‘of
the order of magnitude of that found by Baba and would seem to con-
firm the mechanism of free-surface disruption assumed by us.

The effect is nonlinear since only when taking into account the
second order term does the steepening of the free-surface depend

strongly on Fr,. There is no bow drag in the small Fr, limit.

The present method suggests a possible way for determining

613

Dagan and Tulin

the influence of the bow shape on the breaking wave inception, and
therefore serves in selecting shapes which retard the phenomenon.
Its application to three-dimensional bows is left for future studies.

III. THE HIGH FROUDE NUMBER APPROXIMATION (THE JET
MODEL)

(a) The Outer Expansion

As Fr. increases the breaking wave develops. Energy is
dissipated there and the momentum loss is associated with an induced
bow drag. To treat analytically this free-surface nonlinear problem
we have to adopt a flow model which: (i) permits an ideal fluid
representation of the phenomenon and (ii) lends itself to the lineari-
zation of the free-surface condition.

We adopt here the jet model, well known from planing theories,
and neglect the returning jet flow, such that the jet momentum loss
is equivalent to that of the breaking wave (Fig. 4a). Moreover we
expand the exact equations of flow in an € yi jE small parameter
expansion, consistent with the usual nea sioed ship wave theory.

The expansion, and the associated model, are probably valid
for sufficiently high Fr,, and we only may hope, like in other
asymptotic solutions, that the results are sufficiently accurate even
for moderate Fr,.

We consider now an outer expansion with the rss variables
made dimensionless by referring them to U' and U' */e (see Nota-
tion). Using a procedure followed in similar problems in the past
(Tulin [1965], Wu[1967]) we carry out the expansion in an auxiliary
{ plane (Fig. 4c) related to the complex potential plane (Fig. 4b)
through the transformation

aS ee (15)

We assume now that the jet thickness t and the distance to
the stagnation point b are both o(1), such that under the outer
process €—~0, (= Of1) they coalesce with the origin of the ¢
plane (Fig. 4e).

Under these conditions we obtain by a systematical expansion
of the complex velocity

w(t) = 1 + §,(e)w,(t) + Sgle) walt) +... (16)

after taking into account that by definition the body profile has the

614

Bow Waves and other Non-Linear Ship Wave Problems

A

ate 7

| v
Bee

¢

t

J ty

m
p20) ub) A —_—
= ee

=

(c)

A

by(x) 2-0 + (1a) (e™"/#- 1) z Bas

“

€

tf

Uh 77 (b)

(f)

Fig. 4. High Fr, flow past a blunt body. (a) The physical plane;
(b) the complex potential plane; (c) the auxiliary ¢ plane;
(d) flow past a completely blunt shape (physical plane,
outer variables); (e) the body first-order boundary condition
in the outer expansion; (f) the zero order boundary conditions
in the inner expansion.

equation
h(x) = €h,(x) (17)
at first order
6p=ue (18)
Re (w, + ik,) = 0 (6 <0, p= 0) (19)
Im k,(t) = - h, (6) (6>0, p= 0) (20)

where

615

Dagan and Tulin

-0
and
cs
a= be | w, do (21)
-0
at second order
é3= «?
Re (Siz + iw,) = - Re[ (w, + 26,) S| (E<0, p= 0) (22)
Im w,=-5Imw,” (6>0, >= 3) (23)

We determine now the first order solution by replacing the
body along § >0 by an unknown pressure distribution (equivalent
to a vortex distribution, see Stoker [1957]) of strength g,(§).

The function k \() satisfying Eq. (19) and the radiation conditiea
becomes

peo -v)
Keg) en BILL Vi] go) (25)
Eq. (20) becomes now
@ .
=) Re e*”) Fi [i(é - v)] g(v) dv = - (6) (26)
(@)

The integral Eq. (26), with a displacement kernel, may be
solved by the Wiener-Hopf technique.

The Fourier transform of Eq. (27) reads

+ 1 - +
M(A)G, (A) = —— [N, (4) + H,'(\)] (27)
l > ! |

where M, G,; H,, and N, are the transforms of the kernel, 8)? h,
and the freee paises profile, respectively.

The kernel's transform has been factorized by Carrier et al.

616

Bow Waves and other Non-Linear Shtp Wave Problems

[1967]
1 1
(20)'* 1+[a|

M(A) = =[M()]-M(\)]

(28)

The separation of Eq. (27) can now be accomplished, pro-
vided that we select a given body shape h,(x).

We limit ourselves here to the case of the completely blunt
shape of Fig. 4d. The forebody length &£, is of order €, such that
at the limit € ~ 0 the bow degenerates at first order into a point
singularity at the origin. Any shape with the same length scale of the
forebody will yield the same first order body condition.

With (Fig. 4e)

(20)
1 1 1-a 1
mes (2m) In ie (2m)”? 5 ES A (29)

we obtain from the separation of Eq. (27)

eu

Le ed { 1 1 :
6°00 = a X- ba) geze| een actor GP 89

where the last term, representing eigensolutions, results from the
application of Liouville's theorem, cj, being arbitrary.

Equation (30) cannot be inverted exactly, because of the
integral appearing in M*(\), but the inversion can be carried out
for large \ by expanding M*(A). After carrying out this process
(see Dagan and Tulin [1970]) we arrive at the following expression
for g,(6) in the vicinity of the origin

n a
Be eae GE’ eyes Soe (E> 0) (3 1)
| (né)/2 » plane

where d,, are related in a unique manner to Cy, +
|

617

Dagan and Tultin

From Eq. (31) we obtain

n
BOM a V2 ans
w, (6) = ey + OVC TSA art )ar 2, vi Pe (C0) (32)

which is the central part of our analysis.

The expression of the second order solution, satisfying
Eqs. (22), (23), and (32), was found to be

at

do,
w(t) = sap + Olln £) + aa (33)

Summarizing the results for the outer expansion, we have,
with the estimate t = O(e?)

€a_ 4 bd, Pe cat bs + €2 ide,

we (me) pir3/e ont pivsre
+ €O(t' fn f) (34)
: 1/2 e\. 2
Zi Gae € [ai + 2a(£) J+e) cine +S-fine
1
t 52}
2 ter) vie (35)
'

er, and eo, being again constants related to Cis Coe
The velocity has the familiar square root singularity at first
order and a source singularity at second order. The free-surface
is continuous and attached to the bottom at first order, while at
second order it rises at infinity. The eigensolutions of the problem,
which represent in fact the linearized solutions of a free-surface
flow past a flat horizontal plate, as well as the flow details near the
bow will be subsequently determined with the aid of an inner solution.
It is worthwhile to mention here that only at second order are the
details of the adopted model (i.e. the jet) manifested in the solution.
Any other model attached to the bow will produce an identical first
order solution.

618

Bow Waves and other Non-Linear Shtp Wave Problems

(b). The Inner Expansion and Its Matching with the Outer Solution

We stretch now the coordinates and adopt the following inner
variables,

t = t/e; ee: z= 2/€; t= t/e; b = b/e (36)

and expand the function Q = £n(1/w) = 7 +i@ ina perturbation
series

~

Q

Q, + A,(e)Q, +... (37)

For the body of Fig. 4d we obtain from the inner expansion of the
exact equations the boundary conditions for %, specified in Fig. 4f,
which represent a nonlinear free-surface flow without gravity. The
conditions at infinity are provided by the matching with the outer
expansion. Only in the case of the straight bow of Fig. 4e are the
inner conditions so simple. Ina general case we have to solve an
integral equation for 6 [ Wu, 1967] orto start with a given 6(€).

The solution of Wo is readily found in the form

B/t ~
= ~ 72 7 ~ (/2 Ale am NG d...
=. Oem oad exp ( ) Tae ) (38)

(Eye ete sale th eb

where the exponential represents the eigensolutions of the problem,

ce being arbitrary constants.

Expanding W, for large { we obtain

~

d
posh He37.6(89)

654 Ge ae p/n) + 2[ (E /m)? - b/w] ae

is
t

z=( Lt/mb af =F - l(t /n)® - pB/u] + 2 /a)”® - pb/l® tn?

Wo

t jp td Nie fee (72
-cénG-ait2dt +... (40)

Before proceeding to the matching we rule out the eigen-
solutions appearing in Eqs. (38), (39) and (40) because they lead
either to an infinite velocity in the jet or to an infinite jet thickness,
depending on whether doi are positive or negative. The matching
of wo and z (Eqs. (39) and (40)) with w and z (Eqs. (34) and
(35)) now gives

619

Dagan and Tulin

we ea eee

eat b= Se Oy da, =O (41)

and both inner and outer solutions are uniquely determined.

Our estimates of t and b are confirmed, Eq. (41) showing
that t = O(e2) and b = O(e3/4). The nonlinear character of the prob-
lem is manifest in the inner expansion.

We have now a uniform solution which can be written by
adding the inner and outer solutions and subtracting their common
part.

(c) The Bow Drag

The horizontal force acting on the bow is found by the pressure
integration in the inner zone (Fig. 4f)

J J 0 ~
B=! p dy = 5 Im (+89 dz=tim! (2+%)(1- 5) dt
B B b w 1

Since w is analytical in the lower { half plane the integra-
tion along BJ may be replaced by integration at infinity (at H) and
around the origin J. After expanding 1/w, +w. at infinity and near

ae 0
the origin, we get for D

D

fae (1 + cos B /m) (43)

The same result, excepting the cos B/m term, may be
obtained directly from the first order outer expansion.

To roughly compare the result of Eq. (43), with Baba's
findings, lets assume that the bow is completely blunt with B= 1/2
and,.a-=,l...For’¢€ = Wy ures = 0.34 we have

Gu= 2Di= 0.34 (44)

which is roughly four times larger than the value estimated by Baba.

At this stage it is difficult to find which of the following
factors explain this discrepancy: The asymptotic character of the
solution, the lack of details on the bow shape or, may be the most
important, the crude representation by Baba of a three-dimensional
flow by a two-dimensional equivalent (Fig. ta). Future experiments

620

Bow Waves and other Non-Linear Shtp Wave Problems

and theoretical developments will give the answer to this question.

The method presented here is applicable to other bow shapes,
like blunt round bows. In this latter case the bow drag appears at
higher order than in the completely blunt case. The extension to
other shapes, as well as to three-dimensional bodies is left for
future studies.

IV. CONCLUSIONS

Theoretical models of breaking wave inception and of a free-
surface bow drag have been derived for the case of a two-dimensional
gravity free-surface flow past a blunt body. In both cases the effects
are nonlinear and are related to the important role played by the
inertial term of the Bernoulli equation in the vicinity of the bow.

The results are of the order of magnitude of those found by
Baba [ 1969] , but an improved verification has to be done by carrying
out two-dimensional experiments. The theory presented here may be
extended, with additional approximations, to three-dimensional flows.

ACKNOWLEDGMENT
The present work has been supported by the Office of Naval

Research under Contract No. Nonr-3349(00), NR 062-266 with
HYDRONAUTICS, Incorporated.

REFERENCES

Baba, E., "Study on Separation of Ship Resistance Components ,"
Mitsubishi Tech. Bul. No. 59, pp. 16, 1969.

Carrier, F. G., Krook, M., and Pearson, C. E., Functions of
a Complex Variable, McGraw-Hill, pp. 438, 1966.

Dagan, G., and Tulin, M. P., "Bow Waves Before Blunt Ships,"
HYDRONAUTICS, Incorporated Technical Report 117-14,1969.

Dagan, G., and Tulin, M. P., "The Free Surface Bow Drag of a
Two-Dimensional Blunt Body," HYDRONAUTICS, Incorporated
Technical Report 117-17, 1970.

Eggers, K. W. H., "On Second Order Contributions to Ship Waves

and Wave Resistance," Proc. 6th Symp. of Naval Hydro-
dynamics, 1966.

621

Dagan and Tultin

Ogilvie, T. F., "Wave Resistances: The Low Speed Limit,"
The Univ. of Michigan, Dept. of Naval Arch., Rep. No. 002,
pp. 29, 1968.

Stoker, J. J., Water Waves, Wiley, New York, 1957.

Taylor, G. I., "The Instability of Liquid Surfaces' When Accelerated
in a Direction Perpendicular to Their Plane, I," Proc. Roy.
Soc. ; London, A, 201, pp. 192, 1950.

Tuck, E. O., "A Systematic Asymptotic Procedure for Slender
Ships ," de Ship Res. ; Vol. 8, No. 1s, pp. 1.523, 1965.

Tulin, M. P., "Supercavitating Flows -- Small Perturbation Theory,"
Proc. of the Internat. Symp. on the Application of the Theory
of Functions in Continuous Mechanics, 2nd Vol., pp. 403-
439, 1965.

Wagner, H., "Uber Gleitvorgange an der Oberflache von Flussig-
keiten,". Zammi,. Vol.. 12, No. 4, pp. 193-216; 1932.

Weinblum, G. P., Kendrik, J. J. and Todd, M. A., "Investigation
of Wave Effects Produced by a Thin Body," David Taylor
Model Basin Report No. 840, pp. 19, 1952.

Wu, T. Y., "A Singular Perturbation Theory for Nonlinear Free-
Surface Flow Problems," Int. Shipbldg. Prog. Vol. 14,
No. 151, pp. 88-97, 1967.

* * F RF FR

DISCUSSION

L. van Wijngaarden
Twente Institute of Technology
Enschede, The Netherlands

I would like to ask a question about the authors' interpretation
of Taylor's instability. Th -y use as a criterion for marginal sta-
bility that the normal component of the pressure gradient at the
interface between gas and fluid vanishes. This is indeed the case
for a plane interface.

622

Bow Waves and other Non-Linear Shtp Wave Problems

For a plane interface and y Vv
variables indicated in Fig. 1 we
pave fluid
dv __ ap interface
Pp a e
ot oy gas
For negligible gas density Taylor's Figs 4
result is:
: Ov . Op
stable: SE = 0; En Ors
instable: UNE > 0s Sp <0.
Ov oy
marginal Op _ 0
stability: oy

The question is whether this criterion (ap /8y = 0) holds also for
more general interfaces. As an example consider the spherically
symmetric implosion of an empty gas bubble (Fig. 2). The equation
for the radius R is

2 fluid

RR +32R =-- Po
2 p

where Pg is the pressure far away
in the fluid, pw > 0. From this
relation

ee o2
RR= -Pm_>R'<o. His. 62

The inviscid equation for the only velocity component v is

OV ON = \ 7 op
ot "br ~ p br

Since v(r) = R°R/r’, it follows that at r=R

623

Dagan and Tulin

This vanishes never and according to the foregoing criteria the
motion is stable.

However Plesset and Mitchell [ 1956] proved that in fact this
spherically symmetric implosion is unstable. This demonstrates
that for interfaces which are appreciably curved like in the author's
case, the criterion for marginal stability is not always that the
normal pressure gradient vanishes.

REFERENCES

Plesset, M. S. and Mitchell, T. P., "On the Stability of the Spheri-
cal Shape of a Vapor Cavity in a Liquid," Quart. of App.
Math., 13, 4, 1956.

DISCUSSION

Prof. Hajime Maruo
Yokohama Nattonal Untverstty
Yokohama, Japan

I congratulate the success of Dr. Dagan's beautiful analysis
of the wave-breaking resistance discovered by Baba. As in the
discussion at the 12th ITTC, this resistance component has been
regarded so far, as a portion of the wave resistance. However the
present analysis indicates that it is not the case. The wave-breaking
resistance seems to have a nature akin to that of the spray resistance.
Which is the better classification, the author considers, whether it
belongs to the wave resistance or to the spray resistance?

624

Bow Waves and other Non-Linear Ship Wave Problems

REPLY TO DISCUSSION

Gedeon Dagan
Technton-Israel Institute of Technology
Hatfa, Israel

The authors have indeed applied Taylor's instability criterion
to a flow field which is different from that considered originally by
Taylor: The free-surface is curved, the basic flow is not uniform
and the instability is local.

For a plane free-surface Taylor [1950] has shown that any
disturbance, of arbitrary wave length, is unstable for &p/dy < 0;
moreover, the rate of growth of the disturbance amplitude becomes
large for small wave lengths. It is reasonable, therefore, to assume
that for wave lengths which are much smaller than the radius of the
curvature of the free-surface, Taylor's criterion is locally valid.

The radius of curvature of the free-surface (r') may be
quite large. At instability V'*/gr'21, i.e. r'/T' 2 V'2/gT'. For
the marginal Froude number characterizing instability, Fr, = 1.5,
it was found that V'/V,, = 0.7. Hence, inthis case, r'/T' =
(vi*/vi7) Free 1. For adraft T' of order of a few meters, dis-
turbances of wave lengths much smaller than r' are physically
conceivable.

Although the example suggested by Prof. van Wijngaarden,
of a collapsing spherical bubble, shows undoubtedly that Taylor's
criterion cannot be applied indiscriminately, its resemblance with
the case of a steady free-surface is questionable. The inspection of
Plesset and Mitchell [1956] article reveals that only for R/R,— 0
(R being the actual radius of the bubble and Ro its initial radius)
the disturbances are unstable, although the pressure gradient is
positive. The two cases are quite different.

We agree in principle with Prof. van Wijngaarden's criticism
concerning the need for a more rigorous treatment of the local
stability of a steady curved free-surface. Unfortunately, the basic
flow itself has been determined only approximately and further
refinements of the stability criterion do not seem justified at this
stage. The aim of the computations presented in the paper was
limited to offering a model of the free-surface breaking and the order
of magnitude of the marginal Froude number.

625

Dagan and Tulin

REPLY TO DISCUSSION

M.-P. Tutin

Hydronauttes, Incorporated
Laurel, Maryland

The wave-breaking resistance is undoubtedly similar to the
spray resistance rather than to the wave drag. It is a local phenome-
non related to the shape of the body. The energy is not radiated,
but dissipated locally. On this ground, the theoretical analysis pre-
sented in the paper is based on a model of a semi-infinite body,
with neglect of the downstream conditions.

626

SHALLOW WAVE PROBLEMS IN SHIP
HYDRODYNAMICS

E. O. Tuck and P. J. Taylor
Untversity of Adelaide
Adeltatde, South Australia

ABSTRACT

In this paper we discuss two basic problems in shallow
water ship hydrodynamics, namely the squat problem
and the problem of wave force due to beam seas, Squat
is an important phenomenon in very shallow water
because of the danger of scraping bottom. Apart from
reviewing and extending existing work on sinkage and
trim in canals and in a wide expanse of shallow water,
we indicate here how the shallow water results can
be obtained from a finite depth theory as the depth
becomes small. Unsteady problems associated with
motions or of forces on a ship in beam seas are alsc of
practical importance, as for a ship standing at an ex-
posed mooring facility. We provide here sample com-
putations of the side force on a tanker hull in regular
beam seas.

I, INTRODUCTION

It must be stated at the outset that this paper is concerned
principally with shallow water, and not with finite depth of water,
except in Section 4. The distinction between these two cases is
important from the theoretical point of view and needs to be em-
phasized here since those more accustomed to dealing with water
of effectively infinite depth tend to refer to "shallow water" when-
ever the effect of the bottom is considered.

We take the expression "finite depth" to classify a range of
water depths in which there are significant changes to the flow prob-
lems as compared with infinite depth, and significant but not neces-
sarily dominant effects of the water bottom on the behavior of the
ship. On the other hand, the "shallow water" regime is one in which
the depth is so small that the narrowing of the field of flow has a

627

fuck and Taytor

dominant effect on the ship hydrodynamics.

A degree of quantitativeness may be attached to this concept
of shallowness in a number of ways. Physically and most significantly
we require the depth to be so small that the hydrodynamic part of the
pressure distribution in the field of flow not too close to a disturbing
influence such as a ship is to a good first approximation independent
of the vertical coordinate z. Thus the only pressure variation with
z is hydrostatic.

If there are waves present, ambient or made by the ship, it
is consistent with the above that their wavelength be much greater
than (say 5-10 times) the water depth, and hence the resulting
shallow-water theory is sometimes called a "long wave theory."
However, this terminology can be misleading, since most of the
results obtained also apply when free surface effects are negligible,
as for a ship moving at very low Froude number, when the wave-
length is vanishingly small. An additional requirement for significant
effects of shallowness in the ship hydrodynamics context is that the
draft of the ship be comparable with the water depth, so that the
water bottom does have a profound effect on the ship.

In this paper we are concerned principally with two major
problems, which are apparently quite distinct from each other, but
in which similar phenomena appear. The first problem is the classi-
cal squat problem, associated with steady forward movement of the
ship in calm water, while the second is an unsteady problem associ-
ated with the response of, or forces on, a ship without forward motion
in beam seas.

Squat is the change in draft and trim of the ship as a result of
hydrodynamic pressure variations over its hull. Acceleration of
fluid particles as they pass the middle sections of the ship tends to
produce a diminution of pressure there, and hence a downward force.
There is also an upward force near the bow and stern stagnation
points, but the effect of these forces is small since there is a much
smaller area over which the positive hydrodynamic pressure can act.
In any case, the developing boundary layer and ultimate flow separa-
tion tends to eliminate the upward force at the stern.

Thus we expect a net downward force, which leads to a down-
ward displacement, an increase in draft, or sinkage. At the same
time we might expect a much less significant angular trim effect, in
which the presence of the small upward force at the bow and the lack
of such a force at the stern may give a bow-uptrim. These conclu-
sions are confirmed qualitatively at reasonably low speeds.

In fact, of course, being a simple Bernoulli effect, squat is
present in any depth of water. An interesting and immediate conclu-
sion we may draw is that, like all Bernoulli effects, squat depends
predominantly on the square of the ship's speed. This seems not to

628

Shallow Water Problems in Shtp Hydrodynamics

have been noticed by pilots and ship's captains, who are generally
aware of the problem of squat as a danger to the vessels under their
control, but are prepared to adopt a linear rule-of-thumb such as
"a foot for every 5 knots"!

In shallow water, squat is obviously a most important prob-
lem, if only because it may cause a ship to actually scrape the bottom.
But not only is this danger present, but the magnitude of the sinkage
is actually increased by the proximity of the water bottom. This is
clear physically, since the effective channel of flow is constricted by
the additional boundary, leading to even greater acceleration of fluid
particles across the middle sections of the ship.

Another interesting additional phenomenon in shallow water is
a result cf a close analogy between long-wave theory and linear aero-
dynamics, in which the Froude number F = U/Vgh based on water
depth h plays the role of the Mach number. Thus we expect extra-
ordinary effects in the neighborhood of F = 1, c.f. the "sound
barrier," and indeed both theory and experiment confirm that this
critical speed is of crucial importance. Theory, at least in its
linearized form, predicts infinite sinkage at F = 1, while in both
experiments and in observations at sea we obtain a dramatic increase
in draft associated with the generation of a type of permanent wave
or bore accompanying the ship near F=1.

The squat problem is discussed in Sections 2- 4, each section
presenting a different aspect of the problem. In Section 2 there isa
lateral as well as horizontal constriction to the field of flow, anda
hydraulic-type theory applies, while with the removal of the lateral
boundaries in Section 3 the true shallow-water theory can be used.

In Section 4, for the only time in this paper we use a finite-depth
approach and present calculations of sinkage and trim which are
close to the shallow-water results when the water depth/ship length
ratio is reasonably small.

Problems of ship motions in shallow water, i.e. of flows of
an unsteady nature, are of special interest again because of the
dangers inherent in large motions when there is very little water
beneath the keel. However, even if this danger of grounding were
not present, one might be wary of using present theories of ship
motions for cases when the water depth is known to be significant,
since theories such as strip theory deal (successfully) with infinite
water depth only.

For definiteness we concentrate here on a particular mode of
fluid flow, that involved in pure sideways (sway) motion of the ship
or of force on it. This mode is of interest for a number of reasons,
some practical, some theoretical. From the practical point of view
we expect this mode of motion or force to be of great significance
when a ship is berthed or positioned in such a way that the dominant
seas are from abeam, or when it is being manoeuvred sideways by

629

Tuck and Taylor

tugs or bow thrusters.

From the theoretical point of view, this mode is interesting
in that it provides a transition between the case of a fully-grounded
ship where the clearance is zero and the whole flow must pass around
the ends of the ship, and the case when the depth is sufficiently large
compared with the draft of the ship to allow nearly all the flow to pass
beneath the keel. At draft/depth ratios intermediate between zero
and unity (but close to unity) the ship acts like a porous wall, some
fluid particles passing "through" (i.e. under) it, while others are
diverted toward the bow or the stern,

Just as the steady problem has an aerodynamic analogy, so
the unsteady problem is shown in Section 5 to be analogous to an
acoustic scattering problem, with the ship playing the role of a partly
permeable acoustic barrier of negligible thickness (assuming the ship
is thin). Some results may be obtained directly from the acoustic
literature for such ribbon-like barriers, but new computations are
needed for the general porous case.

In Section 6 we describe techniques for obtaining the effective
acoustic porosity of the ship, i.e. the extent to which each section of
the ship blocks (or rather, fails to block) the flow of fluid particles
beneath it. This porosity is then used in Section 7 to obtain sway
exciting forces on a Series 60 ship at zero speed.

II ONE DIMENSIONAL THEORIES OF SQUAT IN SHALLOW,
NARROW CANALS

Perhaps the most easily treated shallow-water problem in-
volving ships is that for ships moving in a waterway so restricted
in both width and depth that the problem may be treated as if one-
dimensional, The effect of the ship is then little more than that of
an obstruction in an (open) pipe.

This approach clearly has very important applications to
canals and river traffic, and it is not surprising that a number of
similar analyses have been made in response to actual squat prob-
lems arising in the use of such restricted waterways. For instance
Garthune et al. [ 1948] and Moody [ 1964] , following the method of
Lemmerman [| 1942], derive a squat formula for use in the Panama
canal, Constantine [ 1961], following Kreitner [ 1934] , was concerned
with the Manchester ship canal, Sjostrom [1965] with the Suez canal,
Tothill [1967] with the St. Lawrence seaway and Sharpe and Fenton
[1968] with the Yarra river, Australia. No doubt every important
shallow and narrow waterway has had its independent squat investi-
gation.

The theoretical development is in the main quite elementary,
once we accept the one-dimensional hypothesis, which can itself be

630

Shallow Water Problems tn Shtp Hydrodynamics

justified either by careful asymptotic analysis or by physical reason-
ing. Suppose the canal has cross-section area A(x,Z(x)), when the
water surface at station x is defined by z= Z(x), z being a co-
ordinate measured vertically upward from the equilibrium water sur-
face. If, similarly, the ship has section area S(x,Z(x)) at station x,
and the water has only an x-component of velocity u(x), then continuity
requires u(A-S) = constant, while Bernoulli's equation applied at the
free surface gives Paves gZ = constant.

If we take Z=0 far upstream, where u= U, S=0 and
A = Aj, we have

u(A - S) = UA, (2.4)
and
au + gZ=3U, (222)

In this formulation the ship is fixed in position and the fluid streams
past it in the x-direction. Elimination of u gives

U2 - 2gZ (A(x,Z) - S(x,Z)) = UA (2a3)

0?

a transcendental equation from which the water surface elevation Z(x)
at station x may be determined in principle, for a canal of arbitrary
section (not necessarily uniform or vertical sided), and a ship which
occupies any proportion of the available canal area at any station.

In this most general case we should then return to the Bernoulli
equation to obtain the pressure on the hull (which turns out to be hydro-
static) and integrate to obtain the force and moment onthe ship. In
principle the net vertical force would exactly balance the ship's weight
and the trim moment would be zero, if we had started with the ship in
its correct squatting position. In practice we should have to devise
some kind of iteration procedure to move from an initial guess to the
correct position. Such a general and exact study seems not to have
been carried out, although it would be of some considerable interest.

Most investigators avoid this problem by treating an idealized
ship which is a straight-sided cylinder, and ignoring end effects. In
that case Z is constant over the length of the ship, and it follows
that the ship simply rides up (Z > 0) with the water, maintaining
constant displacement. If at the same time we restrict attention to
the case when the canal is constant in section area, and in the region
of interest at the free surface has a width W independent of Z
(locally vertical sides), then we have from (2.3) that

(U* - 2gZ (A, t WZ - S) = UA, (2. 4)

6314

Tuck and Taylor

where S is now constant. On squaring, (2.4) gives a cubic equation
which may be solved directly for Z. Alternatively, following
Constantine [ 1961], we may treat the problem in an inverse manner,
solving for the speed as a function of Z and obtaining in non-dimen-
sional form

|e

me [calcd 3 (2.5)

{= ("d=
where

F = U/Vgh (2.6)

d= "2h (257)
and

s=S/A,, (2.8)
with

h= A,/W (2.9)

as the mean depth.

Constantine [| 1961] discusses the nature of the flow predicted
by (2.5) and presents curves of F against d. Equation (2.5) permits
only a restricted range of Froude numbers F for any given blockage
coefficient s, namely

O<F<F(s) and F,(s)<F<oo, (2.10)

where F, (s),..F.(s)).are. critical Froude numbers* shown in Fig. 1.
No steady flow is possible in the "trans-critical" region F, < F< F
and Constantine [1961] discusses how an unsteady bore fonea ches,
of the ship if it attempts to exceed F,. Notice from Fig. 1 that the
trans-critical regime becomes narrow if s is small, and as the
blockage tends to zero there remains a single critical Froude number
F,=F,= 1.

The last result is relevant to an alternative linearized ap-
proach to solution of (2.3) not utilized by the previously referenced
investigators, but described in a somewhat different context by Tuck
[1967]. Instead of specializing the shape of the ship, one now makes
the approximation that its section area S is everywhere small com-
pared with the canal section area A. If we again take for definite-
ness the case of a canal whose undisturbed section area A is inde-
pendent of x and equal to A, the water elevation Z will likewise

rl w

a
Roots of the equation s = 1 -

632

Shallow Water Problems tn Shtp Hydrodynamics

SUPER-CRITICAL

TRANS-CRITICAL

SUB-CRITICAL

0 0.2 0.4 0.6 0.8 1.0 1.2

s = SHIP SECTION AREA/CANAL SECTION AREA

Fig. 1 Critical zones for squat in a canal

be small relative to the mean depth of water, and we can replace (2. 3)
by its Taylor series expansion with respect to Z, writing

(U - 82 Pose ACs O)) Al (x 0) Zt" nr, tas (yO) = 5 ae) = CALA ft)

On collecting terms of leading order in (2.11) and setting

A, = A(x, 0) (ni)
S(x) = S(x, 0) (2.13)
W(x) = A,(x,0), (2.14)
we have
- 82 a +UWZ - US=0,
ise.

633

Tuck and Taylor

Fe
d=s77 (2°15)

where F,d,s,h are again defined by (2.6) - (2.9) respectively,
although all these quantities may now in principle vary with station
coordinate x. However, if they do not, (2.15) can easily be shown
to be the result of direct approximation of (2.5) for small s.

The most interesting feature of (2.15) is of course the singu-
larity at the critical Froude number F = 1, which is to be expected
from the fact that the former transcritical region F, < F< F, has
shrunk down to an isolated "forbidden" Froude number at F=1. We
shall make use of linearized results like (2.15) throughout the re-
mainder of this paper; however, it is well to bear in mind in each
case that we may expect singularities at the critical Froude number
and that, should these be of concern, they may be explained, studied
or removed by non-linear considerations similar to those of the
present section.

Ill TWO-DIMENSIONAL THEORY OF SQUAT IN WIDE, SHALLOW
WATER

The theories of the previous section are useful only in widths
of water comparable with the beam of the ship. Since the important
blockage parameter is the ratio of the maximum ship cross-section
area to the cross-section area of the channel, naive use of these
theories for very wide channels leads to the conclusion that the squat
effect tends to zero for a given ship as the channel width tends to
infinity. But of course the basic Bernoulli effect must still be present,
even in an infinite expanse of water, so that there will still be squat,
and indeed substantial squat in this case.

Analysis of shallow-water flow past ship-like bodies in in-
finitely-wide water was first attempted by Michell [ 1898] in his
famous wave-resistance paper. The relatively greater importance
of Michell's infinite depth formula, the derivation of which consti-
tutes the first part of his paper, has perhaps led to little interest
being taken in the second part of the paper, where he treats a shallow
water problem. This is unfortunate, since Michell's approach is
what we might now call an aerodynamic analogy, even though his
paper ante-dates aerodynamics!

The problem treated by Michell concerns steady flow at speed

U inthe x-direction past an obstacle of thin cylindrical form, with
equation

yor sped, x) <2; (3.4)

634

Shallow Water Problems in Shtp Hydrodynamics

extending from the bottom z=-h tothe top z=0 ofthe water. It
is apparent right from the outset that with this model of a ship we can
expect no prediction of squat, for the "ship" has vertical sides every-
where, and no fluid passes under it. Michell's only concern was with
wave resistance,

The mathematical problem is specified by a disturbance veloc-
ity potential such that the fluid velocity is V(Ux + 4), satisfying
Laplace's equation a

) a>, 9d _
Be ot pee -~ h <2 < 0, (3:52)

O19 2 Oulart ae, (3,3)

the linearized free surface condition

2
C) 290
Bye TU ge H Oo on =O, pair

and the linearized hull boundary condition

3 +2 AUb Gd eeon. 'y=80,. (3.5)

Both equations (3.4), (3.5) are linearized on the basis that the ship
is thin, i.e. that its slope b'(x) is everywhere small, so that $ and
its derivatives are small, as is the free surface elevation.

We now apply the assumption that the depth h is small. The
corresponding approximate equations may be obtained formally by
stretching the z-coordinate with respect to h, then carrying out an
asymptotic expansion in terms of the small parameter h/L, see
Wehausen and Laitone [1960]. However, the leading terms are
easily obtained by simply expanding @$ ina Taylor series with re-
spect to z, about the bottom value z= -h, i.e.

(x,y, Z) = (x,y, -h) a (z th) 6,(x,y ,-h)

a (2th) ,.(x,y 5h) Trewin 6 (35 6)

The second term in the expansion (3.6) vanishes by (3.3),
and we use (3.1) to express $,, in terms of $,, and $y, , writing

635

Tuck and Taylor

(x,y,z) = o(x,y,-h) - 3(z th V7b(x,y,-h) +... (3.7)

2
where V = (a°/ax°) + (a /ay*). On substitution in (3.4) we obtain
immediately to leading order in h the equation

- ghV°o(x,y,-h) + U2, (x,y,-h) = 0,

or

2 ra

(1 - F) Fa + Fa] olx,y,-n) = 0, (3.8)

where F = U/ygh.

Equation (3.8) is formally identical to the equation describing
linearized aerodynamics in a two-dimensional flow of a compressible
fluid, with the Froude number F playing the role of the Mach number
(see e.g. Sedov [1965]). Indeed, the problem of solving (3.8) subject
to (3.5) is identical to that for subsonic (F < 1) or supersonic (F > 1)
flow over a non-lifting wing of thickness b(x), and we may use
directly the results obtained in aerodynamics. Of course Michell was
not so fortunate, and we should say that aerodynamicists could have
used Michell's results, the first solution of any boundary-value prob-
lem for a non-trivial general boundary.

The character of Eq. (3.8) is different according as F< 1
when it is elliptic and F >1 when it is hyperbolic, and different
mathematical properties and solution techniques apply in these two
cases. Here we quote only the final result for the hydrodynamic part
of the pressure distribution over the body surface, namely

- pu? (C” v\(é) aé
2mj1- F* 4-0 x= 6

p= (3.9)

itt <4

2
eee on Ei ce ty
2m F - 1

the bar denoting a Cauchy principal value. Note that the pressure
given. by (3.9) is a function of x only. The z-dependence has been
neglected as part of the shallow-water approximation and there is no
y-dependence because of the thin-ship approximation. The complete
pressure distribution is obtained by adding to (3.9) the hydrostatic
pressure.

The only possible force on this cylindrical body is in the x-
direction, and there is no net moment. Michell found by integration

636

Shallow Water Problems in Shtp Hydrodynamics

of p times the slope b'(x) that the net force (wave resistance) is

Oy “Pi<iais
ie (3. 10)

n
Sl Leica} dx, F>t.

Palate

No doubt Michell was disappointed in his conclusion of zero
wave resistance in the more important sub-critical regime, and
indeed this conclusion may have contributed to the neglect of his
shallow-water results. However, we can expect no other result
from the present theory, which lacks a dissipation mechanism in the
sub-critical regime to leading order. This feature it has in common
with linearized aerodynamics. However, in aerodynamics the drag
vanishes even according to nonlinear theory for Mach numbers every-
where less than unity, whereas in the present water-wave problem
it is only to leading order that the wave-making dissipation mecha-
nism disappears. No second-order calculations seem to have been
carried out to find the non-zero subcritical wave-resistance, and
this is a problem which merits attention.

Michell's analysis for a wall-sided "ship" was extended to
ships of arbitrary cross-section by Tuck [ 1966]. In this case we
can expect to predict a squat effect, and, although the analysis in the
1966 paper is rather complicated, the main conclusion is quite
simple. By the method of matched asymptotic expansions (Van Dyke,
[1964]), Tuck showed essentially that Michell's result (3.9) for the
pressure still holds, providing we interpret the function b(x) as the
mean thickness of the ship at station x, averaged over the full depth
of the water, i.e. set

“bis =< s(x) (3.11)

where S(x) is, as in Section 2, the cross-sectional area of the ship
at station x. Thus, for example, we obtain again Michell's wave
resistance formula (3.10) but with (3.11) used to rewrite it in terms
OE. FO)

On the other hand, the modified geometry of the ship does now

allow non-zero vertical-plane forces and moments, and we find an
upward heaving force

637

Tuck and Taylor

v2 £ £
ue) ax ( dEB'(xjS'(E)log |x-£|, F241 (eee
2thy 1 - £ -f

4
S'(x) BGe)-dx3, F > 1 (32:43)
-f

ee ip
2h/ F*- 4
and a bow-up pitching moment

ig £ L
- —— ax f dé (xB(x)) *S'(S) log |x-€ | ’ F< 1,
jal 2th: 1i-F -f -f (3.14)

S'(x)xB(x) dx, F> 1, (3515)

Suc
2hyF -1 -2

where B(x) is the width of the ship at the waterline at station x. In
fact the force written down in (3.12) is invariably negative at sub-
critical speeds so that a sinkage is to be expected rather than a lift.

Tuck [ 1966] also gives formulae for the actual sinkage and
trim displacements of the ship in response to these forces, assuming
equilibrium with hydrostatic restoring forces, and provides some
computed results which are in reasonable quantitative and excellent
qualitative agreement with experiments of Graff et al. [1964]. There
is a need for more experiments, especially in the very low water
depth range, but it would appear from the comparisons so far made
that the theory is quantitatively accurate so long as the depth is less
than about one eighth of a ship length, and the Froude number based
on depth is less than about 0.7.

It may be worth observing here that the integrals in (3.12) -
(3.15) are fairly insensitive to the shape of the section curves
B(x), S(x). For instance, the ratio

° fi ax fi a6 B'(x)S'(§) log |x-6|

ve n n
iy B(x) dx = hi S(x) dx

(3.16)

is nearly an absolute dimensionless constant, taking values between
2.0 and 2.4 over a very wide range of B(x), S(x) curve shapes,
including actual ships and mathematically defined curves. Thus a
nearly universal approximation to the subcritical vertical force is

638

Shallow Water Problems tin Shtp Hydrodynamtcs

ou? Q Q
ee rf B(x) ax f S(x) dx (3.17)
: 2mhVi - Fs? J -¢ -t

with a fixed value of \. From this follows a similar approximation to
the actual sinkage, say a displacement of 6 downwards, where

re J) Sx) dx
aa poe ee (3216)

6=2-
Ti or

Finally, introducing the displaced volume

£
a { S(x) dx (3.19)

and making a further assumption (justified in most practical situations)
that F << 1, we have

z
nae Wy
a eh ie (3. 20)

where L = 2£ is the ship length.

In practical terms, if 6, L and h are infeet, V _ in cubic
feet and U in knots, and if we insert reasonable (conservative)
values for X and g, (3.20) implies

6=0.13572- (3321)

We put forward this formula (3.21) quite seriously for practical use
by anyone interested in a quick estimate of squat in a wide expanse of
shallow water. One should note the quadratic dependence on forward
speed, the inverse dependence on water depth, the proportionality to
displacement (at fixed length) and inverse square dependence on length
(at fixed displacement).

In a subsequent paper, Tuck [1967] extended the 1966 work to
the case where the ship is moving along the center of a rectangular
channel of width w, considering only the sub-critical case. The
assumption made was that w is comparable with the ship length L;
however the results obtained were uniformly valid, in the sense that
the infinite- width results were reproduced as w/L — o, while as
w/L— 0 we obtain predictions which may also be obtained by ele-
mentary (linear) one-dimensional theory as in Section 2. An inter-

639

Tuck and Taylor

esting mathematical feature of this small width limit is that the singu-
larity at F = 1 becomes stronger as w/L— 0, changing from in-
verse square root (e.g. (3.12)) to inverse first power (e.g. (2.15)).

Another conclusion in the 1967 paper was that the ratio between
the sinkage at width w and that at infinite width was almost independ-
ent of the shape, size or speed of the ship, depending only on the
parameter (w/L)yi-F*. Thus, starting with any estimate (even
(3.21) !) of the infinite width sinkage, we may further estimate the
effect of finite width by use of the universal curve given in the 1967
paper. For example, at low values of F, a channel width of two
ship lengths increases the sinkage by 10%, one ship length by 33%,
over the infinite width values. For channel widths less than one ship
length a one-dimensional theory as in Section 1 is sufficiently accurate
and probably to be preferred.

IV. THREE-DIMENSIONAL THEORY OF SQUAT IN INFINITE
WIDTH, FINITE DEPTH

We begin the present section by presenting the solution!
Suppose S(x) is the cross-section area curve of a slender ship
moving at velocity U in water of finite constant depth h, and let

4
S*(k) = \ S(x) e!®* ax, (4.1)
-£
Then consider
; 00 , : 00 -idy
Bea oe dk kS"(k) cin ( d=
4 q
-00 - 00

: [e* 4 eo cosh QZ k* cosh q(z th) ] (4, 2)
sinh qh(k® - Kq tanh qh) :

sinh gh

i
where K=g/U* and q= (k* +22,

Although the expression (4.2) is extremely complicated, it
has the following properties, easily checked:

1

Gd 0. tk eer = Ayo Jace, (4,3)
Gay OG) Oz = Oh om tz =. in, (4. 4)
(iii) °k'(8d/dz) +(0'6/dx = 0 “on z=0, y #0, (4.5)

640

Shallow Water Problems tn Shtp Hydrodynamics

(iv) o-4 S (x) dog 2's f(x), P O(e tos +) as’ r= 0. (4. 6)

The physical interpretation of $ is as follows. The contribution
from the first term "e%" inside the square brackets is just the
potential of a line distribution of sources, of strength proportional
to S'(x), in a fluid which extends to infinity in all directions. The
contribution from the second term inside the square brackets cor-
rects for the presence of a bottom wall at z= - h, while the last
term in the square brackets corrects for the presence of a free
surface at z=0.

The last property (4.6) indicates that the given solution (4. 2)
can serve as an outer approximation (see Tuck [ 1964]) and will
match an inner approximation which satisfies the correct boundary
condition on a slender hull surface. Thus (4.2) gives the disturbance
potential for flow around a slender ship in finite depth of water, no
shallowness assumptions having been made.

The function f(x) in (4.6) is of crucial importance, and may

clearly be considered in three parts, arising from the three terms
in the square brackets in (4.2). Let us write

f(x) = fop(x) + g(x) (4.7)

where

L
US'(x) Aa? U S'(x) - S'(E)
ns -

Pa) = log 4(z x ) a i: dé aaa (4. 8)
(Tuck and von Kerczek [ 1968]) is the corresponding function for the
double-body flow in an infinite fluid (no bottom or free surface),
while g(x) is the contribution from the second two terms in the
square bracket of (4.2), and takes the value

. 00 ae
g(x) = ae ak ks Hic) tK* A* (1) (4.9)

where

eee ee (4.10)

k“ - Kq tanh qh

In the integral (4.10), if Kh< 1 there is a pole on the real q-axis
at q = 4q,(k), where

641

Tuck and Taylor

2
k = Kq, tanh qoh (4.11)

and this pole must be avoided by passing beneath it in order that the
waves are behind the ship.

Thus the real part of A*(k) may be written as a Cauchy
principal value integral, which can be evaluated by standard numeri-
(

cal quadratures,whereas the imaginary part of A‘“(k) can be obtained
from the residue at the pole, and we have

wi do So , xh <1

JA “(k) = (4.12)

0. Rae's s

Once A* is determined, g(x) follows by further numerical quad-
ratures from (4.9), if actual numerical values of g(x) are required.

However, our main aim is to find the forces on the ship,
which follow from g(x) via the pressure distribution, given by

P(x,y»Z) = Polx,y,Z) - pUg'(x) (4.13)

where Pog (Xs ¥ > Z) is the pressure on a double body in an infinite
fluid. Hence the vertical force is

£
F, = F, - oul , dx B(x)g'(x)
20° ——
= FS - eu dk k?S"(k) B*(k) A*(k), (4.14)
-00

and the trim moment is

L
F =F +pul dx xB(x) g'(x)

2 eee
Fe +h, \ dk k?S*(k) xB*(k) A*(K), (4.15)

-00

where F%, Fo are the corresponding quantities for the "submerged"
half of the infinite fluid double body, B(x) is again the waterplane

642

Shallow Water Problems in Shtp Hydrodynamtes

width curve, B*(k), xB “(k) are Fourier transforms (cf. (4.1)) of
B(x), xB(x) respectively, and a bar denotes a complex conjugate.

The quantities Ber He must be computed separately, e.g.
by computer programs such as those of Hess and Smith [ 1964] or
Tuck and von Kerczek [1968]. Alternatively, one may estimate them
experimentally. It is important to note that F%, F® are independent
of water depth and of Froude number; indeed, when divided by euU-
these are constants which are a property of the hull geometry alone.

An interesting special case isa hip with fore-and-aft sym-
metry, where (neglecting viscosity) FS = 0. In addition, since
S(-x) = S(x) and -xB(-x) = -xB(x), s* is real and even with
respect to k whereas xB” is imaginary and odd. Asa result,
only the imaginary part (4.12) of A*(k) contributes to the integral
in (4.15), and we have

ish Ga
Bie ae \ dk k°S*9(xB*)9A* (4.16)
OF = (Kh) = <1

7 (4.17)

(ee)
ae dapack S*(K)IxB"(k), F> 1.
27r 2 2

(@) do = k

Similar, but more complicated, results are obtained for F on he

and ¥F.- es when the ship does not possess fore-and-aft symmetry.
Evaluation of the supercritical trim moment F,; from (4.17) re-
quires only a single numerical quadrature (apart from the prior
estimation of the Fourier transforms S*, xB*). However, in the
general case an additional numerical quadrature is needed to deter-
mine the real part of A *(k) from (4.10).

The shallow-water limit of the above finite-depth results
corresponds to letting kh ~ 0, i.e. we let the depth tend to zero
relative to a typical effective wavelength 2n/k. In particular, from
(4.11) we have qgh > 0 as kh—O and hence k—~ykKkhq, or
do—~ Fk. Thus for F >1, (4.17) gives for ships with fore-and-aft
symmetry that

poe Pe ES *k *k
= dk kS* 9xB" (4.18)
2 Q
on, eee S'(x)xB(x) dx, (4.19)

643

Tuck and Taylor

in agreement with (3.15). It is quite straightforward to show ina
similar manner that all of the shallow-water results of Section 3 are
reproduced in the corresponding limit, even when fore-and-aft
symmetry is not assumed.

In carrying out this shallow-water limit, one may wonder
what happens to the "double-body" terms F® and F2. The answer
is that they are of course quite independent of water depth and hence
nothing happens to them, and in principle they remain in the formulae,
However, when the depth is small the shallow-water terms formally
dominate the total expression for F, or F,, so that Bg and ee
may be neglected.

In Fig. 2 we present computed finite-depth sub-critical sink-
age and super-critical trim for a ship with parabolic waterline and
section-area curves, a length of 600 ft, beam of 60 ft, draft of 20 ft,
and block coefficient 0.533. This geometry and size was chosen for
analytical convenience, but is not unlike a destroyer hull. The super-
critical trim was calculated directly from (4.17) by a single numeri-
cal quadrature (since this hull has fore-and-aft symmetry), whereas
the sub-critical sinkage required an extra numerical integration of
(4.10), and, furthermore, required separate estimation of the infinite-
fluid contribution td in (4.14).

DEPTH = 30 FT.

SINKAGE (FT.)
TRIM (DEGREES)

A

O
oo SHALLOW

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

FROUDE NUMBER BASED ON DEPTH

Fig. 2 Finite depth squat for a ship 600 ft long and 20 ft draft

644

Shallow Water Problems tn Ship Hydrodynamics

Instead of using sophisticated numerical techniques for F®,
in the present case we estimated ro by assuming that the under-
water hull could be approximated by an equivalent spheroid, with the
same length and displacement. If we define the slenderness e€ by

{2 V
€ = (ee. (4. 20)

which is equal to the beam/length ratio of the equivalent spheroid,
the exact infinite fluid force on the submerged half of the spheroid
can be obtained from the formula given by Havelock [ 1939] in the
form

h
FO: -pu'- ( B(x) dx + C_(e) (4, 21)

where

-2

2 mee
S log == ) ‘ (4. 22)

2y 1-2 {=./4-«2

In fact, since € is generally small, an adequate slender -body approxi-
mation to (4.22) is

bole 2
C.(€) => tap tt - 3€ ee) &

2 jo! es
Cele) = - € “(log 5€ +5) + 3 + O(e4log €). (4. 23)

The result (4.22) is of course in exact agreement with
Havelock's [| 1939] more general formula for ellipsoids whose sections
are not circular. On the other hand (4.23) is within 10% of computa-
tions based on Havelock's formula for general ellipsoids, providing
€ < 0.2 and the half-beam/draft ratio of the general ellipsoid lies
between 0.65 and 7.0, a range of parameters which includes the
usual ship dimensions. Some preliminary numerical computations
using the theory of Tuck and Von Kerczek [ 1968] have shown that
(4.23) is a good estimate for non-ellipsoidal geometries, while
Havelock [ 1939] himself made satisfactory comparisons between
his ellipsoid estimates and experiments of Horn [ 1937] on actual
ship models, so that there are grounds for believing that (4. 21)
subject to (4.23) gives a useful prediction of the infinite fluid zero
Froude number sinkage force.

The finite depth computations were carried out for water
depths of 100, 60 and 30 feet. The results for the smallest of these
depths are in very close agreement with the shallow-water theory
of Section 3, shown dashed on Fig. 2, over the complete range of
Froude numbers shown. This indicates that a water depth/ship length

645

Tuck and Taylor

ratio of 1 in 20 is quite adequately shallow for the use of shallow-
water theory. But the results at twice this depth are also in reason-
ably good agreement with shallow-water theory, the latter theory
underestimating sinkage by about 20%. The corresponding under-
estimate at 100 ft depth is about 40%, so that one should consider a
water-depth/ship length ratio of 1 in 6 as too great to use shallow-
water theory for sinkage.

However, it must be pointed out that the difference between
the finite depth and shallow water predictions of sinkage is pre-
dominantly due to the influence of the term F® in (4.14). If this
(positive) term is left out of (4.14) the finite depth computations at
all depths merely oscillate about a mean which is quite close to the
shallow-water curve. These oscillations are clearly visible in
Fig. 2; they are quite similar to the humps and hollows in theoretical
wave resistance curves, and have the same explanation, as an inter-
ference effect. One may thus speculate that, by ignoring these oscil-
lations, we may obtain a useful empirical scheme for computing
finite depth sinkage by adding the shallow water estimate (e.g. (3.21))
to the infinite fluid zero Froude number estimate computed by (4. 23).
Further work needs to be done to test this suggestion, which is of
some significance since computations based on the theory of the pres-
ent section are too complicated and expensive of computer time for
general use.

No direct comparisons of the finite depth computations with
experiment have yet been made, but the differences between the finite-
depth and shallow-water results appear to be in the right direction
to explain most of the discrepancies already noted by Tuck [ 1966]
between shallow-water theory and the experiments of Graff et al.
[1964]. In particular, more detailed computations for a depth of
100 ft, (h/L = 0.167) show that the peaks in the sinkage and trim
curves occur at about the right Froude numbers, 0.94 and 0.98
respectively, and that the trim starts to become significant at a
Froude number as low as 0.8.

V. THE ACOUSTIC ANALOGY FOR UNSTEADY LATERAL FLOW

In the remainder of this paper we shall be concerned with a
very special aspect of the problem of ship motions in shallow water,
namely computation of the exciting force on a stationary ship under
the influence of regular beam seas. A more general formulation
and partial solution of the problem of ship motions in shallow water
is given by Tuck [1970]. Most other work on ship motions in shallow
water, e.g. Freakes and Keay [1966], Kim [ 1968], concerns finite
depth rather than shallow water. An exception is a number of papers
by Wilson (e.g. [ 1959]) on responses of moored ships in harbors,
but no account is taken there of ship geometry. Mention must also
be made of the thesis by Ogilvie [1960], in which the shallow-water
asymptotic expansion was developed rigorously for a class of two-

646

Shallow Water Problems tn Shtp Hydrodynamics

dimensional diffraction problems.

We now suppose that, except for the scattering effect of the
ship, the flow field is described by an incident plane wave moving in
the y-direction, with wavelength 21/k, frequency o, and amplitude
A, where k and o are related by the shallow-water dispersion
relation

o =7ghk. (5.1)

The potential of this wave will be taken to be the real part of beiet F
where

>= A en: (5.2)
0 h ik * :

In fact, (5.1) and (5.2) are of course already approximations to the
exact formula (e.g. Wehausen and Laitone [ 1960]) for small-ampli-
tude waves in finite depth h; for instance the exact expression for
do is that given by (5.2) multiplied by cosh k(z +h) /cosh kh, which
tends to unity as kz —~ 0.

This incident wave is modified by the presence of the ship.
We suppose that the total field is then the real part of (od) + d)e'*,
where = $(x,y,z) is the disturbance potential, which is to be

found. The exact equations satisfied by » are (3.2), (3.3), an
unsteady free surface condition analogous to (3.4), namely

gse-o%}=0 on 2=0 (522)
and a boundary condition on the ship's hull of the form
Fa (bo + 4) = 0 (5.4)
ony 2 i

where 8/8n denotes differentiation normal to the hull.

We construct first an outer shallow-water approximation to
@ in the same way as in Section 3, i.e. by expanding in a Taylor
series’ with respect to (z th). Equation (3.7) still applies, but on
substitution of (3.7) in (5.3) we now find

2 2

647

Tuck and Taylor

i.e. (x,y,-h) satisfies the Helmholtz equation
Vb +k} = 0 (5.5)

in the (x,y) plane.

The Helmholtz equation is of course simply the "reduced"
wave equation, and so applies to any scalar wave problem in two
dimensions, for sinusoidal time dependence. In particular, it
describes linear acoustics in two dimensions (e.g. Morse [ 1948]),
and many results obtained in solving acoustic problems may be
utilized.

For example, we may treat immediately the scattering of a
thin cylindrical ship, as in Michell's model of Section 3, which
extends from top to bottom of the water. An important difference
from the theory of Section 3 is that, even in the limit as the thick-
ness tends to zero, the thin "ship" is capable of scattering beam
waves. Thus, to leading order, the problem is independent of
thickness, and reduces to acoustic scattering by a ribbon or strip
of zero thickness placed broadside on to the waves, with a "hard"
boundary condition

ao A [2 = constant on y = 0,, [ao {<0 (5.6)

The exact solution can be written down as a series of Mathieu
functions (Morse and Rubenstein [1938]). Results for the scattering
cross section, the far-field polar diagram and the force on the strip
can be computed from this series, but only with some difficulty,
especially for high frequency. Alternatively, integral equation for-
mulations of the problem can lead to useful high and low frequency
asymptotic solutions (Hénl, Maue and Westphal [1961] ) or even to
efficient numerical solutions (Taylor [1971]). Such numerical
results are included with the discussion of the general case in
Section 7.

Once again this idealized ship is deficient from the practical
point of view. In particular, it allows no account to be taken of flow
beneath the keel of the ship. In any situation of real interest, wave
energy is not only scattered, diffracting around the ends of the ship,
but also transmitted underneath the ship if there is any reasonable
amount of clearance. The most interesting situation is that which
applies when the amounts of disturbance scattered and transmitted
are of the same order of magnitude; we shall see later that this is
true for draft/water depth ratios in the range 0.5 to 0.95.

We shall retain the approximation that the ship is thin, and
hence slender, since it must have small draft. However, the

648

Shallow Water Problems in Ship Hydrodynamics

possibility of fluid passing underneath the ship means that we must
replace the "hard" boundary condition (5.6) by a more general con-
dition, expressing in effect a relationship between the velocity
(8¢/dy) + AVg/h of fluid passing "through" (i.e. under) the strip

y = 04, |x|< £ and the pressure difference (proportional to potential
difference) across the strip, which causes this underflow. Thus we
write

C)
a tA [R= FPS on y = 0s, acl ek, (5.7)

where P = P(x) is the "porosity" of the ship section at station x.
If the ship is actually touching the bottom, then P=0 and (5.7)
reduces to (5.6); at the other extreme, if there is substantial
clearance, P—- oo andthe jump in potential $ across the strip
tends to zero, leading as expected to zero force on the ship.

In the following section we indicate how to obtain the porosity
P(x) for any given ship and sea bottom geometrical configuration.
The problem of solving (5.5) subject to (5.7) is then identical to that
for acoustic scattering by a "semi-soft" or porous ribbon with finite
acoustic impedance, see e.g. Morse [1948]. However, no general
procedure seems to be available in the acoustic literature for solving
this type of problem, and we present in Section 7 a numerical
approach based on an integral equation formulation.

It should be remarked that as k—~ 0 the present problem
reduces to uniform steady streaming flow across the ship, the free
surface being replaced by a rigid wall. This problem was discussed
by Newman [ 1969] , who presented solutions for the added mass of
the ship in such a flow. The present theory can be considered a
generalization of Newman's theory to allow for waves, and gives
results which agree with Newman's in the limit as ki ~ 0, i.e. as
the waves become long compared with the length of the ship.

VI. THE DETERMINATION OF THE EFFECTIVE POROSITY

The problem formulated in the previous section is to be inter-
preted as an outer problem, which provides a solution for the scattered
field everywhere except within a beam or two of the center plane
y = 0 of the ship. In this latter region, the outer solution must match
an inner approximation which describes the detailed flow field beneath
the hull. This flow can easily be shown (Tuck [ 1970] , Newman [ 1969])
to be locally two-dimensional in the (x,z) plane, and to satisfy the
two-dimensional Laplace equation

S$ + SF =0 (6.1)

649

Tuck and Taylor

in that plane. Furthermore, the free surface condition reduces toa
rigid- wall boundary condition

SP = 0 or z=0. (6.2)

Thus the inner problem is identical to that treated by Newman [ 1969],
who assumed that (6.2) was valid everywhere.

The boundary condition at "infinity" for this inner solution is
that the inner solution should match the behavior of the outer solution
in a common domain of validity, say many beams away from y=0,
but not so far away that y is as large as £ or 2m/k. In effect,
this simply means that the inner boundary condition (5.7) for the outer
solution becomes the outer boundary condition for the inner solution.
Thus the inner approximation to the disturbance potential ® must
satisfy

fia [k= Ps neha aries (6.3)
which is satisfied if $ is asymptotically independent of y, i.e.
p+ Avg/h as y->+o (6.4)
implying

LL pig as yi (0. (6.5)

The boundary condition on the hull is (5.4) but where now
8/8n denotes differentiation normal to the hull cross section IT at
station x, and where, since ky is small in the inner region, we
may replace the incident wave 9, by

b—- A JE (xz ty), (6.6)

i.e. by an incident stream of speed Ayg/h. Then (5.4) becomes
Sat ogy (ey
ae A Fon on Fr. (6.7)

Thus the inner approximation to $ is the potential for flow due to

650

Shallow Water Problems in Ship Hydrodynamics

motion of the section I as if it were an infinitely long rigid cylinder
moving in the y-direction with velocity Ayg/h, the fluid at y =+ 00
being at rest.

The problem specified by (6.1) subject to (6.2), the bottom
condition (3.3), the hull condition (6.7) and the rest condition (6.5)
is a classical Neumann boundary value problem, and @ is deter-
mined by these conditions apart from an additive constant. If in
addition we prescribe the natural symmetry condition that o is an
odd function of y, in conformity with (6.4), then $ is uniquely
determined, and the actual limit of ¢ as y—~+0o must, via (6.4),
provide a determination of P.

A number of techniques are available for solving this two-di-
mensional boundary-value problem. If the section is sufficiently sim-
ple (e.g. rectangular) a solution may be found by conformal mapping
methods (Flagg and Newman [1971]). For actual and quite general
ship sections we have (Taylor [1971]) developed a computer program
based on the methods used by Frank [1967] for a similar problem.

In this method one represents the flow by a distribution of
sources around the section I, with variable but unknown density.
These sources individually satisfy the "free" surface and bottom
conditions (6.2) (3.3), but not, of course the hull boundary condition
(6.7) on IT. We now attempt to choose the source density function
in order to satisfy (6.7), thereby obtaining an integral equation from
which the source density is in principle obtainable.

Since analytic solutions for general TI are out of the
question, we adopt a numerical approach in which [I is first approxi-
mated by a set of straight line segments, on each of which we assume
the source density to be constant. The integral equation then reduces
to a set of linear algebraic equations for these unknown constant
source strengths, and this set of equations is solved by direct matrix
inversion.

It is convenient to define a blockage coefficient

Cx) = pry: (6.8)

as the inverse of the porosity. With this definition, we see from (6.4)
that to obtain C(x) we merely divide the potential by minus the
speed Ayg/h of the motion of the section I and take the limit as
y—~> too. Thus, if our aim is solely to determine C(x) or P(x),

we may, without loss of generality, take A=-vh/g for the purpose

of the present section only, and identify C as the limit of $ as

y —~ + oo, a quantity which is readily evaluated from the numerical
solution for the generating source strengths.

651

Tuck and Taylor

The program has been tested by comparison with Flagg and
Newman's [ 1971] computations for a rectangular section, and gives
good agreement over the range of dimension of interest. For instance,
with a rectangle of total width 0.25 and a (submerged) draft 0.1 in
water of depth 0.125, Flagg and Newman's [ 1971] computations give
C = 0.598, while our program with 24 segments on the bottom of the
rectangle and 12 segments on each side gives C = 0.603. Although
this accuracy (1%) is already very good in the present application,
it can easily be and has been improved by use of a larger number of
segments, especially in the neighborhood of the corners.

Another check is by means of asymptotic estimates for small
clearances (Taylor[1971]). A formula which is valid for arbitrary
sections, providing they have substantially vertical sides 2b units
apart and a substantially flat bottom c (<<h,b) units from the water
bottom, is

clacBPy (6.9)

For strictly rectangular sections, this formula may be improved by
estimation of the next term in an asymptotic expansion for small
c/h, giving

~bh,2h,,, bh 42h.
C= 2 +P loge + - b + O(c). (6. 10)

For the rectangular section used as an example above, (6.9) gives
C = 0.625 while (6.10) gives C = 0.597.

Indeed, it must be noted that rectangles are not a fair test for
the computer program, since the generating source strength becomes
infinite at the sharp corner. We should therefore expect far better
accuracy for smooth ship-like sections. For instance, the program
gives results with accuracies of better than 1% when applied to the
oval-shaped sections generated by a single isolated dipole in a channel
(Lamb [{ 1932]).

Figure 3 shows computations of C(x) /f for a Series 60,
block 0.80, tanker hull (Todd [ 1963]), with beam/draft ratio of 2.5
and length/beam ratio of 8.0, the ship length being 2£. The results
are for two depths of water only, with draft /depth ratios of 0.8 and
0.9. In neither of these cases is there a great deal of water beneath
the keel, but this is the interesting range, since it is necessary that
the clearance be relatively small to achieve significant flow blockage.
Thus at a draft/depth ratio of 0.4, the typical values of C/f£ are
already below 0.125 over the whole length of the ship, which leads
to a maximum force (see the following section) less than a quarter
of that for the full-blocked situation.

652

Shallow Water Problems in Ship Hydrodynamics

1.4

1.2

1.0

0.8
C/l DRAFT/WATER DEPTH = 0.9

0.8

0.6

0.4

0.2

0
-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0

(BOW) NORMALIZED STATION CO-ORDINATE x (STERN)

Fig. 3 Blockage coefficient C(x) for Series 60, block 0.80 ship

VII. THE SIDE FORCE DUE TO BEAM SEAS

Once we have obtained the blockage coefficient C(x) or its
inverse the porosity P(x) for any given ship-water bottom geometry,
we have all the information about the ship that is necessary to solve
the outer acoustic-like problem to determine the wave force on the
ship. Numerical techniques for solving the problem formulated in
Section 5 are described by Tuck [1970] and by Taylor [1971] and will
not be discussed in detail here.

It is sufficient to observe that the outer problem can be re-
duced to solution of an integral equation, using methods analogous
to those described by Honl, Maue and Westphal [1961], in which
P(x) appears as an input quantity. This integral equation can be
solved by direct numerical quadrature, followed by matrix inversion,
leading to numerical values for the basic unknown potential
o(x,0,,-h).

This potential is proportional to the pressure difference
across the ship, and hence we may obtain the net force F, on the

653

Tuck and Taylor

ship in beam seas in the form
£
Bei="- 2ieph | ; o(x,0,,-h) dx. Wie 1)

Figure 4 shows computations of | Fo |/2pghLA for the Series 60,
block 0.80 ship whose blockage coefficient C(x) was given in

Fig. 3. This particular scaling of the force was chosen so that the
high frequency or short wave limit kf —~ o is 2.0. This limit cor=
responds physically to the case when the ship acts as a perfect
reflector many wavelengths long, so that a pure standing wave exists
in its neighborhood. This is true for all values of P(x), i.e. for all
draft/water depth ratios, because as the waves get shorter and
shorter they are less able to penetrate beneath the hull.

HIGH FREQUENCY
ASYMPTOTE

IFI/2pghlA

DRAFT/WATER DEPTH = 0.8

/ 0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0
kl = mL/r

Fig. 4 Side force on Series 60, block 0.80 ship, due to beam seas

654

Shallow Water Problems in Ship Hydrodynamics

The results for zero clearance show pronounced wobbles as
a function of frequency. This is to be expected, and is due to inter-
ference between the waves diffracted around the two ends of the ship.
For non-zero P(x) there is a similar but much reduced effect,
since part of the wave energy is transmitted directly beneath the
ship, and a less strong diffraction pattern produced.

The force decreases markedly as the clearance is increased
and the ship presents less of a barrier to passage of wave energy.
If in fact the clearance is large, P—~+co or C ~ 0, it follows from
(5.7) that $(x,0,,-h) — - Avg/h C(x) and hence that

f
Fo > Zipghka | C(x) dx. (7. 2)
i)

The physical interpretation of this limit is that the effect of the free
surface on the disturbance flow field about the ship diminishes as the
clearance increases, until the flow is effectively the same as the low
frequency limit in which the force is in phase with the fluid particle
accelerations. This is in line with an interpretation (Newman [ 1969])
of C(x) as related to the added mass of the section at x, with the
free surface replaced by a rigid wall. The resulting force Fg, is of
course numerically small, if C(x) is itself small.

On the other hand, the only cases shown in Fig. 4 correspond
to clearances small enough to give significant blockage C(x), and
hence a net force comparable with the standing-wave value 4pghlA.
It would appear that this limiting value is a useful, usually conserva-
tive, estimate for clearances of this order, but that it should be used
with caution for low frequencies (very long swell) or for non-small
clearances. In the latter case, e.g. with draft/water depth < 0,5,
the standing wave limit may be a substantial over-estimate of the
total force.

The computations presented here are samples only. They
may be considered extensions of similar computations given by Tuck
[1970] for a mathematically idealized ship with a blockage coefficient

C(x) = Cy 0? - 50 723)

In fact the results for the Series 60 ship are not greatly different
from those given by Tuck [1970], a reflection of the fact that (7. 3)
is not an unreasonable approximation to the shape of the curves in
Fig. 3. Further computations, including other ship geometries and
clearances, and treating the case of incident seas from directions
other than abeam, are given by Taylor [1971].

655

Tuck and Taylor

REFERENCES

Constantine, T., "On the movement of ships in restricted waterways,"
J. Fluid Mech., vol. 9, pp. 247-256, 1961.

Flagg, C. and Newman, J. N., Unpublished manuscript, 1971.

Frank, W., "Oscillation of cylinders in or below the free surface
of deep fluids," N.S.R.D.C. Report No. 2375, Washington,
D.C.', 1967.

Freakes, W. and Keay, K. L., "Effects of shallow water on ship
motion parameters in pitch and heave," MIT, Dept. of Nav.
Arch. and Mar. Eng., Report No. 66-7, Cambridge, Mass.,
1966.

Garthune, Rosenberg, B., Cafiero, D. and Olson, C. R., "The
performance of model ships in restricted channels in relation
to the design of a ship canal," N.S.R.D.C. Report No. 601,
Washington, D. C., 1948.

Graff, W., Kracht, A., and Weinblum, G., "Some extensions of
D. W. Taylor's standard series," Trans. S.N.A.M.E.,
vol. 72, p. 374, 1964.

Havelock, T. H., "Note on the sinkage of a ship at low speeds, m
Z. angew. Math. Mech., vol. 19, pp. 202-205, 1939.

Hess, J. L. and Smith, A. M.O., "Calculation of non-lifting potential
flow about arbitrary three-dimensional bodies," J. Ship.
Res. ’ vol. 8, Ppp. 22-44, 1964.

Honl, H., Maue, A. W. and Westphal, K., "Theorie der Beugung,"
Handbuch der Physik, vol. 25, pp. 218-573, S. Flugge (ed.),
Springer, Berlin, 1961.

Horn, F., Jahrbuch der Schiffsbautechn. Gesellsch., vol. 38,
p."L77; 1937.

Kim, C. H., "The influence of water depth on the heaving and pitching
motions of a ship moving in longitudinal regular head waves,"

Schiffstechnik, vol. 15, pp. 127-132, 1968.

Kreitner, J., "Uber den Schiffswiderstand auf beschranktem Wasser,"
Werft Reederei Hafen, vol. 15, 1934.

Lamb, H., "Hydrodynamics," (6th ed.) C.U.P. and Dover, 1932.

Lemmerman, In "Resistance and propulsion of ships," Van Lammeren,
W. (ed.), H. Stam, Holland, 1942.

656

Shallow Water Problems tn Shtp Hydrodynamtes

Michell, J. H., "The wave resistance of a ship," Phil. Mag. (5),
vol. 36, pp. 430-437, 1898.

Moody, C. G., "The handling of ships through a widened and asym-
metrically deepened section of Gaillard cut in the Panama
canal," N.S.R.D.C. Report 1705, Washington, D. C., 1964.

Morse, P. M., "Vibration and sound," (2nd ed.) McGraw-Hill,
New York, 1948.

Morse, P, M. and Rubenstein, P. J., "The diffraction of waves by
ribbons and slits," Phys. Rev., vol. 54, pp. 895-898,
1938.

Newman, J. N., "Lateral motion of a slender body between two
parallel walls," J. Fluid Mech., vol. 39, pp. 97-115, 1969.

Ogilvie, T. F., "Propagation of waves over an obstacle in water of
finite depth," Ph.D. thesis, University of California,
Berkeley, 1960.

Sedov, L. I., "Two-dimensional problems in hydrodynamics and
aerodynamics," (translation), Interscience, New York,

1965.

Sharpe, B. B. and Fenton, J. D., "Report of investigation of a
proposed dock at Yarraville," University of Melbourne,
Dept. of Civil Eng., 1968.

Sjostrom, C. H., "Effect of shallow water on speed and trim,"
Paper read to N.Y. section, S.N.A.M.E., 1965.

Taylor, P. J., Unpublished thesis, University of Adelaide, 1971.

Todd, F. H., "Series 60, methodical experiments with models of
single-screw merchant ships," N.S.R.D.C. Report No.
1712; "Washirigton,’ D. C., £963.

Tothill, J. T., "Ships in restricted channels," Marine Technology,
vol. 4, pp. 111-128, 1967.

Tuck, E. O., "A systematic asymptotic expansion procedure for
slender ships," J. Ship. Res., vol. 8, pp. 15-23, 1964.

Tuck, E. O., "Shallow-water flows past slender bodies," J. Fluid
Mech. ;’vol. 26, pp. ‘81-95, 1966.

Tuck, E. O., "Sinkage and trim in shallow water of finite width,"
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657

Tuck and Taylor

Tuck, E. O., "Ship motions in shallow water," J. Ship. Res.,
vol. 14, no. 4, 1970.

Tuck, E. O. and von Kerczek, C., "Streamlines and pressure
distribution on arbitrary ship hulls at zero Froude number,"
J. Ship Res., vol. 12, pp. 231-236, 1968.

Van Dyke, M., "Perturbation methods in fluid mechanics,"
Academic Press, New York, 1964.

Wehausen, J. V. and Laitone, E. V., "Surface waves," Handbuch
der Physik, vol. 9, S. Flugge (ed.), Springer, Berlin, 1960.

Wilson, B. W., "The energy problem in the mooring of ships ex-
posed to waves," Bull. No. 50, Perm. Int. Assoc. of Nav.
Cong., Brussels, 1959.

DISCUSSION

Prof. Hajime Maruo
Yokohama Nattonal Untversity
Yokohama, Japan

In the present analysis, the problem of the shallow water
effect is discussed on the basis of a self-consistent linearized
theory. According to the investigations into the same problem both
theoretical and experimental, which were carried out at the University
of Tokyo several years ago, nonlinear phenomena appeared remark-
ably in the trans-critical speed range. This problem can be analyzed
by a similar way to the nonlinear theory of the transonic flow ofa
compressible fluid. At that time, however, the theory of the tran-
sonic gas flow had not been well developed, and the former investi-
gations were obliged to confine themselves in the analysis by the
analogy with the simple one-dimensional duct flow. Nowadays the
theory of the transonic flow around a body has been developed to a
great extent. The mathematical technique used in it may be available
to the problem of the nonlinear shallow water effect.

Shallow Water Problems in Shtp Hydrodynamtes

REPLY TO DISCUSSION

B20. Tuck and°P. J. Paylor
Untversity of Adelaide
Adelatde, South Australia

We certainly agree with the comments of Professor Maruo
regarding the importance of nonlinear phenomena in the trans-critical
speed range. Indeed, the transonic analogy was discussed in the
paper by Tuck[1966] and further work on this aspect of the problem

was suggested.

659

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

Friday, August 28, 1970

Morning Session

Chairman: Adm. R. Brard
Marine National Bassin D'Essais
Des Carenes, Paris, France

Page
Singular Perturbation Problems in Ship Hydrodynamics 663
T. F. Ogilvie, University of Michigan
Theory and Observations on the Use of a Mathematical
Model for Ship Maneuvering in Deep and Confined
Waters 807
N. H. Norrbin, Statens Skeppsprovningsanstalt,
Sweden
The Second-Order Theory for Nonsinusoidal Oscillations
of a Cylinder in a Free Surface 905
C. M. Lee, Naval Ship Research and Development
Center
661 ( 62 BLANK

L04 FOLLOWS

SINGULAR PERTURBATION PROBLEMS IN

g(x,z3€)
G(x,z)
h(x, z3€)

H(x,z)
H(x)

SHIP HYDRODYNAMICS

T. Francis Ogilvie
Untverstty of Michtgan
Ann Arbor, Michigan

NOTATION

added-mass coefficient

damping coefficient

hull offset

restoring-force coefficient

contour of body in the cross section at x
force in j-th mode due to body motion
force in j-th mode due to incident waves
gravity constant

Y offset of camber surface

g(x,zse)/e

half-thickness of body (equal to b(x,z) for a symmetri-
cal body)

h(x ,z3€)/e

part of free-surface deflection in problems in two
dimensions (see Eq. (5=15))

projection of a body onto the y =0 plane (the center-
plane of a ship)

V-1
unit vectors parallel to three Cartesian axes

Fourier-transform variables corresponding tox, y, z@
respectively

modified Bessel function of second kind

length of a ship, or the segment of the x axis between
bow and stern cross sections

663

v(x,y»Z)

xy Z

De Ga’ 4

Z,(xsy3€)

a,(x»Z3€)

Yp(%2Z3€)

6)
€
C(x .9,t)

Ogtlvite

G=1, ... 5 6) aset of functions defined over the sure
face of a slender body (see (2-75))

added mass per unit length of a slender body

(j= 1, .-., 6) a set of functions defined over the surface
of a slender body, equal to the components of the unit
normal vector for j=1, 2, 3 (see (2-72))

damping coefficient per unit length of a slender ship

unit vector normal to body surface (usually taken positive
into the body)

unit vector in a plane x = constant, normal to body
contour in that cross section

pressure

radius coordinate in cylindrical coordinate system
radius coordinate of a slender body

xj +t yj + zk

half-span of a horseshoe vortex or a lifting line

local half-span in wing cross section at x, or cross
section area of a slender body (taken as the cross section
area of the submerged part, for a ship)

half-span of a wing of large aspect ratio
keel depth of a ship at cross section at x

transfer functions between motion variables and forces
on body

speed of a body, or speed of an incident stream (the
latter invariably being taken in the positive x direction)

fluid velocity at (x,y,z) with uniform stream at infinity,
U = 1, flowing around a body

Cartesian coordinates

stretched Cartesian coordinates, e.g., x= X,y= €Y,
z=eZ for a slender-body problem, xyz being far-
field coordinates, XYZ near-field

terms in a near-field expansion of €(x,y;e) (cf.
€(x,yse))

6 (x,0,z;€) in thin-body problem

normal velocity component in the plane of a sheet of
dipoles

motion-amplitude parameter in ship-motion problems
small parameter in most problems considered

displacement of the free surface

064

Singular Perturbation Problems ?n Shtp Hydrodynamics

C(x, y3e)
C(z)
n(x,y)

7(x)
No(x)

8
Q(x,y,t)

K
d,(z)
Hy (Z)
n(x, z3€)

gj (t)

p
o,, (x3€)
o,(x»Z3€)

x,y,z >t)

$x y)

$(x,y »z3€)
$j (x,y »Z)
G(x,y,z3€)
}i(x,y »z)
X (x,y,z)

bj (x,y »Z)
W(x,y, Zz ,t)

Wj(x,y»z)
WwW

Q: Gciyss)

terms in a far-field expansion of €(x,y;e) (cf.
ZA(Xsy3€) )

function mapping the complex variable z onto an
auxilliary (¢) plane (in Section 5)

steady part of free-surface deflection in ship-motion
problem

free-surface deflection in problems in two dimensions

part of free-surface deflection in low-speed problem in
two dimensions (see (5-7) )

angle variable in cylindrical coordinate system

time-dependent part of free-surface deflection in ship-
motion problem

g/U*, a wave number in steady-motion problems
density of dipoles on a line (see (2-40) )

density of dipoles on a line (see (2-40) )

density of dipoles on a surface

w*/g, a wave number in oscillation problems

displacement in the j-th mode of motion (see Section
2232)

water density
density of sources on a line
density of sources on a surface

velocity potential (the arguments may vary, but 6
generally denotes the complete potential function ina
problem)

in Section 5.4, potential for the problem in which the
free surface is replaced by a rigid wall

terms in a far-field expansion of $(x,y,z;e)
normalized potential functions (see (2-73), (3-28) )
terms in a near-field expansion of (x,y, ze)
normalized potential functions (see (3-44) )

velocity potential for the perturbation of a unit-strength
incident stream by a slender ship

normalized potential functions (see (276) )

time-dependent part of velocity potential in ship-motion
problem (with forward speed)

normalized potential functions (see (3-45) )
radian frequency of sinusoidal oscillations

normalized potential functions (see (3-46) )

665

Ogilvie

MISCELLANEOUS CONVENTIONS

1) —— Potential: The velocity is always the positive gradient
of the potential function.

2) Coordinates and Orientation: In problems involving a steady
ncident flow, that flow is always inthe positive x direction.
The vertical axis is the y axis in 2-D problems, the z
axis in 3-D problems.

3) Time Dependence: In problems of sinusoidal oscillation, the
time aependen ce is always in the form of the exponential
function, e!#f, In such problems, the real part only is

intended to be used, but we do not indicate this explicitly in
general.

4) Fourier Transforms: These are denoted by an asterisk. For
example,

foe)

@
* A
o (k) -{ dx es (x) : o(x) = + dk ae oe (ils
-@ ue -@

tok sak Gee ~i(kx +f y)
6 (k,£;z) -{ iY dx dy e o(x,y,Z)-
-00 “-00

5) Principal- Value Integrals: These are denoted by a bar through
the integral sign:

6) Order Notation: There are three symbols used: O.0,”.
a) "“y = O(x)" means: |y/x|< M as x0, where M is
a constant not depending on x.
b) "y = 0(x)" means: ly/x| > 0 as x0.

y ~ £(x) " means ly - £(x) | = o(f(x)) as x0.

666

Singular Perturbation Problems in Ship Hydrodynamics

I. INTRODUCTION

This paper is a survey of a group of ship hydrodynamics
problems that have certain solution methods in common.

The problems are all formulated as perturbation problems,
that is, the phenomena under study involve small disturbances from
a basic state that can be described adequately without any special
difficulties. The methods of solution make explicit use of the fact
that the disturbances of the basic state are small. Mathematically,
this is formalized by the introduction of one or more small param-
eters which serve as measures of the smallness of various quantities.
The solutions obtained will generally be more nearly valid for small
values of the parameter(s).

However, the problems will also be characterized by the
fact that they are ill-posed in the limit as the small parameter/(s)
approaches zero. Thus, we call them singular perturbation prob-
lems. Special techniques are needed for treating such problems,
and we have two which are especially valuable:

4) The Method of Matched Asymptotic Expansions, and

2) The Method of Multiple-Scale Expansions.

The first has a well-developed literature, and it has been made
particularly accessible to engineers by Van Dyke [1964]. The
second, which has a longer history, is perhaps less well-known,

but we now have a textbook treatment of it too, thanks to Cole [ 1968].
Because of the availability of such books, my treatment of the
methods in general will be extremely terse.

The necessity for treating ship hydrodynamics problems as
perturbation problems arises most often in the incredible difficulty
of handling the boundary condition which must be satisfied at the free
surface. Even after neglecting viscosity, surface tension, com-
pressibility, the motion of the air above, and a host of lesser
matters, one can still make little progress toward solving free-
surface problems unless one assumes that disturbances are small
-- in some sense. Historically, it has commonly been assumed
that the boundary conditions may be linearized; in fact, this has so
commonly been assumed that many writers hardly mention the fact,
let alone try to justify it.

The two methods emphasized in this paper can also be applied
to problems involving an infinite fluid. In fact, neither method was
applied specifically to free-surface problems until quite recent
times. Section 2 of this paper is devoted to several infinite-fluid
problems. My justification, quite frankly, is almost entirely on
didactic grounds. The methods can be made much clearer in these
simpler problems, and so I include them here, although in some

667

Ogilvie

cases the infinite-fluid problems can be treated adequately by more
elementary methods.

Most of the material in this paper has appeared in print
elsewhere. My intention has been to present a coherent account of
the treatment of singular perturbation problems in ship hydrody»*
namics, and so I have reworked solutions by other people and put
them into a common notation and a common format. In some cases,
I have made conscious decisions to follow certain routes and to
ignore others. I am sure that I have made many such decisions
unconsciously too. I have tried to give credit where it is due, but
Iam also sure that I have committed some sins of omission in the
references. I apologize to those whom I may have slighted in this
way.

1.1. Nature of the Problems and Their Solutions

We never really derive the perturbation solution of the exact™
problem; we derive, at best, an exact solution of a perturbation
problem. That is, we formulate an exact boundary-value problem,
simplify the problem, solve the simplified version, and then hope
that that solution is an approximation to the solution of the exact
problem,

Thus, there will almost always be open questions about the
validity of our solutions, and these questions can only be resolved
through comparisons with exact solutions and experiments. We can
have little hope of being rigorous. In fact, it is difficult to provide
completely convincing arguments for doing some of the things that
we do; in many cases, our approach is justified by the fact that it
works! Much progress has been made in this field by people who
try approaches "to see what will happen."

This does not imply that we shoot in the dark. It does sug-
gest that we often depend more on intuition (or experience, which is
the same thing) than on mathematical logic in deciding how to solve
problems. The small disturbance assumptions by which free-surface
problems have traditionally been linearized must have been tried
first on this basis. The predictions which result from making such
assumptions agree fairly well with observations of nature, and so we
are encouraged to go on making the same assumptions in new prob-
lems. We may expect to be successful sometimes.

There are also open questions about the uniqueness of solutions.
Engineers do not often worry about such matters, but they should
certainly be aware of certain situations in which the dangers of

ae Vissi ae
"Exact" means only that nonlinear boundary conditions are treated
exactly. I neglect viscosity, surface tension, compressibility,
etc., and still call the problem "exact."

668

Singular Perturbation Problems in Ship Hydrodynamics

non-uniqueness are especially great. The history of the study of
free-surface problems provides numerous examples of invalid solu-
tions being published by authors who were not sufficiently careful on
this score. We have learned to be careful about imposing a radia-
tion condition when necessary, although newcomers to the field are
still occasionally trapped. ~ Questions about stability of our solu-
tions are not so well appreciated, but of course solution stability is
just one aspect of solution uniqueness. A particularly startling
example has been pointed out in recent years by Benjamin and Feir
[1967]: Ordinary sinusoidal waves in deep water are unstable. This
has now been demonstrated both theoretically and experimentally.

It comes as no great surprise to those experimenters who had tried
to generate high-purity sinusoidal waves for ship-motions experi-
ments, but it was certainly quite a surprise to the theorists, who
apparently did not suspect any such phenomenon before its discovery
by Benjamin and Feir.

Since we shall be considering small-perturbation problems,
we may expect the solutions to appear in the form of series expres-
sions (not necessarily power series!) Often, we are content to
obtain one term in sucha series. Practically never do we face the
question of whether the series converges. In fact, we usually just
hope that the series has some validity, at least in an asymptotic
sense.

The question will arise from time to time, "How small must
the small parameter be in order that a one- (or two- or three- or n-)
term expansion give valid predictions?" In ship-hydrodynamics
problems, it is quite safe to assert that the only answer to sucha
question must be based on experimental evidence. In fact, even in
simple problems, the knowledge of a few terms is not likely to help
much with this question. For example, suppose that one tries to
solve the simple differential equation: y"(x) + y(x) = 0, by means of
a series of odd powers of x. How does one know that a two-term
approximation is accurate to within one per cent even if x is as
large as unity? One might compute the third term, of course, and
compare it with the second term, hoping to guess what the effect
of further terms would be. If it were too difficult to compute that
third term, one could only hope that the solution had some validity,
and perhaps one would try to find some experimental evidence on
which to hazard a guess about validity. So it is in our ship-hydro-

* Within the last few years, a leading German journal published an
article on wave resistance in water of finite depth, in which it was
concluded that a body had identically zero resistance if it were
symmetrical fore and aft. The author was, I believe, primarily
a numerical analyst, not familiar with the pitfalls of free-surface
problems. He did not impose a numerical condition equivalent
to a radiation condition. (This is one reference that I intentionally
omit. )

669

Ogilvie

dynamics problems. It will be necessary to discuss this point further
at an appropriate place.

A related question concerns the precise definition of the small
parameters that we use to formulate the approximate problems. In
this paper I avoid defining the small parameter quantitatively. It is
usually unnecessary and it is dangerous. I shall return to this
point also.

1.2. Matched Asymptotic Expansions

For most of our problems, the approach advocated by
Van Dyke [1964] is entirely adequate. I shall assume that the reader
is familiar with (or has access to) Van Dyke's book. Only a few
definitions and concepts will be mentioned here.

Perhaps the simplest problem that demonstrates the applica-
bility of the method of matched asymptotic expansions is the following:
Find the solution of the differential equation,

ey + 2y ty =0,
subject to the initial conditions:
y(0)=1; y(0) =0.

The parameter e€ is to be considered small, and, in fact, we want

to know how the solution of this problem behaves as € ~0. Now,

if we set € = 0, the order of the differential equation is reduced, and
two initial conditions cannot be satisfied. Therefore, one cannot
obtain a series expansion for the solution by a simple iteration scheme
which starts with the solution for the limit case, € =0.

The exact solution for this problem is:

apie a Pst
y (t) = =_P28 ___ Pie
Pine

’

where

ae)
N
mm

670

Singular Perturbation Problems in Ship Hydrodynamics

If we consider that t = O(1) as € —~ 0, then the following approxi-
mation is valid for y(t):

y(t)~e Mh +E (2-0 4S Slee ee et.
This approximation could be obtained step-by-step, iteratively:

2¥n tT Yn = - €Yp-1 »

where y(t) ~ y,(t). However, it is not uniformly valid at t =0,
and the constants cannot be determined. On the other hand, we
could consider that t = O(e€) as € ~ 0 and rearrange the exact
solution accordingly. This is most easily done if we set t = €T
and rewrite everything in terms of T. The approximation for y(t)
is then:

2
yi 1 te(Z-Fy +g -F+ Bye...
€
we te er + g) ted

This approximation could be obtained completely from the differ-
ential equation by an iteration scheme in which we let y(t)~ 7, Y,(Ts€),
the individual terms satisfying the equation:

YM(7) + 2¥' (7) = - €¥,, (7) = [ ¥ = a¥/d7]
and the conditions:
¥,(0)= 4; ¥,(0)=0, n>14; ¥(0)=0, n21

However, this solution is not uniformly valid for 7 00; in fact,
one would hardly suspect that it represents a solution decaying ex-
ponentially with time.

The difficulty arises because the problem is characterized
by two time scales, 1/p, and 1/p, , and the two are grossly differ-
ent. One of the two exponentials in the exact solution decays very
rapidly and the other decays at a moderate rate. The contrast in
these two time scales, along with the fact that each has its dominant
effect in a distinct range of time, allows us to apply the method of
matched asymptotic expansions to this problem. The Van Dyke
prescription for doing this is as follows:

671

Ogilvte

Define the n term outer expansion of y(t) as [y,(t) +...
+ yn(t)] ; define the m term inner expansion of y(t) as
[Y,(7) +... + Ya(7)]. Inthe n term outer expansion, substitute
t= €7T and rearrange the result into a series ordered according to
€; truncate this expression after m terms, which gives the m
term inner expansion of the n term outer expansion. Similarly,
in the m term inner expansion, substitute 7 =t/eé and rearrange
the result into a series ordered according to €; truncate this
expression after n terms, which gives the n term outer expan-

sion of the m term inner expansion. The matching rule states
that:

The m term inner expansion of the n term outer expansion

= the _n_ term outer expansion of the m_ term inner expan-
sion.

In the example discussed in the previous paragraphs, the
outer solution could not be obtained by a simple iteration scheme.
The matching principle can now be used to determine the constants
in the outer solution, and so an iteration scheme is now available,
requiring, however, that inner and outer expansions be obtained
simultaneously. In the example, the inner solution could be ob-
tained completely and independently of the outer, but this is an
accident which occurred because of the simple nature of the prob-
lem above. Ordinarily, in cases in which one might consider using
the method of matched asymptotic expansions, one must proceed
step-by-step to find first a term in one expansion, then a term in
the other expansion, and so on.

It is worthwhile to be fairly precise about certain definitions.

We use the equivalence sign, "~," frequently. For example, we
write:

N
d(x, y,Z3€) ~ > b,(x,y »Z35€)

n=O
This means that:
N
| - . o,| = o(d) as e —~ 0 for fixed values of (x,y,z).
n=O

Also, it implies that py = 0(¢,) as € 7 0. The qualification that
(x,y,z) should be fixed is very important. In the example above,
we would have the equivalent statement for the outer expansion:

N
| y (tse) - , yp(tse) | = oly,) as a>, 0 for fixed t,

n=l

and, for the inner expansion:

672

Singular Perturbatton Problems in Shtp Hydrodynamics

N
ly(tse) - » ¥itrie) p= o(Yy) as e—~0 for fixed T.

n=l

In the latter, we evaluate the difference on the left-hand side for
smaller and smaller values of t (= €T) as e€ —~ 0; in other words
we restrict the range of t more and more as e~0O. This is in
contrast to the interpretation of the outer expansion, in which we
simply fix t at any value while we let € ~ 0. In even more physi-
cal terms, we may say that the inner expansion describes the solu-
tion during the time when the e?!t term is Maree rapidly, and the
outer expansion describes the solution when the e it term has
effectively reached zero and the eP2! term is varying significantly.
This separation into two distinct regimes is characteristic of prob-
lems in which we apply the method of matched asymptotic expan-
sions. Of course, the real key to the success of the method is in
the procedure by which the two aspects of the solution are matched
to eachother. After all, they do represent just two aspects of the
same solution.

Usually, we insist that our asymptotic expansions be
consistent. A precise definition of this term is awkward, but per-
haps it is clear if we state that each term in sucha series depends
on € in asimple way that cannot be broken down into simpler terms
of different orders of magnitude. For example, the following two
series are equal:

2 3 1 1 2 ee!
= = _ = 1
[ie ite ee” Fo] ise ae Way ae
1 f.2 5 ih 3
ae ra tee Te ]
: ey ae
se a= ea Ts ]
1 3
+ [ie + arate
ic i
On the right-hand side, let:
il il 2 1.2
fle)=itszetze tee Taereuel 3p
Pa
f (e) = = fp(e), for mo 0%

Then we can write:

673

Ogilvte

N

us N
> é€! ie >. f,(e) as e= 0.

n=O n=O

These happen to be convergent series (if € <1), but we can inter-
pret them as asymptotic series just as well. The series on the left
is "consistent"; the one on the right is not, because individual terms
have their own e€ substructure.

The striving for consistency can become a religion, but it is
not a reliable faith. Consistency (or the lack of it) tells us nothing
about the relative accuracy of otherwise equivalent asymptotic
expansions. In fact, we could define a third asymptotic series with
terms given by:

BOE) = 1/(1-€) ; g,(€) = (9) for n.> 0}

This series is grossly inconsistent, but one term gives the exact
answer for the sum of the previous series! Occasionally one can
make educated guesses about such things, replacing a few consistently
arranged terms by a simple, inconsistent expression having much
greater accuracy in practical computations. Mathematically, these
different asymptotic series are equivalent, and, if € is small
enough, they will all give the same numerical results. But we want
in practice to be able to use values of € that are sometimes not
"small enough. "

We shall work with consistent series, for the most part, in
spite of such possibilities of improvement through the use of incon-
sistent series. Most newcomers to this field of analysis find that
there is a considerable element of art in the application of the
method of matched asymptotic expansions, and I personally consider
that the improvement of the expansions through the development of
inconsistent expansions is the highest form of this art. Except in
one respect, I do not intend to pursue the possibilities of inconsistent
expansions in this paper.

The exception that I make is the following: Many singular
perturbation problems lead to asymptotic-expansion solutions of the

form:
N n
» » anm€ (log ey

n=0 m=O

where a,,, does not dependon €. We can, of course, write this
out in a long string of terms quite consistently arranged. However,
my practice will be to treat the sum:

674

Stngular Perturbation Problems in Shtp Hydrodynamics

n

€” > anm(log €)"

m=0

h,(e)

as a single term (albeit inconsistent) in the series Dy bal €) « An
alternative way of describing this practice is to say that I consider
log € = O(i) as € ~ 0! I have encountered some practical prob-
lems which could apparently not be solved by the Van Dyke matching
principle unless treated in this way, and I have never seen or heard
of a problem in which this practice led to difficulties. There are
some good arguments for proceeding in this way, but I know of no
proof that either way is the correct way. (Some of my colleagues
will call this a cheap trick, rather than a higher expression of an
art form.)

The classical example in physics of this kind of mathematical
problem is the boundary layer first described by Prandtl in 1904. The
thickness of the boundary layer becomes smaller and smaller as the
small parameter, 1/VR approaches zero (R is the Reynolds num-
ber), but the presence of the boundary layer cannot be neglected,
because then the governing differential equation becomes lower order,
and the body boundary conditions cannot all be satisfied. Unfor-
tunately, Prandtl did not realize the generality of the analysis which
he introduced into the viscous-fluid problem, and, lacking the
modern formalism for treating such problems, he could not obtain
higher-order approximations.

Perhaps I should include a discussion of Prandtl's problem
in this paper, since it might be considered as a "singular perturba-
tion problem in ship hydrodynamics." However, I shall not do this,
for several reasons. Van Dyke's coverage of the problem is excel-
lent, I think. Also, the analysis concerns only laminar boundary
layers, and they are really of quite limited interest in ship hydro-
dynamics. Finally, the formal procedure breaks down completely
at the leading edge of a body, and the singularities that occur there
cause major difficulties in all attempts to use the formalism to
obtain higher-order approximations.

One final point should be emphasized, even at the risk of
insulting the intelligence of readers who have read this far. When-
ever we write, "e ~ 0," we are implying the existence of a se-
quence of physical problems in which the geometry of some funda-
mental parameter varies. For example, in Prandtl's boundary-
layer problem, we may consider that viscosity changes as
e=1/¥R—0. Inthe simple ordinary-differential-equation example
presented above, we may think of a spring-mass system in which the
mass is changed systematically from one experiment to the next.
Later, when we treat slender-body theory, we consider a sequence
of problems in which the body changes eachtime. The theory always
implies the possible existence of such a series of problems, and the
quality of the predictions improves as the problem more nearly fits

675

Ogtlvte

the limit case. Thus, we shall be able to apply the results of
slender-body theory to bodies which are not especially slender. In
such cases, we may expect that the predictions will be less accurate
than the predictions that we woutd make for a much more slender
body. But we never know a priori how slender the body must be for
a certain accuracy to be realized, and it would be wrong to assert
that the theory applies only to needle-like bodies. All that we can
say is that it would be more accurate for such bodies than for not-
so-slender bodies.

1.3. Multiple-Scale Expansions

In the problems of the previous section, we had two greatly
contrasting scales for the independent variable. The fact that enabled
us to obtain two separate expansions was that each of the scales
dominated the behavior of the solution in a particular region of space
or aparticular period of time. The major practical concern was to
ensure that the separate expansions matched, because they really
represented just different aspects of the same solution.

The present section is devoted to problems in which there
are again two greatly contrasting scales. However, in these prob-
lems, it will not be possible to isolate the effects of each scale
into a more or less distinct region of space or time. The effects
of the two scales mingle together completely. However, we may
still expect to be able to identify these effects somehow, just because
the two scales are so different.

There are classical problems of this kind, the most famous
being related to nonlinear effects on certain periodic phenomena.
Cole [1968] discusses a number of these problems. Perhaps the
simplest example of all is alinear one: Find approximate solutions
for small € inthe problem of a linear oscillator with very small
damping, where the differential equation might be written:

y t2ey ty =0.

To be specific, let the solution satisfy the initial conditions: y(0) = 1
and y(0) =0. Physically, we expect that the system will oscillate
with gradually decreasing amplitude. It would be desirable if the
approximate solution at least did not contradict this expectation.

We might try representing y(t;€) by an asymptotic expansion
with respect to €: y(t;e) ~ 7, ypltse). We would find immediately
that the first term in this expansion is just: yg(t;e) = cos t. This
seems quite reasonable, since it represents a steady oscillation at
the frequency approximate to the undamped oscillator. The second
term in the expansions would be obtained from:

676

Stngular Perturbation Problems tn Ship Hydrodynamics

y + y, == 2€Y5 = 2€ sin Gs with y, (0) yt (0) = 0.

It is impossible to obtain a steady-state particular solution of this
problem. In fact, the solution is:

y, (tse) = e[ sin t - t cos t].

Thus, we obtain an expansion in which the second term grows linearly
with time. One might expect that succeeding terms will grow even
faster. This expansion is correct, and, for small values t, it could
be used for numerical predictions. But we would certainly prefer

to obtain an expansion which is uniformly’ valid, even for very large
t.

The exact solution is easily found, of course. It is:

y(tse) = en [ cos Vi-e*)t trae sin y(1-e°)t] .

The approximate solution becomes worse and worse with increasing

t because the frequency is wrong and because the exponential factor
is expanded in a power series in t. If we watch the oscillating mass
on atime scale appropriate to the period of the oscillation, we do
not see the exponential decay and the slight shift of frequency caused
by the damping. On the other hand, if we watch for a very long time,
the effects of damping accumulate gradually. Thus, the effects of the
"slow-time" scale, 1/e, persist throughout the history of the motion
as observed on a real-time scale, but these effects never occur
suddenly. It is this fact which enables us to separate them out of

the real-time problem.

There seems to be less reliable formalism available for
handling such problems than in the case of the method of matched
asymptotic expansions. More is left to the insight and ingenuity of
the individual problem solver. In the example discussed above, the
procedure is fairly clear: Expand y(tje) in a series such as this:

y(ts€) ~ yolt, Tse) Fy (t,73€) +... ,
where we define:

“~

t. = €t 3 tam FE (rs€) Fete) tees

*
Strictly speaking, the series really is uniformly valid except at
t =o,

Ogilvie

and the functions f, are to be determined in such a way that the
approximation is uniformly valid for all t. In treating this parti-
cular problem, Cole immediately assumes that t~ 7 and further
that t/7 = 1 + O(e2). These extra assumptions speed the solution
considerably, but it is not clear how one would know to make them
if the exact solution were not available. The exact solution takes
the form:

y(t;e) = at [iss Pur (¢ /7) sin T],

in terms of the new variables. (The factor (t /T) does not depend
on t.) Here it is clear how the two time scales enter into the
solution as well as the problem. One may expect the relationship
between t and 7 to be equivalent to the expansion of the quantity

(1-e€2). The reader is referred to Cole's book for further discus-
sion of the solution of such problems.

One problem that will be discussed later is a close relative of
the classical problems mentioned above. The solution by Salvesen
[1969] of the higher-order problem of the wave resistance of a sub-
merged body leads to a situation in which the first approximation is
periodic downstream and that period is modified in the third-order
approximation. (Otherwise the waves downstream in the higher
approximation would grow larger and larger, without limit.) A
similar problem involves the oscillation of a body on the free sur-
face, in which the wave length of the radiated waves must be modified
in the third approximation. For example, see Lee [1968].

A quite different application of this method is the problem of
very low speed motion of a body under or ona free surface. The
simplest such case has been discussed by Ogilvie [1968]. Fora
translating submerged body, there are two kinds of length scales:
length scales associated with body dimensions and submergence,
and the length scale U Sas which is associated with the presence
of the free surface. Presumably, the latter has effects primarily
near the free surface, ina "boundary layer" with thickness which
varies with u*/g as that variable approaches zero. But the
effects of the body dimensions are also important near the free
surface (or at least near a part of it). Thus the effects of the two
length scales cannot be separated into distinct regions. A brief
discussion of this problem appears in Section 5.42 of the present

paper.

There may be many other problems of ship hydrodynamics in
which this approach would be valuable. For example, many authors
have obtained approximate solutions of problems involving submerged
bodies by alternately satisfying a body boundary condition, then the
free-surface condition, then again the body condition, etc. At each
stage, when one condition is being satisfied, the other is being
violated, but it is assumed that the errors become smaller and

678

Stngular Perturbation Problems in Shtp Hydrodynamics

smaller with each iteration. Such a procedure is discussed, for
example, by Wehausen and Laitone [1960], who point out the use-
fulness of Kochin functions in such procedures. However, there is
often a question about the precise nature of such expansions. In the
first approximation, for example, the effects of the free-surface
are likely to drop off exponentially with distance from the surface.
This makes it inappropriate to treat depth of submergence as a large
parameter in the usual manner, because exponentially small orders
of magnitude are either trivial or exceedingly difficult to handle.

I do not believe that anyone has yet shown how to treat this problem
systematically.

II. INFINITE-FLUID PROBLEMS

It is mainly the presence of the free surface in our problems
that forces us to seek ever more sophisticated methods of approxi-
mation. However, the nature of the approximations can often be
appreciated more easily by applying those methods to infinite-fluid
problems. In this section, I discuss a number of problems that are
geometrically similar to the ship problems that are my real concern.
In some cases, it must be realized that the methods used here are
not necessarily the best methods for the infinite-fluid problems.
However, without the complications which accompany the presence
of the free surface, one can better understand the significance of
the coordinate distortions, the repeated re-ordering of series, and
the matching of expansions.

The reader who feels comfortable with matched asymptotic
expansions is invited to skip this chapter.

Zed) hin Body

A "thin body" has one dimension which is characteristically
much smaller than the other dimensions. In aerodynamics, the
common example is the "thin wing," and, in ship hydrodynamics,
one frequently treats a ship as if it were thin. In such problems,
the incident flow is usually assumed to approach the body approxi-
mately edge-on, and so the thinness assumption allows one to
linearize the flow problem.

In this section, thin-body problems are treated by the method
of matched asymptotic expansions. This is not the way thin-body
problems are normally attacked, and, in fact, I do not recall ever
having heard of suchatreatment. At the outset, I must point out
that there are good reasons why this has been the case. If the body
is symmetrical about a plane parallel to the direction of the incident
flow, one does not need inner and outer expansions for solving the
problem. And if the body lacks such symmetry, the lowest-order
problem cannot be solved analytically, and so the method of matched

679

Ogtlvte

asymptotic expansions does not offer the possibility that one may be
able to obtain higher-order approximations.

In fact, the problem of a thin body in an infinite fluid is not
a genuine singular perturbation problem (although it may contain
some sub-problems that are singular, such as the flow around the
leading edge of an airfoil). However, I believe that the problem of
a thin ship is singular; I shall discuss this in Section 4. There has
been a considerable amount of misunderstanding as to what consti-
tutes the near field and what constitutes the far field in the thin-ship
wave-resistance problem, and the rectification of such misunder-
standing requires a careful statement of the problem.

It is conceivable that this interpretation of the thin-ship
problem may be useful in formulating a rational mathematical
idealization of the maneuvering-ship problem.

For convenience, I separate the thin-body problem into two
parts: a) the symmetrical-body problem, and b) the problem ofa
body of zero thickness. To treat an arbitrary thin body, with both
thickness and camber, one should certainly consider both aspects
at once. It is not really difficult to do this, and indeed the problem
of an unsymmetrical body of zero thickness actually involves thick-
ness effects (at higher orders of magnitude than in the symmetrical-
body problem). I have kept the problems separate here only for
clarity in discussing certain phenomena that occur.

2.11. Symmetrical Body (Thickness Effects). Let the body
be defined by the equation:

+th(x,z3e) for (x,0,z) in H>,
a (2-1)

0 for (x,0,z) not in H,

where | is the part of the y = 0 plane which is inside the body.
(It is the centerplane if the body is a ship.) The "thinness" of the
body is expressed by writing:

h(x, z;€) = €H(x,z), (2-2)

where € is asmall parameter and H(x,z) is independent of e€.
The body is immersed in an infinite fluid which is streaming past it
with a speed U inthe positive x direction. The flow, in the
absence of the body, can be described by the velocity potential: Ux.

It will sometimes be convenient to say that the body is defined

by the equation: y = + h(x,z;e), implying that the function h(x,z;€)
is identically zero if (x,0,z) is not in H. Also, note that we shall

680

Singular Perturbatton Problems in Ship Hydrodynamics

frequently drop the explicit mention of the € dependence.

As € ~ 0, the body shrinks down to a sheet of zero thickness
aligned with the incident flow. Thus, the first term in an asymptotic
expansion of the velocity potential in the far field is just the incident-
stream potential. In general, let the far-field expansion be expressed
as follows:

N
(x,y ,Z;€) ~ » $(x,y,z;€), where $,) = o(¢) as
n=O
e—>0 for fixed (x,y,z). (2-3)
Then we have:
$y(xsy»Z5€) = Ux. (2-4)

The far field is the entire space except the y =0 plane.
Since the potential $(x,y,z;€) satisfies the Laplace equation through-
out the fluid domain, the individual terms in the above expansion
satisfy the Laplace equation in the far field:

on. t Pnyy t Pis= 0 for ly|> 0. (2-5)

At infinity, we expect (on physical grounds) that:
V(o - Ux) > 0. (2-6)

Therefore, for n>0O, every $, must be singular onthe y = 0
plane or be a constant throughout space. The latter would be too
trivial a result to consider, and so we assume that 6, is indeed
singular on the y = 0 plane.

But what kind of singularities will be needed? Because of
the symmetry of the problem, it is not difficult to show that a sheet
of sources will suffice. One can use Green's theorem to show this.
Alternatively, one can use transform methods for solving the Laplace
equation, which is practically equivalent to solving by separation of
variables. Whatever method is used, the result is the same}
$(x,y,z3€) has a representation:

ve og (& ,o3€) d& do (2-7)

1 0
( rV¥ 2Z3€) = - it ’
OMX ry + 40 J Yo [ tx-é)" ty" 2 (z-t)"] 172

where o,(x,z;€) is an unknown source-density function. The outer

681

Ogilvie

expansion is just the sum of these:

n(SsG3€) d& dt
ee ere eee
[ (x-8) yiot (ze oy!

This is the most general possible outer expansion for this problem.

It will be necessary presently to know the inner expansion of
the above outer expansion. To find it, define an inner variable:

y'=y/e, (2-9)

substitute for y in the outer expansion, and re-order the resulting
expression with respect to €. A direct approach to this process is
difficult, but the following method, in four steps, allows us to obtain
the desired results to any number of terms ina fairly simple way:

1) Take the Fourier transform of 4, with respect to x:

2 © -ikx
) 1 * dx e
> (ky ,2;€) =-— dG o. (k;G3e) te ns
i [x ty? + (2-67)?

co
SV ag ot tastse) Ky [kl vy? + 2-01)
-@

where - 0 is the modified Bessel function usually denoted this way,
and of *(k;z;€) is the Fourier transform of the function o,(x,Z5€)-
The convolution theorem was used in the first step above.

2) Take the Fourier Transform next with respect to z:
40% 1 ** a
-i 2 2,V2
>, (ksy;m;e) = - san (mie) | dze ™ KO lKILy +z iv )
-@

* 2 _ 21/2
Uk, mje) _-fk+m)

- ca lea ,

21" +m )

where o mie, mje) is the double transform of oa,(x,z;€).

3) Substitute: y = €Y and expand the exponential function
into a power series:

682

Singular Perturbation Problems in Shtp Hydrodynamtes

agin €) f- -2eG> 2 2
Tn »™M,; oe
2,2 [1 top Pv tk soak

**
an (iyim;€) = = ——>
2(k +m )

4_4 2
+ Fe Vile +m) +...]

+5 o. (ky mje) [e |y| + e°| ¥ |? (k* + m?)

5 Sruie 22
tore ly|(k +m) sane P
4) Note that:
eK
: Kk KI
_on (Ksmje) , (k;0;mj;e) = a, (k,m;e) . (2-10)
2(k? oO m_?)!/2
*%* ; P
Also, we observe that, if f (k,m) is the Fourier transform of

f(x,z), then (k? + m*)f**(k,m) is the Fourier transform of
- (f,, + £,,). Defining the inverse transform of a**(k,m;e€):

a(x,z3;€) = $,(x,0,z5€), (2-11)
and inverting the above series term-by-term, we obtain:
( z3€) ~ a ( se) to ely] ( zie) - oy €° (YP le ta)
>, XsVsZys nh%sZs 2 O,\X 24s 21 Nyy Nes

1 3 3
Ty 2s3it : [| (ony ss Tnzz )

{2h ite
aos + + Se Ee
; IY | ene olor 5 leas

(2-12)

This is the inner expansion of a typical term in the outer expansion.

In order to combine the expansions of the separate terms into
a single inner expansion of the outer expansion, let us assume that
n :
go, and a, are both O(c). (It is not necessary to assume this, it is
merely convenient.) Then we have for the desired expansion:

683

Ogilvie

g(x, y,z;€) ~ Ux O(1)
+ a (x,z5€) O(e)
+ a5(x,Z;€) +S ly |o, (x, z;€) O(e?)

1 1 2
+ a3(x,z5€) a) ly |on(x,z3€) -5 ly| (a + a.) O(e?)

+ O(e*). (2-13)

Note that we have reverted to far-field variables. We must here
consider that y = O(e€) in order to recognize the orders of magnitude
as indicated above.

Next we must find the inner expansion of the exact solution.
Substitute y = €Y in the formulation of the problem. The Laplace
equation transforms as follows:

2
byy= - Udy + by,)- (2-14)
The kinematic condition on the body is:
+ ph, - $y $h,=0 on y= + h(x,z),

which transforms into:

o,=+ e°(6H +H) on Y= + H(x,z). (2-15)

We assume that there exists a near-field asymptotic expansion of the
solution:

N
(x,y »Z3;€) ~ >, @,(x,Y,z;€), where py =0(@,) as en 0,
n=0
for fixed (x,Y,z). (2-16)
We could show carefully that:
@)(x,Y,z;€) = Ux.

(Perhaps it is obvious to most readers.) We then express the con-

684

Singular Perturbation Problems tin Ship Hydrodynamics

ditions on the near-field expansion as follows:

~~ e7[ ie 7 oie +@o + e., iF paral (2-17)

[H] 4 +@,, +3 +...~ + @[ UH, + @ Hy, + OH, +...]

on Y =+ H(x,z) . (2-18)

Solution of the @, problem. From the [L] condition above,
it is clear that:

Pi yy= 0 (2-19)

in the fluid domain. Therefore ®; must be a linear function of Y.
In view of the symmetry of the problem, we can set:

® (x, Y,z;€) = A, (x,25€) + B, (x,z3€) hy; for ly | > H(x,z).
(2-20)
The body condition reduces to:

(x, +H(x,z),z3€) =e e° UH, (x,z) ee B, (x,z3€) = O(e?). (2-21)

It appears that we have determined the value of B,(x,z;€) -- but this
is wrong, as we shall see ina moment. The two-term inner expan-
sion appears to be:

x,y,z;€) ~ Ux + A,(x,z;€) + B, (x, z5€) ly |.

Its outer expansion is obtained by setting Y = y/e:

o(x,y,z;€) ~ Ux + <8, (x,z;€)|y| + A (x, z5€) .
O(1) Ofe) O(e*)

The order-of-magnitude estimates were obtained as follows: B, is
O(e*), from (2-21). If our expansion is consistent (as we insist),
then A, is also O(e®), by (2-20). Now, in the outer expansion of
the inner expansion, the B, term is lower order than the A, term.

685

Ogilvie

The two-term outer expansion of the two-term inner expansion is:

o(x,y,z;€) ~ Ux +4 B (x,zse) ly| .
O(1) Ofe)

On the other hand, the two-term inner expansion of the two-term
outer expansion is, from (2-13),

o(x,y,z3€) ~ Ux + a, (x,z3€).

There is no linear term here at all, and it seems that we cannot
match the two expansions.

It is a very comforting feature of the method of matched
asymptotic expansions that things go wrong this way when we have
made unjustified assumptions. Our mistake was this: When we
found that apparently B, = €°UH, = O(e?) » we eliminated the possi-
bility that there might be a term which is O(e) in the inner expan-
sion’. Now we rectify this error. Once again, let ®, be given by
(2-20), but suppose that both "constants" are, in fact, O(e). The
body boundary condition immediately yields the condition that:

B,(x,z3€) = 0,
and so we have:
®, (x, Y,z;€) = Aj(x,z3€) .
The inner expansion, to two terms, is now given by:
$(x,y,z3€) ~ Ux t+ A; (x,z3€).

When we match this to the inner expansion of the outer expansion,
we find that:

A, (x,23€) = a,(x,z5€) = $(x,0,z5€).

(See 2-11.) Now we have matched the expansions satisfactorily, but

*This trouble would have been avoided if I had started by assuming
that the expansion is a power series in €, as many people do in such
problems. However, that procedure can lead to even greater diffi-
culties sometimes.

686

Singular Perturbation Problems in Shtp Hydrodynamics

the result is not yet of much use, since we do not know either function,
A; or a,. It is worth noting, however, that the inner expansion can
be rewritten:

(x,y »Z;€) ~ Ux + $,(x,0,z3€).

Thus, to two terms the inner expansion is determined entirely by the
far-field solution, the latter being evaluated on the centerplane. In
other words, in the near-field view, the fluid velocity (to this degree
of approximation) is caused entirely by remote effects.

Solution of the @,; problem: This is much more straight-
forward, and the results are more interesting. We may expect that
@p = O(e*), since we still have the nonhomogeneous body condition
to satisfy. In this case, then,

@,(x,Y,z3€) = A,(x,z3e) + Bo(x,zse)|Y|,

and the body condition requires that B,(x,z;€) = €°UH,(x,z) « The
three-term inner expansion is:

o(x,y,z;€) ~ Ux + a (x ,2;€) 2 A,(x,z5€) + UH, (x,z) |Y
O(1) Oe) O(e*) Ole")

The two-term outer expansion of this three-term inner expansion is:

P(x,y,z3€) ~ Ux t+ a,(x,z;€) + Uh,(x,z3€) ly
O(1) O(e) O(e)

The three-term inner expansion of the two-term outer expansion is,
from (2-13):

1
o(x,y,z5€) ~ Ux + a,(x,z5€) +5 ly |o,(x,z3e).

(The a, in (2-13) is not carried over to the above expansion, since
it originates in the third term of the outer expansion.) These two
match if:

o, (x,z3€) = 2Uh,(x,z;e€) = O(e). (2-22)

Thus, finally, we have found o,(x,2;€), the source density in
the first far-field approximation, as a function of the body geometry.
It is the familiar result from thin-ship theory. In addition, we can

687

Ogilvte

now also write down a,(x,z;€) by combining (2-7) and (2-11):

tee ae Uh Cre) de de _
| ae aS Sy [(e-8) + (tye

We have the two-term outer expansion -- with everything in it known
-- and the three-term inner expansion -- with the "constant"
A,(x,z;€) not yet determined.

Solution of the higher-order problems: From the [L] con-
dition, (2-17), it can be seen that &, (x, Y,z;€) is not linear in Y.
However, the differential equation for @) is easily solved, the body
boundary condition can be satisfied, and matching can be carried out
with the outer expansion. The result is:

$3(x,Y,z;€) = A,(x,z;€) + B,(x,z3€) ly | - 5 Ya, + a.)
where
B3(x,z3€) = €[ (a, H) + (aH), ],
A,({x,23€) = a(x, Z5€).
We also obtain o,, through the matching,
o(x,z3€) = 2[ (ah), + (a, hj],

and this information also gives us ap and Ap.

Summary: Symmetrical Body. The results for both near- and
far-fieId expansion are stated in terms of the far-field coordinates
(the natural coordinates of the problem) in Table 2-1. Ina sense,
the results are rather trivial. There could be difficulties near the
edges of H, but, barring such possibilities, the inner expansion
could be obtained from the outer expansion and then matched to the
body boundary condition. This is actually the classical thin-ship
approach. The outer expansion is uniformly valid near the thin
body, except possibly near the edges.

In the classical approach to the thin-body problem, there is
usually a legitimate question concerning the analytic continuation of
the potential function into the region of space occupied by the body.
Sometimes one avoids the problem by restricting attention to bodies
which can legitimately be generated by a sheet of sources, but this
is not very satisfying. The method of matched asymptotic expansions
avoids the question altogether by eliminating the need to ask it. What

688

Stngular Perturbation Problems in Ship Hydrodynamics

TABLE 2-1

SYMMETRICAL THIN BODY

Near-Field (Inner) Expansion y = O(e)

(x, yszse)~™ Ux + o,(x,23e) + ,(x,z3e) + $a, (x, ze) ly|
oo cuEEESESEEEEENES aumeenEmEeeet
O( ) Ole) O(e*)

i i 2
tas(x,z3€) + > o,(x,23€) ly] - > lyl (a + 1.)
ER ee

O(e°)

+ Ole’)

Far-Field (Outer) Expansion y = O(1)

N
e ~ = Jt. Gnlesose) d& dG
o(x,y,Z3€) ~ Ux = » Op Tleeb)e+ yy? + (zt) =

~ an e A Gary! 2

O(1) O(e")

From Matching

o,(x,z;€) = 2Uh,(x,z5€) ;

o,(x,z3€) = 20 (qh), + (a, h),] 5

etc. s

t n(S,G3€) do d
@,(x,25€) = - Fe aie Pex-f)” + (n-tr] ate = : z

689

Ogilvie

we are really saying is this: From very far away, the disturbance
appears as if it could have been generated by a sheet of sources,
but close-up we allow for the possibility that this observation from
afar may be somewhat inaccurate. In fact, there is no analytic
continuation presumed in the present method.

One can show by the use of Green's theorem that the far-
field picture is valid even if the analytic continuation is not possible.
A particularly appealing (to me) version of such a proof has been
provided by Maruo [1967] for the much more complicated problem of
a heaving, pitching slender ship moving with finite forward speed on
the surface of the ocean.

I suppose that the uniformity of the thin-body solution is the
result of the fact that a well-posed potential problem can be stated
by giving a Neumann boundary condition over a surface. The situ-
ation will be quite different when we consider slender-body theory:
in the far field, it would be necessary to give boundary conditions
on aline, and that does not lead to a well-posed potential problem
in three dimensions. Similarly, we may expect trouble at the con-
fluence of two boundary conditions, and this indeed occurs when we
try to treat a ship problem by the method discussed above. The
free-surface conditions cannot be satisfied, and the difficulty can
be traced back to the behavior of the far-field potential near the
intersection of the centerplane and the undisturbed free surface.

2.12. Unsymmetrical Body (Lifting Surface). For the sake
of simplicity, Tet the body have zero thickness. Then it can be

represented as follows:

y = g(x,z3e) = €G(x,z) for (x,0,z) in H, (2-23)

where H is now the projection of the body onto the y=0 plane.
Again, there is a uniform incident flow in the positive x direction.

The analysis is quite similar to the symmetrical-body case,
at least in the near field, and so most ot the details will be omitted
here. In the near field, let there be an expansion:

N
(x,y ,235€) ~ > @n(x,Y,z5€),
n=O
just as in (2-16). The first term is, again, @o(x,Y,z;€) = Ux. The
terms again satisfy the transformed Laplace equation, (2-17):

+ i: + ead
fel Porch Dean ier ae

yy ‘4yy
2
T= Cf Oi) ) TOD ot Oost ia sti abel >

690

Singular Perturbation Problems tn Ship Hydrodynamics

the body boundary condition is now:

[H] &, +2 +3, +O, +...

Y
a
ex { S [ UG, + (dG, + ie G;) aR (Bp Gy I: ®, G,) lj oe =|

on Y =1G (x77). (2-24)

The solution for ®, is generally an expression linear in Y, but,
for the same reasons as in the symmetrical-body problem, only the
"constant" term can ultimately be matched to the far-field solution,
and so we take for 9®,:

&,(x,¥,23€) = Aj(x,zse) = Ole).

The superscript + has been attached to the solution to indicate that
this quantity may be different on the two sides of the body. This was
not necessary in the previous problem, because of the symmetry, but
in the present near-field problem the body completely isolates the
fluid on its two sides and there is no reason to assume that A; is

the same on both sides of the body. (It turns out, in fact, that

Aj = - Aj -)

One next obtains:
+ +
$,(x,Y,2;€) = A,(x,z3€) + Ba (x,zse) Y .

From the body boundary condition, the following is true:

+
$2,(x,G,z3€) = Ba (x+z5€) = "UG, (x,z) : (2-25)
Thus, we find that
Ba(x,z3€) = Bg(x,z;€) = B,(x,z3e).

Similarly, one can proceed:

1 2.2
$,(x,Y, ze) = A, (x,z3€) + B sboze)¥ -5e YA, + Ay),

where

+ ‘ 2 se oe
B,(x,z;€) = €[(GA,) + (GA, ),].

691

Ogtlvie

It is interesting to note the following about the symmetry:
It turns out that ®, and @®» are odd with respect to Y, but 3
is neither even nor odd. The linear term in $3 , namely,
Ba(x,2e)¥, is even, since it turns out that B; = - Bs. Careful
study of the ®, problem shows that it actually implies that there is
a generation of fluid in the body, but the rate of generation is higher
order than the ®,; term. Physically, of course, there can be no
fluid generated, and so a compensating source-like term appears
in 5.

The far field is again the entire space except for the plane
y = 0. The relations (2-3) to (2-6) are again valid, as well as the
discussion of them. But now it will not suffice to provide only source
singularities on the centerplane; clearly we must also provide
singularities which lead to antisymmetric potential functions. In
fact, since the body has zero thickness, weshall expect the leading-
order approximation to be strictly antisymmetric. These require-
ments are all met by a distribution of dipoles which are oriented with
the y axis. The potential of such a sheet of dipoles can be expressed:

f(x,y,z) = ee eS rey
ae aa (ees Pia sie

The inner expansion of such an integral can be obtained by the same
Fourier-transform technique that was used before. One finds that:

Lyi(ke* mey2

**(k3y3m) = = (sgn y) pb” “(k,m) e :

The exponential function can be expanded into a series, which is then
inverted term-by-term. Define a new function (cf. (2-10)):

2 ye a 388

Oe eau ¥ mn (kc in)iee~ek Ee — ee

The following relationships exist between the two functions ,(x,z)
and y(x,z):

béee) = ie oa (0) a de (2-28)

“0 [(x-€) + (z-t)*]

aos Lem + Hee) do oe E
y(x,z) = a0 ie 7 ee re ae Vv ; (2-29)

692

Singular Perturbation Problems tn Shtp Hydrodynamics

(Note the comparison between (2-28) and the relation between a,
and on in Table 2-1. In fact, (2-29) gives the inversion of the
formula in Table 2-1.) The inner expansion of f(x,y,z) can now be
written in terms of these two functions:

f(x,y ,z) = F{wlx,2(sgn y) - y(x,z)y - + y(sgn y) (jy, + Hg2)

1 1
7 31 y7(Yxx + Vz) + 41 y(sgn Y) (pexxxx + 2uxx22 + zzz) +.. “ °
(2-30)
This may be compared with (2-12).

Now let us assume that the two-term outer expansion is:

00

° °
$(x,y,z;€) ~ Ux ES (€,0;€) d& d&

-0 [ (x-§)° ty” + (x-t)*] 7

Furthermore, assume that p; and y, are both O(e). (If these
assumptions are too restrictive, that fact will become clear in the
subsequent steps of the method of matched asymptotic expansions.)
Then the inner expansion of the two-term outer expansion is:
4
Ax,y ,2z3€) ~ Ux + p(x, z3€) - y¥,(x,23€) = 5 yh, +p)
O(1) O(e) O(e?) O(e>)

I have kept four terms, as indicated by the order-of-magnitude
notes under the terms. (Recall that y = O(€) in the inner expan-
sion.)

Matching with the appropriate forms of the outer expansion
of the inner expansion, we find that:

=
A, (x,25€) = + p,(x,z5€)3 (2-31)

B5 (x,z3€) = - ey, (x,zZ5€). (2-32)

From (2-25), we find that:
¥; (x,z3€) = - €UG,(x,z) =a= Ug,(x,z3€). (2-33)
It appears now that we could use this knowledge of y, in (2-28) for

determining p,. But this is wrong. Note from (2-30) that yj,(x,z)

693

Ogilvie

is the normal velocity component on the y = 0 plane caused by the
distribution of dipoles, pw,(x,z), over the same plane. Now we
would presumably restrict the dipole distribution to the region H,
and so (2-29) is valid if the range of integration is reduced to just
H, since the integrand is identically zero outside H. But the same
is not true in (2-28). There is a generally non-zero normal com-
ponent of velocity, y,(x,z), over the entire plane, and the range of
integration in (2-28) cannot be reduced to just H. Unfortunately,
we know y;(x,z) only on H, from (2-33), and so we have solved
nothing.

This difficulty is hardly surprising, since we are really
formulating here the classical lifting-surface problem, and its
solution requires either the solution of a two-dimensional singular
integral equation or the introduction of further simplifications --
which will be discussed presently.

In the lifting-surface problem, we really should distribute
dipoles over two regions, the centerplane H and the part of the
plane y=0 which is directly downstream of H. Let the latter be
called W. Pressure must be continuous across W, since there is
no body there to support a pressure jump. In the usual aerodynamics
manner, one can then show that dp,/8x must be zero on W. In
this way, the integration range in (2-29) can be reduced to an integral
over just H.

Of course, lifting surface theory is usually worked out in
terms of vorticity distributions. I happen to prefer using dipole
distributions, mainly because then I do not have to worry about
whether a vortex line might be ending in the fluid region. The con-
nection is fairly simple between the two versions, of course. A
single discrete horseshoe vortex extending spanwise between z=s
and z=-s and downstream to x = oo corresponds to a sheet of
dipoles of uniform density, spread over the plane region bounded
by the vortex line. The potential function can be written, for unit
vortex strength,

6 @
us -\ d ae = (sdb xh A
eisebc Lc! aaa [ (x-£)° + y? + (2-271?
Ss
it LGN ere se
Seis y? + (2-0)? [ ety roan |

2

‘ Af a / 2 Leije
=| tan! - tan Ce... [xt ty" tlassy]
Z-s Zak x(zZ-s

Singular Perturbation Problems in Ship Hydrodynamics

The normal velocity component in the plane of the vortex is:

1/2

Meme at (ye tee) fi oe ee)

Z-Ss ZTS

A lifting line can be described in a similar way if we allow
the dipole density to vary with the spanwise coordinate, z. For
simplicity, let us assume that p(z) = p(-z), and that p(s)=0. The
potential for a lifting line is:

ul

s 00 at
(xy +2) x5 dt ute) f eC era ICT (2-34)

(x-€)* + y? + (2-2)? ]¥?

s s' rr)
cae Ah ds* u's! d hoe VEeitole ness
oa s' p(s f 4 \ [ (x-6)2 ty? + (2-EF]2”
(2-35)

and the normal velocity component is:

s t !
$)(x,0,2) = - maf ds" p'(s!) E + Lx? + (z-s'P ee als | mi(easa}
-$

Z-S

Note that this reduces to the result for the single horseshoe vortex

if a) we set p'(z) = 6(zts) - 6(z-s)™, and b) we integrate over a span
from -s-f to s+®, where B is a very small positive number. This
may lend some credibility to the procedure frequently advocated by
aerodynamicists in wing problems, viz., when integrating by parts
in the spanwise direction, extend the range of integration slightly
beyond the wing tips so that quantities which become infinite at the
tips do not yield infinite contributions that cannot be integrated.

(This is terrible mathematics, but apparently the physics is sound,
since the results seem to be correct.)

Finally, we can use the above procedures to derive the cor-
responding expressions for a lifting surface. The important quantity
is the normal velocity component, given by:

s(€) 2 21 1/2
$,(x,0,2) = J at ee: wal fy 4 LosnBik + (2-oF)*)

(2-37)
*
6(z) is the usual Dirac delta function.

695

Ogilvte

where L is the range of x covered by the lifting surface (the
length of L being generally the chord length), and s(x) is the half-
span at cross section x. On H (the projection of the wing on the
plane y = 0), the normal velocity component, dy, is known, either
by direct application of the body boundary condition or by matching to
a near-field solution, and we obtain the usual integral equation for a
lifting surface.

We shall not be concerned here with the various methods of
attempting directly to solve this integral equation, either by analyti-
cal or numerical methods. In fact, analytical methods do not exist,
so far as I know, except for a few special geometries, such as
elliptical planforms. The pair of equations (2-28) and (2-29) forms
a remarkable analogy to a standard boundary-value problem in two
dimensions which is analyzed thoroughly by Muskhelishvili [1953].
One three-dimensional case has been solved analytically by a method
that has some similarity to the standard methods for the 2-D prob-
lem; this was done by Kochin [1940]. Even his circular-planform
wing led to so much difficulty, it seems unlikely that it will be
generalized to other planforms.

Analytical solutions have also been obtained for circular and
then elliptic planforms by formulating the problem in terms of an
acceleration potential in coordinate systems appropriate to such
shapes of figures. This was all done long ago. See Kinner [ 1937]
and Krienes [1940].

There are many numerical techniques for obtaining approxi-
mate solutions of this problem. However, I ignore these and proceed
to analyze a special configuration which can be treated approximately
as a limiting case of the general lifting-surface problem.

2.2. High-Aspect-Ratio Wing

It is an interesting historical fact that Prandtl's boundary-
layer solution really contains the essence of the method of matched
asymptotic expansions, but Prandtl failed to observe that the same
technique would work in his lifting-line problem. In the boundary-
layer problem, he really required the matching of two complementary,
asymptotically valid, partial solutions. It was probably Friedrichs
[1955] who first recognized that the high-aspect-ratio lifting-surface
problem could be treated the same way. Van Dyke [1964] discusses
the derivation of lifting-line theory in some detail from the point of
view of matched asymptotic expansions. My presentation is not
different from Van Dyke's in any startling ways. There are some
differences, partly because I have in mind applications to planing
problems eventually, partly because I am not an aeronautical (or
aerospace) engineer at heart.

The conventional approach to solving the problem of a wing of

696

Stngular Perturbation Problems in Ship Hydrodynamics

high aspect ratio is to simplify (2-37) by arguments that relate the
sizes of the terms involving (x-&)© and (z-t)*. (Quite comparable
arguments are used in the conventional approach to the theory of
slender wings.) If the radical in (2-37) can be simplified, then the

€ integration can be performed, and one is left with just the integral
over ¢. In this way, the 2-D integral equation is reduced to a one-
dimensional integral equation, which is of a standard form.

Using the method of matched asymptotic expansions, we
return to the original formulation of the problem and derive a
sequence of simpler problems, rather than try to work out approxi-
mate solutions of the integral equation. The large-aspect-ratio
wing is "slender" in the spanwise direction. This means that cross
sections parallel to the z=0 plane vary gradually in size and shape
as z varies; in particular, the maximum dimension inthe z
direction, say 2S (the span), is much greater than the maximum
dimension in the cross sections. We shall make whatever further
assumptions of this kind that we need in order to keep the solution
well-behaved. The small parameter can be defined as the inverse
of the aspect ratio, that is,

e€ = 1/(AR) = (area of H)/4S*,

where H_ is the projection of the wing onto the y=0 plane. As
before, it is not necessary to be so specific about the definition of
€, and in fact it may be misleading. A wing with aspect ratio equal
to 100 might be slender in the required sense if, for example, there
were discontinuities in chord length in the spanwise direction. In
any case, the wing shrinks down to aline, part of the z axis, as

€ 7; 0..

Let the body be defined by the following relation:
y= g(x,z) = h(x,z), (2-38)

for (x,0,z) in H. See Figure (2-1). It is not necessary that the
body be a thin one, in the sense of the previous section. I do,
however, specify that it should be symmetric with respect to z,
for the sake of simplicity in what follows. Both of the functions
g(x,z) and h(x,z) really depend on €, of course*, but we shall
generally omit explicit mention of the fact.

There is an incident flow which, at infinity, is uniform in
the x direction. Let the far-field solution be represented by the
asymptotic expansion:

*
In fact, g and h are both O(e).

697

Ogtlvie

Y= 9 (*,2)

2h(x,2)
Fig. (2-1). Coordinates for the High-Aspect-Ratio Wing

N
o(x,y,z) ~ Ux + yS $,(x,y,Z), where $a = o(4,) as e~0O,

n=l
for fixed’ '(x,y;z) + (2-39)

(Again, the dependence on € is suppressed in the notation.) Since
the body shrinks to a line (x= 0, y=0, |z|<S) in the limit as

€ > 0, the terms denoted by $, all represent flow perturbations
which arise in the neighborhood of this singular line. They can be
expressed in terms of singularities on that line, and the strengths

of such singularities should be o(1) as € ~ 0. In an ideal fluid,

we could expect the occurrence of dipoles, quadripoles, etc., on the
singular line. We also take the realistic point of view that viscosity
cannot be completely neglected and that there may be some circula-
tion as a result. In the usual aeronautical point of view, this implies
that there may be a vortex line present, complete with a set of trail-
ing vortices. Inthe point of view adopted in the previous section,

I assume that there may be a sheet of dipoles behind the singular line.
I also make the usual assumption that these wake dipoles (or vortices)
lie in the plane y =0. This part of the y=0 plane (0<x<o,

|z| <S) will be denoted by W. (Note that H_ has all but disappeared
in the far field view. It is only a line.)

We can now write the outer expansion in the following form:

698

Singular Perturbation Problems tn Ship Hydrodynamics

N i
Ss 00 :
Hxsy.2)~Ux+Z ) ae (o) dé ae
n=l

“SO [(x-6) ty + (x-€)]

+L #2 \ tal te pee 1s ees dg Sf

“S[x> ty? + (z-£)

N
SO i An(6) at ;
ht Set (2-40)
ae i. [ x? ty? + (z- pie

The first sum contains terms which are exactly of the form given
in (2-34), that is, they represent a lifting line with a strength

yn(z). The second and third sums represent lines of dipoles
Oriented vertically and longitudinally, respectively. It is implied
above that the sums are asymptotic expansions, in our usual far-
field sense.

We shall presently require the inner expansions of these

ferme. We obtain the inner expansions by assuming that
r= (x? + y 2) Ve = O(e€), which implies that both x and y are small.

Inner expansion of the lifting-line potential: Each of the
double integrals containing a y can be rewritten as a single integral:
4)
y { i ( y(t) dé dg
TJ Jo 2 2 2, 2

[(x-§) ty + (z-¢)]

i) VE x2 + y2 + (z-0)*]
== AY” at yen (ton! ty + tant! what ey BF] ) |

(2-41)
Now break this into two parts:
1) The first term in brackets on the right-hand side does not

depend on x. As yO (i.e., for y = O(€) ), its contribution can
be represented:

Ps '
ezved- By SVE [1+ of),

where the double sign is chosen according to whether y>O or

y < 0, respectively, and the special integral sign indicates that the
Cauchy principal value is intended. This representation is valid only
for |z|<S, but that is no restriction here. It may be noted that

699

Ogilvie

Every term in (2-51) must be of the same order of magnitude
at a point in the near field, that is for r= O(e). If aterm were
of some other order of magnitude with respect to €, the definition
of "consistency" would eliminate it from this series. The orders
of magnitude of most of the unknown constants can then be written
down. Since the first term, Ux, is O(€), we can make the follow-
ing statements:

2 4
Nor Agg= Ol€)s Agr» Boy= O(€)3 Aon» Bon = Ole i:

The term containing the logarithm does not fit the pattern quite so
well -- unless we follow my arbitrary practice of saying that

log € = O(1). (See the discussion of "consistency" in Section 1.2.)
Then we can say that:

So = O(e).

The Laurent series expression for the near-field expansion
is very convenient when it comes to finding the outer expansion of
the inner expansion. All we need to do is to interpret r differently
and rearrange the terms according to their dependence on €. Thus,
if we consider that r = O(1), the outer expansion of the one-term
inner expansion is:

(x,y,z) ~ Oo(x,y3z) ~ Ux + S69 log r + No fan y + Aoo
O(1) Oe) O(e) Ole)
+ Aor cos © 4 Sorsin’ + of). (2-51')

O(e*) O(e*)

This obviously matches the one-term outer expansion, with an
asymptotically small error which is O(e).

We could keep two terms in (2-51'); that would be the two-
term outer expansion of the one-term inner expansion, which would
have to match the one-term inner expansion of the two-term outer
expansion. From (2-51') and (2-45b), we thus construct the equality:

-| 4 4 “I
Ux + 6, log r + m)tan Y + Ago= Ux +5 y, (2) [1- > tan x].
This can be true only if the following are separately true:

1 1
55 = 0; NO = Y,(z)3 Aoo =F y,(z)- (2-52)

700

Singular Perturbation Problems tin Ship Hydrodynamics

N
x,y .z)~ y @.(x,y»Z), yy = 01%) ‘as -Ee= 0,
n=0
with (x/e, y/e, z) fixed. (2-48)

The first term in this expansion satisfies the conditions:

fo. 7 20, = 0 inthe fluid region, (2-49)
Eto = 0 onthe body. (2-50)

From (2-45a), it is clear that the one-term inner expansion,
$(x,y,z) ~ &o(x,y,z), must match the one-term outer expansion,
6(x,y,z) ~ Ux. Thus @o(x,y,z) is the solution of a two-dimensional
potential problem, and a rather conventional problem at that: Ina
section through the body drawn perpendicular to the spanwise axis,
the potential satisfies the Laplace equation in two-dimensions, a
homogeneous Neumann condition on the body, and a uniform-flow
condition at infinity. The direction of the uniform flow is the same
as the direction of the actual incident stream as viewed in the far
field.

Since @) does satisfy the Laplace equation in two dimensions,
the methods of complex-variable functions are available for deter-
mining its properties. In particular, if we assume that V@p is
bounded everywhere in the fluid region and single-valued too, then
@9 can be expressed as the real part of an analytic function of a
complex variable, the analytic function being such that its derivative
can be expressed by a Laurent series. Thus, we can write for
@o(x,y3z):

a re)
@)(x,y3z) = Ux + 6) log r + nm) tan £ 2 Ago + Aoi ces

+ Bo sin 0 + Az cos 20 + Boz oa 20 Itt oe Sah
r
r r

where r = (x? +y?)'/2, The "constants" are all unknown functions of
z, the spanwise coordinate. The first term represents a uniform
stream at infinity, and I have already performed one matching to
determine this term. The second and third terms represent a source
and a vortex, respectively; the fourth term, a constant, is included
for generality; the fifth and sixth terms represent a dipole; etc.

Such an expansion as (2-51) is valid outside any circle about the
origin which encompasses the body cross section.

701

Ogtlvie

I have taken the trouble of writing out the inner expansion of the
outer expansion in three ways just to point out how, in this problem,
there is an additional term in the lowest-order expression each time
we add another term of higher order in the outer expansion. Each
of the three terms metoded In (2-44) contributes to the € term in
(2-45c). This phenomenon occurs frequently, and its occurrence is
the reason that one must proceed step-by-step in the matching. In
the present problem, one would be in some difficulty if he tried to

write down an arbitrary number of terms in each expansion and
immediately start matching.

Next we formulate the near-field problem. Instead of making
the formal changes of variable, x= e€X and y = €Y, we shall simply
understand now that, in the near field,

x= O(e) and y=O(e); | also 0/ax= O(e!) and 8/8y = Ole").

Of course, differentiation with respect to z does not affect orders
of magnitude.

The Laplace equation can be written in the form:

9 TF Syy = - $27, (2-46)

where the right-hand side is e* higher order than the left-hand side.
The boundary condition on the body is:

0 = 6,(g,+h,) - 6+ $(g,#h,) on y=gth. (2-47)

The last condition is equivalent to requiring that 8¢/én =0 onthe
body, where 8/8n denotes differentiation in the direction normal to
the body surface. An alternative statement is the following:

) S (+ gx + hy) by Fy _ _ (hz + gz)$z =gth. (2-47!)
aR v1 + (gy + h,)*] v[ 1 + (gy + h,)] en P

where 986/@N is the rate of change in a plane perpendicular to the
z axis, measured in the direction normal to the body contour in
that cross section plane. Note that the left-hand side is O($/e),
since differentiation in the N direction has the same order-of-
magnitude effect as differentiation with respect to x or y. The
right-hand side, on the other hand, is O(ge), since g and h are
both O(e).

Now let there be an inner expansion:

702

Singular Perturbation Problems tn Shtp Hydrodynamics

y,(z) = Ole); w(z), Aylz), y9(z) = O(€*);

also, all other terms in (2-40) are o(e?). These statements can
all be proven. The description of the problem is greatly simplified,
however, by their being assumed now.

We can write the three-term outer expansion now:

(x,y,z) ~ Ux + $\(x,y,z) + (x,y,z), (2-44)

where
Gay. )=% a AG de ; (2-44a)
OAXsy +z oh ae Me ve ae a i

S z. Y2(6) dé dt
Gea S, i [ (x-€)? +y? + (2-0F]¥?

fa Lyp, (6) +xa,(0)] do . (2-44b)
Se ey leet

The inner expansion of the one-term outer expansion is, of course:
$(x,y,z) ~ Ux, [ Ofe)] (2-45a)

to any number of terms. (Recall that x = O(e) in the near field.)
The inner expansion of the two-term outer expansion is:

(x,y,z) ~ Ux “s y, (z) [1-= 2 4an ¥ | [ O(e)]
(2-45b)

£5 sx ob seo [ O(e?)]
Finally, the inner expansion of the three-term outer expansion is:
+
(x,y,z) ~ Ux +5 y,(2)[ 1 - = tan -| x | a. (z) xr (z) [ O(e)]}

2n(x? + y%)
(2-45c)

Ss
dé yilS) 4 1 Laren 2
-£5. ME) 4 Sy tey[1-Ltan' ZL]. [ove

703

Ogilvie

this term represents a distribution of vorticity extending to infinity
both upstream and downstream. Thus, it leads to a discontinuity
across the y=0 plane, even upstream. The second term must
compensate for this behavior, since there can be no discontinuities
in the region x <0.

2) The second term in brackets on the right-hand side of
(2-41) must be considered carefully with respect to the branches of
the square-root function. With a bit of effort, one can show that,
as r— 0, its contribution is:

x2) (1 - 2 tan! XY) (1 + Ofe*)), 0 < tan’ L<n;

Vie) (3 - = tan’ Y) ( 4 + O(e*) ) 5 Ge << tank Y< ae

Combining this result with the previous one, we find that the

inner expansion of a lifting-line potential function can be written as
follows:

ye q ie ¥(6) dé dg
4m Je YO [ (x-8)° ty? + (2-4)?

et tinsel ¥) ove (Se xile) | 2
[ 5 wer [1-4 tan x}-xS" z-t [1+ oe],
1 y
for O< tan = <2n. (2-42)
Inner expansion of the dipole-line potential: An integration
by parts with respect to € transforms these integrals into an
appropriate form so that one can let r—~O and thereby obtain the

first terms in the desired expansions. Typical terms in the second
and third sums of (2-40) have the following inner expansions:

)
es 29) OO ere cal
ao iF [x* + y° t (2-4) e* [ 2n(x* + y*) J [ As: 198 e| ( )

Note the occurrence of the logarithm of ¢€|!

Inner expansion of the outer expansion: In order not to con-
fuse the picture, I shall make more as sumptions now, namely:

704

Singular Perturbation Problems in Shtp Hydrodynamties

The first of these three equations means only that there is no net
source strength in the 2-D problem. The second relates the 2-D
vortex strength, 1, to the dipole density, y,, in the far field.

The latter can obviously be interpreted also as a vortex strength.
The third quality relates the "constant" term, Ago, in the near-field
solution to the far-field solution's dependence on z, the spanwise
coordinate. It is important to include such a term as this in the
near-field solution, because it provides a three-dimensional effect
in the otherwise two-dimensional problems.

Presumably, the near-field problem can be solved somehow.
If the body is simple enough, an analytic solution may be obtainable;
with the available powerful methods of the theory of functions of a
complex variable, it is even reasonable to hope to find such solu-
tions. However, even if numerical methods must be used, the
solution can be found. Then all of the constants in (2-51) except
Apo are known. The constant of most interest at this moment is
N,; it will be non-zero only if some mechanism has been included
that can generate and determine a circulation around the body. I
shall assume that a Kutta condition is available for this purpose,
since the present section is concerned with wings. Then, with tq
known, we can find the first approximation to the vorticity (and
dipole density) in the far field, by means of (2-52). At the same
time, Ago is determined.

Nothing more can be done now unless we find a higher-order
term in either the near- or far-field expansion. It is interesting
to pursue the near-field solution further first.

When we substitute the expansion, (2-48), into the Laplace
equation, (2-46), and keep only leading-order terms, we obtain the
partial differential equation for 9:

G1, + Piyy = - &o,,, in the fluid domain.

Now @9 was found to be O(€), and we might reasonable expect that
@, would be O(e"). In fact, this turns out to be quite correct. In
the equation just above, this means that the left-hand side is O(1)
and the right-hand side is O(e). Asymptotically, then, we have
that:

Gm +O. =0 in the fluid domain.
Xx yy

We again have a purely two-dimensional boundary-value problem
to solve, if we can state the boundary conditions appropriately.
From (2-47'), we find by the same arguments that:

O®,
aN

0 on the body.

705

Ogilvie

We do not know the conditions at infinity yet, but let us assume that
the condition on ©, is similar to that on 4p, i-e., the gradient of

G, should be bounded.

This problem is identical to the @g9 problem, and so we can
represent its solution outside of some circle by another series like
the one in (2-51). We have not determined yet what the coefficients
of the increasing terms are like, and so we allow two more arbitrary
terms (the first two terms in the following):

-| A 8
@, (x,y;z) = @,x +B,y + 6,log r + n, tan x + Aio + Sle

ueaae pee Ses SS + ale Se Se too neaees)
r 1g:

All terms must be the same order of magnitude if r = O(e).
Assuming that order to be €', we have:

ai, B, = O(€); nn Ty Ajo = O(e*); Ai By = Ole); ete.

With this information in hand, we combine the first two terms
in the inner expansion and then we obtain the outer expansion of the
two-term inner expansion:

(x,y,z) ~ Ux O(1)

+ Notan™ Z + Ago + %x + Bry Ole)
+ Soi cos 8 + Bor sin 9, 6, logr +n, tan’ x + Aig. o(e*)
(2-54)

First we can keep just the first two orders of magnitude and match
them with the two-term inner expansion of the two-term outer ex-
pansion, given in (2-45b). Using (2-52), we see that everything
already matches except for the terms a,x + B,y and the integral in
(2-45b). For these to match, we require that:

Fear pC Sete) (2-55)

Physically, this means that the ©, problem should have had as the
condition at infinity:

706

Singular Perturbatton Problems tin Shtp Hydrodynamics

ld - Biy| +0 as r=" oo;

that is, there is a uniform stream at infinity, moving at a right angle
to the actual incident uniform stream. This is the downwash velocity.
With this condition at infinity known, the ©, problem can be solved
by the same method used for the 9 problem, and all of the terms

in (2-53) are then known, except Aig

We have all of the information available to match the three-
term outer expansion of the two-term inner expansion with the two-
term inner expansion of the three-term outer expansion. Using
(2-45c) and all of the terms in (2-54), we obtain the equation:

-|
Ux + No tan x + Aoot Biy + Aa_cos 6 sae g + osin? 0

+6, logr +n, tan” zt

- a et Belly, ypy(z) + xd) (z)
+ Aig = Ux + 7) y, (z) [1-4 tan x | a ance ; 2)

ES See! 5 velz) [1 - + tan x].

The unknown quantities are: Ajo, HW,» hy, Yee This equation is satis-
fied only if:

1 i
6, = 0; Rar hs vn Yo(z); Aio = = Yal2);

p,(z) = 27Bo); A(z) = 217A).

From this matching step, we see that all quantities introduced so
far are now Sp EN known. There is no source strength in the
second approximation”; there is a correction to the vorticity in the
far-field description; there is a correction to the "constant" in the
near-field problem; and the density of both vertical and longitudinal
dipoles in the far field is known. It is interesting to note thatthe
last were determined entirely from the lowest-order near-field
solution, that is, from Pp. When quadripoles first enter, it will be
found that they too are determined in strength from 9% 9 solely.

The next term would be much more difficult to obtain, since,
in the near field, it entails solving a Poisson equation in which the

*
It would have been possible to eliminate the 6 log r terms in both
problems above by noting that the body boundary conditions allow for
no net source strength.

Ogtlvte

nonhomogeneous part depends on ®o, which we have not obtained
explicitly. (The right-hand side of (2-46) finally has an effect.)
Also, spanwise effects occur in the body boundary condition, (2-47),
for the first time. Therefore I shall put this problem to rest at this
point. Table 2-2 shows the sequence of steps that we have followed
in this problem.

TABLE 2-2

HIGH- ASPECT-RATIO WING -- SUMMARY

eras Far-Field Near-Field Quantity Determined
Expansion Expansion by Matching
4
4 $= Ux =——> 14 Condition at infinity for
®, problem
3 {e)
2 Vorticity, y,(z), in far
field
2 , + $,<— +4, + ®, 3 Downwash velocity (condi-
tion at infinity for 9,
problem)

4 Correction to vorticity in
far field; densities of
vertical, horizontal

: Fo #13 Ge dipoles in far! field

One point in particular should be noted: The near-field
problem was not linearized. If one can predict the flow around the
two-dimensional forms which appear in the near-field problem, one
is not limited to consideration of, say, thin wings. All that is
necessary is that the spanwise length be much greater than the
dimensions in the two-dimensional problems and that there be gradual
change in the body and flow geometry in the spanwise direction.
Needless to say, the latter condition is usually violated at the wing
tips, and so the analysis breaks down there. It may be hoped that
the prediction of important physical quantities is not affected too
seriously thereby, but higher and higher approximations certainly
cannot be found until the extra singularities at the tips are removed
somehow.

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Singular Perturbation Problems tn Shtp Hydrodynamics

223+. Slender Body

In the previous section, we considered the flow around a
slender body which was oriented with its long dimension perpendicu-
lar to the incident flow. Now we consider the flow around a slender
body which is oriented with its long dimension approximately parallel
to the incident flow. The same geometrical restrictions will be
applied to the body in this problem, namely, that its transverse
dimensions should be small compared with its long dimension and
that cross-section shape, size, and orientation should vary gradually
along the length.

Although both this section and the previous section concern
slender bodies in an incident flow, convention says that only this
section really presents "slender-body theory."

In ship hydrodynamics problems, slender-body theory has
been applied mostly to nonlifting bodies, i.e. , bodies not generating
trailing vortex systems. * T shall limit myself here to such prob-
lems too. Specifically, I assume that there is no separation of the
flow from the body; furthermore, there are no sharp edges at which
a Kutta condition might be applied. The potential function should be
continuous and single-valued throughout the fluid domain.

This restriction is not generally desirable. Certainly an
important aspect of aerodynamics is the calculation of lift ona
slender body which does generate a vortex wake; modern high-speed
delta-wing aircraft and many slender missiles are genuine slender
lifting bodies. There are several important ship-hydrodynamics
problems which may ultimately be best analyzed by a slender- wing!
approach. Most important, perhaps, is the problem of a maneuvering
ship. An attempt is made in this direction by Fedyayevskiy and
Sobolev [ 1963], but it is not very successful because they use the
conventional methods of slender-wing theory, and these break down
in application to wings which are not more-or-less delta shaped. +
A modern approach to slender-wing theory is given by Wang [ 1968].

*Obviously, a ship is a "lifting body," but I think it is commonly
understood that the term implies a dynamic lift process, and that
is the way I use it.

Tusiender wing," "wing of very low aspect ratio," and "slender
lifting surface" are all equivalent terms in my usage.

* Conventional slender-wing theory can be used for wings in which the
span increases monotonically downstream, ending in a squared-off
trailing edge. If the incident stream is uniform and steady, the
wing does not have to end at the location of the maximum span, but
the part of the wing aft of this location must be uncambered. Not
all of these conditions are satisfied in the interesting ship maneuver-
ing problems.

709

Ogtlvte

uae RADIAL COMPONENT

|
|
| FLUID VELOCITY

AXIAL COMPONENT To \

Fig. (2-2). Fluid Velocity Near a Slender Body
in Steady Motion

The physical ideas behind slender-body theory were developed
fifty years ago, and the original way of looking at this problem is
perhaps still the best way. Take a reference frame which is fixed
with respect to the fluid at infinity. As a slender body moves past,
one may imagine that its greatest effect on the fluid is to push it
aside; the body also imparts to the fluid a velocity component in the
axial direction, but this component should be quite small compared
with the transverse component. Both components should be small
compared with the forward speed of the body.

In modern slender-body theory, we attempt to formalize this
estimate of the relative velocity-component magnitudes. We devise
a procedure that automatically arranges velocities in the anticipated
order:

1) Forward speed
2) Transverse perturbation
3) Longitudinal perturbation

When this pattern comes out of the boundary-value problem, we
then investigate further to see what other patterns follow from the
same assumptions. The whole body of assumptions, results, and
intermediate mathematics constitute what we call "slender-body
theory."

In aerodynamics, the original intuitive approach of Munk was

not completely displaced until the late 1940's. The newer, more
systematic approach which developed then is described well by Ward

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Singular Perturbation Problems in Ship Hydrodynamics

[1955]. For the first time, it was possible to predict with some
confidence how the flow around the various cross sections interacted.
There were some difficulties in principle, even with the new ap-
proach; what we now call the "outer expansion" of the problem was in
effect forced to satisfy body boundary conditions. The difficulty is
somewhat comparable to trying to force a Laurent-series solution
to satisfy prescribed conditions which are stated on a contour inside
the minimum circle of convergence. A readable, refreshing account
Se ripe aoa theory in the 1950's has been provided by Lighthill
1960].

During the early 1960's, slender-body theory was applied to
ship hydrodynamics problems by several investigators. Probably
the earliest to try this on a major scale was Vossers [1962]; he
attacked a variety of steady- and unsteady-motion problems by
slender-body theory. He used a Green's function approach, which
apparently avoids the fundamental difficulty in principle of the
previous method. However, it is really too much to hope to obtain
asymptotic estimates of five-fold integrals -- without making mis-
takes. Apparently Vossers did hope for too much, but Joosen [ 1963]
and [1964] corrected many of his mistakes. Newman [1964] also
advocated the Green's-function approach and produced some inter-
esting results.

The modern (i.e., fashionable) alternative is to use the
method of matched asymptotic expansions. In ship hydrodynamics,
Tuck [ 1963a] first used this method in his doctoral thesis at
Cambridge University. It avoids the difficulties in principle of
Ward's approach, and it is easier to work with than the Green's-
function method. Of course, the method of matched asymptotic
expansions has its own set of difficulties of principle. However,
it is the method that I shall pursue here. *

In any case, the analysis can be no better than the assump-
tions which are made at the beginning. Therefore I shall be (perhaps
painfully) explicit about the assumptions.

2.31. Steady Forward Motion. Let the body surface be
specified by the equation:

*A very recent account of slender-body theory, particularly with
respect to its applications in ship hydrodynamics, has been pub-
lished by Newman [1970]. I think that his presentation and mine
generally complement each other (and perhaps occasionally contra-
dict too). Newman has provided a survey that seems comparable
in intent to the one by Lighthill [1960], mentioned above, whereas
I am trying to place slender-body theory into a hierarchy of singu-
lar perturbation problems. My emphasis is on the development and
application of the method of solution.

ceo

Ogilvie

ra ro(x, 8) x in K,

where r= (y? +t: z2)V2 , and @ is an angle variable measured about
the x axis. It will be assumed that r = O(e). In this section, I
take the most conventional definition of 8, namely, that it be
measured in a right-handed sense from the y axis. (In ship prob-
lems, it is more convenient to measure the angle from the negative
vertical axis.) A is the part of the x axis which coincides with
the longitudinal extent of the body; typically, one might take it to be
the interval, - L/2<x<L/2, butI shall not insist that the origin
be located at the mid-length section. Figure 2-3 shows a typical
cross section.

Fig. (2-3). Cross Section of the Slender Body

As usual, assume that there exists a velocity potential,
$(x,y,Z), which satisfies the Laplace equation. There is an incident
stream which, in the absence of the body, is a uniform flow in the
positive x direction, with the velocity potential Ux. It will be
convenient to use cylindrical coordinates, (x,r,®), in which case
the Laplace equation takes the form:

1 1
a Yr o, + ze P06" Pig Oe + fox ene, To(x, 8). (2-57)

The kinematic boundary condition on the body can be written:

1
9,70, - , + pi $6 og = 0 on r = rQ(x,9). (2-58)

With respect to the physical arguments presented at the

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Singular Perturbation Problems in Shtp Hydrodynamics

beginning of the section, note that our frame of reference is now
moving with the body. Therefore, the velocity components should
be ordered:

86/8x ~ U= O(1); 96/8y, 96/8z, 96/ar = o(1);

8( - Ux) /8x = 0(86/ar).

In each case, of course, the appropriate limit operation is that
€e—~>0, where e€ is the slenderness of the parameter. These order
relations should be valid near the body.

Far away, there will be the uniform stream, which is O(1),
but there is no reason to assume that the perturbation velocity will
have components with differing orders of magnitude.

These order-of-magnitude relations all come about auto-
matically if, in the near field, we define new variables:

r= eR, y=eY, Z=€Z,

and assume that differentiation with respect to x, Y, Z, R, and 08
all have no effect on the order of magnitude of a quantity. Thus,
suppose that the potential in the near field can be written: (x,y,z) =
Ux + @(x,Y,Z). Then the derivatives have the following orders of
magnitude:

Ob _ Od _ 0h _ Ob _ 1 OO _
ge Ut ges Ol) + OG); GEST = Ty = Olb/e);

882 0(6/e); 2= O(6/e).

It will turn out that @ = O(e*). This means that the transverse
velocity components, $y, $2, and 4, are all O(e), that is, they
are proportional to the slenderness parameter. Note also that a
circumferential velocity component would be given by (1/r)8o/80 =
(1/eR)d8@/80 = O(@/e), when we interpret R= O(1) (that is, in the
near field), and so circumferential and radial velocity components
have the same order of magnitude. The perturbation of the longi-
tudinal velocity component is O(®) = O(e*), which is, appropriately,
a higher order of magnitude than that of the transverse velocity
components.

In the far field, we assume that differentiation with respect

to any of the natural space variables has no order-of-magnitude
effect. Thus, we use the Cartesian coordinates (x,y,z) and the

(13

Ogilvte

cylindrical coordinates (x,r,®) in a very conventional manner.

As € ~ 0, the slender body becomes more and more slender,
shrinking down to a line which coincides with part of the x axis.
(This is the line segment that I defined as A previously.) In the
limit, there is no body at all and thus no disturbance of the incident
uniform flow. In the far field, the disturbance is always o(1).
Therefore the far field consists of the entire space except the x
axis, and the potential function must satisfy the Laplace equation
everywhere except possibly on the x axis.

At infinity, it is reasonable to require that the perturbation
of the incident flow should vanish, which implies that the pertur-
bation potential must be regular even at infinity. A velocity potential
cannot be regular throughout space, including infinity, unless it is
trivial. Therefore the velocity potential must be singular some-
where, and the only place in the far field where such behavior is
permitted is on the x axis. Our far-field slender-body problems
all reduce to finding appropriate singularity distributions on the x
axis.

The Far-Field Singularity Distributions. In the far field,
the first term in the asymptotic expansion for the potential function
will be Ux. All of the following terms must represent flow fields
for which the velocity approaches zero at infinity; they represent
distributions of singularities on the x axis. The nature of the
singularities can only be determined in the matching process, and
so we must generally be prepared to handle all kinds of singularities.

One of the easier ways of doing this is to apply a Fourier
transform to the Laplace equation, replacing the x dependence by
a wave-number dependence. The resulting partial differential
equation in two dimensions can be solved by separation of variables
in cylindrical coordinates. When we require that the potential
functions be single valued, we find that the solutions must all be
products of:

K,(|k|r) or 1,(|k[r) and sin nO or cos n@,

where Kn and I, denote modified Bessel functions. Since I, is
poorly behaved when its argument is large, we reject it, so that the
solution consists of terms:

K,(|k|r)[@ cos nO + B sin né].

The quantities @ and B are constants with respect to r and 6,
but they are both functions of k. They also depend on the index n,
of course. The general solution is obtained by combining all such
possible solutions. Any term in the far-field expansion of the
potential function might be of the form:

714

Singular Perturbation Problems in Ship Hydrodynamics

oo
ik
bq(X2¥ 12) = = » i dk e™ K,(|k|r)[az,(k) cos n@ +b, .(k) sin n6] ,
no 0

(2-59)

where ama(k) and be Ak) are unknown functions. The most general
far-field expansion comprises the incident-flow potential, Ux, and
a sum of terms like the above, that is,

M
x,y,z) ~ Ux + b(x yz) - fortixed (x,y,z) as € => 0.
y. m y
m=l

(2-60)

It will be necessary to have the inner expansion of the outer
expansion. This means that we must interpret r to be O(e) in
the above expressions, instead of O(1) as heretofore, and re-
arrange terms according to their dependence on €. The easiest
procedure is to replace any of the Kp, functions in (2-59) by its
series expansion for small argument. We obtain formulas such as
the following:

ra, FS (OE

oo
1 ikx *
= dk e™ K({k [nya o(k)

-0

00 Co)
_ logr ikx 1 ikx Clk].
—— i. dk e amolk) - aS. dk e a nolk) log Tse

(2- 61a)
n>0:
1 ied ikx *
! cos
x. dk e K,(|k|r)a,,(k) (CP) 0
Coe aie _ikx
~ 2 (n= 1)) (£22) no { Se ama(k)- (2-61b)
2nr” a -00 [1k |

Physically, the n=0 integral represents the potential for a
line of sources. This can be seen directly from (2-61a): As r~ 0,
the function is proportional to log r, which is the potential function
for a source in two dimensions. However, the strength of the
apparent 2-D source is a function of x. In fact, the integral defin-
ing that strength is identical to the integral which gives the inverse
of a Fourier transform. Let ao) be the function having a* (ik)

715

Ogilvie

as its Fourier transform, and further define:
o,(x) = - 27a, (x).
Then the result in (2-61a) can be rewritten:

or)
iS. dk e| * Kol [k |r) aol k) oa Om(x) log r - qa f(x), (2-012')

©
f(x) -{ d€ of (§) log 2 |x-€ | sgn (x-&). (2- 61a")

By manipulating the full integral containing Kp, one can also show
that:

i esate * 1 (° __ om(é) a€
ikx
oa dk e Kol |k | r) anol) =- ae Fee ae » (2-62)

which is easily recognized as the potential function for a line distri-
bution of sources.

Similarly, the other integrals can be interpreted in terms
of dipoles, quadripoles, etc. In particular, we see that for n=l
the inner expansion of the integral reduces to the potential in two
dimensions for a dipole. We may consider the variable x asa
parameter, and then we have a different 2-D dipole strength at each
ora

The Sequence of Near-Field Problems. In the near field,
we can formalize our procedure by making the changes of variables
already mentioned, r=eR, y = €Y, z = €Z, then assuming that
differentiation with respect to R, Y, or Z does not affect orders
of magnitude. Instead of doing this, I shall simply retain the ordi-
nary variables, r, y, and z, and Iask the reader to recall that
differentiation with respect to any of these three variables causes
a change in order of magnitude. Thus, for example, 06/dr = O(¢/e)
in the near field.

In cylindrical coordinates, the Laplace equations and the body
boundary condition can be written as follows:

716

Singular Perturbation Problems in Ship Hydrodynamics

[L] 9er +2 , ty $09 = - Pxxi (2-57')
[ H] 86 - Nv = 29,4 (1/20) 70949 = PkTOy
Vv[it (r99 /r0)"] il dt (r099/roF]

on r = ro(x,@). (2-58')

The definition of N is analogous to that in (2-47'). It is a unit
vector lying in the cross section plane at some x, perpendicular to
the contour of the body in that cross section. It has the three com-
ponents:

(0,-1 »TO9/To)
V[4 + (r99/r0)*]
measured in the x, r, and 9 directions, respectively. Equation

(2-58'), like (2-58), expresses the fact that 8¢/8n =0, where n
is the unit vector normal to the body surface.

Let the inner expansion be expressed as follows:

N
o(x,y,z) ~ » @,(x,y,z) as €—~0O for fixed (x,y/e,z/e).
n=0

Substitute this expansion into the [L] and [H] conditions above:

2
(ee) | Vy,2(Po + P + a + 3 +...) = - (Go + Ot eee);

ae@ To(Po, + O, Tec)

8Dp , AD - :
[ H] Sh aon i aT re cere a
[1 + (r96/t9)"]

The operator Vy2 is the 2-D Laplacian inthe y-z plane, that is,

It can be proven that the first term in the expansion, 9,
represents just the uniform stream:

(x,y,z) = Ux.

717

Ogilvte

This appears so obvious that I pass on immediately to the 9,
problem. Fromthe [L] and [H] conditions, we find:

[Lui] Viz % =0 inthe fluid domain;

[H,] ES ee on r=. (2-63)

OR Ee att (r9,/20)

Finding ®, is strictly a problem in two dimensions. In fact, it is
just the problem that the early aerodynamicists put forth intuitively
at the beginning of their slender-body analysis. (It was also the end
of their analysis!) For an arbitrary body shape, we might have to
solve this boundary-value problem numerically; that is not much of
a problem today. However, we are not yet ready to work with num-
bers, because the formulation of the problem is not quite complete:
we have not specified the behavior of #, at infinity. To do so
requires that we match the unknown solution of this problem to the
far-field expansion.

First, note what (2-63) tells us about the order of mag-
nitude of ©. The right-hand member is O(e) and the left-hand
member is O(@,/e) (because of the differentiation in the transverse
direction), which together imply:

®, = Ole’).

Actually, (2-63) says only that ®, cannot be higher order than e*;
it could be lower order if the matching introduced some effect that
required € to be o(®,), but this does not happen.

This , problem is remarkably similar to the @) problem
in Section 2.2. If we can assume that V4, is bounded at infinity,
then we can express ©, ina series just like the one in (2-51).
Whether V®, really is bounded at infinity can only be determined
from the matching, of course, but we go ahead with the assumption,
trusting that our method will show us if we have made unwarranted
assumptions.

It should be noted too that there are important differences
between this problem and the problem of Section 2.2. The Neumann-
type of condition on the body was homogeneous there, but it is not
homogeneous here. Thus, one may expect that there may be a non-
zero net source strength inside the body in the present problem.
What happens at infinity is also different. In the earlier problem,
the potential had to represent a uniform flow at infinity, and we
supposed that there might be the proper circumstances that a circu-
lation flow could occur. In the present problem, the uniform flow

718

Singular Perturbation Problems in Ship Hydrodynamics

at infinity has been included in ), and so we might expect that ©,
will represent a flow with velocity vanishing at infinity, and there
appears to be no reason to expect a circulation in the 2-D problem.

It would be tedious to go through the same arguments that
were used previously, and so I shall only summarize the results that
would be obtained after a careful matching process. In the near
field, , does indeed yield a velocity field which is bounded in
magnitude at infinity, and there is no circulation. Thus, it can be
represented by the series:

Ai cos @ + By sin 6 4

a wee » (2-64)

A
® (x,y,z) = C, sis is log r t+

The "constants" are all functions of x. Inthe near field, all terms
must be the same order of magnitude, by definition, and so Aj

and Bit are O(€Ajg). (Iam, as usual, ignoring quantities which
are O(log e).) In the matching, the 1/r terms are lost in the first
round, and the log r and constant terms are forced to match the
inner expansion of the outer expansion.

In the outer expansion, (2-60), only a line of sources in the
$, term of (2-60) can match the near-field expansion properly.
That is, in (2-59) and (2-60), we have the following:

*
a," (Ik) = by, (k) =0 except for n=0.

The two-term outer expansion and its two-term inner expansion are:

[e6)
ik *
(x,y +z) ~ Ux + i. dk e Ko [k| r)a,o(k) (2-65a)
UUs oe tiloge tote) (2-65b)
on 9 8 an 1%! »

where (2-61a') has been used to express the latter.

Matching between the near-field and far-field then shows that:

Aig = 0, (x) (2- 66a)

Cy ge f(r). (2-66)

In obtaining an actual solution, one proceeds through the
following steps: 1) Matching shows that ® represents a flow with
bounded velocity at infinity. 2) Thenthe #, problem is completely

719

Ogilvie

formulated and can be solved. 3) From the solution of the ®,
problem, the function Ajo(x) can be determined, which, through the
matching, gives o,(x), and the far-field two-term expansion is known,
4) From the matching relation for C,(x), along with formula (2-61a"),
the near-field potential is known completely to two terms, and the
C,(x) term includes the most important effects of interaction among
sections. This sequence of steps shows what an intimate relation-
ship exists between near- and far-field expansions.

The source strength, o, (x) = Aio(x) » can be computed without
the necessity of solving the flow problem. In the near-field picture,
draw a circle which encloses the body section. The net flux rate
across this circle is just Ajg. From the body boundary condition,
(2-63), one can show that there is a net flux rate across the body
surface, and it is given by Us'(x), where:

er

1
s(x) -3\ dé ré(x, 0) = cross section area at x. (2-67a)

The two fluxes must be equal, and so we find that:
o, (x) = A\o(*) = Us'(x). (2-67b)

Thus, the source strength is proportional to the rate of change with
x of the body cross sectional area.

I shall not pursue the solution to higher order of magnitude,
although there is no insuperable difficulty in doing so. Rather, I
prefer to point out several interesting facts about the solution and
then close this section.

In the far field, the solution to two terms is axially sym-
metric, although the body is not a body of revolution. The near-
field two-term expansion is not symmetric in this way unless the
body is circular and is aligned with the incident flow, However,
the near-field solution can be represented by the series,

Ai; cos 8+ By sin 0

x 4
o(x,y,z) ~ Ux + AO) jog x - aa f(x) + er

rd os r

and, at large r, the axially symmetric terms dominate this series.

If the far-field expansion is carried to three terms, it will be
found that the third term can be interpreted in terms of a line of
dipoles, both vertically and horizontally oriented. Such terms will
be of the form given in (2-61b), with n= 1; they contain unknown
functions a3,(k) and b3,(k), which must be determined through
matching. These unknown functions will depend entirely on the

720

Singular Perturbation Problems in Ship Hydrodynamics

solution of the ®, problem discussed above. In fact, one finds
explicitly that:

ikx *

atic): a Sele is = ike

oo
dk *
Ay (x) = he. Teh e bo, (k) °

Thus, the two-term inner expansion contains enough information to
determine the strength of the dipoles which appear in the third term
of the far-field expansion. The same inner expansion would deter-
mine the strengths of quadripoles in the fourth term of the far-field
expansion, etc. , étc.

On the other hand, the far-field expansion (even at the second
term) contains much information about three-dimensional effects,
information which is largely lacking in the near-field expansion. I
have already pointed out that only the "constant" term contains im-
portant information about 3-D effects in the two-term near-field
expansion. The rest of the ©, solution depends on just the shape
of the local section and the local rate of change of section shape and
size. If higher-order near-field terms are found, it will be seen
that they are influenced even by the two-term outer expansion. In
fact, the "constant" term in ©, can be interpreted as a modification
to the incident stream, caused by the presence of all the other cross
sections of the body. The effects of this extra incident flow on the
transverse velocity field are not perceived until one finds a higher
order expansion of the solution in the near field.

The briefest account of slender-body theory would be seriously
lacking without mention of the possibly catastrophic effects of body
ends. If a body has a blunt end, then s(x) increases linearly in
some neighborhood of the end. Accordingly, s'(x) is discontinuous,
jumping from a value of zero just beyond the end to a finite value at
the end. This is an obvious violation of our assumptions about
"slenderness." But trouble develops even without a blunt-ended
body. For example, if the tip is pointed (but not cusped), there will
still be a stagnation point right at the point. Thus this case violates
the assumption that longitudinal perturbation of the incident flow
velocity is a second-order quantity.

Sometimes these end effects can be overlooked with impunity.
There are major examples later in this paper. However, even when
we have such luck, we must be prepared to have higher-order expan-
sions go awry.

2.32. Small-Amplitude Oscillations at Forward Speed. In
this section, we consider the same Kind of body as in Section 2.31,
namely, a slender body which is aligned approximately with an
incident stream. However, now we formulate a time-dependent
problem in which the body performs small-amplitude oscillations
while it moves through the fluid.

G24

Ogilvie

It would be entirely feasible to consider the general problem
in which the body oscillates with the six degrees of freedom of a
rigid body. (We could even include more degrees of freedom by
allowing deformations of the body.) However, the major concepts
should be clear if we allow only two degrees of freedom, a) a
lateral translation, comparable to the heave or sway of a ship, and
b) a rotation, like the pitch or yaw of a ship.

In this section, I shall depart from my usual approach and
first treat the problem for a perfectly general body, then introduce
the slenderness property at the very end. This introduces a bit of
variety, but more important is the fact that some general properties
of the physical system can be pointed out, without any confusion
over the effects of assuming slenderness of the body.

We use two coordinate systems: Oxyz is fixed in the body
with its origin at the center of gravity, and O'x'y'z' is an inertial
system which moves with the mean motion of the body center of
gravity. With respect to the stationary fluid at infinity, the mean
motion is a translation at speed U inthe negative x!' direction;
thus, inthe O'x'y'z' system, there appears to be a flow past the
body in the positive x' direction.

The two reference systems differ because the body oscillates
inthe z direction, the instantaneous displacement being denoted
by §,(t), and rotates about the y axis, the angular displacement
being denoted by & (t). In a more general problem, we could let
& (t), E,(t) and Es t) denote surge, sway, and heave (displacements
along the x, y, and z axes, respectively) and &4(t), E5(t), and
E,(t) denote roll, pitch, and yaw (rotations about the x, y, and z
axes, respectively). It will be assumed explicitly that §j(t) is a
small quantity, so that squares and products can be neglected in
comparison with the quantity itself. Furthermore, it will be assumed
that €; (t) varies sinusoidally in time and it will be represented by
the real part of a complex function varying as e!t, We shall not
usually bother to indicate that only the real part of a complex
quantity is to be implied. Thus we can write:

E(t) = iw (t). (2-68)

The relationship between the two coordinate systems is as
follows (see Figure (2-4) ):

x = x' cos & - (z'-§,) sin€,@ x' - z', ;
yes yes (2-69)

z =x' sin &. + (z'-€,) cos &, = ee Ss .

122

Singular Perturbatton Problems in Ship Hydrodynamtes

Fig. (2-4). Two Coordinate Systems for
Oscillation Problem

The absolute velocity of the center of gravity is:
DR OE te: iw6,k! = (- Ucos && - iwé, sin €,)i
+ (- U sin &,+ iw€, cos ey) k (2-70)
=- Uit (iw6., - UE.) k (2-70')

where (i,j,k) are unit vectors in the Oxyz system, and (i', j',k')
are unit vectors in the O'x'y'z' system.

Let the body surface be defined by the equation:
S(,y,z) = 0. (2-7 1)

Denote the unit vector normal to the surface, inwardly directed,
by n:

n = ni + nj + n,k- (2=TZa)

It is convenient to make a number of other definitions, as
follows:

1) nj: Extend the above definition of ny to j =4,5,6 as
follows:

ACS

Ogilvie
rXn =n,i tn,j + nk (2-72b)
where

= Xj + yj + zk.
In particular, note that:
n,= n'k and n,= zn, - xn,. (2-72')
2) j: This is a normalized velocity potential. It satisfies:

Ping | Pix, + oj,, =0 in fluid region;
Dd;
sel = nj on S(x,y,z) = 0; (2-73)

|v,| > 0 at infinity.

3) v(x,y,z): This is a normalized fluid velocity, equal to the
fluid velocity at (x,y,z) when an incident stream flows
past the body, the stream having unit velocity, i , at
infinity. It can be represented as follows:

v(x,y,z) = V[x - $,(x,y,z)] : (2-74)

4) mj: This quantity is related to the rate of change of
v(x,y,z) in the neighborhood of the body, as follows:

mi +m,j tm3k= m= - (n'V )v; (2- 75a)
magi + msj +t mgk= - (N-V)(r Xv). (2- 75b)

In particular, note that:

a,
_ z = ° = !
sae on (2-75a')
mM, = - ae r Xx v) = - on (2% - XV3)

i

Fi io [z(1-4, ) +x, =) ee +e (26, = xo). (2-75b')

724

Singular Perturbatton Problems tn Ship Hydrodynamics

5) Wj: This is another useful normalized velocity potential.
It is related to mj; the way 9; is related to nj. It
satisfies:

Hiyy + iyy + hi,, = 0 in fluid region;

It

oi mj; on S(x,y,z) = 0; (2-76)

Ivy; | > 0 at infinity.

In particular, it can be seen that these conditions are
satisfied for i=3, 5 if:

a(x sy »Z) = (x,y 2Z)3 (2-76')
b.(X,y +2) =e $(x,y»Z) = (29, I x9, )- (2-76")
x z

(The last term does satisfy the Laplace equation.)

Now we can write down the velocity potential for the combined
translation and oscillation in terms of the above-defined quantities.
It is a well-known fact of classical hydrodynamics that the fluid
motion can be expressed as a superposition of six separate motions,
each of which would be caused by the motion of the body in one of the
rigid-body degrees of freedom. However, it is essential for the use
of this fact that the description be made in terms of a coordinate
system fixed with respect to the body. Note that there is no lineari-
zation implicit in this superposition, in the sense that there is no
requirement that motions be small in any way. In the body-fixed
reference frame, the velocity potential is:

[- Ucos E. 2 iw, sin E.] , (x,y +2)
+[-Usin & + iwé,cos .]ofx,y,z) + iw6, (xy »Z)-

The nature of the superposition is obvious when we compare the
first two coefficients here with (2-70). However, it must also be
recalled that the velocity potential obtained in this way gives the
absolute velocity of the fluid, that is, the gradient of this potential
is the velocity in a reference frame fixed to the fluid at infinity.
Thus, we must add to this potential an extra term to provide for the

x
This can be concluded also by recalling the definition of qj: its
gradient vanishes at infinity.

725

Ogilvie

apparent incident stream in the observation reference frame. The
latter has the velocity potential Ux', and so the complete potential
is:

x', y',z',t) = Ux''='(U cos E.t iw6;sin €) (x,y »z)
+ (- U sin & + iwS; cos &,)bfx,y,z) + iwSsoe(x,y,z)
(2-77)

= Ux'- Uo (xsy>2) t [ iw (x,y »z)] 6 5(t)
+ [iwo(x,y,z) - Ud lx,y,z)] &,(t) . (2-77')

The potential @ has been defined basically in terms of the inertial
reference frame, although most of the right-hand side here is ex-
pressed in terms of the body-fixed system. Note that not only the
incident stream is defined in terms of primed coordinates, but also
the body motion is really defined in those coordinates as well; in
particular, heave motion is a translation of the body along an axis
fixed with respect to the fluid at infinity.

The Bernoulli equation must be used for computing the pres-
sure:

ind 2 2 2
- = >; + = ($y + $y: ar $,')«
The linear approximations of the derivatives here are as follows:
2
6, = [(ia) 3 + (io) ,,] £5(t)

+ [ (iw) $, - (io) $, + (io) (24, - 26, )] 5 (5

$. = UL1 - $,] + Livdglés(t) + [ings - Uds,- Ud, ]Ep(t)s
by = - UL 4] tind. ésit) + [log - Udy] Es(t)s
$,, = - UL4,] +[inbs,] 83(t) + Linge, - Uds, + U4] E5(t).

Some simplification has been done through the dropping of quadratic
terms in §|. Substituting these expressions into the Bernoulli
equation and simplifying somewhat, one finds that:

726

Singular Perturbation Problems in Ship Hydrodynamics

2

v? +[ (iw), + (iwU) (bs + V+ $9] S5(t)
+ [ (iu)?$5 + (ioU) (Ys + V+ 7 $5) - Ulds + V+ 7 4)] a(t)

In the terms containing §j, one can use primed and unprimed co-
ordinates interchangeably, since the difference leads to terms of
higher order.

The force (moment) corresponding to the j-th mode of
oscillation is given by:

Fj (t) = if ds njp(x,y»zZ,t) = Tji g (t) + Fio»
s 7

where S is the surface of the body at any instant and Fjg is the
steady force component. (For j =1,2,3, the latter is zero.) The
"transfer functions" Tjj are:

T33=- P i ds nf (ia) $5 + (iwU)(¥, t+ vi¥ ,)] 5
ae of dS nj (iu), + (iwU)(Y, + V9 4) - Ul, + ¥-V OJ];
Ty3 = - of ds nl (ia), + (iwU) (3 + v-.V $3) | ;

Ty5 = - | ds n,[ (1c), + (iwU) (Hs + v:V $,) - urw,+ v-V $3].

These formulas can be simplified considerably, even before
we introduce the slenderness approximation. We use two theorems:
One is an extension of Stokes’ theorem, proven by Tuck (see
Ogilvie and Tuck [ 1969]):

f. dS nj(v-V $)) = J dS mj4j -

The other theorem is Green's theorem; in applying it, we note that
all of the functions decrease sufficiently rapidly far away that there
is no need to account for effects at infinity.* Thus, in T33 and
Tz, we have:

*
>, and $, appear to represent dipoles at infinity; thus, both are
proportional to 1/ r@ as r—~ oo. $5 appears to represent a quad-
ripole, thus is proportional to 1/r> at infinity.

727

Ogtlvie

J, asagiigt vv) =) a5 (4,,45~ 5,6) =o
Similarly, in T 55)
f. dS n(p, + ve V $,) = 0)4

In T 35° we manipulate one integral as follows:

dS + “V = dS -
i nal, + v°V $5) \ (nsb; - m5$,)
© ‘ a dS (ngs - 45,4, + ts%5, - 4305)
a i ds n3$, ay ds z(n3,_ - n\$,,)-

Similarly, in Ts, and T,,., we find:

i ds ns (5 + v-V $,) = i ds n3$3 - f. dS 2(n,$, - n>, )-

S

The last integral in the last two expressions can be rewritten:

i‘ dS z(n3$, - n,9, ) =j- \ dS znxV 9,

j- { dS [n Xv (zo) -n X ($k)]
)

\ dS n, 9;
S

the last equality following from application of Stokes' theorem to the
first term. Combining all of these results, we find for the Tjj:

2
T33 = - p(iw) { dS n,$33
s

T35= - p (ie)" {. dS nz, + p(iwU) i ds (n3$, - n, $,);

728

Sitngular Perturbation Problems in Ship Hydrodynamics

Ts3 = - p(iw)* i ds n39, - p(iwU) { dS (nz - n,,);
s
Ty, = - pliw)” i dS ng, + ul dS (n3¢, - n,$,)-

These results have been obtained with no assumptions made
about the shape of the body. The only assumption was that the sinu-
soidal oscillations had very small amplitude.

Now, finally, let us assume that the body is slender. The
only effect is that we lose the terms containing n,$,. For a slender
body, n3 and n, are O(1) as the slenderness parameter, €,
approaches zero, whereas n, is O(e€). From (2-73), we see that
oly is therefore higher order than $, and 9, by a factor of €.

E

us:
r7r
lS. dS n, 4, /S. ds nsés| = O(e?).

Seldom in practical problems do we ever retain terms with sucha
great difference in orders of magnitude, and so we neglect the terms
containing n,$, if the body is slender.

In the ship-motion problem, the quantity corresponding to
T33 will be

(iw) [ ag, + é bgsl »

where a3, and b,, are the heave added-mass and damping coeffi-
cients, respective viet The other Tjj's have a similar interpreta-
tion in terms of pitch added-moment-of-inertia and damping coef-
ficients, cross-coupling coefficients, etc. We note that there are
three kinds of terms here:

a) Terms independent of U. These are all of the same form:

Tt =- Gal f. dS nj $j. (2-78)

b) Terms proportional to U. These occur only in the cross
terms, Tij » with i#j. For a slender body, we have:

*
In the ship-motion problem, $j is complex. Here, of course,
>; is purely real, and so there is no analog to bij.

729

Ogilvte

71) (0)
Tg digs r (U/iwl bans (2-79)

7) (0)
ee + (U/ia) Ty (2-80)

c) A term proportional to U'. This occurs only in T 55

pO)
T55

(0)

sa + (Ufa)? TS). (2-81)

Even at zero forward speed, there is coupling between the
heave and pitch modes, unless the body is symmetric ) fore- an o
aft. Ifthe body is symmetrical, one can show that T3, and T,
are zero. But even in this case, the existence of forward meen.
causes aloss of symmetry, and so a pure-heave motion causes a
pitch moment, and a pure-pitch motion causes a heave force. The
symmetry between T,, and T,, should be noted: The speed-
independent parts are ‘equal, whereas the speed-dependent parts are
exactly opposite.

One remarkable fact is that there is no interaction between
the oscillatory motion and the perturbation of the uniform stream
by the steady forward motion. If the above formulas are derived
from the kinetic-energy formula by use of the Lagrange equations,
this fact is perhaps obvious. When we derive expressions for force
and moment on an oscillating ship, it is anything but obvious.

For the sake of completeness, I write out here the final
formulas for the ti s for a slender body in an infinite fluid. We

note first that, by the same procedures used in the steady-forward-
motion problem, the following is true to a first approximation:

iy, +j,,=9, in the near field.
From (2-72'), it is rather obvious that, for a slender body,
ees sl (| + O(e*)]
5 3 4

and thus, from (2-73):

$,= - xpi + O(e*)].
Now let:

m(x) = mat déin,$,= added mass per unit length, (2-82)
C(x)

730

Singular Perturbation Problems in Ship Hydrodynamics

where C(x) is the contour around the body in the cross section at x.
Then clearly,

(0)

2
T33 = T33 = - p(iv) i

ax ( df nz, = - (ie) if dx m(x),
L C(x) L

where L is the domain of the length of the body. Similarly, we
obtain:

Ole 2 - 2 ‘
wee? Tags p (iw) f. dx «J df no, = (iw) \ dx x m(x) ;

= (ih , dx x*m (x) .

ae
a
uN

Collecting these results, we have:

ve
I

a3 = - (el i" dx m(x) ;

ou
w
|

ae a U
(iw) A dx xm(x) - To T 3s ;
(2-83)

FI
N

53 Gaye ih dx xm/(x) + ~ T33 $

ry
1

2
so (ay dx x*m(x) + (=) Tg-6

Ill. SLENDER SHIP

Of all the problems discussed in this paper, the slender-ship
problem has led to the most important practical consequences.
Therefore it is not unreasonable to devote the longest chapter to the
problem. Even so, some aspects will not be covered; perhaps the
most important missing example is the case of sinkage and trim of
a ship.

In the four sections, two steady-motion and two unsteady-
motion problems are discussed. The first steady-motion problem
is the wave-resistance problem, that is, the problem of a ship in
steady forward motion on the surface of an infinite ocean. In the
second section, the problem treated is essentially the same, but the
Froude number is assumed to be related to the slenderness param-
eter in such a way that Froude number approaches infinity as slender-
ness approaches zero; this rather unnatural relationship is discussed
at some length. In the third section, I discuss in some detail the
problem of heave and pitch motions of a ship at zero forward speed;
the results are not at all surprising, but the method is quite clear
in this case, which helps one in approaching the final section. It is
concerned with the problem which is the combination of the first and
third problems: heave and pitch motions of a ship with forward speed.

W3i

Ogtlvie

3.1. The Moderate-Speed, Steady- Motion Problem

The theory presented here is due to Tuck [ 1963a] *. The
analysis -- as far as I carry it here -- is not very much more diffi-
cult than the analysis of the infinite-fluid problem, and so it will
only be sketched here.

The theory is attractive for its simplicity and its elegance,
but unfortunately it has not been successful in predicting wave
resistance. The reasons are not entirely clear, although they have
been discussed for many years. See, for example, Kotik and Thomsen
[1963]. The difficulty could very well be that real ships are just
not slender enough for a one-term expansion (or perhaps any number
of terms) to give an accurate prediction of wave resistance. This is
the old question, "How small must the 'small' parameter be?"
Another possibility is that the error arises because the lowest-order
slender-body theory places the source of the disturbance precisely
on the level of the undisturbed free surface, and so there are no
attenuation effects due to finite submergence of parts of the hull.
(These two possible causes of error are not entirely separate.)

Still another possible cause is considered in Section 3.2.

The hull surface will be specified by the equation:
r = ro(x,@). (3-1)

Now it will be convenient to measure 9 from the negative z axis,
since most ships are symmetrical about the midplane. We assume
that rg = O(e) and that 8°r,/8x" = O(€), as needed.

There is a velocity potential satisfying the Laplace equation
and the same kinematic body boundary condition, (2-58), as in the
infinite-fluid problem. The incident stream is again taken in the

positive x direction, that is, with velocity potential Ux. The two
free-surface conditions are:

ebtslertoytdd = 50%, on z= Ulxyls (3-2)
C,o+ by Py- >, = 0, on z= C(x,y). (3-3)

Finally, there is a radiation condition to be satisfied.

* This reference is not readily available, but the material which is of
interest here can also be found in Tuck [ 1963b] , Tuck [ 1964a], and
Tuck [1964b] , all of which are gathered into Tuck [1965a] .

732

Singular Perturbation Problems in Ship Hydrodynamics

As usual, we assume that there is a far-field expansion:

N
(x,y,z) ~ » $(x,y,z), where dng = (by) ase. == 0,

“ee for fixed (x,y,z). (3-4)

and a near-field expansion:

N
(x,y,z) ~ bt @(x,y,z), where @, =o0(@) as e-0,

n=O for fixed (x,y/e,z/e). (3-5)

These expansions are substituted into all of the exact conditions,
from which we obtain two sequences of problems which must be
solved simultaneously.

In the far field, the first term in the expansion for ® must
be just the incident uniform-stream potential, Ux, since the body
vanishes as € —~ 0 andthe asymptotic representation $ ~ ¢9= Ux
satisfies the free-surface conditions (trivially). The second term
represents a line of singularities on the x axis. One really ought
to allow the most general possible kind of singularities on this line,
but it is no surprise to find that just sources are sufficient at first,
and so we consider the special case of a line of sources on the free
surface. One can show that higher-order singularities could not be
matched to the near-field solution. Alternatively, one can construct
a far-field solution using Green's theorem and show that it really
represents just a line distribution of sources. See, for example,
Maruo [1967].

One can use the classical Havelock source potential to ex-
press the desired potential for a line of sources, but Tuck's pro-
cedure is more convenient in the slender-body problem: Apply a
double- Fourier transform operation to the Laplace equation, re-
ducing it to an ordinary differential equation with z as independent
variable:

~ (ke? + 07) 6 *(e,£5z) + bee (k,£3z) = 0,

where k and &£ are the transform variables, and the asterisks
denote the transforms. Assume for the moment that the line of
sources is located at z =Z,)< 0. The above differential equation
can be solved generally, with a different solution above and below
Z=Zqg- The solution in the upper region is forced to satisfy the
linearized free-surface condition, the solution in the lower region
must vanish at great depths, and the two must have the discontinuity
at Z= Zp appropriate to the source singularities. Finally, one may

433

Ogtlvte

allow Z—~ 0. In physical variables, the result is:

foe)
1 5
$(x.y +2) = - 5 dk e! 6 *(k) Ko (|k |r)
@
2 ro : 00 r ify +2-V(k2+L%)
ime \ dle J c2a*(K) § aie
2 2492 eT, 2,’
rOeT “-m 00 f(x? +02)[ (2402) -(Uk- ip /2)"]

(3-6)
where yp denotes a fictitious Rayleigh viscosity, puaranteeing that

the proper radiation condition is satisfied, and o (k) is the Fourier
transform of o(x), the source density.

The two-term outer expansion is:
(x,y, 2) ~"Ux + $,(x,y»z),

which has the two-term inner expansion:

(x,y,z) ~ Ux +2 o(x) log x - 3 f(x) - glx), (3-7)
where
@
f(x) =a dé o'(&) log 2|x-&| sgn (x-&) (3-8a)
ee) .
= + a dice o diiloc ielLs ; (3- 8b)
-@
2 po od
U ikx, 2 * dé
g(x) = lim —— dk ek Go J SE
pO 4m? vm : 00 (8 +8) [ eV (e487) - (Uk- ip /2)*]
(3-9)

The expansion should be compared with the corresponding expansion
for a line of sources in an infinite fluid, as given in (2-65). We now
have an extraterm, g(x), and the terms containing o(x) and f(x)
differ by a factor of two from the earlier result. The latter variation
is not important; it results from the fact that the line of sources was
taken at z = zg <0, and those sources merged with their images
when we let zo 0.

*
Define o(x) =0 for values of x ahead of and behind the ship.

734

Singular Perturbation Problems in Ship Hydrodynamics

The most interesting feature of this inner expansion of the
two-term outer expansion is that the wave effects are all contained
in g(x) -- a function of just x. In the infinite-fluid problem, all
3-D effects in the near field were included (in the first approximation)
in the single function of x, f(x) - We now have a generalization of
this for the free-surface problem.

In the near field, it is easy to show that the first term in the
asymptotic expansion of the potential is again just the uniform-stream
potential, Ux. The next term, %,, must satisfy the Laplace
equation in two dimensions (in the cross-plane) and the same body
boundary condition as before, (2-63):

Ur, (x, 98)

oar ae eae on r= To(x, 6). (3-10)
a ty r
Op 0

As in the infinite-fluid problem, this conditions suggests that

® = Ole*),

since ro = O(e) and 9/9N = Ole”).

Now consider the free-surface conditions. In the Bernoulli
equation, note the orders of magnitude:

1 2 a 2
gf + UG, Feo NG Cy, i) Pees © on z= (x,y).

O(t) O(e%) Ofe*%) Ofe?) Ofe?)

The term containing or, can be dropped, but the others containing

@, are all the same order of magnitude, and we have no reason to
suppose that the € term is higher order. In the kinematic condition,
note the orders of magnitude:

Us, +O bx tO by - Ot... =0 on z= O(x,y).

O(t) O(te?) O(f) Ole)

Clearly, we can drop the term containing ),, but no others.

Now we must relate the order of magnitude of ¢ with the
order of magnitude of @,. From the kinematic condition, one might
suppose that ¢ = O(e€). However, the dynamic condition then implies
that ¢ ~ 0, which means only that ¢ is higher order than we
assumed. In fact, the only assumption which is consistent with both
conditions is that:

735

Ogilvie

ae O(e?).
The kinematic condition then reduces to:
®, = 0 on z= 0; (3-11)

thus, ©, represents the flow which would occur in the presence of
a rigid wall at z=0. From the dynamic free-surface condition,
we can compute the first approximation to the wave shape:

g(x,y) ~ - (UG, +5 é,) Ie 9° (3-12)

It may appear to be a paradox that we have a flow without waves,
from which we compute a wave shape! But, like all paradoxes, it
is a matter of interpretation and understanding. We shall return
to this point presently.

Since ®, satisfies the Laplace equation in two dimensions and
a rigid-wall condition on z = 0, it can be continued analytically into
the upper half space as an even function of z. All of the arguments
used in the infinite-fluid problem can then be carried over directly.
In particular, at large distance from the origin, we can write, as
in (2-64),

o ~C, + Mog x + O(1/r), as ri 00.

The two-term inner expansion can be matched to the two-term outer
expansion. We obtain

Aio 20(x);

u

C,

- f(x) - g(x).

Note that there is again a factor of 2 difference from the infinite-
fluid results, (2-66). Of course, the term g(x) is new here.

We can again determine Ajg andthus o in terms of body
shape, without the necessity of solving the near-field hydrodynamic
problem. By the simple flux argument, we find that:

Aio = 2Us '(x) ? (3- 13a)
where s(x) is the cross-sectional area of the submerged part of
the hull. With this convention, we find that

736

Stngular Perturbation Problems in Shtp Hydrodynamics

o(x) = Us'(x), (3-13b)

just as in the infinite-fluid problem. Again, we have been able to
determine the complete two-term outer expansion without explicitly
solving the near-field problem. This occurs because the source-
like behavior which dominates far away from the body (still in the
near-field sense) can be found simply in terms of the rate of change
of cross section, and it provides all the information needed for
determining the two-term far-field expansion.

Enough information is now available to determine a first
approximation of the wave resistance. It can be computed in either
of two ways: 1) integrate the near-field pressure over the hull
surface, or 2) use the far-field expansion and the momentum
theorem. In either case, one obtains:

Dy = wave resistance

© 0
4
~ p do(x) da(&) Y, (k|x-& We
ee
where

o(x) = source density, given in (3-13b),

[a g/U*,
dete ae ae

YQ(z) = Bessel function of the second kind, of order zero,
argument z.

This is the slender-body wave-resistance formula which is so
notoriously inaccurate. At speeds for which one would hope to use
it, it gives values that are too high by a factor of 3 or more.
Generally, one could not (and should not) expect to correct such
errors by including higher-order terms, and so it is rather futile
to pursue this analysis further.

Streamlines, Waves, Pressure Distributions. I mentioned
previously the apparent paradox of prescribing a rigid-wall free-
surface condition, then using the solution of that problem to compute
wave shapes, as in formula (3-12). Such a procedure really can be
quite rational.

Once a velocity potential is known everywhere, it is a fairly

137

Ogilvie

simple task for a computer to figure out the velocity field and to pro-
duce streamlines. Figure (3-1) shows the streamlines around a
Series 60 hull, calculated from the near-field slender-body solution
by Tuck and Von Kerczek [1968]. The upper boundary of the figure
is the rigid-wall streamline. Figure (3-2) shows the same stream-
lines in two other views. These drawings are accurate (in principle)
to order €. This means, loosely speaking, that they show the
streamlines on a scale which is appropriate for measuring beam

and draft of the ship. Thus, we see that some of the streamlines
start near mid-draft, pass under the bottom, then return to approxi-
mately their original depth. These are variations which show ona
scale intended for measuring quantities which are O(e).

The wave height, on the contrary, is O(e*), as we found
earlier. Therefore it should not show in these figures. Our
assumptions have led to the conclusion that wave height is small
compared with beam and draft. Thin-ship theory, on the other hand,
predicts that wave height and beam are comparable -- without being
very explicit about the ratio of wave height to draft.

In the section of Fig. (3-2) showing hydrodynamic pressure
along streamlines, only the waterplane curve (denoted byW) is really
consistent. On any streamline, the pressure will vary mostly because
of the changing hydrostatic head along the streamline. Such pressure
variations are O(e). If we were to work out a second-order theory
and plot the streamlines, the shift in streamline position from first-
order theory to second-order theory would lead to a hydrostatic pres-
sure change which is O(e*). This is the same as the order of magni-
tude of the hydrodynamic pressure, but it is ignored in the figure.

On the other hand, if we were inside the ship measuring
pressure at a point on the hull, we would not care which streamline
went past that point. We could use the Bernoulli equation to esti-
mate the pressure at any point, and the estimate consistent to order
€* would be found from the equation:

. DP i 2 2
O= ry + gz fu + 2 (2), + @,)-

3.2. The High-Speed, Steady - Motion Problem

In the preceding analysis, we have said nothing explicit
about the speed other than assuming that it was finite. The first
term in the velocity-potential expansions was Ux, and all other
terms were assumed to be small in comparison.

In principle, there is no reason to provide or allow a con-
nection between Froude number and our slenderness expansion
parameter. However, the practical manner in which a perturbation
analysis is used may justify our making such an unnatural assumption.
In practice, we work out an asymptotic expansion, which provides

738

Singular Perturbatton Problems tn Shtp Hydrodynamics

Sia Se
a ~
mx Ba
~
NX Se
\Y et
* \
\ \
) }
/
eS :
STREAMLINES -————-—
SECTIONS

Fig.(3-1). Steady-Motion Streamlines on Ship Hull According to
First-Order Slender-Body Theory (Body-Plan View).
From Tuck and Von Kerczek [ 1968].

0.2 HYDRODRODYNAMIC PRESSURE ALONG STREAMLINES

STREAMLINES IN SIDE VIEW
Ww

Fig. (3-2). Steady-Motion Streamlines and Hydrodynamic Pressure
on Ship Hull According to First-Order Slender- Body
Theory (Side and Plan Views). From Tuck and
Von Kerczek [ 1968].

139

Ogilvie

a description that becomes approximately valid (in a certain sense)
as the small parameter approaches zero. But we use the expansion
under conditions in which the small parameter is quite finite, and
we just hope that the resulting error is not too big. The size of that
error may depend on other parameters of the problem, and we may
possibly reduce the error by allowing such other parameters to vary
simultaneously with the basic slenderness parameter.

In the steady-motion problem that we have been considering,
the small parameter e€ could be thought of as the beam/length
ratio. There is a completely different length scale in the problem,
namely, U 2/g = = F*L, where F is the Froude number and L is
ship length. This length is proportional to the wavelength of a wave
with propagation speed equal to ship speed. When we assume that
F=O(1) as € ~ 0, we imply that the speed is such as to produce
waves which can be measured on a scale appropriate for measuring
ship length, and we imply that this speed is unrelated to slenderness.

If we are interested in problems of very-low-speed ships or
very-high-speed ships, in which the generated waves are, respectively,
much shorter or much longer than ship length, it is entirely con-
ceivable that our severely truncated asymptotic expansions may be
made even more inaccurate by the extreme values of Froude number.
We may increase the practical accuracy by assuming, say, that
wavelength approaches zero or infinity, respectively as €~ 0.

This is not to imply that there really is a connection between speed
and slenderness. It is done only in the hope that wavelength and
ship length may be more accurately represented when we use the
theory with a finite value of e€.

Formally, the low-speed problem may be treated simply as a
special case of Tuck's analysis, as described in Section 3.1. One
finds that the appropriate far-field problem contains a rigid-wall free-
surface boundary condition (in the first approximation). Thus, both
near- and far-field approximations are without real gravity-wave
effects. However, this formal approach is quite improper. The diffi-
culty is so serious that we devote a special section later to the low-
speed problem. It is perhaps the most singular of all of our singular
perturbation problems. The difficulty, in essence, is that we have
treated all perturbation velocity components as being small compared
with U, and this leads to nonsense if we allow U to approach zero.

At high speed, a slender-body theory can be developed along
lines paralleling Tuck's analysis. This has been done by Ogilvie
[1967]. The resulting near-field and far-field boundary-value prob-
lems are quite different from Tuck's however. No numerical
results have been obtained yet from this analysis.

Near-field and far-field regions are defined just as in the

previous slender-body problem. In the far-field, the velocity-
potential expansion starts with the uniform-stream term, Ux,

740

Stngular Perturbation Problems in Shtp Hydrodynamics

followed by a term representing a line of singularities. The near-
field expansion also starts with the uniform-stream term, followed
by a term which satisfies the Laplace equation in two dimensions.

The differences appear first in the boundary conditions
satisfied by these expansions. The proper way of setting up these
conditions is to nondimensionalize everything and then assume that
Froude number, F, is related to the slenderness parameter, e,
in such a way that _ F—-o as €~0O. Itis easier just to let the
gravity constant, g, approach zero inthis limit. The only inter-
esting new case, it turns out, is: g = O(e€). We now assume this to
be’ the’ case.

Since g appears only in the dynamic free-surface boundary
condition, the body boundary condition will be the same as in the
moderate-speed problem, Eq. (3-10), and in the infinite fluid prob-
lem, Eq. (2-63).

In the far field, the disturbance vanishes as € ~ 0. There-

fore the free-surface disturbance is o(1). If we let the expansion of
the velocity potential, (x,y,z), be expressed:

N
blxsy,z) ~ Ux + » (x,y,z), for fixed (x,y 2)4
n=l

the dynamic and kinematic free-surface conditions are, approxi-
mately:

O= UL, - 4,
on maa (3-14)

Ue go + Ud), >

We do not know the relative orders of magnitude of € and 9,

a priori, but a study of the possibilities shows that only one combi-
nation is possible, namely, that ¢ and @, are the same order of
magnitude. Then, in the dynamic condition, the term containing

g is higher order than the other term, and it can be neglected in
the first approximation, that is,

9), =O on Ze Os (3-14)
which implies also that
$, = 0 on z=0Q, (3-15)

Thus, the free surface acts like a pressure-relief surface, with no

741

Ogilvte

restraining effect of gravity (to this order of magnitude).

This condition points to a fundamentally different kind of
solution from that of the previous problems. If we continue the
function 4, analytically into the upper half-space, it must be odd
with respect to the surface z=0. Thus 9, cannot represent a
line of sources. The least singular solution represents a line of
dipoles, oriented vertically. Assuming that 9, will consist only
of such dipoles, we can write it:

6, (xy 2) f. sin 0 ." d& (6) | 1+, | «4 (Biz4 6)

grb TE [ (x-8)) + 29]

where y=rcos@and z=rsin®. The two-term outer expansion
and the two-term inner expansion of the two-term outer expansion
are, respectively:

x,y,z) ~ Ux t+ $(x,y,z)

~ ux + 2sine (ae we). (3-17)
0

I am now assuming that the bow of the ship is located at x = 0; then,
in matching to the near-field solution, we can show that the dipole
density must be zero upstream of the ship bow. This expansion is
unaffected by the downstream dipoles.

In the near field, we assume the usual expansion:
N

Deaive zp Ux + ®, (x,y ,Z) for fixed (x,y/e,z/e).

n=l

The term @®, satisfies the 2-D Laplace equation:

b +6 =0.
yy zz

The body boundary condition suggests that ®, = O(e*) , just as it did
before in Eq. (3-10).

From the dynamic free-surface condition,
a wee 2
O= gh + US, +5 (G+ ,) on 2= (x,y),

we see that { = O(e) (since g = Ole) ). This causes a new problem.
We would like, as usual, to change this condition at z = C(x,y) toa

142

Singular Perturbatton Problems in Ship Hydrodynamics

modified condition at z=0. But this is not possible. For example,
the term oe would be transformed:

© (xsysO(xsy)) = &, (x,y 10) + Slxsy)@,, (x+y0) +... -

O(e") O(e?) O(e) Ole)

Every term, in fact, will be the same order of magnitude, and so this
ordinary kind of expansion fails. We must continue to apply the con-
dition on the actual (unknown) location of the free surface.

The kinematic free-surface condition is also nonlinear and
must be satisfied on the unknown location of the free surface:

O- UC, + 2, Cy- Pi, on z= C(x,y).
Each term here is O(€), and so none can be ignored.

We are left in the rather uncomfortable position of having to
solve a nonlinear problem just to obtain a first approximation to the
near-field potential function. However, that nonlinear problem is a
two-dimensional problem, which is not an insignificant advantage,
and, as we shall see, it is possible in principle to predict the loca-
tion of the free surface, thus avoiding the necessity of searching for
its

We do not have a condition to apply at infinity in the 9,
(near-field) problem. It is not so straightforward in this case to
predict the form of the solution as r— oo, but Ogilvie [ 1967]
showed that:

[1 + O(1/r)] as roo,

A 6
®, (x;y ,z) = lle ah

where Aj,; is a constant to be determined. There is no source-like

behavior. This might have been expected, of course, since the inner
expansion of the outer expansion, (3-17), showed the characteristics

of a two-dimensional dipole. An intermediate expansion can be

used to show that these statements are correct.

A numerical procedure for solving this problem may be the
following: Suppose that at some x we know the value of ®, onthe
free surface, z = €(x,y), and that we also know (C(x,y) at that x.

*T¢ we expand: ¢ ~ ae we could apply the condition on z=(,,
then apply the usual kind of transformation, as above, so that
conditions on higher-order terms would be applied on a priori
known surfaces.

743

Ogtlvite

Using Green's theorem, we can write:

1 ( 7 8@ fe)
@, (xsy,z) = a) [=o log r' - ® 5p (log r') | alt,

where r!*=[(y-y')? + (z-z')*], and the integration is carried out in
the cross section, with (y',z') ranging over the body contour, the
free-surface contour, and a closing contour at infinity. The last of
these contours contributes nothing and can be ignored. We assumed
that ®, is known on the free surface, and, from the body boundary
condition, we know 0,/8N onthe hull. If we let the field point,
(x;y,Z), approach the hull surface, we obtain an integral equation,
with @, unknown on the hull and 984,/8N unknown on the free
surface. This is not quite the usual form for an integral equation,
but it should be possible to solve it approximately by essentially
standard numerical methods. Then the Green's-theorem integral
can be used to express ©, at all points in that cross section. Thus,
the solution of an integral equation in one dimension allows the
potential to be found.

This procedure has not used the information contained in the
free-surface conditions. Usually, we look on the free surface con-
ditions as complications that cause tremendous difficulty in the
finding of solutions. Now we take an opposite point of view:
Supposing that we have solved the above problem at some x, we use
the kinematic conditions to predict the value of ¢ just downstream:

b(x + Axsy) = E(xsy) + Ax (x,y) toe.
= C(x,y) +o [®,, (xyysGl<sy)) = P, yl hid Ea

Similarly, we predict the value of ®, on the free surface just
downstream:

Bil ceany 721 TMS, FEB] teen

where the right-hand side is evaluated at (x,y,€(x,y)), and the
dynamic boundary condition is used to evaluate 9®),.

Now we are ready to start over. Presumably having solved
the problem at some x, we have used the free-surface conditions to
formulate the equivalent problem at x + Ax. The most serious
difficulty may very well be in starting the whole process, and there
seems to be no elegant prescription for carrying out that essential
first step; in some problems, it is possible that a linearized solution
may suffice for a start, but this is not certain. Another serious
difficulty may be the stability of the method.

744

Singular Perturbation Problems tn Shtp Hydrodynamics

This analysis has led to the possibility of predicting waves
with amplitude which is O(e), that is, waves comparable in ampli-
tude to ship beam and draft. Such a possibility makes the analysis
worth further investigation, but itis also the cause of the major
difficulty, viz. , the necessity of solving a nonlinear problem in the
near field.

When the above analysis was offered for publication in 1967,
one of the referees called attention to the fact that the conclusions
seemed to be quite at variance with those of Rispin [1966] and Wu
[1967]. Simple observation shows that, at very great distance, the
dominant fluid motion should be gravity-related free-surface waves,
whereas my high-Froude-number analysis predicts no true wave
motion in the far field. Actually, all aspects of the problem are in
complete harmony if we consider a "far-far field" in which distance
from the ship is O(e'), that is, much greater than ship length.
The two free-surface conditions then fall into the usual linearized
format, and we would expect to find progressive waves in sucha
region.

This is quite reasonable. At very high Froude number, one
expects typical waves to be very long -- in this case, considerably
longer than the ship. The appropriate distortion of coordinates is
an isotropic compression in scale far, far away, in contrast to our
usual anisotropic stretching of coordinates in the cross-plane near
the body. In their two-dimensional planing problems, Rispin [1966]
and Wu [1967] performed just such a distortion. Their problem is
discussed at some length later, when we come to two-dimensional
problems.

The present problem is an interesting case in which an
inconsistent expansion might be useful in the far field. Suppose that
we arbitrarily replace the free-surface condition, (3-15), by the
usual moderate-speed condition,

U*s,, i Soy = OF on Zw,

If Froude number is indeed very high, then this condition is quite
equivalent to (3-15). But the potential function which satisfies this
condition does not represent just the simple line of dipoles implied
by (3-16). There will be all of the well-known extra terms involving
the free surface. If such an inconsistent far-field solution can be
matched to the near-field solution, then the waveless far-field solu-
tion obtained previously can be avoided. Perhaps this is worth
further study.

3.3. Oscillatory Motion at Zero Speed

A. systematic study of the zero-speed ship-motions problem
by means of the method of matched asymptotic expansions does not

745

Ogilvie

yield any results that were not obtained previously by simpler means.
However, it is instructive to consider this problem by this method
because the results are rather obvious and it is then clear how the
formalism is used in place of some common physical arguments. Then,
in the more complicated forward-speed problem, in which physical
insight is less reliable, the same formalism can be applied with
reasonable faith in its predictions.

Only the slender-body idealization of a ship has led to useful
prediction methods in the ship-motion problem.” The thin-ship
model, which was intensively studied from the late 1940's until the
early 1960's, was useful for certain restricted aspects of the prob-
lem. For example, the damping of heave and pitch motions, as
predicted by thin-ship theory, is fairly accurate. But the complete
theory is deficient. A straightforward one-parameter analysis
leads to the prediction of resonances in heave and pitch with no
added-mass or damping effects, as shown by Peters and Stoker
[1954]. (See also Peters and Stoker [1957] and Stoker [ 1957].)

A multi-parameter thin-ship analysis is apparently satisfactory in
principle, as demonstrated by Newman [1961], but no one has used
it for prediction purposes. It is too complicated.

Slender-body theory at one time appeared to have comparable
difficulties, but these have been largely removed in recent years,
and a theory which is essentially rational now exists and is fairly
successful in predicting ship motions.

In early versions of the slender-body theory of ship motions,
all inertial effects (both ship and fluid) were lost in the lowest-order
approximation, along with hydrodynamic damping effects. The theory
was even more primitive than the classical Froude-Krylov approach.
Excitation was computed from the pressure field of the waves, un-
disturbed by the presence or motions of the ship, and the restoring
forces were simply the quasi-static changes in buoyancy and moment
of buoyancy. Even the mass of the ship was supposed to be negligible
in the lowest-order theory.

These deficiencies are removed by assuming that the fre-
quency of motion is high, in an asymptotic sense. That is, jf gne
assumes that the frequency of sinusoidal oscillation is O(e! 2) a
then the ship inertia force is the same order of magnitude as the
excitation and the buoyancy restoring forces. The hydrodynamic
force and moment also enter into the calculation of ship motions at
the lowest order of magnitude. This was all recognized, for example,
by Newman and Tuck [1964]. However, correcting the slender-body

*
Note that "strip theory" is a special case of "slender-body theory."

Re
€ is the usual slenderness parameter.

746

Singular Perturbatton Problems tn Ship Hydrodynamics

theory in this way was rejected by many workers on the ground that
the resulting theory would be valid only for very short incident waves,
whereas the most important ship motions are known to occur when

the waves have wave lengths comparable to ship length.

The choice was this: 1) Follow the reasonable usual assump-
tions of slender-body theory and obtain a rather useless theory”.
2) Accept the formal assumption that frequency is high and obtain
a much more interesting theory -- which turns out to be very similar
to the intuitive but quite successful "strip theory" of ship motions.
In what follows, I make the second choice.

The reasons for the success of this choice have become clear
in the last few years. In one of the most important practical prob-
lems, namely, the prediction of heave and pitch motions in head
seas, we can truly say that we are dealing with a high-frequency
phenomenon. Because of the Doppler shift in apparent wave frequency,
fairly long waves are encountered at rather high frequencies; the
waves are long enough to cause large excitation forces, and the
frequencies are high enough to cause resonance effects. At zero
speed, on the other hand, incident waves with frequency near the
resonance frequencies of a ship are likely to be much shorter in
length than the ship, and so their net excitation effect is much re-
duced through interference. For typical ships on the ocean, most of
the heave and pitch motion at zero speed is caused by waves with
length comparable to ship length, and so the frequencies of such
motion are well below the resonance frequencies. Thus, at zero
speed, prediction of ship motions can be treated largely on a quasi-
static basis; the system response is "spring-controlled" rather than
"mass controlled."

The problem is very much like the simple spring-mass
problem discussed in Section 1.2. Ifthe mass of a spring-mass
system is very small, we can ignore inertia effects at low frequency.
Thus, if the system is described by the differential equation:

iwt

my tky = Fe,

the exact and approximate solutions, given by:

*Newman and Tuck showed, for example, that the lowest-order
perturbation potential resulting from ship oscillations satisfies a
rigid-wall free-surface condition, even with forward speed in-
cluded. Maruo [1967] has the same result for the forced-oscilla-
tion problem. Newman and Tuck performed calculations with a
second-order theory for the zero-speed case and found practically
no change in their predictions due to second-order effects. They
did not make such calculations in the forward-speed problem.

747

Ogtlvte

Viegas et) /(is mw’), Yap= Felt /c,

respectively, are approximately equal if w is small enough. If we
solve this equation on the understanding that w is very large, we
must keep all quantities in the exact solution. But that solution will
reduce numerically to the approximate solution if we evaluate it with
a small value of ow.

We could say that the solution obtained on the assumption
of high frequency becomes inconsistent if we apply it to problems at
low frequency, but, if the appropriate small parameter is small
enough, an inconsistent approximation is no worse numerically than
a consistent approximation.

Once more I would warn against trying to make absolute
judgments of what is "small" and what is "not small." I avoid
careful definitions of my small parameters largely for this reason}
if the definition is not precise, one can never be tempted to put
numbers into the definition! In the problem ahead, we cannot
possibly judge analytically how "slender" the ship must be or how

high" the frequency must be for the results to have some validity.

In all of the discussions of ship motions, I use the same
notation as in the study of oscillatory motion in an infinite fluid.
See Section 2.32.

The ship in its mean position will be defined by the equation:
Solxsy +z) =-zt d(x,y) = 0, (3-18)

where d(x,y) = O(e); the instantaneous hull position is defined by the
following equation:

SGc.y.2,t) = - 2.t d(x,y) + €,(t) - x6, (t) =O. (3-19)

The ship is heading toward negative x (although it does not matter
in the zero-speed case). Upward heave and bow-up pitch are con»
sidered positive.

We assume that all motions have very small amplitude.
Symbolically, we write that:

€,(t) = O(€6) as either € or 6 approaches zero.

where € is the usual slenderness parameter, and 6 is a "motion-
amplitude" parameter. This convenient assumption allows us to

748

Singular Perturbatton Problems in Ship Hydrodynamics

vary the motion amplitude for a given ship (i.e. , for fixed e€), and it
also guarantees that the motions are small compared with the ship
beam and draft, even as the latter approach zero as €~ 0. Velocity
potential, wave height, motion variables, and all other dependent
variables may be expected to have double asymptotic expansions,
validas €—~+0O and 6~0. We shall consistently carry terms
which are linear in 56. The steady-motion problems already treated
correspond to the 6 =0 case; at zero speed, the 6=0 case is
trivial. The problem ahead is to solve the linear motions problem
-- "linear" in terms of motion amplitude. With respect to the
slenderness parameter, we shall consistently carry up to € terms.

It should be noted that the slenderness assumption is not
needed in formulating a linear motions problem at zero forward
speed; it is convenient, however, in practical application of the
theory.

All motions are assumed to be sinusoidal at radian frequency
w. Iuse a complex exponential notation, so that: §j;(t) = iw6;(t). Also,
it is assumed that w = O(e7”2), and so symbolically we can write:
a/at = O(e/2),

The potential function, $(x,y,z,t), satisfies the Laplace
equation and the following boundary conditions:

[A] O= gto tole teste], on 2 = Llxsyst)s (3-20a)

ben on |
td

—
°
Hl

0,0 i yb, = b, Ls Cy ? on aa Cts3yat)s (3-20b)
LG] 0 = $,S, + bySy - 9, Tey on So, y.2,t) = 04 -(3=21)

We consider first the problem of a ship which is forced by
some external means to heave and pitch in calm water. In the far
field, the slenderness assumption leads us to expect that the potential
function can be represented by a line of singularities on the x axis.
From previous experience, we might hope that a line of sources
would suffice in the first approximation; this turns out to be correct,
Since these sources represent an oscillating ship, the strengths of
the sources will also vary sinusoidally. Suppose that there is a
source distribution on the x axis:

Re {o(x)e} , - 0 <x<oo.

Define o(x) to be identically zero beyond the ends of the ship.
Obviously, o(x) = o(1) as 6—~ 0, since there is no fluid motion at
all for 6=0. Therefore, in the first approximation, we may

749

Ogilvie

linearize the free-surface conditions. More precisely, we could
assume the existence of asymptotic expansions, #~ /7,d, and
%~ )'Uq, and let the first term in each be o(1) as 60. The
linearized free-surface conditions take their usual form:

a
>
—
(2)
if)

go t+ o, on z=0;
[ B] O=-o, tf, on Z = 04
These can be combined into the following:
$,- vo=0, on Zz = 0, (3-22)

where v = w°/g = O(e7'). In the far field, it is very difficult to
guess how differentiation alters orders of magnitude. If the oscilla-
tion frequency is very high, then the resulting waves are very short;
it would be reasonable, perhaps, to try stretching the coordinates,
and there would be no obvious basis for doing this anisotrophically.
The approach which I take here is somewhat different: Solve the
above- stated linear problem exactly, then observe the behavior of
the solution for high frequency of oscillation. In other words, the
problem is not stated in a consistant manner, but when we have the
solution we rearrange it and make it consistent.

The desired potential function can be written in the following

form:
o(x,y,z,t) = Re {$(x,y,z)e"}, (3-23)

where:
$(x,y,zZ) = sea dé ot — ekZ 3 p(k V llx-£) + 1) (3-23a)

7 gi tvezy (it?) (k VK 02)
eee “ak elk Bae — (3-23b)

4n ee ores

The form in (3-23a) can be obtained readily by superposing a distri-
bution of free-surface sources: Jo is the ordinary Bessel function
of order zero, and the wiggly arrow shows that the integral is to be
interpreted as a contour integral, indented at the pole in the obvious
sense indicated. Form (3--54b) is obtained by a transform method;
o*(k) is the Fourier transtorm of o(x); details may be found in
Ogilvie and Tuck [1969]. Again, the inner integral is to be inter-
preted as a contour integral; there are two poles in this case. In
both formulas, the path of the contour has been chosen so that the

750

Stngular Perturbation Problems in Shtp Hydrodynamics

solution has a satisfactory behavior at infinity, viz., it represents
outgoing waves.

We need the inner expansion of this potential function, that
is, we must find its behavior as r= (y2+ 2*)”2 +0. The basic idea
here in finding the inner expansion is to use the second form of
solution, convert the contour integral into an integral along a closed
contour, and use the calculus of residues. The integrand of the
inner integral has four singularities, located at £=+ 29 and at
#=+tilk|, where fy = (v? - k?)!/”#, The first two are simple poles,
but the second two are branch points. We "connect" the latter via
the point at infinity; see Fig. (3-3). It is drawn for the case that

k| <v; if |k| > v, all four singularities are purely imaginary.
The contour is closed as shown if y>0O. (Otherwise, the contour
is closed below.) The integrals along the large circular arcs
approach zero as the radius of the arcs approaches infinity. Then
the inner integral in (x,y,z) is equal to 2mi times the residue

at £ = - £5, less the value of the contour integral down and back up
the imaginary axis. The latter can be shown to be O(e€), and so the
inner integral in (x,y,z) is:

00 i Ry+z+ ke+e P : 2.2
{ dZe 2tTiv evz-iyvV-k + O(e)

wo (Ke + H2y2_ y (v2 — «2/2

Next, we assume that the source distribution is smooth
enough that o(x) does not vary rapidly on a length scale comparable
with ship beam. This assumption implies that o'(k) decreases
rapidly with increasing values of k, and so the value of the above
inner integral -- a function of k -- does not really matter except
when k is small in magnitude. Accordingly, we expand the above
expression in a manner appropriate for small lk]. We obtain

Fig. (3-3). Contour of Integration Defining the
Velocity Potential of a Line of Pul-
sating Sources: Zero-Speed Case

151

Ogilvie

oO
i i Be a
(x,y s2) = a clic en ge (ie ae Pl

e . @ .
Eee ge iy dk Ag o*(k) {14 ieeeataay
27 og

=iieee yg dicheh a oe (3-24)
With the time dependence reintroduced, we have:
d(x,yszst) @ Re {lo(xjeZell@t) p40 551. (3-25)

This approximation represents a travelling wave; for y > 0, in
particular, the wave is moving away from the line of sources. For
y <0, we must start over, closing the contour for the £ integration
on the lower side of the £ plane. It turns out that the result is the
same if only we replace y by ly|. Thus, we have an outgoing wave
for y <0 also. In both cases, the outgoing wave has the form
appropriate for a gravity wave in two dimensions.

In the approximations above, it is necessary to require that
r be not extraordinarily large; if one assumes that r = O(i) and
w= O(e-/2), then the above results follow logically. Thus the very
simple approximation above is valid even in part of the far field. It
is an example of the well-known physical principle that nearly
unidirectional waves can be generated if the wave generator is much
larger than a wave length.

If we let r = O(€), no change occurs in this approximation.
Since v = O(e7'), it is not permissible to expand the exponential
functions even when y and z are O(e). The only effect of passing
from far field to near field now is to change the scale of the observed
wave motion.

This far-field analysis has provided information that was
probably quite obvious intuitively: In the near-field, the condition
at infinity is that there should be outgoing, two-dimensional, gravity
waves*. With this information in hand, we can move on to the for-
mulation and solution of the near-field problem.

In the near-field, we make the usual slender-body assump-
tions:

“7 cannot imagine that anyone would ever have doubted this fact,

even without the above analysis to show it. But in the forward-speed
problem, the condition at infinity in the near field is not at all ob-
vious, and such an analysis seems necessary.

752

Stngular Perturbation Problems in Shtp Hydrodynamies

) ce) Cue -I

By? Ba’ or = Ole )-

To a first approximation, the potential function satisfies the Laplace
equation in two dimensions:

Pyy + $22 ~ 0,
and the linear free-surface condition
$,- vo~ 0 on Zz =. 04 (3-22)

With the assumptions made above, the two terms here are of the
same order of magnitude. (If we did not assume high frequency, we
would obtain just the rigid-wall boundary condition, @,=0.) This
condition implies that we shall be solving a gravity-wave problem in
two dimensions. At infinity, we know from the far-field solution
that the appropriate condition is an outgoing-wave requirement. All
that remains is to put the body boundary condition, (3-21), into the
appropriate form.

Let 8/8N denote differentiation in the direction normal to
the body contour in across section. Then, from (3-19) and (3-21),

aN (ita? )¥/2 (ita2)”?
= és - xts - Es, + dydy _ Es - xés : (3-26)
(1+a5 yi”? (140% yV2

The last simplification involves an error which is O(e?) higher
order than the retained terms. To the same approximation, we can
write (see (2-72')):

(14a2yv2' ge ees
y

Thus, the boundary condition is:

o3 5o5 ? on Z= (x,y) » (3-27)

oo _
DN. *.03

As in the infinite-fluid problem (cf. (2-73)), we can define
normalized potential functions, 4;(x,y,z):

153

Ogtlvte

ie + $j, = OR, in the fluid region; (3- 28a)
te =n, on z = d(x,y); (3-28b)
6, - ¥% =O, on z=0, (3-28c)

where v= w*/g. In the present case, the functions satisfy the 2-D
Laplace equation and a 2-D body boundary condition, and they must
satisfy the linearized free-surface condition. Instead of the previous
simple condition at infinity, we must impose the 2-D outgoing-wave
radiation condition and a condition of vanishing disturbance at great
depths. Thus, the boundary-value problem is much more compli-
cated than in the infinite-fluid case, but, thanks to the slenderness
assumption, we have only 2-D problems to solve, and, thanks to the
small-amplitude assumption, the problems are linear.

The actual velocity potential function can now be expressed:

(x,y ,Z st) “Re | » 1w6j (A Cx,y,2) | ° (3-29)
j=35

It must be observed that each j is complex, because of the radi-
ation condition. It is necessary to devise an appropriate numerical
scheme for solving these problems. Both mapping techniques and
integral-equation methods have been successfully applied. Note,
incidentally, that the heave/pitch problem requires solution of just
the 3 problem, since the slenderness assumption allows the
approximation to be made that $,~ - XOz0

The result of this analysis is a pure strip theory, that is,
the flow appears to take place in cross sections as if each cross
section were independent of the others. It is consistent to follow
the solution of this problem with a computation of the pressure field
at each cross section, from which force-per-unit-length, then force
and moment on the ship can be found after appropriate integrations.
We obtain the following formulas for the force and moment on the
ship resulting from the motion of the ship:

m 2
F(t) = - | ds njl aby ~ xf) + al (Eg, +S 4)], 3-30)
So
where j =3 for heave force and j = 5 for pitch moment, and the
symbol Sg denotes that the integration is to be taken over the hull

surface in its mean or undisturbed position, which is specified by
Eq. (3-18). The first term, involving g, is just a buoyancy effect.

154

Singular Perturbatton Problems tn Ship Hydrodynamics

The following terms are purely hydrodynamic; they will be expressed
in terms of added-mass and damping coefficients, as follows: Let:

iw

m(x) +=—n(x) = p i vat nyt, (3-31)
C(x

where C(x) is the contour of the immersed part of the cross section
of x. Cf. (2-82). We call m(x) the "added mass per unit length"
and n(x) the "damping coefficient per unit length." Using the
slender-body approximations that $¢,™ - x, and ng@ - xn,, we
find for Fj; (t):

FF (t) = - val dS n(&5-x&) - iw)” ‘ dx (€-x6,)[ m(x) + n(x)/io] ;
So
(3-32)

EM(e) = pg | dS xnglty-xf) + (iol | ax x(Eg-n6)[ mb) + (v0) /iel «
So E

Finally, we abbreviate these formulas:

Fi(t) = - > [ (iw)*a,, + (lalby, +o; 18; (t) » (3-33)
i=3,5
where
as = : dx m(x); bss= J dx n(x);
Ags = ag, - i dx xm(x); ba. = by, == { dx xn(x);
E iE
hye eee eceri(an) b -{ dx x?n(x) ;
age i. x°m(x) 55 . x x°n(x

c= ral dS n, = 208) dx b(x,0);
33 So "3 7
Crt “na 7 RE is ds Sons zoe dx xb(x,0);
ce)
Cog = al ds x, = ee | dx x“b(x, 0);
So E

b(x,z) is the hull offset at a point (x,z) on the centerplane.

95

Ogilvie

The wave-excitation problem can be formulated as a singular
perturbation problem, but such a problem has never been satis-
factorily solved, even for the zero-speed case. Fortunately, another
approach is available for obtaining the wave excitation; this is the
very elegant theorem proven by Khaskind [1957]. It allows one to
compute the wave excitation force, including the effects of the diffrac-
tion wave, without solving the diffraction problem. Since we thus
avoid the singular perturbation problem altogether, only the final
results are presented here. (Reference may be made to Newman
[1963] for details of the zero-speed case.) Let the incident wave
have the velocity potential:

yz+ilwt-vx) 5

~ igh
Oo(x,z,t) es € >

the corresponding wave shape is given by:

€o(x.t) = Hele e

This is the head-seas case, For an arbitrary body, the heave force
due to the incident waves is:

F5(t) = pghe'™ ( ds ec" {(1 - vg)ng t ivdgn,} .
So

If the body is a slender ship, with axis parallel to the wave-propa-
gation direction, this formula simplifies to the following:

FY (t) = pghe's! ( dx “6 | dé ne” (1 - vo,). (3-34a)
3 (_ C(x) "3 we

The corresponding expression for pitch moment on a slender body is:

FS (t) = pghe@? ‘ dx Pile aS dl ne". (4 - vo). (3-34b)

C(x)

In the expression (I- v,) in the integrand, the first term leads to the
force (moment) which would exist if the presence of the ship did not
alter the pressure distribution in the wave; in other words, it gives
the so-called "Froude-Krylov" excitation. This fact can be proven
by applying Gauss' theorem to the integral, Dynamic effects in the
wave ("Smith effect") are properly accounted for. The second term
gives all effects of the diffraction wave.

A final rewriting of the wave-force formula is worthwhile.

The above approximate expression for FS (t) can be manipulated
into the following:

756

Singular Perturbation Problems in Ship Hydrodynamics

20
M v

+ ipw | dx (xe) | df n_d,e””.
L Fo, C(x) 33

The first term shows the Froude-Krylov force quite explicitly; the
product of € (x,t) and the quantity in brackets is often called an
"effective waveheight," the second factor being a quantitative repre-
sentation of the Smith effect. The second integral term has been
expressed in terms of the vertical speed of the wave surface,
Co,(x,t). This term should be compared with the force expression
for the calm-water problem, (3-30). For a slender body, the hydro-
dynamic part of the latter can be written, for j = 3,

soa) dS n,[&,(t)3;+6,(t) 5] = -tpo dx [ £,(t)-xé,(0)] Sn! N35.

The last quantity in brackets is the vertical speed of the cross section
at any particular x. Comparison with the second term of FY (t)
shows that the latter is almost exactly the same as the hydrodynamic
force that we would predict if each section of the ship had a vertical
speed - Co,(x,t)- This analogy would be exact, in fact, if the expo-
nential factor, e”*, were not present in the By (t) formula.

Except for that factor, what we have found is that Korvin-
Kroukovsky's well-known "relative-velocity hypothesis" is approxi-
mately correct according to the analysis above. The hypothesis is
particularly accurate for very long waves, in which case e”*= 1
over the depth of the ship, but it is less accurate for short waves.
Again, it should be noted that we have no absolute basis for saying
whether a particular wave is short or long in this respect. In com-
puting the Froude-Krylov part of the force, it is well-known that the
exponential-decay factor must be included in practically all cases of
practical interest; this has been amply demonstrated experimentally.
It suggests that one should be wary of dropping the exponential factor
in the diffraction-wave force expression.

Summary. In the far field, we assumed that the effects of
the heaving/pitching ship could be represented by a line of pulsating
singularities located at the intersection of the ship centerplane and
the undisturbed free surface. For a first approximation, we tried
using just sources, and these were sufficient to allow matching with
the near-field solution. In particular, the inner expansion of the
outer expansion showed that the near-field expansion would satisfy a
two-dimensional outgoing-wave radiation condition, at least in the
first approximation. With this fact established, we formulated the
near-field problem; it reduced ultimately to the determination of a
velocity potential in two dimensions, the potential satisfying a linear

tot

Ogilvie

free-surface condition and an ordinary kinematic body boundary con-
dition, as well as the outgoing-wave condition. This is a standard
problem which must generally be solved numerically with the aid of

a large computer; such programs exist. The force and moment were
expressed as integrals of added-mass-per-unit-length and damping-
per-unit-length, both of which could be found from the velocity
potential for the 2-D problem. Finally, the determination of the wave
excitation force and moment was carried out by application of the
Khaskind formula, which permits us to avoid the singular perturbation
problem involved in solving for the diffraction wave.

3.4. Oscillatory Motion with Forward Speed

The problem of predicting the hydrodynamic force on an
oscillating ship with forward speed is not fundamentally much differ-
ent from the same problem in the zero-speed case. It is considerably
more complex, to be sure, but no new assumptions are needed.

The approach here is that of Ogilvie and Tuck [1969]. Alter-
native approaches have been devised by numerous other authors;
some of these were mentioned in the last section. The distinguishing
characteristics of the Ogilvie-Tuck approach are: 1) application of
the method of matched asymptotic expansions, and 2) assumption that
frequency is high in the asymptotic sense that w= O(e"/@), while
Froude number is O(1i). Also, the problem is broken down into a
series of linear problems by the use of a "motion-amplitude" param-
eter, 6, whichis a measure of the amplitude of motion relative to
the size of ship beam and draft.

The reference frame is assumed to move with the mean motion
of the center of gravity of the ship. Thus it appears that there is a
uniform stream at infinity, and we take this stream in the positive
x direction. The z axis points upward from an origin located in
the plane of the undisturbed free surface, andthe y axis completes
the right-handed system. (Positive y is measured to starboard. )

Let the velocity potential be written:
(x,y 5Z ,t) = Ux 7 Ux (x,y 32) + w(x, y»Z st) s (3-35a)

where Ul x + x(x,y,z)] is the solution of the steady-motion problem
discussed in Section 3.1. For the moment, we simply assume that
W(x,y,z,t) includes everything that must be added to the steady-
motion potential so that &(x,y,z,t) is the solution of the complete
problem. We shall also divide the free-surface deformation function
into two parts:

C(x,y,t) = n(x,y) +t O(x,y,t), (3-35b)

758

Singular Perturbatton Problems in Ship Hydrodynamics

where 1(x,y) is the free-surface shape in the steady-motion prob-
lem (the {(x,y) of Section 3.1), and 6(x,y,t) is whatever must be
added so that €(x,y,t) is the complete free-surface deformation.

The body surface is defined mathematically just as in Section
3.3 for the zero-speed problem; see (3-18) and (3-19). The same
assumptions are made about orders of magnitude:

E(t} = O(ed); w= Oe").

From these assumptions and the subsequent analysis, it turns out
that

W(x,y,t) = Ofe%28), O(x,y,t) = O(es),

as either € or 6—~ 0. Wecan look on the complete solution as a
double expansion in € and 6. From this point of view, the expan-
sion for the potential can be written:

Gov e2st) = 1 Ux + UX, Gay 52) + «ee }

Ol6°e) “Ol S-e-)

tAWi(cry zt) + U(x, yee, t) tees } P'0(6).1 (3536)

o(s'e*?)

O(8' é*)

The order of magnitude of the term Uy,(x,y,z) was found in Section
3.1. The order of magnitude of ~, may be somewhat surprisinge
Physically, it implies that the effects of ship oscillations dominate
the effects of steady forward motion -- in the first approximation.
These orders of magnitude were derived by Ogilvie and Tuck. Here,
I shall not prove them, but I hope to make them appear plausible.

It should be noted that the high frequency assumption was made just
so that the orders of magnitude would come out this way. (Cf. the
discussion in Section 2.3, in which it was pointed out that the for-
malism for the steady-motion slender-body problem is established to
force certain expected results to come out of the analysis. We are

doing the same here, forcing strip theory to come out as the first
approximation. )

The linearity of the ~, problem permits us to assume that
the time dependence of , and of the corresponding first term ina
@ expansion can be represented by a factor e'™’.

In order to find any effects of interaction between steady
motion and oscillatory motion, it is necessary to solve for the term

159

Ogtlvie

WAx,y,z,t). Thus, we must retain two terms in the time-dependent
part of the potential function. (The problem is still linear, however,
in terms of 6.) It is not convenient to be repeatedly attaching sub-
scripts to the symbols, and so I shall simply write out equations
and conditions which are asymptotically valid to the order of mag-
nitude appropriate to keeping e€° terms in the expansion of
W(x,y,Z,t).

In the far field, the effect of the oscillating ship can be repre-
sented in terms of line distributions of singularities. Again, we
try to get along with just a distribution of sources, and we are
successful if we allow for the existence of both steady and pulsating
sources. The steady-source distribution is exactly the same as in
the steady-motion problem. Let the density of the unsteady sources
be given by o(x)e'“t; define o(x) =0 for the values of x beyond
the bow or stern. The corresponding potential function must satisfy
the Laplace equation in three dimensions, a radiation condition, and
the usual linearized free-surface condition:

(iw) + 2ioUY, + UY, tgu,=0 on z=0. (3-37)

Then it can be shown that:

eat fan ikx dé exp[ il + 2/k* +27]
W(x, y »2,t)~ ~ 2 ) dk e o (k) 2
4m J« Cee 2 + Uk /w)

(3-38)

where o*(k) is the Fourier transform of o(x), and the contour C
is taken as in Fig. (3-4), where k, and kg, are the real roots
(k, < kp) of the equation:

*There are two real roots if T= wU/g > 1/4; the other_two roots
are a complex pair. Since we assume that w= Oltaler then also
1 Olive , and we are assured that T >> 1/4. However, if
Tt | 1 7ay the complex pair come together, and our estimates are
all very bad. Of course, it is well known that the ship-motion
problem is singular at T = 1/4, For still smaller values of T,
there are four real roots of the above equation, and the solution
can again be interpreted physically and mathematically. From
experimental evidence, it appears that our final formulas can be
applied for any forward speed, at least in head seas, but the
presence of a singularity at 7= 1/4 shows that this is accidental.
Our theory is a high-frequency, finite-speed theory, and it really
should not be possible to let U vary continuously down to zero.

750

Singular Perturbation Problems in Ship Hydrodynamics

cia

k < ky
®

k> ko

Fig. (3-4). Contour of Integration Defining the
Velocity Potential of a Line of Pul-
sating Sources: Forward-Speed Case

[+]. Bf =o.

and the contour is indented as shown at the poles on the real axis
inthe £ plane. The contour C extends from - oo to too. The
poles inthe £ plane all fall on the imaginary axis if k,; <k<k,,
and then C is the entire real axis, with no special interpretations
being necessary.

The above expression for (x,y,z,t) is a one-term outer
expansion, but it is not a consistent one-term expansion. It is
shown by Ogilvie and Tuck that a much simpler expression is pos-
sible if r= (y*+z°)'* is O(1) as €—0; emphasis should be
placed here on the restriction that r is not extraordinarily large.
If o (k) is restricted in a rather reasonable way, it follows that:

. fo) : ; 2
Nee Yio 22) ~All | die! otters PUNT Sr ay eso)
-00

We can take this as our one-term outer expansion of (x,y,zZ,t).

The inner expansion of this expression is obtained by letting
r = O(e). Then we find that:

W(x ,y oz pt) ~ 18 ef ote) - 2i(oU/g)(z - ily |Jo')]. (3-40)

Since vr = O(1), it is not appropriate to expand the exponential
function further. This is a two-term inner expansion of the outer
expansion of ; the first term represents an outgoing, two-dimen-
sional, gravity wave, just as in the zero-speed problem (see (3-25)),
but the second term represents a wave motion in which the amplitude

761

Ogilvte

increases linearly with distance from the x axis. The latter is a
rather strange kind of potential function; it represents a wave which
becomes larger and larger, without limit, at large distance. How-
ever, one must remember that this is the inner expansion of the
outer expansion of (x,y,z,t); it means that there are waves near
the x axis which seem to increase in size when viewed in the near
field. At very great distances, one must revert to the previous
integral expressions for (x,y,zZ,t).

We must next find an inner expansion which satisfies con-
ditions appropriate to the near field and which matches the above
far-field expansion. One finds readily that:

Wyy + bz = 0 in the fluid region,

to the order of magnitude that we consistently retain. Thus, the
partial differential equation is again reduced to one in two dimensions,
and so we seek to restate all boundary conditions in a form appropri-
ate to a 2-D problem.

The body boundary condition must be carefully expressed in
terms of a relationship to be satisfied on the instantaneous position
of the body. This condition can then be restated as a different con-
dition to be applied on the mean position of the hull. It can be shown
that:

Oy —~ §3 - x65 , - USs + U(3-x85) (hoxyz - X22) a
ON (40a) + e) ee wey” : on Z= d(x,y).

[ €'/25] [€6] (3-41)

The derivative on the left has the same meaning as in the previous
slender-body analyses: It is the rate of change in a cross section
plane, in the direction normal to the hull contour in that plane. The
first term on the right-hand side is the same as in the zero-speed
problem; see (3-26). The quantity - U& has a simple physical
interpretation: it is a cross-flow velocity caused by the instantaneous
angle of attack. The remaining terms all arise as a correction on the
steady-motion potential function, Uy; the latter satisfies a boundary
condition on the mean position of the ship, which is not generally the
actual position of the ship, and so it must be modified.

Intuitive derivations of strip theory usually omit the terms
involving x. However, in a consistent slender-body derivation,
they are the same order of magnitude as the angle-of-attack term.
(This says nothing about which is the more nearly valid approach!)

The free-surface condition reduces ultimately to:

762

Singular Perturbation Problems in Shtp Hydrodynamics

Wie t Bb, ~ - 2UUy - 2ZUXy Uy - Uxyyy on z=0,. (3-42)
[e¥? 6] [ €6§]

The orders of magnitude are noted, again on the basis of information
not derived here. This condition can be compared with the linear
condition used in the far field, (3-37). The two terms on the left
here are obviously the same as the terms (iw) |W + gw, in (3-37), and
the first term onthe right here, - 2U\),, is the same as the term
2iwUb, in (3-37). The other two terms on the right-hand side here
are basically nonlinear in origin; they involve interactions between
the os cillation and the steady perturbation of the incident stream.
The term U ey which appears in (3-37) is missing here because
itis O(e%/%6) in the near field by our reckoning.

Again it is worthwhile to compare this boundary condition
with its nearest equivalent in other versions of slender-body theory
or strip theory of ship motions. If we did not assume that frequency
is very large, slender-body theory would require in the first approxi-
mation that ~,= 0, since the other terms are all higher order. This
is just the free-surface boundary condition obtained in this problem
by Newman and Tuck [1 964] and by Maruo [1967]. Higher order
approximations would involve nonhomogeneous Neumann conditions
on z=0. Onthe other hand, in most derivations of ane theory,
it is assumed that the free-surface condition is: Wy, + glb,=0 on
z=0. This agrees with the lowest-order condition peed by
Ogilvie and Tuck, as given above. However, the assumption of this
boundary condition in the usual strip-theory derivation is quite
arbitrary, and no means is available to extend it to higher-order
approximations. The assumptions made by Ogilvie and Tuck were
chosen explicitly so that the simplest approximation would be just
strip theory, and we see here that that goal was achieved. This
basis for choosing assumptions was selected only because strip
theory had proven to be the most accurate procedure available for
predicting ship motions.

The method of solution used by Ogilvie and Tuck is to find
several functions each of which satisfies some part of the nonhomo-
geneous conditions. In particular, let the solution be expressed in
the following form:

b(x,y,z,t) -) [ind + UW, + (eo) Uj ]E, (t), (3-43)

*
In other words, we stopped fretting about how irrational strip theory
was and set out to derive it formally]

763

Ogtlvte

where j =3 and 5, and 9%), Wj, and &j satisfy the following con-
ditions, respectively:

2j a pi =. Gj, ae on = d(x; y); Pj, - vj =0 on ze 0;
(3-44)
Jyy jzz = 0; 2; Him On. 42 = d(x,y); Y ~ vo =0 on z=0;

Qj, + Qj =0; Q); =0 on gz =4d(x,y);

(3-46)

The quantities nj were defined previously, in (2-72), as the six
components of a generalized normal vector. Also, the quantities
mj were defined earlier, by (2-75). In the present notation, let
v(x,y,z) (see (2-74)) be defined by

v(x,y+Z) =i E% aX (x,y,z)] °

Then mj is again given by the previous formulas. Now it requires
just a bit of manipulating to show that the assumed solution above
indeed satisfies the body and free-surface boundary conditions; I
omit the proof.

The above near-field solution must match the far-field solu-
tion, which has an inner expansion given in (3-40). In connection
with the latter, a comment was made earlier that the near-field
solution would have to represent a wave motion in which one com-
ponent grows linearly in amplitude as ly | — oo. Now we can see
that just such an interpretation must be given to the Qj functions,
for otherwise we cannot possibly find solutions to the problems set
above for {92}. The nonhomogeneous free-surface condition on Qj
can be compared to the free-surface condition that would result if a
pressure distribution were applied to the free surface. In fact, if
a pressure field were applied externally on z = 0, the pressure
being given by

p(x,y,t) = ipwU6; (t)[ 20). + 2X y Pj, + X yy Fj | >
then the potential function would have to be tay UN) Ej (t), with

Qj) (x,y,z) satisfying the conditions stated previously. This

764

Singular Perturbatton Problems tn Shtp Hydrodynamics

"pressure distribution" is periodic in time, and it is also periodic in
y as ly| — oo; the latter comes from the term containing 4)j,.
Furthermore, the time and space periodicities are related to each
other in just the way that one would expect for a plane gravity wave.
This can be proven by studying the boundary-value problem for 9).
Thus, there is an effective pressure distribution over an infinite
area, and it excites waves at just the right combination of frequency
and wave length so that we have a resonance response. In an ordi-
nary two-dimensional problem, there would be no solution satisfying
all of these conditions. However, our solution need not be regular
at infinity; it must only match the far-field expansion. And the far-
field expansion predicts an appropriate singular behavior at infinity.
It is shown by Ogilvie and Tuck that the solution of this inner prob-
lem does exactly match the above far-field solution. The way the
pieces of the puzzle all fit together is rather typical of the method
of matched asymptotic expansions, and it indicates at least that the
manipulations of asymptotic relations were probably done correctly!
(It still says nothing about the correctness of the assumptions.)

There is no benefit to be derived by repeating here the solu-
tion of the above detailed problems. Rather, we jump to the results
for the heave force and the pitch moment, and we do little more than
compare these results with the comparable formulas in two previous
problems:

CASE 1: The oscillating slender body, with forward speed,
in an infinite fluid (Section 2. 32)

CASE 2: The oscillating slender body (ship), at zero forward
speed, on a free surface (Section 3. 3)

In all cases, let the force (moment) be expressed in the form:

m Ne :
sO cia \ [ (iw) aj; + (io)by +), JE; (t) .
i=3,5
We define cjj to be independent of frequency and of forward speed.
(We must make some such arbitrary convention, or the separation
into ajj and cj; components is not unique.) With this convention,

cjj represents just the buoyancy restoring force (moment). Thus,

cj, =O forall j,i in case 1; in cases 2 and 3, cj, is given by:

[oj] = 208) dx {1-3 (4) bl, 0) .

765

Ogilvie

Table 3-1 shows ajj and bjj for the three problems. InCases 1 and
2, the results have been obtained from Sections 2.32 and 3.3, re-
spectively. ForCase 3, the present problem, the lengthy derivation
will be found in Ogilvie and Tuck [1969]. Some points should be
noted:

1. All of the terms’ in Case 3 include the corresponding
Case 2 terms, i.e., the added mass and damping at forward speed
can be computed in terms of the added mass and damping at zero
speed, plus a speed-dependent component. Formally, we could also
say that Case 1 includes all of the Case 2 terms, with n(x) set
equal to zero. From this point of view, the only differences among
the three cases are the forward-speed effects.

2. The coupling coefficients b35 and bsz include a forward-
speed term +¥Ua3z3 in both Case 1 and Case 3. This means, first
of all, that there can be some damping even in the infinite-fluid
problem. Secondly, it means that this contribution to the damping
coefficients is not altered by the presence of the free surface. Note
that in neither case is it necessary to ignore the steady perturbation
of the incident stream (the x terms in (3-41), for example) in order
to obtain this result.

3. The other coupling coefficients, az, and a,., contain
similar speed-dependent terms in Case 3; they arise at the same
point in the analysis as the terms discyssed in 2 above. We could
arbitrarily include such terms, +(U/w )b3z, in Case 1 too, without
causing any errors since bg, is zero anyway in Case 1.

4, In Case 1, there is a speed-dependent term in age which
is lacking in Case 3. The reason for the lack is that such a term
is higher order in terms of € in the ship problem, because of the
assumption that w= O(e"/2). There was no need for a high-frequency
assumption in Case 1, and so the extra term could legitimately be
retained.

5. If, in Case 3, one arbitrarily includes the forward-speed
term, =(W/a)aans in the ag, coefficient, making it identical to the
Case 1 coefficient, then it is consistent to modify b,, in a similar
way, namely by changing it to:

2
b =f dx x*n(x) - (U/w) b,,

55

The relationship between these forward-speed effects is quite the
same as that discussed above in paragraphs 2 and 3. Inthe bes
coefficient of Case 1, we could also introduce an extra term,
-(U/w)*b33, without causing any error, since b,, is zero anyway in
this case. Thus we can maintain the symmetry between Case 1 and
Case 3.

766

Stngular Perturbation Problems in Ship Hydrodynamics

TABLE 3-1

ADDED-MASS AND DAMPING COEFFICIENTS IN THREE PROBLEMS

CASE 1 - Body CASE 2 - Ship with CASE 3 - Body
with Forward Speed Zero Forward Speed with Forward Speed
in Infinite Fluid on Free Surface on Free Surface

dx m(x)

—7

dx n(x)

ae)

dx xm(x) + (U/w')b,, - (2pvU/u) Im {1}

'
7

dx xn(x) - Ua,, - (2pvU) Re {1}

a

dx xm(x) - (U/w)b,, + (2pvU/u) Im {1}

:

dx xn(x) + Ua,, + (2pvU) Re {1}

ca

iy dx x*m(x) - (U/w)"a5, dx x*m(x)

L

—

dx x?n(x)

oo

NOTE 1) In all cases, m(x) and n(x) are defined:

1
m(x) + y> n(x) = ef df n,$,,
Clix)
where C(x) is the wetted part of the cross section contour at x, and n, and 4, have the

same meaning as in Section 2.32 and 3.3. In CASE 1, $5 ia a real quantity, and so n(x) = 0.

NOTE 2) The quantity I in Case 3 is defined as follows: Let $= $4, and let ¢q be a 2-D potential
function which is sinusoidal in y, such that |¢- $,|—> 0 as y— oo. Then:

co
2 2 1 2
I “J dx [So [¢ (x,y ,0) = $a(x»y»0)] s oy. oitxsb(x0),0) ’

where b(x,z) gives the hull offset corresponding to the point (x,0,z) on the centerplane.

767

Ogilvie

6. The only forward-speed terms not yet discussed are those
in Case 3 which involve the integral I. They arise from the inclusion
of the functions $j; in the potential function, as in (3-43), and the
necessity for including those functions is a consequence of the fact
that the right-hand side of (3-42), the free-surface condition, is not
zero. Now, the right-hand side of (3-42) represents an interaction
between the forward motion and the oscillation. One might try to
simplify matters by assuming that one can neglect the effects of x,
the perturbation of the incident stream by the body. But this reduces
(3-42) to the following:

Ves + gus = oa 2UYy,; on YT Oi. (3-47)
O(c”? §) O(e 6)

The right-hand side is still not zero, and we would still have the
Qj functions to contend with. In fact, it may be recalled that this
remaining term on the right-hand side was the one that caused the
major trouble in interpreting the 92} problems. Neglect of the x
terms leads to the condition on Q) (cf. (3-46)):

Q) - vQj = - (2/g)4j,, on z=0, (3-48)

jz
and it is the one remaining right-hand term which causes the solution
for &j to diverge at infinity. The usual procedure at this point is to
set Q; = 0, turn the other way, and just ignore these problems. The
results are in remarkably good agreement with experimental obser-
vations, and one still wonders how this can be rationalized mathe-
matically.

Finally, we should at least mention the problem of predicting
wave excitations in the forward-speed problem. The singular per-
turbation problem involved in solving for the diffraction waves has
not been satisfactorily worked out yet, at least, not in a manner
compatible with the approach presented above.

One might hope to avoid the diffraction problem by using the
Khaskind relations, as in the zero-speed problem. (See Section 3.3.)
In fact, Newman [1965] has derived what I call the Khaskind- Newman
relations. These provide a generalization of Khaskind's formula,
relating the wave excitation on a moving ship to the problem of forced
oscillations of the ship when the ship is moving in the reverse direc-
tion. Unfortunately for our purposes, Newman's derivation i based
on an a priori linearization of the free-surface, in the sense that our
terms involving x can be neglected. Therefore, the appropriate
diffraction problem cannot really be avoided in this way. Also, it is
necessary to have available the potential function for the forced-
motion problem, and this includes at least a part of the 92} functions
even if the x dependence is ignored.

768

Stngular Perturbatton Problems tn Ship Hydrodynamics

In a not-yet published paper, Newman has applied the
Khaskind-Newman relations in the forward-speed problem by arbi-
trarily ignoring the 9j functions in the forced-motion potential
function. He finds for the heave excitation force:

‘et zg :
1 (t) = pgh(1 + Uw)/g)e- dx e cag dfn ene
L
Uv Uv
i E pes ay oe - al,

where, as before, w is the frequency of oscillation (that is, the
frequency of encounter) and v = w‘/g; the frequency measured in an
earth-fixed reference frame is denoted by wo, and we define

Vo = wo/g. The actual wave length of the incident waves js N= 2n/v.
The two frequencies are related as follows: w= wot Uwy/g- These
formulas are all valid for the head-seas case only.

This formula should be compared with (3-34a), which was
the corresponding result in the zero-speed problem. The first
term in brackets yields the Froude-Krylov force, and the second
term yields a pure-strip-theory prediction of the diffraction wave
force, which can be interpreted approximately in terms of the
relative-motion hypothesis. The remaining terms represent an
interaction between forward speed and the incident waves.

Again, it should be pointed out that more than just nonlinear
effects have been neglected in setting 2j equal to zero. In fact,
the usual linear free-surface condition for ship-motions problems
can be written:

by t BH, = - 2UW, - Udy, on 220,

(Cf. (3-37) and (3-47).) Even the inclusion of the Qj) terms still
omits some effects usually considered as linear, namely, the effects
of the term - Ud,, inthis boundary condition. These effects are
higher order in the theory presented here solely because of the high-
frequency assumption.

IV. THIN-SHIP THEORY AS AN OUTER EXPANSION

It has already been shown how one can view a symmetrical
thin-body problem in terms of inner and outer expans ions; the usual
description of the flow around such a body is really just the first
term of an outer or far-field expansion. It was not at all obvious that
one had to use such a powerful method on such a problem, but it was
clear that one could do this. Probably the only advantage of doing

769

Ogilvie

so in the infinite-fluid case was that one could avoid possible ques-
tions about the validity of analytically continuing the potential
function inside the body surface. On the other hand, one had then to
face all kinds of difficulties in principle in justifying use of matched
asymptotic expansions. It was a rather academic exercise.

The situation may be quite different in the thin-ship problem.
The purpose of this chapter is to show one can obtain the first
results of thin-ship theory in the same way as for the infinite-fluid
problem but that a second-order solution leads to fundamental
difficulty. The latter appears to suggest that a combination thin-
body/slender-body approach may be appropriate. A limited amount
of other evidence may be cited to support this idea.

I wish to emphasize that there are no new results in this
chapter. It is all a mater of interpretation. Perhaps someone will
be able to show that the problem discussed here has a trivial expla-
nation. On the other hand, perhaps someone will be stimulated to
do further research on the subject. In either case, I shall be happy
with the outcome.

The problem may be partially stated just as the infinite-

fluid, thin-body problem was stated. Let there be a velocity potential,
(x,y,z), which satisfies the Laplace equation,

[Li] xx + dyy + b22 = 0,

everywhere in the fluid domain and the body boundary condition,
[ H] O = dyhy F dy + $7h,, on y= h(x,z) = t€H(x pa):
Now we add on the two free-surface conditions:

[ A] 5 U'= gt +S Lont oy + 42), ony sz )=e6 (x53
[B] 0 = bySxt byby - $2 on 2 = £(x,y).

Also, we must specify a radiation condition.

In the far field, where y = O(1), we assume the existence
of the expansions:

N
o(x,y,Z)~ >) b(x,y>Z)>

n=O for fixed (x,y,z);

C(x.) = os C,(x2y)>

nsl

770

Stngular Perturbatton Problems in Ship Hydrodynamics

N
(x,y,z) a » Orcs vin 2) +

n=0 .
N for fixed (x,y/e,z).

C(x,y) ~ > Zy(X,y) »
n=l
We assume right away that:

bo(x,y,z) = @)(x,y 52) = |Upien

In the far field, the ship vanishes as €—~ 0, and so we take
the entire outside of the plane y = 0 (below the free surface) as the
far field. It is easily seen that the second term in the outer expan-
sion must be of the form:

$(x,y.Z) = - #S. o(&,6)G(x,y,z35,0,0) d6 dt, (4-1)

where His the portion of the centerplane of the ship below z=0,
o,(x,z) is an unknown source density, and G(x,y,z35 »1,6) is the
usual Green's function for a linearized problem of steady motion
with a free surface. It has the important property:

Gy tT KG) = 0), on Ze 0; (4-2)
where K = g /U*. Of course, the potential @;, also has this property:
Pie VOC, on z ="0.

For later convenience, we define
a,(x,z) = (x,0,z), (4-3)
and so a,(x,z) has the property too:
Te + Ka), = Or on Zia Ole (4-4)

With $ (x,y,z) given by (4-1), the two-term outer expansion
is:

(x,y,z) ~ Ux + $ (x,y,z),

Teh

Ogtlvte

and the inner expansion of the two-term outer expansion is:

1
Oi ny .z)i~ Ux.t.o,(x,2), + = ly | oy(x,2) een

O(1) Ole) O(é)

I have taken my usual liberty of indicating unproven orders of mag-
nitude. I am not really assuming these orders of magnitude; I am
saying that one can prove that these are correct, and I display them
here now simply as an aid to the reader.

Now consider the near field. Just as in the infinite-fluid
problem, one may stretch coordinates, y =€Y, and follow through
the consequences. This is effectively what I do, without writing the

change of variable explicitly. Thus, the Laplace equation yields
the condition:

Plyy = 9;

and so 4, must be a linear function of y. The same analysis as
used in the infinite-fluid problem, Section 2.11, leads to the con-
clusion that @; is even more restricted than this. It must bea
constant with respect.to. .y.. Thus, let:

®, (x,y,z) = A, (x,z).
The two-term inner expansion is then:
x,y,z) ~ Ux + Aj(x,z).
Matching gives the unsurprising result that:
A,(x,z) = a,)(x,z). (4-5)

In other words, once again the inner expansion starts out simply as
the inner expansion of the outer expansion; it is not necessary to
formulate a near-field problem to obtain this result.

The same arguments lead to the prediction that:

@o(x,y,z) = Ao(x ,z) + UbAx)Z) ly 3 (4-6)

Thus the three-term inner expansion is formed, and it can be matched
with the three-term inner expansion of the two-term outer-expansion,
yielding the familiar result once again that:

772

Stngular Perturbation Problems in Shtp Hydrodynamics

o Ge ,z) = 2Uh,(x,z).

(See (2-22).) This obviously had to come out this way, since we have
not yet introduced any effects of the presence of the free surface.

It should be noted that only the function Ao(x,z) is not already deter-
mined. (Knowledge of o, (x,z) allows us to express a(x,z) ex-
plicitly, from (4-1) and (4-3).)

A systematic treatment of the free surface-conditions leads
to the following:

[ A] 0 = gZ, + Ud, O(e)
+ gZp + Ud: + UZ, %,, +S (Gi, - 61.) +S (€3,) O( €%)
GPiaracs, 5 Onez = 10;

[ B] 0 = UZ, - O, O(e)
+ UZa, - @g, - 2,9, + ®,21, + &2 Ze, O(e*)
ter isheverce on 2 = 0.

The lowest-order conditions in [A] and [B] together require that:
+ KO. =70), on Zz =O

We see that this is automatically satisfied by our ©, (x55) = A\(x,z) =
a,(x,z).- (See (4-4).) The first term in the expansion for wave shape
in the near field is also determined:

Z Un
Z, (x,y) =- ra @) (x, 0).

This really says only that the free surface appears in the near field
to be raised (or lowered) by just the limiting value (as y — 0) of
C(x,y) in the far field. Again, a rather trivial result.

When we consider the (a terms in the free-surface conditions,
it is a different matter. The two conditions can be combined into the
following:

tts

Ui 1 2 2
O= & + Kd2. - gl lee T Oy l@ly F yh
(4-7)
Wi7.2 i}
+ > (hy)x - FU aX, ‘lj TU Wy ly F +a h,Zay-

2
In condition [| A] , we note that differentiation of the € terms with
respect to.y yields:

0 = Za, + Uhy.

Therefore, in the complicated free-surface condition above, (4-7),
only the first two terms involve y; all of the other terms are func-
tions of just x. From (4-7) and (4-6), we can thus write the follow-
ing:

Ora 1h x, 0) Kh, (x,0)] Uly| + (a function of , x).

XXX

This must be true for any y, and so we obtain the condition:
O7= hyyy ote Khyz; on Z= 0.

If the ship is wall-sided at z= 0, the second term is separately
zero, and so we would have to require that h,,,=0 at z=0.

Now this is clearly unacceptable. Why should our theory
work only for such a special case? (The waterline is made of circu-
lar arcs inthis case.)

As a result of our having stretched the coordinates, we came
to the prediction that the fluid velocity near the thin body consists
of a tangential component which is essentially independent of the
local conditions plus a normal component which depends only on
local conditions. Near the free surface, such results are simply
untenable.

I present here a formalism which apparently avoids this
difficulty. Again, I point out that no new results are obtained.
However, it does seem possible that the procedure might be fruitful
if studied further,.

The idea is to define a third region, complete with its own
asymptotic expansions of @ and ¢. This region will be essentially
the same as the near field in a slender-body analysis, that is, it is
a region in which y = O(€) and z= O(e) as € ~ 0. It follows from
this assumption that 8/8y and 8/8z both have the effect of changing
orders of magnitude by a factor 1/e. What is most important is
that this region is interposed between the thin-body near field and

114

Stngular Perturbatton Problems in Ship Hydrodyanamics

the free surface. Thus, it is no longer necessary or even proper to
try to make the previous inner expansion satisfy the free-surface
conditions.

We expect, as usual, that the first term in the expansion of
@ in this new near field will be just Ux. Furthermore, we can
expect the next term to be rather trivial, since the second term in
the previous near-field expansion did actually satisfy the free-sur-
face condition. Using the usual arguments of slender-body theory,
we find in fact that the three-term expansion of $ in this new field
is:

(x,y,z) ~ Ux + a,(x,0) + Uh,(x,0) ly | - = ZQ,,(x,0).
Olt) Ole) O(e*) O(e*)

The corresponding wave shape is found to be:

Gixry) ~ - Fay, (40) Of<)
We
- Z[ onyx, 0) ly | ecu ay, (x, 0) iggy (Xs 0) |
g : O(e*)
2
- 35[, (x,0) + U hilx, 0) +(Z ai cx.0)|
Pies

It can be shown in straightforward fashion that these results match
the far-field expansion as /(y2 + z2) ~ oo and they match the pre-
vious (thin-body) near-field expansion as z—~ - oo. Furthermore,
they satisfy the free-surface conditions without the necessity for
imposing unacceptable restrictions on the body shape. There is
just one aspect that requires special care: The free-surface condi-
tions cannot be satisfied on the surface z=0 inthis near field.
The reason is that the first term of the ¢ expansion is O(e), and
differentiation with respect to z is assumed to change orders of
magnitude by 1/e. Thus, suppose that we want to evaluate some
function f(z) on z=6€ in terms of its value (and values of its
derivatives) on z =0. The usual procedure is to write:

HCh= 0) 4 CEO), Fai SAO) Pawn
O(f) Ole)» O(f/e) O(e*)- O(£/e*)

With our set of assumptions, this expansion is useless; we cannot

C15

Ogtlvte

terminate it. The one simplification which is admissible here is to
evaluate f(z) and its derivatives on z=Z,, where € = Z, + o(e).*

I have not worked out any more terms in any of these expan-
sions, but I suppose that the next term in this near-field expansion
will be much more interesting. In the far field, it is well-known
that the third term in the expansion of the potential function will in-
clude the effects of what appears to be a pressure distribution over
the free surface. It was shown by Wehausen [ 1963] that at the inter-
section of the undisturbed free surface and the hull surface the solu-
tion is singular, and he represented the singular part by a line
integral taken along this line of intersection. From the point of
view of the method of matched asymptotic expansions, it should be
possible to represent the far-field effects of that line integral in
terms of an equivalent line of singularities onthe x axis. The
strength of the singularities would be determined, as usual, by
matching the solution to the near-field expansion. At this stage,
thin-ship theory will have become a singular perturbation problem.

V. STEADY MOTION IN TWO DIMENSIONS (2-D)

Sometimes we study two-dimensional problems with the intent
of incorporating the solutions into approximate three-dimensional
solutions, as in the treatment of high-aspect-ratio wings and in
slender-body theory. And sometimes we investigate two-dimensional
problems simply because the corresponding three-dimensional prob-
lems aré too difficult.

The problems discussed in this section are in the second
group. It is not likely that any of these problems and their solutions
will have practical application before several more years have
passed, even in the context of strip theories. Here are some of the
most fundamental difficulties related to the presence of the free
surface.

The first two subsections concern a 2-D body which pierces
the free surface. Such a problem is intrinsically nonlinear. We
might try to formulate the problem as a perturbation problem, in
this case involving a perturbation of a uniform stream. However,
there must be a stagnation point somewhere on the body, and at that
point the perturbation velocity is equal in magnitude to the incident
stream velocity. It is not small! If the stagnation point is near the
free surface, the free-surface conditions cannot be linearized. We
must find methods which are adaptable to highly nonlinear problems.

Such a method is the classical hodograph method, used since

*The same difficulty arises also in Sections 3.2 and 5.42.

776

Stngular Perturbation Problems tn Shtp Hydrodynamics

the nineteenth century for solving free-streamline problems. But it
introduces a new difficulty: It cannot be used to treat free stream-
lines which are affected by gravity, which means that only infinite-
Froude-number problems can be treated directly. This leads to a
further great difficulty, which is discussed in some detail in
Section 5.1.

In Section 5.3, a brief discussion is presented of the problem
studied by Salvesen [1969]. It contains two aspects of interest: It
is a case in which the free-surface conditions can be linearized
because of the depth of the moving body, and I have already commented
in the Introduction that there are very interesting fundamental ques-
tions involved in such procedures. Also, it presents a clear example
of the classical phenomenon discussed in the section on multiple scale
expansions: The wave length obtained in the first approximation must
be modified in subsequent approximations, or the solution becomes
unbounded at infinity -- where we know perfectly well that the waves
are bounded in amplitude.

Finally, Section 5.4 describes two recent attempts to approach
the problem of extremely low-speed motion. The difficulty is basically
this: In the usual linearization, we assume that all velocity components
(at least in the vicinity of the free surface) are much smaller than the
forward speed -- which becomes nonsense if we subsequently decide
to let U, the forward speed, approach zero. What is needed is a
perturbation scheme in which somehow the small parameter is pro-
portional to U. Then it is certainly permissible to allow U to
approach zero. Section 5.41 shows a very straightforward procedure
for doing this; however, it leads to a sequence of Newmann problems,
and so the wave nature of the fluid motion is lost. In Section 5.42,
an alternative method is discussed. It is an application of the multi-
scale expansion procedure to which Section 1.3 was devoted.

5.1. Gravity Effects in Planing

Before we try to treat this problem properly, let us consider
briefly a well-known approach to the 2-D planing problem and deter-
mine why it is not completely satisfactory. In the middle 1930's,

A. E. Green wrote several papers on the subject, and the essence of
his approach is well-presented by Milne-Thomson [1968]. A flat
plate is located with its trailing edge at the origin of coordinates, as
shown in Fig. (5-1). There is an incident stream with speed U
coming from the left, and, at infinity upstream, there is a free sur-
face at y = h. The effects of gravity are neglected. The fluid is
assumed to leave the trailing edge smoothly (a Kutta condition), and
a jet of fluid is deflected forward and upward by the plate. In the
absence of gravity, the jet never comes down to trouble us again.

In the figure, A marks the leading edge of the plate and C marks
the stagnation point.

The physical plane shown in Fig. (5-1) is also the complex

ett

Ogtlvie

Fig. (5-1). Planing Problem in Fig. (5-2). Planing Problem in
the Physical Plane the Plane of the
Complex Potential.

z=xtiy plane. Let F(z) = $(x,y) + id(x,y) be the complex
velocity potential for this problem, Then F(z) effects a mapping of
the z plane onto an F plane, as shown in Fig. (5-2), in which
points are marked to correspond to Fig. (5-1). It is assumed that
@=0 and w=0 atthe stagnation point. Furthermore, we have set
w= Ua onthe upstream free-surface streamline, IJ, which implies
that a is the thickness of the jet and that Ua is the rate at which
fluid leaves in the jet. Of course, F(z) is not known yet.

We can also consider that the z plane is mapped by the
function w(z) = dF/dz. w(z) is the "complex velocity," that is,
w=u-iv, where u and v are the velocity components inthe x
and y directions, respectively. The entire fluid region is mapped
by w(z) into the region bounded by a half-circle and its diameter,
as shown in Fig. (5-3). Again, points are marked to correspond to
Fig. (5-1). The diameter is the image of the planing surface, on
which the direction of the velocity vector is known, and the circle is
the image of the entire free surface, on which the magnitude of the
velocity vector is known (from the Bernoulli eyactiani: Again, we
note that the mapping function itself is not yet known.

Fig. (5-3). Planing Problem in Fig. (5-4). Planing Problem in
the Plane of the the Auxiliary (¢)
Complex Velocity. Plane.

778

Stngular Perturbation Problems in Ship Hydrodynamics

The functions w(z) and F(z) are, of course, very simply
related, although neither is known explicitly yet. In order to obtain
another relationship, one introduces the ¢ = € t+ in plane, in which
the fluid domain is mapped into the lower half-space, as shown in
Fig. (5-4). We can write out the explicit expressions for mapping
the F and w planes into the € plane. The first is accomplished
by means of the Schwarz-Christoffel transformation:

deo Ua. tite.
a) eateieic) Gat

which can actually be integrated, yielding:

ricien = B( £58 tog £22).

The second mapping can be shown to take either of the equivalent
forms:

ia G=¢

Ue
(1 - gc) ¢ivd - Avie? - 1)
Jt Gee) ale - 2 v(t? - 1)
- Cc

w(z(f))

(5-1)

The solution is then completed by using the relationship between F
and w, along with these expressions, to obtain the relationship
between z and ¢. Since:

dz _ dz dF _ _Hi(s)
C dF dt © wlz(t)) ’
(C) = H(6') dt!
20) = | wetery a
= _— -c(G-1) + (1 tbc) log ea7 + if(1-cWi(2-1)
- thy (1-c) log [¢ + vit®-1)]

- iV(1-c?)y(br-1) log ee 1)

T79

Ogilvie

So now we have z as a function of €, as wellas F and w as
functions of ¢.

There are three parameters inthis solution, a, b, and c,
none of which has been determined yet. By letting 4 | “> o.,7Green
came to the conclusion that the flow far away is a uniform stream
as required only if:

¢ = = cos @ and v(i - c*) = sina. (5-2)

(Both statements are necessary to avoid an ambiguity in sign.) Also,
one can use the z(€) formula to evaluate z at the leading edge of
the plate:

z(-1) = - be

(Compare Figs. (5-1) and (5-4).) This provides a relationship
among a,b, and c. Buc there are no more conditions to be found
unless we introduce more information about the physical problem.
For example, we could use the solution with unspecified values of
a and b, and work out the formula for lift on the plate. (Milne-
Thomson gives the formula.) If then we fix the value of lift, we
have another condition on a and b. However, this is rather a back-
wards way of going at the problem. We are most likely to want to
solve the entire problem just to find the lift and other interesting
physical quantities, and so we have not gained much if we must
assume the value of the lift as a given datum.

There is another anomaly in this result: The value of h

(See Fig. (5-1)) has not been used in any way. In the formula for
z(t), let G = &€, with |€| very large. Then every value of z com-
puted in this way gives a point on the free surface far away from the
planing surface. With a considerable amount of tedious algebra,
one can eliminate § and express y asa function of x (at least
asymptotically, as |x| oo). The first term is the most inter-
esting:

E ay (1 =e%)
‘er ors [log |x| + constant].

Thus, far away from the planing surface, the free surface apparently
drops off logarithmically to - oo. The slope of the free surface
approaches zero (« 1/|x|) and so there is no violation of our assump-
tion that the flow at infinity is simply a streaming motion parallel to
the x axis. But obviously the assumption that the trailing edge was
located at a height h below the free-surface level at infinity was
quite meaningless, and it cannot be enforced in the solution.

780

Singular Perturbation Problems in Ship Hydrodynamics

There are thus two difficulties: 1) The above solution is not
unique (a common difficulty in free-streamline problems); 2) It has
unacceptable behavior at infinity.

These difficulties were resolved by Rispin [1966] and Wu
[1967] , who recognized that the solution of Green's problem is part
of a near-field (inner) expansion of the complete solution. An inner
expansion does not necessarily satisfy the obvious conditions at
infinity; it must only match some outer expansion in a proper way.
Rispin and Wu produced the appropriate outer expansions and showed
that matching does occur. The effects of gravity appear first in the
far field, which is hardly surprising, for two reasons: 1) Far away,
one expects to find gravity waves as the only disturbance. 2) The
divergence of the free-surface shape in Green's solution is so weak
that one might expect the smallest amount of gravity effect to bring
the free surface into the region where we expect to find it: thus,
the small effect of gravity eventually would have a large consequence,
but only far away from the planing surface.

Rispin defines the small parameter:
B= gf/U°=1/F*,

where F is the usual Froude number. Inthe near field, the natural
coordinates are used, which means effectively that £ is considered
to be O(1). Smallness of B is achieved by allowing g—~0 or

U— oo. Rispin treats his small parameter properly by nondimen-
sionalizing everything, so that he then does not have to specify
whether U-~oo or g~O. Rather than change all variables now,
I shall treat g as a small parameter, as in Section 3.2; the results
are the same as Rispin's, of course.

In the far field, typical lengths are assumed to be O(1/) in
magnitude, or O(1/g), in my loose notation. We could define new
coordinates, say,

z2=Pz; x= 6x; y= By,

and consider that z = O(1) as g—> 0 inthe far field, while z= O(1)
as gO inthe near field. Rather than do this, we shall just keep
in mind that such orders of magnitude are to be assumed. Also, we
note that d/dz = O(1) inthe near field and d/dz = O(8) in the far
field.

This problem is reversed from the most common kind of
stretched-coordinate problem: The inner problem is solved by
natural coordinates, and the outer coordinates are compressed.
Note, however, that there is no distortion of coordinates between
near- and far-fields. There is just a change of scale.

781

Ogilvie

In the far field, the planing surface appears to vanish in the
limit, and so the first term in a far-field expansion must represent
just the incident uniform stream. That is, if the outer expansion is
represented:

N N
F(z;B) ~ : F, (zB), w(z;B) ~ 3 W,(z38), for fixed Bz
n=O n=O asB—O,

then clearly we have:
Fo(z;B) = Uz, and Wo(z;B) = U.

This one-term outer expansion must match the one-term inner ex-
pansion, the latter being just Green's solution. This much of the
matching procedure is rather obvious, and Green already used this
fact to determine the value of c, as given in (5-2).

The next term in the outer expansion is not quite so obvious.
In order to facilitate the matching process, Rispin solved the problem
inthe ¢ plane, just as we did above for Green's problem. The
free-surface boundary condition on W, is not much different from
the familiar linearized condition. One can show fairly simply that:

dW, , igA 3 A,
Re[ Sr! + -B5 wi] =0 on = :0';

where A=a/n(b tc). (The factor A is just the value of dz/dt

far away from the planing surface.) Note that the first term is
O(BW,) because of the differentiation, and the second term is the
same order because of the g factor. The solution for W, must

be analytic in the lower half-space and satisfy this condition on 7 =0,
|€ | > 0; note the exclusion of the origin, where singularities may
Occur.

As usual, we try to restrict the singularities to the simplest
kind possible. In this case, we would find nothing in the near field
to match with if we allowed all kinds of singularities in W,. A
sufficiently general solution” is the following:

-igat/uz°$ igAt/U*TC
W, (638) = te aa { dt e" [St +S ,
[e.e)

where C; and Cg are real constants yet to be determined (in the
matching).

SG Gales sae
Rispin discusses more general solutions, which are needed in con-
structing higher-order solutions.

782

Stngular Perturbatton Problems in Ship Hydrodynamics

The two-term outer expansion is now:

w(z3B) ~ U + W, (S38),

with W, given as above. Its inner expansion to one term is easily
found:

w(Z3p)°o u- + °

We cannot really say positively that these two terms are the same
order of magnitude, but it turns out that they must be if this expres-
sion is to match the two-term outer expansion of the one-term inner
expansion. The latter is obtained readily from Green's solution for
w(z(€)) which was given in (5-1). It is:

wipe + iU sina
C
Then, obviously, we find that:
C, = - U sina.

We cannot determine the other constants, C,, from the solu-
tions so far obtained. It is necessary to solve for the second term
in the inner expansion, and Rispin carries this through. Then, he
matches the two-term outer expansion of the two-term inner expan-
sion with the two-term inner expansion of the two-term outer expan-
sion, finding that C, = - aU/n. Thus, C; is proportional to the
rate at which fluid leaves in the jet; the C, term represents a sink,
infact. (The C, term represents a vortex.)

Rispin obtains estimates for h as well, but the results are
rather complicated, and it would add no perspicuity to the present
section to repeat them. The important point in principle is that it is
possible now to specify the value of h and not come to a contradic-
tion as a result. The far-field description has effectively provided
a height reference, because of the effect of gravity. This effect
does not change the first-order inner solution, but it does modify the
second-order term. (The velocity magnitude is not constant on the
free surface in the second approximation. )

In the second-order term of the inner expansion, there is
another interesting phenomenon, namely, the apparent angle of
attack changes. This means, physically, that the occurrence of
gravity waves modifies the inflow to the planing surface. In the

783

Ogtlvte

near field, it is still not possible to see the waves that exist far
away, but the latter have the effect of making the incident stream
appear to be rotated somewhat. It is like a downwash effect (although
the physical origin is quite different).

If one were given a planing problem such as we formulated
early in this section, with the incident stream and all geometric
parameters prescribed, it would be necessary to solve for the
parameters a and b. One equation relating these parameters has
already been mentioned, namely, the equation relating the length of
the plate to these parameters. The other equation comes from the
expression (which was not written out here) for has a function of
a and b.

Rispin avoided much tedious algebra by solving the inverse
problem. He assumed that a, b, and c were given, then solved
to find h. He also had to treat the angle of attack as an unknown
quantity, and he found an asymptotic expansion for it. (Note that
only two of the basic parameters can be prescribed arbitrarily,
unless we are prepared also tolet £ be an unknown quantity.)

One final comment on Rispin's work must be made. He finds
terms of six orders of magnitude: O(1), O(f8 log B), O(f), 0(p? log? B) ;
O(B* log 6), and O(8*). But he finds also that they cannot be deter-
mined one at atime. Rather, they must be taken in groups: a) the
O(1) terms, b) the terms linear in B (the logarithm being ignored),
and c) the terms involving 8°. This is the same kind of matching
procedure that would have been used if he had adopted the working
rule that logarithms should be treated as if they were O(1). (See
Section 1.2.)

5.2. Flow Around Bluff Body in Free Surface

A problem related to that of Rispin [1966] and Wu [ 1967]
has been studied by Dagan and Tulin [1969]. They have concerned
themselves with the flow at the bow of a blunt ship, where any kind
of linearization procedure must be completely wrong. In order to
handle such a situation, they have adopted essentially the same pro-
cedure that the previous authors used, namely, they set up inner-
and outer-expansion problems in which the nonlinearity is confined
initially to the near field and the effects of gravity are confined
initially to the far field. Then, by limiting their study to a two
dimensional problem, the nonlinear near-field problem can be solved
by the hodograph method, and the far-field problem is a simple vari-
ation of a well-studied problem in water-wave theory. The geometry
of their problem is shown in Fig.(5-5), which is reproduced from
their paper. They argue that at very low speed there will bea
smooth flow up to and then down under the bow, with a stagnation
point at the location of highest free-surface rise, but that that flow
becomes unstable as speed increases, until finally a jet forms, as

784

Stngular Perturbatton Problems in Shtp Hydrodynamics

Fig. (5-5). Bluff Body in the Free Surface

sketched in Fig. (5-5). Regardless of whether their description of the
flow at very low speed is correct , this jet model appears to be
entirely reasonable physically; a barge-like body usually causes a
region of froth just ahead of the bow, and this froth is probably

caused by such a jet being thrown upward and forward, then dropping
downward (which the theory overlooks). Thus it seems appropriate

to study the formation of such a free-surface jet by the use of free-
streamline theory, and one may expect that the details of the formation
of the jet are not terribly sensitive to the effect of gravity.

The body, as shown in Fig. (5-5), extends downstream to
infinity. (In a sense, the whole problem is part of the inner expan-
sion of a much larger problem, in which the stern of the body would
be visible and in which waves would follow the body.) Thus, there
is no Kutta condition or equivalent which can effectively cause a
circulation type of flow in the fluid region. In Green's problem, for
example, the flow at great distances appears to have been caused by
a vortex. It is this property that causes the apparent logarithmic
deflection of the free surface far away from the body, and it is this
property that requires the far-field description (as in Rispin's
problem) to contain a logarithmic singularity at the origin. Dagan
and Tulin have no such logarithmic solutions.

They find that the jet appears, from far away, to be caused
by a singularity of algebraic type. Specifically, the outer expansion
of their inner expansion shows the complex velocity behaving like
Zaye: where Z is the complex variable defined in the physical
plane, shown in Fig. (5-5). Thus, their far-field expansion must
exhibit a singularity at the origin of this same type.

This result, if correct, is most interesting, for, as Dagan
and Tulin point out, it means that the far-field expression for

*
Their Section III. 2 has some questionable aspects.

785

Ogtlvte

pressure is not integrable, and so one must use the near-field ex-
pansion for any force calculation. Furthermore, it is a disturbing
result, because it suggests that many previous attempts to incor-
porate bow-wave nonlinearities into linear-theory singularities
have been futile exercises.

Personally, I am not yet willing to admit that the possibility
of having the complex velocity behave like Z™ is really to be re-
jected, as Dagan and Tulin claim. Wagner [ 1932] analyzed the
region of the jet and the stagnation point for the flow against a flat
plate of infinite extent downstream, and he showed that this flow,
from far away, has the behavior of a flow around the leading edge
of an airfoil, that is, the velocity varied with nee Physically it
seems rather difficult to imagine that, by curving the body around
just behind the stagnation point, one causes such a drastic change in
the apparent singularity.

Dagan and Tulin present a figure (their Fig. 2) in which they
have placed many symbols showing beam/draft ratios of more than
a hundred ships, and it is quite evident that most ships have values
of this ratio considerably greater than unity. They then use this fact
as an alleged justification for claiming that their 2-D model of the
bow flow (as in Fig. (5-5)) will have some validity in describing the
flow around the bow of an actual ship -- since most ships are pre-
sumably of the "flat" variety. However, this claim is completely
misleading. The theory might apply to a scow, but not to a ship.
After all, beam/draft ratio is measured amidships, and even ships
with the largest block coefficients have entrance angles less than
180°.

Also, it is appropriate to mention again the warning against
defining a small parameter precisely and then trying to interpret on
some absolute basis whether a particular value of the parameter is
"Small enough." For example, it is conceivable that a thin-ship
analysis would be valid for a ship with beam/draft ratio of 10,
whereas a flat-ship analysis might fail for the same ship. I am not
saying that this is likely, but it is possible. In one problem, a
value of 10 might be "small," whereas in another problem a value
of 1/10 might be "not small."

Notwithstanding these objections, the paper by Dagan and
Tulin has provided a refreshing change in outlook on the bow-flow
problem, and perhaps it will be more fruitful eventually than the
usual attempts to place complicated singularities at the bow in the
frame-work of linearized theory.

786

Singular Perturbatton Problems in Shtp Hydrodynamics

5.3. Submerged Body at Finite Speed

Since the principal difficulty in solving free-surface prob-
lems follows from the nonlinear conditions at the free surface, we
are always seeking new arguments to justify linearizing the condi-
tions. One possible basis for linearizing is that a body is deeply sub-
merged. Then its effect on the free surface will presumably be
small, even if it is not appropriate to linearize the problem in the
immediate neighborhood of the body itself.

Such problems were discussed by Wehausen and Laitone
[1960], where the previous history may also be found. Tuck [1965b]
introduced a more systematic treatment for the case of a circular
cylinder. Salvesen | 1969] solved the problem for a hydrofoil (with
Kutta condition and thus with circulation), and he compared his
results with the data from experiments which he conducted. In the
earlier studies of such problems, the approach was usually an itera-
tive one in which the body boundary condition was first satisfied,
then an additional term was added to the solution so that the free-
surface condition would be satisfied; the latter would cause the body
boundary condition to be violated, and so another term would have to
be added to correct that error, but then there would again be an error
in the free-surface condition. And soon. The free-surface con-
dition that was satisfied once during each cycle was generally the
conventional linearized condition. Thus, if the procedure converged,
one obtained a solution which exactly satisfied the body boundary
condition and the linearized free-surface condition. The contribution
of Tuck seems to have been in systematizing the procedure in terms
of a small parameter varying inversely with depth of the body and in
pointing out that a consistent iteration scheme involves using the
exact free-surface conditions as a starting point. Then, as the
boundary condition on the body is corrected at each stage, so also is
the free-surface condition made more and more nearly exact.

Tuck concluded, in fact, that it was more important to include
nonlinear, free-surface effects than to improve the satisfaction of the
body boundary condition if one were most interested in certain free-
surface phenomena, e.g., predicting wave resistance and near-
surface lift. Salvesen agreed with this conclusion only on the con-
dition that the body speed be not too large. At fairly high speed,
his results indicated that precision in satisfying the body boundary
condition was just as important as precision in satisfying the free-
surface condition. Figure (5-6) is taken from Salvesen's paper; it
shows the theoretical wave resistance of a particular body as a
function of (depth) Froude number, the resistance being calculated
by three different approximations: 1) linearized free-surface
theory, 2) theory in which the free-surface condition is satisfied to
second order, and 3) theory in which both the free-surface condition
and the body boundary condition are satisfied to second order. The
differences are quite apparent.

787

Ogtlvie

0.03

aw?

0.02
06

001
+——40.5

03 }

= 04 = 04
03 J+. los

RESISTANCE

J
03 05 07 09 1
FROUDE NUMBER, U/Vgb)

—___._._ ,_ first-order theory;

————_,_inconsistent second-order theory
(neglecting body correction effects);

———-— ' consistent second order theory

(From Salvesen (1969))

Fig. (5-6). Theoretical Wave- Resistance
Gurves for € = t/b = 0.30.

The figure is a very interesting one. The difference between
the linear-theory curve and either of the other two curves is pre-
sumably a second-order quantity, and yet that difference is -- in
one case -- of the same order of magnitude numerically as the
linear-theory curve itself. The problem is worth further discussion.

Salvesen defines his small parameter as follows:
= t/b;,

where t is the thickness (or some other characteristic dimension)
of the body, and b is the submergence of the body below the undis-
turbed free-surface level. It is not assumed that the body is "thin"
in any sense; it could be a circular cylinder (Tuck's problem), for
example. Salvesen's calculations and experiments were carried out
for a rather fat, wing-shaped body with a sharp trailing edge. The
body was symmetrical about the horizontal plane at depth b. Ifthe
free surface had not been present, there would have been no lift on
the body.

A complex velocity potential, F(z) = $(x,y) + iu(x,y), can be

defined for the problem, with z =x tiy measured from an origin
located in the body at a depth b below the undisturbed free surface.

788

Stngular Perturbation Problems in Shtp Hydrodynamics

Salvesen expands the complex potential in a series which he groups
in two alternate ways:

F(z)

il

[ Uz + Fol pa be + Fp] eas (5-3)

Uz +[ Fp, + Fe] Pho, + Ff] ae (5-4)

These terms are defined in terms of the iteration scheme already
mentioned. The grouping in (5-3) is to be used near the body, and
the grouping in (5-4) applies far away from the body; in particular,
the latter applies on and near the free surface. Salvesen points out
that this distinction means that: a) near the body, we are consider-
ing the zero-order flow to be that flow which would occur in the
presence of the body and the absence of the free surface, and b) near
the free-surface, the basic flow is just the uniform a stream,
Thus, in (5-3), we must determine Fpo so that [Uz + F satisfies
the kinematic boundary condition on the body and ae that Wore, => 0
as |z| — oo (in any direction).

Next, Salvesen assumes that Fpo is O(e) far away from the
body. The two terms so far obtained do not satisfy a free-surface
condition, and so F¢, must be determined so that, when it is added
to the first two terms, the sum satisfies the appropriate free-surface
condition, which is:

Re (Fy + Fe, tikFy) tixF,}=0 on y=b (5-5)

where K= g/U*. Since Fpop is assumed to be O(e€) near the sur-
face, then the same should be true for Fe, as

Now the three terms in the series do not satisfy the body
condition, and so Fp, is determined so that, when it is added to
the first three terms, the sum satisfies the condition properly.
Then Fp, is assumed to be O(e*) near the free surface, and a new
function F¢, is found to provide a further correction needed near
the free surface.

It is in this last step that the Tuck-Salvesen approach differs
from the previous treatments of such problems. If Fp, is really
O(e?) , then the free-surface condition ought to be gafiatied to that
order of magnitude. It can be shown that this implies the following
condition on Fe

Re {Fy f FY, +ikFp, + ikF4,}

= 7, {Fp + FY, + iKFy + iKFy J : (1/2U)| Fy. - (5-6)

789

Ogtlvte

The right-hand side of this equation takes account of the nonlinearity
of the free-surface conditions , since obviously it involves just the
potential function from the previous cycle of the iteration. 1, is the
free-surface elevation from the previous approximation; it is given
by:

n(x) = - (U/g) Re {Fo + Fy};

with the right-hand side evaluated on y= b. One might try to cut
corners in (5-6) in either of two ways, namely, 1) ignore the right-
hand side by setting it equal to zero, 2) Drop the terms involving

Fp, on the left-hand side. The first is equivalent to retaining just

a linear free-surface condition. The second is equivalent to neglect-
ing the effect of the second-order body correction at the free surface};
this is the "inconsistent" second-order theory to which Fig. (5-6)
Ferrers.

Apparently , Salvesen did not prove one important step in his
development, namely, his claim that Fbpg is O(1) near the body
and O(¢e) far away from the body. In fact, with his definition of
e = t/b, it appears that the statement is wrong. The potential Fbo
represents just a thickness effect, since it is the solution of the
problem of a symmetrical body ina uniform stream. Although the
body can be replaced by a distribution of sources, the disturbance
will appear from far away to have been caused by a dipole, and
so it must have the form: Fbo~ C/z. If the body were a circular
cylinder, we could evaluate C: C= Ut’, where t is the radius of
the cylinder. The complex fluid velocity on the free surface caused
by the body is, in the first approximation, - C/z*= O(e*), since
z =x +tib onthe level of the undisturbed free surface. This con-
clusion contradicts Salvesen's assumption that the free-surface
disturbance is O(e€), but perhaps it does not matter. At this point,
the results would presumably be just the same if he had defined:
€= (t/b)! e (The argument above for a circular cylinder agrees
with Tuck's conclusions.)

When the first free-surface correction is found, namely,
Fe,» its effect in the neighborhood of the body is not diminished by
an order of magnitude, since at least one part of F¢, involves an
exponential decay with depth, the exponent being K(y - b). Near the
body, y= 0, and so the exponential-decay factor is e“", and it has
been assumed that Kb is O(1). (See Salvesen's paper.)

Since Fy, is O(e*) near the body, the order of magnitude
of the next correction term, Fp); must be the same. This time,
however, the nature of the body disturbance is quite different from
a dipole disturbance. The effective incident flow corresponding to
Ff, is not a uniform stream, and so the presence of a sharp trailing
edge on the body requires that a Kutta condition be imposed, and
then a circulation flow occurs. From far away, it appears that Fp,

790

Singular Perturbation Problems in Shtp Hydrodynamics

is caused by a combination of a vortex and a dipole. If the strengths
of the two apparent singularities were comparable, the vortex
behavior would dominate the dipole behavior far away, and the
induced velocity would diminish in proportion to 1/z, rather than
1/z*, which was the case for the dipole. Thus, Fp, would be O(e>)
near the free surface. In the absence of a sharp trailing edge which
can cause the formation of a vortex flow, the corresponding Fp,
would be O(e€4%). This matter remains to be resolved.

There are other interesting aspects to this problem. One
relates to the interpretation of the small parameter, € =t/b. In
defining such a dimensionless perturbation parameter, one nor-
mally assumes that the smallness of € can be realized physically
either by letting t be extremely small or by letting b be very
large. In the present problem, this choice is not really available
tous. The reason is that there is another length scale in the prob-
lem, namely, 1/xk = U‘/g, and this length scale appears generally
in combination with the dimension b. It has been assumed that
Kb = O(1) as € ~ 0. Therefore, if we want to consider the problem
of a body which is more and more deeply submerged, (b —~ oo),
then we must also restrict our attention to higher and higher speeds.
This is awkward.

Finally, one more important aspect must be mentioned. The
relation between wave number, xk, and forward speed, U, namely,
K = 2/U-, is based on linearized free-surface theory. In general,
if one seeks to find the nature of nonlinear waves which can propa-
gate without change of form, the wave length of those waves is not
related to their speed in this simple fashion. To be sure, the
relationship is approximately correct if the waves are not terribly
big in amplitude, and so one might expect that the wave length or
the wavenumber can be expressed as an asymptotic series in €

K~ Kot K, + Ky toes F

with Kg = g/U*. This can indeed be done, but it turns out to be

much more convenient to assume that kK is precisely given and then

to find the value of forward speed that corresponds to that wave
number. Thus, one expands the forward speed, U, into an asymptotic
expansion:

Ut ug Fy Pech ss

This procedure is discussed by Wehausen and Laitone [1960], and
Salvesen uses it in his hydrofoil problem. I was able to omit mention
of it in writing Eqs. (5-5) and (5-6) because it turns out that u, = 0,
and so the effect of this speed shift (or period shift) does not enter
the problem until the third approximation is being sought. However,
this is a classic example of the kind of expansion described in

ioe

Ogilvie

WAVE ELEVATION,FT

~~

BODY LOCATION ne,

ae, he FIRST—ORDER THEORY
a 4 SECOND— ORDER THEORY
THIRD— ORDER THEORY
es EXPERIMENT
FROUDE NUMBER =0.79; &=t/ b=0.30
(FROM SALVESEN (1969))

wwe eee eee |

Fig. (5-7). Third-Order Effect on Wave Length

Section 1.3. If one did not allow for a variation in either K or U,
the third approximation would not be valid at infinity, and so one
would have great difficulty in predicting wave resistance, since that
quantity depends explicitly on the wave height at infinity.

Figure (5-7) is taken from Salvesen [1969]. It shows very
clearly the change in wave length that arises in the third-order
solution. In fact, it appears in this case that the change of wave
length is practically the only third-order effect. This figure also
speaks well for Salvesen’s experimental technique!

5.4. Submerged Body at Low Speed

Salvesen [ 1969] computed the wave height behind a hydrofoil
up to the third approximation, as already mentioned in Section 5.3.
Although his third approximation is not really consistent, he gives
what appear to be sufficient arguments to demonstrate that the con-
sistent result would not be much different from the results presented
in his paper. Figure (5-8), from Salvesen [ 1969], presents the wave-
height computations in a way that shows the relative importance of the
first-, second-, and third-order term. Let the wave amplitude be
expressed by the series:

He yh Bett ss where Has = 0(H,) as t = 310.

192

Singular Perturbatton Problems in Ship Hydrodynamtes

F = U/y(gb)
E=t/b

fo)
°

WAVE -HEIGHT RATI

2 3
gt its = €bk
—-— H, / (Hy +Ho*Ha)
—---- Hb/ Wane)
———_—= Ha/(H)+HotH 3)
From Salvesen (1969)

Fig. (5-8). First-, Second-, and Third-
Order Wave Heights at Low
Speeds.

(t is the thickness of the foil, as in the last section.) Then the
figure shows the three ratios, H,/(H, + H2+H;), for n=1,2,3;
that is, each curve shows the relative contribution to the wave
height of one of the first three terms in the wave-height expansion.
As speed decreases (toward the right-hand side of the figure), the
second-order part comes to dominate the linear-theory part, and
then the third-order part dominates the first two. It seems quite
likely that the fourth-order term would take over if the graph were
extended, then the fifth-, sixth-, ... order terms.

Salvesen's analysis is based on the condition that t (or,
more properly, t/b, where b is the body depth) is very small; the
Froude number is simply a parameter unrelated to t, which is
equivalent to saying that Froude number = U/V(gb) is O(1) as
t/b + 0. Perhaps it is not surprising if Salvesen's expansion is not
uniformly valid with respect to Froude number. That is all that
Fig. (5-8) really says.

The reason for its nonuniformity has already been mentioned:

In the expansion of the solution near the free surface, it has been
assumed that the lowest-order approximation is just the uniform-
stream term, Ux; all other terms in the expansion of the potential
must be very small compared to this term. And this is nonsense if
we consider the limit process U~0. Of course, we might have
been lucky: It could have turned out that the velocity perturbation
approached zero more rapidly than U. But it does not. And so we
have here a genuine singular perturbation problem.

Let us consider a sequence of steady-motion experiments,

each lasting for an infinite length of time. We arrange the sequence
of experiments according to decreasing values of body speed, U,

T93

Ogilvie

and we suppose that all conditions except forward speed are identical
in all experiments. We shall discuss what happens when "UO,"
and we shall understand by the limit operation that we are passing
through the sequence of experiments toward the limit case in which
there is no forward speed at all. In each experiment, U isa
constant. *

As U~>0O, we certainly expect all fluid motion to vanish.
But we would like to know to what extent the velocity field vanishes
in proportion to U (that is, what partis O(U)), what part vanishes
more rapidly than U (that is, what partis o(U)), and what part,
if any, vanishes less rapidly than U.

In an infinite fluid, the velocity everywhere is exactly pro-
portional to U. Far away, the velocity approaches zero; it drops
off like 1/r if there is a circulation around the body, and it drops
off like tps if there is no circulation. But in both cases the
constant of proportionality is O(U). No matter how distant our
point of observation is from the body, the velocity is O(U) as
Ui —= Oe

At very low speed, one expects that gravity will force the
free surface to remain plane. The constant-pressure condition will
be violated to the extent that the magnitude of the fluid velocity on
that plane is not quite constant, but the error in satisfying the dy-
namic condition will be proportional to the square of the fluid
velocity magnitude. The kinematic condition will be satisfied ina
trivial manner. Accordingly, it seems quite reasonable to assume
that the free-surface disturbance is O(U‘) as U0, and so the
velocity potential in the first approximation is the same as if the
free surface were replaced by a rigid wall. Let the rigid-wall
velocity potential be denoted by (x,y). Clearly, it is true that:

$o(x,y) = O(U).

This follows by the same arguments as those used in the preceding
paragraph. The more important problem is to determine the order
of magnitude of [ $(x,y) - $o(x,y)], where (x,y) is the exact
velocity potential for the case of the body moving at speed U under
the free surface.

In order to be specific now, let (x,y) be the velocity
potential in two dimensions which satisfies the conditions:

£90=0, on body; |$o- Ux| +0, as x— - a} Fe = 0 on y= 0.

*
This is the same point that I belabored in the last paragraph of
Section 1.2. Again, I apologize to those to whom it is obvious.

794

Stngular Perturbatton Problems in Shtp Hydrodynamics

The body is at rest in our reference frame.

The rigid-wall solution satisfies all conditions of the free-
surface problem except the dynamic condition on the free surface.
The latter could be used to define the free-surface shape. Thus, if
the free-surface disturbance is expressed by

mix) nolx) tk) ce Gf

the dynamic free-surface boundary condition says that:

nlx) ~ nolx) = 35 LU" - 40,(x,0)] (5-7)

Of course, the kinematic condition is now violated, but an additional
velocity field which is O(U*) can correct that. And so it appears
plausible that:

(x,y) - $o(x,y) = 0(U). (5-8)

One point should be noticed from this conclusion. The limit
process "U-—-0O" implies that Froude number goes to zero. Nothing
has been said about the length scale used in defining Froude number,
but it does not matter so long as all dimensions are fixed. The
submergence and the body dimensions may be quite comparable,
for example. Thus, we are not considering t/b as small, in the
sense that Salvesen did. However, both t and b are supposed to
be large compared with the length U*7s; we imply this if we state
that all dimensions must be fixed as U0.

It would be wrong to take (x,y) as the potential for the flow
around the body in an infinite fluid (without either free surface or a
rigid-wall substitute). The body can be quite near to the free
surface in Salvesen's sense, and so the effect of its image cannot be
neglected. Furthermore, at least part of the effect of the image is
O(U), even if the body is very far away from the free surface, and
such an effect must be included in the first term of the approximation
which is supposed to be validas U~>0O.

The next problem is to find [ $(x,y) - $9(x,y)]. We consider
two possible approaches in the following subsections.

5.41. A Sequence of Neumann Problems. As above, let there

be a velocity potential, (x,y), which provides the solution of the

exact problem:

195

Ogilvie

2

en) +5 ent ey) -5U=0, on y=nlxs (5-9)

OxNx - y= 9, on y = (x); (5-10)
oe = 0, on the body; (5-11)
(2 5.y))-) Ux Fe 105 as xo, = 003 (5-12)

The rigid-wall potential, (x,y), satisfies (5-11) and (5-12) too,
but it does not satisfy the free-surface conditions, of course;
instead, we have

8
od =05 on y =0. (5-13)

Now we introduce one more potential function, the difference between
the above two potentials:

@(x,y) = (x,y) $ oo(x,y) ° (5-14)

It must satisfy the body boundary condition, of course, and it
vanishes far upstream. On the free surface, which we now define as:

y = N(x) = No(x) + H(x) , (5-15)

where Tox) is defined as in (5-7), the new potential satisfies the
two conditions:

0 = gH(x) - 5 66,x,0)
2 2 2
+5140, + $0, + 240% + 2hoydy te + Fy lyeqoy § (5-16)
0 =[nolx) + H'(x)][¢0, + Pl enti - [Hoy # Py] Jenin” (5-17)

These conditions are still exact. An obvious approach to
solving for ®(x,y) and H(x) is to re-express these conditions on
y = n(x) as conditions on, y = 0. Here I shall assume that this can
be done in the usual way.” Then it follows from the exact conditions

*This is the crucial point which distinguishes this section from the
next section.

796

Singular Perturbatton Problems in Shtp Hydrodynamics

that the following are appropriate simplifications:

O= gH(x) + 0,2, » on y = 0; (5-18)

d, = To) $0, - Nolx) Poy on y=0O. (5-19)

The second condition is a Neumann condition; the right-hand side
is known, and the condition is prescribed on a known, fixed surface.
In fact, (5-19) is satisfied by the real part of:

00
*) ds p(s)

s-z’
00

where

z=x tiy,

P(X) = N(x) $9 (x, 0). (5-20)

This follows from the Plemelj formula. (See, e.g., Muskhelishvili
[1953].) The function p(x) can be interpreted in terms of the fluid
velocity which is needed to correct the flow field because of the
error incurred by taking the free surface at y = N(x) while using
the potential function $ x,y) to prescribe the velocity field. This
is the same correction which was discussed above in gonnection
with (5-8). Now we may observe that, since No= O(U) and

G(x,y) = O(U), it follows that p(x) = O(U"). Thus also:

|Vve|=o(u) as UO. (5-21)

This is certainly a much stronger conclusion than (5-8)!

The integral expression given above is not the solution of the
@ problem, even in the first approximation, since it does not satisfy
the body boundary condition. However, since the existence of ®
arises from a defect of #9 in meeting the free-surface conditions,
it is difficult to imagine that the above estimate of the order of mag-
nitude of @ is not correct.

Numerical procedures could readily be worked out for
solving problems of the above type. In fact, all that is needed is
one algorithm which handles the problem of a given distribution of
the normal velocity component on a surface in the presence of a
plane rigid wall. The integral part of the solution given above would

197

Ogtlvie

lead to a non-zero normal velocity component on the body, and this
would have to be offset by a flow which does not change the condition
at the plane y=0. Presumably, all higher-order approximations
would be solutions of problems which are identical in form to this
one,

A variation on this approach has been discussed several
times by Professor L. Landweber, although he has not published
the work. He points out that the usual linearized free-surface
condition,

2
a og 2
—_ + = = =
Ae K 35 0, on z=0, where K= 2/Us,

becomes the rigid-wall condition when U~— 0, and so one might try
an iteration scheme in which @ is expanded ina series, $¢~ hy eo;
and the terms are obtained as the solutions in an iteration scheme:

2
x

In order to test the scheme, Professor Landweber proposed trying

to obtain the potential function for a Havelock source in this way; this
obviates the need to satisfy a body boundary condition, and the known
potential for the source can be expanded in a series in terms of 1/kK.

Neither of the above schemes appears very promising to me.
Salvesen's findings about the singular low-speed behavior seem to
condemn any approach which overlooks the peculiar nature of the
free-surface problem at low speeds. The next section should make
clear why I am pessimistic about these approaches. It should be
obvious even now that the wave-like nature of the problems has been
lost, but the difficulty is more serious than that.

5.42. A Dual-Scale Expansion. According to linearized
wave theory, the wave-like nature of a free surface disturbance
loses its identity exponentially with depth. A disturbance created
at the free surface is attenuated rapidly with depth, and a disturbance
created at some depth causes a free-surface disturbance which de-
creases with the depth of the cause. The depth,effect is essentially
proportional to e*“Y, where, as above, kK=g/U~ and y is measured
as positive in the upward direction.

As U approaches zero, this depth-attenuation factor ap-
proaches zero for any fixed y. In other words, the free-surface
effects are restricted to a thin layer which approaches zero thickness
as U->0O. We might say that the free-surface is separated from
the main body of the fluid by this "boundary layer" in which there is

798

Singular Perturbatton Problems in Ship Hydrodynamics

a rapid transition from conditions at the surface to conditions inside
the bulk of the fluid. From our experience with viscous boundary
layers, we should expect the occurrence of large derivatives in this
region and also some difficulty in satisfying boundary conditions on
a face of the boundary layer.

In a viscous boundary layer, of course, the derivatives are
much greater in one direction than in another, and this fact allows
us to stretch coordinates anisotropically and apply the limit pro-
cesses of the method of matched asymptotic expansions. In the free-
surface boundary layer, however, this does not appear to be a
possible approach. From the linear theory, we expect that there
will be a wave motion with wave lengths which are O(U?/g). Thus,
derivatives will be large in at least two directions inside the boundary
layer -- in the direction normal to the layer and in one direction
parallel to the layer.

When I tried to solve this problem two years ago (see Ogilvie
[ 1968]), I did not apply very systematic procedures. Rather, I
simply assumed that the first approximation to @, as defined in
(5-14), would have certain properties, namely,

= 5 3
B(x,y) = OU); (x,y), Fy(x,y) = O(U);
also, the surface deflection function would be given by (5-15), with:
Hix) =/O(U 1. H'b) =-0(0).

The order of magnitude of, ® was chosen just so that the velocity
components would be o(1y>), and I assumed that differentiation
changes a quantity by 1/uU? in order of magnitude. The arguments
leading up to (5-21) contributed heavily to the conjecture about veloc-
ity components, and the 1/U® effect of differentiation was chosen
just because the free-surface characteristic length is U‘/g. It is
important to note that the rigid-wall potential, 9), is still part of
the solution, and these statements about orders of magnitude and
differentiation do not apply to it. In fact, lassume that ¢) is com-
pletely known, and so it is not necessary to conjecture about the
effects of differentiation.

In terms of the general approach of the multiple-scale ex-
pansion method, I have assumed that an approximation to the solution
can be represented as the sum of two functions. The first depends
only on the length scale appropriate to the body geometry. The
second function depends primarily on lengths measured on a scale
appropriate to U#/g, but it also depends on the first function and
thus on lengths typical of the body. However, it seems to be
possible to keep clear when differentiations are being carried out
with respect to each of the length scales.

799

Ogilvie

Physically, the situation may be described in the following
way: If U is small enough, the body extends over a distance of
many wave lengths of the surface disturbance. The initial dis-
turbance is caused by the body, of course; this is the "rigid- wall"
motion, and its dimensions are characteristic of the body. It
causes a free-surface disturbance, with the result that waves are
created. But these waves are very, very short, whereas the initial
disturbance from the body appears to be just a slight nonuniformity
in flow conditions when viewed on the scale comparable to the wave
length. The method is, in fact, quite similar to classical methods
such as the W-K-B method.

When the assumptions listed above are actually applied, we
find that the approximate free-surface conditions given in (5-18)
and (5-19) must be replaced by the following:

R

BH(x) + bo, 0) (2, Mol) = 05 (5-22)

G(x, No(x)) - 4,(x.0)H"(x) = p'(x);

the function p(x) is the same that was given in (5-20). Note that @
in both conditions here is to be evaluated on y = N(x), rather than
on y=0. The reason is the same that was given in Section 3.2 in
the near-field problem: If we tried in the usual way to expand
@(x,9), say as follows:

4.2
B(x, 1g) = B(x, 0) + nob (x,0) + 5 NoByy(x,0) +...

we would find that every term on the right-hand side is the same
order of, magnitude according to my assumptions. In particular,
No= O(U‘), and, symbolically, we have: 8/dy = O(1 /U‘). So this
expansion procedure is not useful.

The two conditions above can be combined consistently into
the following:

Bylves Mole) +5 4 (2610) By (Mg) = PICA). (5-23)

This is remarkably similar to the free-surface condition for another
problem. In the ordinary linearized theory of gravity waves, sup-
pose that a pressure distribution, p(x), is travelling at a speed U.
The free-surface condition would be:

2
&, (x, 0) + Gyx(x,0) = p'(x) ,

800

Stngular Perturbatton Problems in Ship Hydrodynamtes

if (x,y) were the potential function for the problem. Replace U
by 0,(x, 0), the "local stream speed," and evaluate the condition

on y= T(x) 5 then this condition transforms into the condition found
for (x,y) in the low-speed problem. Thus, ona "local" scale

(in which a typical lengthis U 2/g), the free-surface condition is just
a very ordinary condition; one cannot see that the stream velocity
changes slightly along the free surface, because the change occurs
on a scale in which a typical measurement would be a body dimension;
the change is very gradual. Also, the level of the undisturbed free
surface appears to change gradually, as given by (5-7); this change
also cannot be detected on the "local" scale.

It is now clear that the two length scales are quite distinct.
We cannot separate the fluid-filled region into distinct parts in each
of which only one length scale needs to be considered. Rather, the
gradual changes which appear on the body-size scale appear to
modify the short-length wave motion in the manner of a modulation.

In trying to find a potential akon which satisfies (5-23),
I made a nonconformal mapping: x! =x, y!=y - M(x). Then ©®
soos a complicated partial differential equation in terms of x'
and y', but the terms in the equation can be arranged according to
their Hepentence on U, and it is found that the leading-order terms

are simply the terms in the Laplacian, that is,
Oyye + ®y'y! = 0;

all other terms are higher order. In this new coordinate system,
the free-surface condition, (5-23), is transformed too, but again
the leading-order terms are just the same after the transformation
(but expressed as functions of x' and y'). Furthermore, the
boundary condition is then to be applied on y'=0. Let us now
drop the primes on the new variables, for convenience. Then the
problem is as follows: Find a velocity potential, @(x,y), which
satisfies the Laplace equation in two dimensions and the free-
surface condition:

2
Dy(xsy) += Gof%+ 0). (x0) = p(x),
where

p(x) = No(x) $o,(x,0) .

In addition, the potential must satisfy a body boundary condition;
this has not been carefully formulated yet, and, in any case, the
only solution that has been produced so far is one that satisfies
the free-surface condition but not a body condition. There may be

801

Ogilvte

some good justification (or rationalization) for proceeding this way,
but it is really an open question.

With such restrictions and reservations expressed, we can
write down a "solution" of the above problem. Define:

@ = Re {fp(z)}; O(x,y) = Re {F(z};
-2

Note that:

fifse) = 0,000); ke) = ef doe 00)

Then the solution is given by:
H 00 a2 z
LT a = fa) ds p'(s) { ia exp E if du (u) | é
-© -00 C

The ¢ integral is a contour integral starting at x = - o, located
entirely in the lower half-space. It should pass above the location
of the singularity in k(z). This solution represents no disturbance
at the upstream infinity, as one would expect.

Far downstream, this solution can be approximated:
0 z
F'(z) = 2ie i ds p"(s) exp [ixs = i du[k(u) - id] ;
-00 s

where K= g/U% Then, from (5-22), we obtain the wave shape far
aft of the body:

00
H(x) = - aul ds p"(s) sin[ x(x - s) + K(s)],
& J-o

where

K(s) -{ du [k(u) - kK].

Ss
Calculation of the wave resistance is then very simple in principle.

(In practice, it is a very tedious calculation.) Notethat the expres-
sion for the wave shape downstream does not require knowledge of

802

Stngular Perturbatton Problems in Ship Hydrodynamics

F'(z) (or &(x,y)), that is, the surface disturbance far away is a
real wave, but its shape and size depend only on the solution of the
rigid-wall problem. This is not true of the wave disturbance in the
vicinity of the body.

It would be very useful, I am sure, to formulate this problem
carefully by the method of multi-scale expansions. The approach
described by Ogilvie [1968] is very heuristic and leaves much to be
desired.

ACKNOW LEDGMENT

The preparation of this paper was supported by a grant of the
National Science Foundation (Grant GK 14375).

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806

THEORY AND OBSERVATIONS ON THE USE OF
A MATHEMATICAL MODEL FOR SHIP MANEUVERING
IN DEEP AND CONFINED WATERS

Nils H. Norrbin
Statens Skeppsprovningsanstalt
Sweden

ABSTRACT

This paper summarizes an experimental and analytical
study of ship maneuvering, with special emphasis on
the use of a research-purpose simulator for evaluating
the behaviour of large tankers in deep water as well as
in harbour entrances and canals. In an introductory
Section some new results from full-scale measure-
ments and simulator studies are given to illustrate the
demands put on a mathematical model in the two ex-
treme applications: course-keeping in deep water and
manoeuvring in a canal bend.

Well-known derivations of rigid body dynamics and
homogeneous flow solutions for forces in the ideal case
are included to form skeleton of the mathematical
model, Separate equations handle helm and engine
controls. Coefficients and parameters are made non-
dimensional in a new system — here designated the
"bis" system as different from the SNAME "prime"
system generally used — in which the units for mass,
length and time, respectively, are given by the mass
of the ship, m, the length, L, and the time required
for travelling one ship length at a speed corresponding
to Via oe 1, L/e.

Semi-empirical methods are suggested for estimates
of the force and moment derivatives. Special consider-
ation is given to added mass and rudder forces in view
of their predominant importance to course-keeping
behaviour; the rudder forces measured ona scale
model are corrected for differences in wake and screw

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loading before application to full-scale predictions.
Non-linear contributions to hull forces are included
in second order derivatives, relevant to the cross-
flow concept.

The extension of the mathematical model to the con-
fined-water case is based upon the theoretical results
by Newman and others, and upon relations found from
special experiments. In the model the hydrodynamic
interferences appearing in forces and moments due to
the presence of port and starboard side wall re-
strictions and bottom depth limitations are represented
by additional terms containing higher order derivatives
with respect to three suitable confinement parameters,
N=. tp» N=Ns- Np» and €~ Ina canal the asym-
metrical forces are considered as due to the aaded
effects from port (p) and starboard (s) walls rather
than as the effect of an off-centreline position;
primarily n is a measure of this position, 7 a
measure of the bank spacing.

The mathematical model is here applied for evaluation
of model test data obtained for a Swedish 98 000 tdw
tanker in the VBD laboratories. Oblique towing and
rotating arm tests were performed in "deep" and
shallow water. Oblique towing tests were also run
at various distances from a vertical wall in the deep
tank, and in two Suez-type canal sections. The effect
of shallow water was especially large in force non-
linearities. Missing data for bottom and wall effects
on added mass and inertia are taken from theory and
from test results due to Fujino, respectively.

The deep-water predictions for zig zag test and spiral
loop prove to be in good agreement with full-scale trial
results. Analogue computer diagrams are given to
show the effects of shallow water upon definite
manoeuvres and upon course-change transients follow-
ing auto-pilot trim knob settings. Finally a few results
are included to illustrate auto-pilot position control
of the tanker in free water, in shallow water, between
parallel walls and in a canal.

808

Shtp Maneuvering in Deep and Confined Waters

I. INTRODUCTION

On Course-Keeping in Deep Water

The average depth of the oceans is some 3800 m. Small
native crafts still steer their ways between nearby islands in these
oceans. New ships are built to transport ever larger quantities of
containers or bulk cargoes at a minimum of financial expense
between the continents.

It is not necessarily obvious that the helmsman shall be able
to control a mammoth tanker on a straight course. A few years ago
ship operators were stirred by the published results of an analytical
study, interesting in itself, which in fact did indicate, that manual
control of ships would be impossible beyond a certain size. Upon
request by the shipbuilders a series of real-time simulator studies
were initiated at SSPA in autumn 1967 to investigate manual as well
as automatic control of large tankers then building, [1].

At an early stage of these tests the helmsman was found to
constitute a remarkably adaptive control, which could not be simu-
lated by a simple transfer function. As could be expected a rate dis -
play proved to make course keeping more easy; the rate signal was
even more essential to the auto pilot.

The simulator findings were confirmed in subsequent proto-
type trials. The diagrams of Fig. 1 compare simulator and proto-
type rates of change of heading and yaw accelerations for a large
tanker as steered by the author in a Force 6 following sea. (In the
simulator case the sea disturbance was represented by a cut-off
pseudo-random white noise of predetermined root mean square
strength, that was fed into the yaw loop.) This particular tanker is
dynamically unstable on a straight course, and the steady-state
L(6)-diagram from a deep-water spiral test exhibits a hysteresis
loop with a total height of 0.5 °/s and a total width of a little more
than 3° of helm. If yaw rate is maintained within some 40 per
cent of the loop height value it has been found possible to control the
straight heading by use of small helm only.

The use of the computer-type simulator for the prediction of
ship behaviour implies the adoption of a suitable mathematical model
and the knowledge of a number of coefficients in this model. An
alternative technique that simulates full-scale steering by controlled
free-sailing ship models is still in use. Mostly the steering has been
exercised by manual operation of the controls, and it has been claimed
that at least comparative results should then be valid. It is likely
that the truth of this statement depends on the actual speed and size
(and time constants) of the prototype ship as well as of the model
scale ratio used.

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7 | 7
H
¥ é
z : At |
3

fg 0,01 °/5

40,002 %st
ner

Full scale single-channel record (Helmsman: Author)

BESNE 2 0 cel tment poe
\ \ p 4 Ip | Af »

VAY

Simulator records (Helmsman: Author)

Fig. 1. Manual steering of an unstable 230 000 tdw tanker in
quartering sea. Prototype and simulator records.

810

Shtp Maneuvering in Deep and Confined Waters

Time is scaled as square root of length. Human response
time may be "scaled" within certain limits only. The w(6)-diagrams
of Fig. 2 demonstrate results of simulated steering of the tanker
prototype already referred to, as well as of her fictive models of
four different sizes. (Note that curves run anti-clockwise with time.)
The smallest "model" is in scale 1:100, i.e. it has a length of 3.1m,
which should permit free-sailing tests in several in-door facilities.
The two helmsmen, which each one seem to represent one kind of
steering philosophy and who were allowed a short training period in
each case, both failed to maintain the proper control of the two
smaller "models."

The control of a ship on a straight course is governed mainly
by the effective inertia, by the yaw damping moment, by the rudder
force available, and by the time this force is applied. A mathemati-
cal model intended for studies of manual or automatic steering may
therefore be quite simple; in contrast to the test basin model it may
include proper corrections for the large scale effects often present
in rudder force data. (Cf. Section VII.)

Figure 3 repeats the original simulator b(5)-curves from
real-time straight running, recorded by use of the "complete"
mathematical model, but it also presents results from tests with a
linear model as well as with a model, which contains no other hydro-
dynamic contributions than those in lateral added inertias and rudder
forces. No major differences were experienced in using these three
models of increasing simplicity.

On Manoeuvring in Confined Waters

Manoeuvring, involving yaw rates and drift velocities, which
are not small compared to the forward speed, demands a mathemati-
cal model of considerable complexity. A useful presentation of non-
linear characteristics has been given by Mandel, [ 2]. One particular
non-linear model designed to include manoeuvres in confined waters
will be more fully discussed in subsequent Sections of this paper.

The average depth of the oceans is some 3800 m. But ocean
voyages start and terminate at ports behind the shallow waters of the
inner continental shelves. Additional confinements are presented
by many of the important gateways of world trading, such as the
Straits of Dover and Malacca, the Panama Canal, and the Suez Canal
now closed.

The maximum draughts of "large" ships have always been
limited by bottom depths of docks and harbours, and of canals and
canal locks. With few exceptions the requirements placed on under-
keel clearances ~ by ship owners or by authorities — have been
chosen solely with a view to prevent actual ship grounding or exces-
sive canal bed erosions. Thus the Suez Canal Authorities accepted

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Helmsman W

Helmsman N

Fig. 2. Simulator tests of manual steering of an unstable 230 000
tdw tanker in real and speeded up time. Yaw rate versus
helm angle. (Numbers along curves indicate minutes in
real time.)

812

Shtp Maneuvering tn Deep and Confined Waters

“Complete” model:

Linear model:

Helmsman N Helmsman W

Fig. 3. Simulator tests of manual steering of an unstable 230 000
tdw tanker using alternative mathematical models. Yaw
rate versus helm angle.

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a nominal blockage ratio of 1:4 for ships in northbound transit at a
maximum speed of 13 kilometres per hour, corresponding to a mean
back-flow velocity of some 1.5 m/s.

Today new limits are imposed by the depths of ocean sills as
well as by the depths and widths of open sea port approaches. The
potential dangers of a large oil tanker navigating in such waters
under, say, the influence of an unexpected change of cross current
must not be denied. Whatever nautical experience the master or
pilot may possess, he is still in need of actual data and of means to
convert this information to helm and engine orders. Automatic
systems on a predictor basis are likely to appear in a near future, [3].

In the planning for dredged entrance channels and harbour
turning basins the maneuvering properties of the ships must no longer
be overlooked. The upper drawing of Fig. 4, reproduced from
Ref. [4], shows part of the plan view and a typical section of the
buoyed channel for 200 000 tdw tankers unloading at a new oil ter-
minal. Before entering the 90° starboard turn the speed is brought
down to less than 2 knots, and the tanker then proceeds under slow
acceleration by own power. Braking tugs are used on quarters,
and forward tugs assist in the S-bend. The lower diagram of Fig. 4
is taken from SSPA records of yaw rates in the passage; the initial
curvature corresponds to r' = 0.175, and the maximum rate of
change of angular velocity is of the order of 0.0005 o/s? at a forward
speed of 2.3 knots.

In general the lateral forces on the ship will all increase as
water depth turns smaller, and the dynamical stability is also likely
to increase. From extensive measurements by Fujino it appears,
however, that the picture is not so simple, and that for some ships
there may be a "dangerous" range of depth-to-draught ratios, in which
the dynamic stability gets lost, [5].

Recent model tests indicate that the large-value non-linearities,
such as the lateral cross-flow drag at high values of drift, do increase
even more than the linear contributions governing the inherent stability
conditions. Whereas these non-linearities may be omitted in the
mathematical model of the ship in a canal the bank effects here intro-
duce destabilizing forces, that are again highly non-linear.

The effects of well-known forces experienced by a ship sailing
parallel to the bank of a canal are clearly apparent in the record from
a Suez Canal transit here reproduced in Fig. 5, [6]. (The positions
in the canal as well as the width between beach lines were derived
from triangulation by use of two simple sighting instruments designed
for the purpose.) Upon approach to the Km 57 bend the ship is slightly
to port of the canal centre line. The pilot orders port helm for two
minutes, by which the ship is pushed away from the near bank and the
desired port turn is also initiated. Back on centre line the ship
mainly turns with the canal. In spite of a starboard checking rudder

Ship Maneuvering in Deep and Confined Waters

Plan and typical section of dredged channel

comer a pave Ser Gine ao
SS Jee te eae a a Che
H i Baad eee ! ee
- c=]
: Bape rears oa ta geen bibs ne Cebets se See Seep cean ae ea th
i N ; :
H N Le :
Si ROG! SL MMe Le ek SO SER ON AE Mest OAT Ree cee oeR Oe” Oe Wm) GD LEG eee
t an ae ee
Er Oe a ee EE ie ae ee ee eee Gy i j
: min ce Ag
‘0,01 %s

ies os 4 Ship Speed 2,3 Knots

Pease es. mS
oh Q}.
: . So
: ae 1 “|:
i sine
3 c 0 eee 2 x ee vb 5 6 7 8 9 0 1 3 4

Part of yaw rate record in transit

Fig. 4. Example of yaw rates recorded on 210 000 tdw tanker in
harbour approach,

815

: ot lirrited to
mancewres initieted by pilot orciers.

ugh Suez Canal on 36' draught.
ds in KM 57 Bend.

uthbound thro

Fig. 5. 60 000 tdw tanker so
Abstract of Recor

Shtp Maneuvering tn Deep and Confined Waters

she again moves closer to the port bank, and again port rudder has
to be applied, etc.

So far analytical studies of ships moving in canals have been
dealing with straight running. It is believed that the mathematical
model which is presented here may also be extended to the case of
slowly widening and bending canals.

II SYMBOLS AND UNITS, ETC.

When applicable the symbols and abbreviations here used
have been chosen in accordance with the ITTC recommendations,
[7]. Some new symbols are introduced to define the position and
orientation of a ship in confined waters. (See also Section X.)

The system of axes fixed in space is 0)x9y9z,, that fixed in
the body or ship is Oxyz. The point of reference O lies at distance
Lpp/2 forward of A.P. of the ship. (Cf. Fig. 6 and Section IV.)

Zo

Fig. 6. Inertia frame and body axes, etc.

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Dimensional numbers are given in metric units unless other-
wise stated. Generally coefficients and relations are expressed in
non-dimensional forms. In addition to the non-dimensionalizing
"prime" system usually adopted use is here made of a new "bis"
system, further presented in Section III.

A dot above a variable stands for a derivation with respect
to time. Partial derivatives of forces and moments are designated
by the proper subscript attached to the force or moment symbol.

TABLE I
Symbol Definition ca deaesk Remarks
A, Channel section area ie
A Section area of hull isp
Ajj Added mass.) Mt= 1.2.53 9 =1,2:3. M
noon i= 4,5,6;)=4,5,6 ML*
: 2 b= 1,52;33j = 495.6 ML

Ay Added mass in horizontal oscill.

inka free surface, neplecting

gravity M
Ay Added mass in horizontal oscill.,

unbounded fluid M
A, Total proj. area of rudder i
Bea Moveable proj. area of rudder ic
B Beam of hull hb
Cp Cross-flow drag coeff., 3-dim. -
D Diameter of propeller ie
F Force vector MLT
Bae Froude number on depth - Pye v/Vgh
Fy Froude number on length - Fat = V/VeL =v"
I Moment of inertia Mi
Ti Mass product of inertia ML* ee Ke
J Propeller advance coefficient - J=u(1-w)MD
K Rolling moment about x axis Mir?
Kg Propeller torque coeff. - Kg = - Q?/pn*p°
K, Propeller thrust coeff. - K,= T?/on*p*

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Ship Maneuvering tn Deep and Confined Waters

Symbol Definition Fy Soe Remarks
SE Te a a a ee ere ee
L Length of hull L L = Lpp
M Pitching moment about y axis Mier
M Moment vector is ag
N Yawing moment about z axis MI?T* N" = N/mgL
Q Torque about propeller shaft ML*T*® Qt= turbine
torque
R Turning radius L ri= 17R
R Resistance MLT~ xX(R)=-R
+ Hull draught Te
EG Propeller thrust MLT™
rs Kinetic energy of liquid ML T~
U Total flow velocity LT”
V Velocity of origin of body axes LT y"= V/VeL
ve Speed of water current LT"!
V; Ship speed over ground LT”
W Channel width in general io
WwW Bank spacing, half of L 2W = We - Wp
X,Y,Z Hydrodynamic forces along body -2
axes MLT
y* Y-force due to rudder MLT™®
ee Y-force on rudder proper MLT
a Depth to top of rudder I
a Water surface elevation L
ay Slope of lift coefficient curve -
b Height of rudder 1
c Flow velocity past rudder ot
Cp Cross-flow drag coeff., 2-dim. -
g Gap between rudder and hull 18
g Gravity vector LT
h Depth of water ae
h Vector in general Undef.
kj; Coefficients of accession to
inertia - rete eee:

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Physical

Symbol Definition Die nnion Remarks
ki, Coefficients of accession to

inertia - ic= 45.56
k, Corr. factor for rudder inflow - Cf. eq. (7.4)
k, Corr. factor for rudder inflow -
ee Non-dim. radius of gyration -
m Mass of body M rm a
n Number of revs. of prop. in

unit time a

-| -
2) Pressure in general ML FT. :
-| _-

q Stagnation pressure Mi... al :
p.q,r Angular velocity components an
Dang Max. radius of equivalent body

of revolution L
s Lateral thrust factor - Cf. Section VII
s Sinkage L
t Time T t"=t//L/g
t Thrust deduction factor -
u,V,;w Components of Vv along body EF

axes Le
w Wake fraction -
X2V 9% Orthogonal coordinates of a right-

handed system of body axes cL
X92Vo2Z%, Orthogonal coordinates of a right-

handed system of space axes

(inertia frame) i
A Weight displacement MLT* 4A=ppgV, =mg
ny Volume displacement L3 Normal approx.!

V=WV

Vo Volume displacement at rest =
A Aspect ratio -
A, Aspect ratio of rudder x A, = b®/A,
Re Do for rudder + plane wall

image - A,= 2h,

Bl is

® Velocity potential Ls wi PM y= ®/LV gb

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Physical

Shtp Maneuvering tn Deep and Confined Waters

Symbol Definition Miimennioen Remarks
ot Angular velocity of ship iT
a Angle of attack -
B Angle of drift - tanB = - v/u
Y Frequency parameter - Y = Vw/g = u"w"
Y Coeff. of heading error term
in proportional rudder control - "Rudder ratio"
6 Rudder angle (deflection) -
6* Rudder angle ordered by auto
pilot -
5e "Effective" rudder angle - Se = 6 for v=r=0
€ Phase lead angle -
ec Restricted water depth parameter - C70 (eee)
n Ship-to-bank distance parameter - n=n, t Np
n Bank spacing parameter = 1 =7s - Np
m1, Port bank distance parameter - Np = L/(Wp - yo)
"1. Starboard bank distance parameter - Ns = L/(W, - Yo)
's] Angle of pitch -
y. Body mass density ratio - ik = m/p V0, or
norm. surface
ships p= l
p Mass density of water ML”?
o Coeff. of rate of change of
heading term in proportional
rudder control ae "Rate (time)
constant"
9? Prismatic coefficient -
> Angle of roll or heel -
Wy Angle of yaw, or heading error -
-|
w Circular frequency T w" = wL/g
w! Reduced frequency - wo = wl /V = w/a"

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Ili, NON-DIMENSIONALIZING BY USE OF THE "BIS" SYSTEM

The use of non-dimensional coefficients is accepted in all
branches of ship theory, and when motion studies are considered
even the variables of the equations are often normalized.

Within the field of maneuvering a unit for time is usually the
time taken by a body to cover the distance of its own length, and the
unit for velocity then is most naturally given by the momentary
speed V = (u* + v*)'*. Ifthe body does not move forward this defi-
nition is less attractive. In the system just mentioned — which is
recommended by ITTC and which in most cases is fully adequate —
symbols for non-dimensional quantities usually are indicated by a
prime.

The unit for length almost always is chosen equal to the length
L of the body, and for the common surface ship more specified
L=L
pp*

The unit for mass is mostly taken as the mass of a certain
volume ofthe liquid, defined in terms of the body or ship geometry.
In the "prime" system already referred to reference volumes are,
say, 3L° [8] or $1°T [9], the latter one used with the reference
area LT suggested by the wing analogy.

In case of bodies, which are supported mainly by buoyancy
lift, the main hull contour displacement V, is perhaps the most
natural reference volume: if body mass thenis m=p* p* Vo the
non-dimensional mass is equal to wp. (When treating heavy aircraft
dynamics Glauert chose ppV in place of pV for the mass unit,
[10]-) In normal ship dynamics = 1, whereas for heavy torpedoes
p=1.3 - 1.5, say; the symbol p will be rejected in certain appli-
cations.

Here a consistent normalization of motion modes and forces
will be made in a new system, the "bis" system, where the unit for
mass is m =ppV,, the unit for length is L and the unit for linear
acceleration is equal the acceleration of gravity. From this
the unit for time is ¥L/g, and it also follows the Table below:

‘Ship Maneuvering in Deep and Confined Waters

TABLE II

Unit for "bis" system "prime" system

P73 Pye
mass (M) LPV, 5 L 5 Pat
length (L) L Ie t
time (T) JL/g L/V L/V
linear velocity Vel V Vv
linear acceleration g v-/L V7/L
angular velocity V¥g/L VE V/L
angular acceleration g/L Maye vin
force upgV> gv SwLt

p «2.3 p 12.2
moment wpgVyL 5 VL Vota
Reference area aM ie Ly

It will be noted that, in the system suggested, a non-dimen-
sional velocity is given by the corresponding Froude number, and
that all forces are related to the displacement gravity load
A =ppgV, of the body. (Cf. quotients suchas R/A, "resistance
per tons of displacement," used in other fields of applied naval
architecture. )

It is customary to form a non-dimensional force coefficient
by dividing by the product of a stagnation pressure (q = (p/2)V*)
and a reference area, and of course the new system will not demand
any different rules. In place of the velocity V, however, here is
chosen that particular velocity which corresponds to F,, = 1, i.e.
the normalized stagnation pressure is q = (p/2)gl. The reference
area then is seen to equal p(2V,/L).

IV. KINEMATICS IN FIXED AND MOVING SYSTEMS

The two orthogonal systems of axes here used, 0,xoyoZp
fixed in space — the inertia frame — and Oxyz fixed in the body,
are shown in Figs. 6 and 7. The orientation of the body axes may be
derived, from an original identification with the inertia frame, by

Norrbin

ah, = bh + Sdech at oy
Lae =F. SR ¥ @

Fig. 7. Graphical deduction of the absolute time derivative of a
vector OF, =h defined in the moving body system

the successive rotations through the angle of yaw, \w, the angle of

pitch, 0, and the angle of roll, $, respectively, defined around the

body axes z, y, and x in their progressively changed positions.
In_a certain moment of time the,relation between the space

vector 0,P = xp and radius vector OP = x,, invariant in the body
system, is given by

= Ax (4.4)

where the orthogonal transformation matrix reads

824

Shtp Maneuvering tn Deep and Confined Waters

coswcos® -sinwcos¢$ +cosWsin@sin®é sinysind tcosWsin®cosd>
Az=|sinJcos® coswcos$+sinWsin®@ sing -cos sind tsinsin® cos >
- sin® cos 8 sind cos 8 cos¢

(4. 2)

When applied in opposite direction the transformation is

— -l => — ~ — =_
xp=A (Xp - X49 = A(x, - X,9) (4. 3)

where A is the transposed matrix, in which rows and columns
appear in interchanged positions.

In particular, note that the gravity vector ‘Bo = QZ, will be
given by the column vector

0 - g-* sin 6
g = A} o} = g cos @ sin (4.4)
g g cos 8 cos

in the moving system.

From Fig. 7 will be seen how the absolute (total) value of the
time derivative of any vector h in the body system may be calcu-
lated from the relation

ap ae —>— —
hy. =H tM xh (4.5)

The angular velocity vector oy may now be expressed in
terms of the Eulerian angles and their time derivatives: For the
vector h there is h,=Ah and

+AAh (4. 6)

and so the column vector 2 is obtained from the carresponding anti-
symmetric angular velocity matrix for the product AA,

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p @- sin @
% = q| = cos @ sin +0 cos (4.7)
r icon Gica-sheue ain

The angular velocity components resolved in the inertia frame
are

$=p +q sin @6tan 0 trcos $tan 6
8 q cos $6- r sind (4.8)

wW=rcos #sec 8 +q sin sec 8

In the special case of motion in a horizontal plane in absence
of rolling and pitching itis Yp=r.

In Section VIII an expression will be required for the absolute
acceleration of amass element dm at station P(x,y,z) in a body
moving through the water with velocity V. From (4.5) then

u O -r q vd u -ry tqz
ape = lv | +t] r O -p y| = |v trx -pz (4.9)
w -q p 0 Zz w -qx tpy

and by a repeated application of the transformation formula

u - rv t+ qw - (q2 + r?)x + (pq - r)y + (rp + q)z
2 = : = + = 2 + 2 + th ce ae ry ° ,
(ap)bs v- pwtru- (r° + p*)y + (qr - p)z + (pq + r)x (4,10)

w-qut+pyv - (p? +q2)z + (rp - q)x + (qr + py

=F

~_,, Inthe presence of a homogeneous steady current Vg aterm
Gig is to be added to the right-hand member of Eq. (4.9). In
practical applications this current may be assumed to take place
in planes parallel to the horizontal, so that Vis fully identified
by u and v,°. It is easy to show that the column matrix for the
acceleration in (4.10) will remain unchanged. To the surface ship

826

Shtp Maneuvering tn Deep and Confined Waters

in horizontal maneuvers, this homogeneous current will only mean
a steady shift of the path; alternatively, if a certain straight course
is required heading shall compensate for the steady drift. The
local finite current, on the other hand, generates varying outer
disturbances and shall be handled by other means.

V. FLOW PHENOMENA AND FORCES ON A SHIP IN FREE WATER
Ideal-fluid Concepts

As a source of reference for further discussions this Section
recapitulates some of the characteristics of the flow past a ship in
free or open water.

When a double-body ship form — i.e., a body which is sym-
metrical about the xy-plane — moves forward in a large volume of
ideal-fluid water the streamlines adjust themselves according to the
laws of continuity. The shape of those streamlines remain the same
at all speeds. The increase of relative velocity past the wider part
of the body corresponds to a back-flow or return flow of the water
previously in rest. This disturbance in the potential flow pattern
extends far into the fluid volume — a beam-width out from the side
of the body the super-velocity still has a value, which is some 80 per
cent of that just outside the body.

From a resistance point of view the steady forward motion
within this ideal homogeneous fluid may lack some realism. Accord-
ing to the d'Alembert's Paradox the body will experience no resultant
force. However, if the body is to be accelerated the kinetic energy
of the fluid must be increased. This energy increase is manifested
by a resistance, which for a given geometrical form is proportional
to the mass of displaced fluid and the amount of acceleration, i.e.
to the product of an "added mass" and the acceleration component in
the direction considered. The resultant force is not necessarily
orientated in the same direction.

In the simple steady motion the total energy certainly will
remain constant, but as the body moves forward through virgin fluid
there takes place in each transverse section a repeated particle
acceleration and transformation of energy. The impuls pressure
distribution thus generated will normally be unsymmetric, and soa
free moment results on the body. This moment may be expressed
by a combination of total-body added mass coefficients.

In the general case of a complex motion in the ideal homo-

geneous fluid all the forces and moments will then be available in
terms of added masses and inertias, according to the theories

827

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originated by Kirchhoff [11] and Lamb[12]. In spite of the fact that
these forces will be modified by the presence of viscosity in the real
fluid, and that new forces will also be generated by the viscous
effects, these ideal results should be considered when formulating
the mathematical model.

If U is the velocity vector of the lqcal fluid element the
total kinetic energy is given by T, = (p/2) U* dr, or in a potential
flow generated by the impuls pressure p®

eh £0Py Vi a
T,=-£{o% ds (5.1)

The integration is to be extended over the total boundary, i.e. over
the wetted surface of the body. Let the potential be written in
linearized form as

$= Gut dv + o,w + &p + O&q + Or (5.2)

with respect to the six component body velocities uj. The six
coefficients @; then are functions of the body geometry and of the
position in relation to the body.

The condition for fluid velocity - 8@/8n at the body boundary

to equal the body normal velocity may be formulated by use of the
directional cosines for the normal inthe Oxyz-system, whereby

or

2T =- X:u*- Y.v*- Z.w?- 2¥. vw - 2X.wu - 2X.uv
i) Vv w Ww Ww Vv

2 2 2
2 Kp ae M,q - N,r - 2M. qr - 2K; rp - 2K,Pq
(5.3)

- 2(X,u + Y¥,v + Z.w)r

Here there are 21 different added masses (Ajj) or "accelera-
tion derivatives." Force derivatives with respect to a linear accelera-
tion are of dimension M, and moment derjvatives with respect to an
angular acceleration are of dimension MIL’, as are the mass moments
of inertia. Cross coupling derivatives such as X,=- Ajq are of
dimension ML.

828

Shtp Maneuvering in Deep and Confined Waters

If the body has a plane of symmetry there remain 12 different
acceleration derivatives , and for a body of revolution generated
around the x axis there are only the three derivatives A,,, A,, and

Ae

The motion of the ideal liquid takes place in response to the
force and moment expended by the moving solid. At any time this
motion may be considered to have been generated instantaneously
from rest by the application of a certain impuls wrench. The rate
of change — cf.Eq. (4.5) — of the impulse wrench is equal to the
force wrench searched for. Again, the work done by the impulse is
equal to the increase of kinetic energy, and as shown by Milne-
Thomson [13] the force and moment on the body may therefore be
expressed in terms of the kinetic energy of the liquid,

F =~ g (7it) -@x 23
oV OV
(5.4)
M=-$ (23) -dx-¥x4
a aQ oV

(The partial derivations shall be considered as gradient operators.)
The complete formal expressions for the inertia forces in the ideal
fluid have been derived from Eqs. (5.3) and (5.4) by Imlay [14], and
they are here given in Eq. (5.5).

=

ia = Xyu + X,(w + ug) + X.q + Zywq + Zq? + Xv + Xp + Xr

2
=~ Y,vr - Yorp - ¥,r° - Xyur - Yywr + Yyvq + Zspq - (Y; - Z,)qr
Yi =Xu+Y,wt Y¥4q +Y¥,v + Yjp + ¥;r + Xwr - Yyvp + X;r?

2
+ (X,- Z;)rp - Zp - X,(up- wr) + Xur - Ziwp - ZaPq + Xsqr

X,,(u - wq) + Z yw t 24 - X,uq - Xx," ee Zp ge Ste tad Gag)

2 .
+Y,rp + Y;p + Xyup + Yywp - Xyq - (X,- Yq)pq - Xqr

829

Norrbtin

Kig = X;u + Z,w + Kya - Xywu + X;uq - Yyw? - (¥, - Z:)wq + Mzq?
+Y,v + Kp + Kx - (¥, - Z,)vr + Zavp - M,r° - K.rp + X,uv
= (Y, = -Z,)vw = (¥.°* Z,)wr - Y, WP Saree oi Z,)v4
+ Kpq « (M, - N;)qr + Yyv"

Mi4= X,(u + wq) + Z.(w - uq) + M,q - X,(u* =P WA) oss (Z, - X,)wu
+Y,v +K,p +M,r + Y,vr - ¥;vp - K,(p® - r*) + (K; - N;)rp
- Yyuv t Xww - (X; + Z.)(up - wr) + (x, - Z;)(wp + ur)
- M;pq + K,aqr
Ni = Xa + Z.w + M,q + Xu + Ywu » (X, - ¥,)uq - Z,wq - Kia"
+¥,v + Kp + N.r - Xv" - X.vr - (X,- Y,)vp + M,rp + K,p?
- (Xx, - Y,)uv - X,vw + (X, + Y,)up + Y.ur ct Z wp
- (X, + ¥;)vq - (K, - M,)pq - K:qr

(5.5)

Forces in Horizontal Motions - General
Especially, for a body which is symmetrical with respect to

its xz-plane and which is moving in the extension of its xy-plane,
there are

Skin Yves Yer, + X(v - ur) + Xr

Yig = Yyv + Xjur + ¥;r + X,(a + vr) + X;r* (5. 6)
Nig = Nr +(¥, - Xuv + ¥,(v tur) [+ Xu? - v4) + X,(a - vr)

By careful application of sound reasoning it is suggested that terms

830

Shtp Maneuvering in Deep and Confined Waters

to the right of the bar may be dropped. Terms containing the coef-
ficient Y; have been retained in view of the fore-and-aft unsymmetry
present particularly in propelled bodies.

The coefficients for u in xX, for v in Y, and for r in N
— with signs reversed — are the most commonly well-known added
masses and added moment of inertia respectively. These inertia
coefficients also appear in some of the cross-coupling terms.

Lamb's "coefficients of accession to inertia" relate added
masses to the mass of the displaced volume V (kjj, i= 1, 2, 3) and
added moments of inertia to the proper moments of inertia of the
same aac volume (ki; > = 4, 5, 6). Lamb calculated k,,,

and kg.= ky, for the sphereoid of any length-to-diameter
seein. ais]. For ellipsoids with three unequal axes the six different
coefficients were derived by Gurewitsch and Riemann; convenient
graphs are included in Ref. [16]. For elongated bodies in general
the total added inertias may be calculated from knowledge of two-
dimensional section values by strip methods, applying the concept
of an equivalent ellipsoid in correcting for three-dimensional end
effects. (See further below.)

Of special interest in Eq. (5.6) is the coefficient Yy - X,
in the "Munk moment," [17]. (See also discussion in [18]. This
free broaching moment in the stationary oblique translation within
an ideal fluid defines the derivatives

. 2v

kee- ki
patel “I ete NaS
’ B L3

(Lie
Nuv= a L

(kao - ky) (5.7)

(Cf. Table II.) The factor kg 9- kj, may be looked upon as a three-
dimensional correction factor.

Due to energy losses in the viscous flow of a real fluid past
a submerged body the potential flow picture breaks down in the
afterbody. In oblique motion there appears a stabilizing viscous
side force. So far no theory is available for the calculation of this
force, but semi-empirical formulas give reasonable results for con-
ventional bodies of revolution. Force measurements on a divided
double-body model of a cargo ship form have demonstrated that some
de-stabilizing force is still carried on the afterbody but that most of
the moment is due to the side force on the forebody, predictable
from low-aspect-ratio wing or slender body theories, [18].

Similar measurements on a divided body in a rotating arm
shall be encouraged. Contrary to the case of stationary pure trans-
lation the pure rotation in an ideal fluid involves non-zero axial and
lateral forces. From Eq. (5.6) the side force is given by Xyur,
whereas the moment here is Y;ur. For bodies of revolution the
distribution of the lateral force may be calculated as shown by Munk

831

Norrbtin

[17] whereas strip theory and two-dimensional added mass values
may be used for other forms. The magnitude of ideal side force as
well as moment are small, however, and ina real fluid the viscous
effects are dominating.

There are reasons to believe that the main results of the
theories for the deeply submerged body will also apply to the case of
a surface ship moving in response to control actions at low or
moderate forward speeds. Potential flow contribution to damping
as well as inertia forces depend on the added mass characteristics
of the transverse sections of the hull, and as long as these character-
istics are not seriously affected by the presence of the free surface
the previous statement comes true. However, an elongated body
performing lateral oscillations of finite frequencies will generate
a standing wave system close to the body as well as progressive
waves, by which energy is dissipated. The hydrodynamic character-
istics then are no longer functions ofthe geometry only. At a higher
speed or in a seaway displacement and wave interference effects
will further violate the simple image conditions.

VI. CALCULATIONS AND ESTIMATES OF HULL FORCES

On Added Mass in Sway and Added Inertia in Yaw

A brief review will here be given of the efforts made to
calculate the added mass and inertia of surface ships in lateral
motions. Four facts will be in support of this approach: The added
masses are mainly free from viscous effects; the added masses
appear together with rigid body masses in the equations of motions,
and relative errors are reduced — this is especially true in the
analytical expression for the dynamic stability lever, which involves
only the small X,y the added masses are experimentally available
only by use of non-stationary testing techniques, and in many places
experimental data must therefore be supplemented with calculated
values; the added masses are no unique functions of geometry only,
and experiments must be designed to supply the values pertinent to
the problems faced.

The velocity potential for the two-dimensional flow past a
section of a slender body must satisfy the normal velocity condition
at the contour boundary as well as the kinematical condition for the
relative depression velocity at the free constant-pressure surface,
In case of horizontal as well as vertical oscillations this latter
linearized conditions is w* + g(9@/8z) = 0 — cf. Lamb[ 12] — or,

pe tape the non-dimensional potential @" = @/Lygl and
w!"! = L/g,
o® iy = n@ u"
at | © ® (6.4)

Shtp Maneuvering in Deep and Confined Waters

For steady horizontal drift at moderate forward speeds one
finds a similar condition

BOM ona? 27) re

e F °
oz" nL aty"

(6.2)

which shall govern the local accelerations of the flow in the trans-
verse plane penetrated by the moving body, [18].

As is seen from the two equations above the vertical velocities
at the water surface are zero in the limit of zero frequency or zero
drift, and negligible for w << V¥g/L or 8 Fir << 1. The water sur-
face may therefore be treated as a rigid wall, in which the underwater
hull and streamlines are mirrored, i.e. the image moves in phase
with the hull.

For high frequencies, where w>>yg/L, the condition at the
free surface is ®=0. The water particles move up and down normal
to the surface, but no progressive waves are radiated. At the juncture
of the horizontally oscillating submerged section contour and the free
surface this condition may be realized by the added effect of an image
contour, which moves in opposite phase. (Cf. Weinblum [19].) The
value of added mass in this case, "neglecting gravity," is smaller
than the deeply submerged value by an amount equal to twice the
image effect.

Added masses Aj for two-dimensional forms oscillating
laterally with very low frequencies in a free surface have been cal-
culated by Grim [ 20] and by Landweber and Macagno [ 21], using
a LAURENT series with odd terms to transform the exterior of a
symmetric contour into the exterior of a circle (TEODORSEN map-
ping). By retaining the first three terms this transformation yields
the well-known two-parameter LEWIS forms [ 22]; other combina-
tions of three terms have been studied by Prohaska in connection
with the vertical vibrations of ships [23]. Two terms (and one single
selectable parameter for the excentricity) define the semi-elliptic
contour as that special case with given draught, for which the added
mass is a minimum. Landweber and Macagno also made calculations
of the added masses Ay in the high-frequency case. For the semi-
elliptic contour A, /A; = 4/n*, which result was first found by
Lockwood-Taylor, [ 24] é

A basic theory for the dependence of the hydrodynamic forces
on finite frequencies was developed for the semi-submerged circular
cylinder by Ursell, [ 25]. By use of a special set of non-orthogonal
harmonic polynomials he found the velocity potential and stream
function that satisfied the boundary conditions and represented a
diverging wave train at infinity. Based upon similar principles
Tasai extended the calculations of added masses (and damping
forces) for two-dimensional LEWIS forms to include the total

833

Norrbtin

practical range of swaying frequencies, [ 26]. His results are con-
densed in a number of convenient tables and diagrams; the added
mass values are seen to vary even outside the limit values cor-
responding to zero and infinite frequencies.

An application of a generalized mapping function technique
to ship section forms of arbitrary shape was performed by Porter,
who studied the pressure distribution and forces on heaving cylinders,
[27]. A way of solving the two-dimensional problem without resort
to conformal mapping was developed by Frank, who represented the
velocity potential by a distribution of wave sources over the sub-
merged part of the contour, now defined by a finite number of off-
sets. The varying source strength was determined from an integral
equation based on the kinematical boundary condition

Vugts [ 29] contributed an extensive experimental and theo-
retical study of the hydrodynamic coefficients for pure and coupled
swaying, heaving and rolling cylinders, based on the previous works
by Ursell, Porter and de Jong, [30]. The coefficients of the
THEODORSEN mapping function were defined by a least square fit
of the geometry of the cylinder contours to off-sets in 31 points. Of
special interest is the good agreement obtained between experiments and
the theoretical predictions for the added mass of a typical midship
section; the oscillation experiments do not cover the very low fre-
quencies, however. Although small the difference in the calculations
for the actual section fit and for an approximate LEWIS form was
mainly confirmed by the experiments.

When used with the strip method the integrated section contri-
butions to total added mass and inertia shall be reduced by the
appropriate "longitudinal inertia factors" for three-dimensional
effects. Following Lewis these factors are usually taken equal to
those derived for the prolate sphereoid in a similar mode of motion.
This is only an engineering artifice, and it is certainly not correct,
say, in case of accelerations in yaw for normal hull forms; thus
these correction factors are mostly omitted in hydrodynamic studies
of sufficiently slender bodies.

In a discussion of the strip theory Tuck [31] included the
results of all the added mass and damping coefficients of a surface
ship at zero forward speed, calculated by use of Frank's close-fit
method with 15 off-sets for each of 23 stations. The total added
mass (Aj) and moment of inertia (Az 69 oF of : Series 60 Block , 70 form
are here represented by full lines in Fig. 8. Tuck also examined the
forward speed corrections to be applieg to, ae a Nee values;

thus, especially, he put Agg = Ace = (U 1 jes age or in present
notation
Ni(u" th .= NI( my es r x" my (6 3)
ran oh ea ae u''=0 wile v5 tO i

834

Shtp Maneuvering in Deep and Confined Waters

-Y¢" Ne

1.6

Speed corr. to Fry = 0.2
0.10
1.4
Theory ( Tuck )
1z2
$Hi2.0
J g Theory ( Tuck )
1.0 Exp. Fry = 0.2
(v Leeuwen )
0.8
0.05
0.6 Exp. F,,=0.2
(v Leeuwen )

0.4

0.2

0 0

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
WwW w"

Fig. 8. Total added mass and added moment of inertia for a
Series 60 Block .70 form according to theory and
experiments.

(Note that the strip theory is not valid for small "reduced frequencies"
w' = w'"/u", where it shall be replaced by a slender body theory, [ 31] .)
The dotted curves in the diagrams indicate predictions for

Fa = ul = 0.20.

The Series 60 Block .70 form was subjected to oscillator
experiments in lateral modes at several frequencies and forward
speeds by van Leeuwen, [32]. The results for the naked hull with
rudder at F,, = 0.20 are compared with the predictions from strip
theory in Fig. 8. The experimental values fall well below these
predictions in the entire range of frequencies, especially in case
of the moments in yaw. Although it is inherent in the testing tech-
nique that very low frequencies could not be included van Leeuwens
results do cover the critical range around w" > u" = 1/4,

Consider a surface body in steady motion along the centre-
line between two parallel walls width W apart; the diverging bow
wave displays an angle to the centreline. If the motion is steady
the reflected wave will pass aft of the body only if W/L > tg,
regardless of the speed. For the simple travelling pressure point
the cusp line angle is equal to 19947 according to the Kelvin theory,
whereas-slightly different values may be observed for real ship
forms. In case the body is oscillating (as in the simple example may

835

Norrbtn

be illustrated by a pulsating source) additional waves will form,
which move with speed g/w. At low frequencies these waves move
faster than the body, so that the diverging wave front folds forward,
and at a certain forward speed there is now a new requirement on
basin width to avoid wall interference. For combinations of w and
V (or u), in which y = w"u" = 0.272, the opening angle equals

B = 90°, and with a further reduction in speed it rapidly reduces
again to 55° as y approaches 1/4, This latter condition is associ-
ated with a special phenomenon of critical wave damping, as has
been shown from theory as well as experiments by Brard, [33].

In model tests with a ship form in lateral oscillations a narrow
range of critical frequencies may be identified by a change of the
distribution of the hydrodynamic forces, which was clearly demon-
strated by van Leeuwen's analysis.

Whereas there is a discrepancy in the absolute values of
added masses compared in Fig. 8 this discrepancy could be reduced
by the application of a three-dimensional corrector; more elaborate
theories of forward speed effects for slender bodies at low fre-
quencies may further improve the comparison. In the main, there-
fore, it may be stated that the variation of added mass with frequency
is well documented.

Added Masses in Maneuvering Applications

The performance problems set up in maneuvering studies
usually involve a short-time prediction of a transient response to a
control action, and it is therefore convenient to be in the position
to use ordinary non-linear differential equations with constant coef-
ficients. This, of course, is in contrast to the linearized spectrum
approach to the statistical seakeeping problem, which will more
readily accept frequency-dependent coefficients. (Frequency- or
time-dependence as a result of viscous phenomena will be touched
upon below.) Which values of added mass are now to be used in the
equations for the manoeuvring ship? It shall be noted that it is hard
to judge from the behaviour of a free-sailing ship or ship model which
is the correct answer unless special motions are carefully examined.

It was early suggested by Weinblum that the low added mass
values of the high-frequency approximation should be adequate for use
in dealing with problems of directional stability, where starting con-
ditions should simulate impulsive motion, [19]. Weinblum also drew
attention to Ref. [ 34] , in which Havelock proved that the high-
frequency values appeared in horizontal translations with uniform
acceleration, regardless of the initial velocity.

The impulsive pressures experienced on the tapered bow and
stern portions of a slender body in oblique translation may be calcu-

836

Ship Maneuvering tn Deep and Confined Waters

lated from the sectional area curve slope and the added mass
characteristics of the transverse sections, as shown by Munk [ 17]
and experimentally verified for the submerged doublebody ship form
in Ref. [18]. The good agreement obtained between total yawing
moments measured on this form and its surface ship geosim suggests
that the deeply submerged added mass values should apply in this
case. It is observed, however, that the water particles in way of a
certain section station here are not repeatedly accelerated from
rest as is the case when considering the cylindrical part of the hull.
Again, if the principle of superposition of damping and inertia com-
ponents to the total hydrodynamic force shall be retained for general
motions it shall be necessary to adopt the zero-frequency added
mass values.

An illustrative discussion of added masses with special
application to the design and analysis of experiments is due to
Motora in Ref. [35]. For the determination of the added mass in
sway to be used in the aperiodic equations of a maneuvering ship he
recorded the direction of the acceleration imparted to a model by a
force suddenly applied in a certain direction. The added mass then
could be found from a reasonable estimate of virtual mass in surge.
To obtain the added moment of inertia in yaw he recorded the angular
acceleration following the impact by a pendulum, the momentum
loss of which was also known. He suggested that the inertia values
so derived should correspond to the impact or high-frequency type,
but the results included from tests with a series of ship models indi-
cate sway mass values of the same order as those valid for the deeply
submerged case, and moments of inertia in yaw of magnitudes cor-
responding to finite frequency surface values.

In a recent paper Motora and co-authors [36] compare the
results of new experiments and calculations of an "equivalent"
added mass for a ship model in a sway motion, which is initiated
by a ramp- or step-form impact input of finite duration. The calcu-
lations are based on Tasai's section values in the frequency domain
[26] , and in agreement with the experiments they confirm that the
value of the equivalent added mass defined is a function of impact
duration. (Cf. Fig. 9.) If the duration is infinitely small only the
equivalent added mass is equal to its high-frequency value, and it
becomes larger the longer the duration. Thus these results help to
explain the earlier findings for added masses as well as for added
moments of inertia, for which latter the impact technique then used
did generate rather short input impulses.

For application to normal ship maneuvers it may now seem
justified to use the low-frequency or deeply submerged values.

In recent years it has been widely accepted that the accelera-
tion derivatives for a surface ship model may be evaluated from a set
of "planar-motion-mechanism" tests in pure sway or yaw. The
acceleration amplitudes are varied by an adjustment of oscillator

837

Norrbtin

Vv cm/s

10

8 Sway acceleration v(t )

6

rl Calculations
— -—O- -— _ Experiments

2

0

Fig. 9. Motora's equivalent added mass coefficient as defined by
acceleration due to step input impact of duration T

amplitudes, whereas the frequency is kept as low as running length
permits, [32]. A typical reduced frequency w' =w-+ L/V willbe

of order 0.5, corresponding to w"=y/u", wo" =o' + u" =0.4. in
Fig. 8. The derivatives so obtained may be expected to be somewhat
higher than the zero-frequency values.

The theoretical zero-frequency added mass values for two-
dimensional LEWIS forms as well as for semi-submerged ellipsoids
of finite lengths indicate the main dependence on principal geo-
metrical characteristics. Especially, for very large length-to-
draught ratios the ellipsoid values tend to those of a semi-elliptic
cylinder, (1/2)pT?, so that - Yj = T/B. Moreover, it will be seen
from [ 24] that for LEWIS eee general - YJ! likewise is rather
close to T/B for fullness coefficients corresponding to midship
sections.

The ellipsoid family has a constant prismatic coefficient

g = 2/3. The correction for finite length involves a slight dependence
on B/T, as may be seen from Fig. 10. In amore general case

838

Shtp Maneuvering in Deep and Confined Waters

a) b)
B wre)
-Yy el Ye - Nr a il
0.4
< 1 0.3
oe
2 » SLCDAY
77 |
° * yf
fos I °
/ Zero freq.
ofA y 0-2 F for B/Te
a & & 31325 20 ., ots
Se | § Weep tyes
y, > A g
| zs
ess . oO
2 : A 4
Ellipsoid 2 2 —_
(Theory ) Fe 0.1 g O afellipsoid (Theory)
H =|
3 4A
Experiments : = Oo Experiments :
4 Motora (B =1.83) of a 4 Motora (¢ = 0.0319)
Oe (2565) 5 igh reg OF a 054n
Ei 9566) i £ ae O —"— ~ (0:067))
* HyA (Misc. ) * HyA ( Misc.)
» vy Leeuwen 0.05 =» vy Leeuwen

0 0.5 1.0 0.1 0.2 0.3 0.4 0.5
Y - a -¥

Fig. 10. Non-dimensional added mass (a) and added moment of
inertia (b) from theory and experiments T

this correction will also depend on g and on lateral profile, etc.

For the inclusion of ship form values in Fig. 10a the diagram is drawn
to a base of g - (2T/L). The ordinates are given by the product

- YJ (B/T)¢, by which the intercepts on the vertical g - (2T/L) = 1
then corresponds to the infinitely long cylinders.

In addition to the ellipsoid and LEWIS cylinder values the
diagram include the experimental results by Motora just referred to
as well as a number of oscillator results, chiefly from tests run for
SSPA inthe HyA PMM. The general character of the three-
dimensional corrector is clearly seen, and it is suggested that the
diagram may be used for approximate estimates.

Non-dimensional added moments of inertia, in terms of

339

Norrbtn

product - N}'+ B/T, are displayed in Fig. 10b, compiling experiment
data from different sources. Here the two-dimensional LEWIS-form
values for high as well as low frequencies are indicated by off-sets
to the left in the diagram. Motora's 1960 impact test data, which
appear on a level close to the high-frequency prediction, do not
indicate any definite dependence on draught-to-length ratio. These
data as well as low-frequency PMM data clearly indicate an increase
of moment of inertia with reduced fullness. This trend may be
expected in view of the deep and narrow bow and stern sections in
fine forms — certainly the deeply-submerged ellipsoid is not repre-
sentative for a ship form in yaw acceleration.

Semi-Empirical Relations for the Four Basic Stability Derivatives

Among the large number of first-order force and moment
derivatives, that are used to describe the linearized hydrodynamics
of the moving hull, only four appear in the analytical criterion for
inherent dynamic stability with fixed controls. These are the stability
derivatives proper, Yyy, Nyy, Yy, and Nyy» From simple analogy
with the zero-aspect-ratio wing theory of Jones [37] they turn out
as in Table III.

TABLE III

Non-dim. system: "Prime"

tr

Ref. area:

Ld

Symbol and analogy
value:

1
ES
°
Mm

N

RAE es es
ee ee
~

ola AW WA NIA

Sa NN 5

oa WA WA MA
Ala Ny Ny

Although this analogy has been verified in principle for a submerged
double-body model as well as for the surface model at small Froude
numbers [18], it shall not be expected to furnish an adequate nu-
merical prediction. It suffices to point on the alternative relation
for a closed body in a perfect fluid, given by Eq. (5.7), and to the
fact that at least some negative lift is still carried on the run of
normal ship-form hull. The bow lift or transverse force is not

840

Ship Maneuvering tn Deep and Confined Waters

concentrated to the leading edge as in case of a rectangular wing but
distributed over the forebody as an effect of fullness and section
shape. Certain modifications to the hull form are known to affect
the force derivatives, but do not appear in the simple form
parameters of Table III. The fin effect of screw and rudder con-
tributes to the derivatives even in the case of vanishing aspect ratio
of the hull.

From the analysis of a large number of derivatives it has
been found that the scatter of data in a plot of, say, May versus the
parameter LT /V is somewhat smaller than the scatter of VG on
base of aspect ratio 2T/L.

The diagrams Fig. 11-12 include stability derivative data
for normal ship form models with normal-sized rudders propelled
at medium Froude numbers on even keels. The dotted lines shown
correspond to the simple wing analogy. The full lines are derived
by linear regression and upon the tentative assumption of a - 1:2
relation of moment and force intercepts at zero aspect ratio. Their
equations are given as

-Yuv -Nuv

1.5 0.6
a
8G
V
1.0 0.4
Number of Number of
Prop. Rudder : Prop. Rudder

0.5 0.2
1
1
2
0 0.2 0.4 0.6 0.8
LT2/V

Fig. 11. Stiffness force and moment derivative data with mean
regression line. (Cubic fit to experimental results.)

841

Norrbtn

Fig. 12. Rotary force and moment derivative data with mean
regression line, (Cubic fit to experimental results.)

2 2
LT gh pes
Yiy=- 2.6647 - 0.04= - 1.69- J+ ae - 0.04
nue - 1.01 £2°+0.022-1.28-7-S2+0.02
uv ° LY, e eo 4 VAR °
(6.4)
72,
tors DS 3 = 7, iT =
vies 1.02 V 0.18 _ 1.29 4 VW 0.18
LT? _ + LT
— 1.88 ro aya +0.09

Ni,= - 0.74 + 0.09

and of the data 100, 86, 67 and 79 per cent respectively, appear
within + 20 per cent of these mean values.

It is obvious that these expressions should be regarded as
guide values only, but they may also be used for comparative studies,
especially when steering on a straight course is of main concern.

In this latter case it is more important to have a proper knowledge
of the control derivatives, whereas Eq. (6.4) may furnish adequate
estimates for the hull forces; they again shall be corrected for
alternative control arrangement alternatives, however.

842

Shtp Maneuvering in Deep and Confined Waters

In the next Section an approximate method will be given for
finding the control derivatives of a rudder of conventional design.
In the hypothetical case of an isolated rudder experiencing the nomi-
nal inflow at the stern of the ship it would be easy to calculate its
contribution to the total "hull + rudder amidship" derivatives from
a knowledge of its control effectiveness. In general the interference
effects in behind condition are much more complicated, and in fact
the contribution searched for mostly is quite small. Even more,
then, the effect of a modification to rudder and control derivatives
comes out as a very small change in the stability derivatives. The
diagram in Fig. 13 is compiled to correlate the effects of such modi-
fications as reported by Eda and Crane [38] and documented in test
results available at SSPA. Obviously new experiments are required.

Reference shall here also be given to the methods of estimating
stability derivatives for surface ships as suggested and successfully
tested by Jacobs, [ 39].

The aerodynamic wing analogy should only be valid for small
Froude numbers as the limit solution of a general lifting surface
integral equation. The effects of finite Froude numbers on the
lateral stability derivatives of a thin ship of small draught-to-length
ratio was studied by Hu, [40]. According to Hu the force and

0,15 Dav. Lab.Exp. with

Series 60 Block.60 Form

HyA Exp. with
SSPA Twin-Screw/ Twin-Rudder
Tanker

0 0,005 0,010 0,015 a(t]
Fig. 13. Change of control force derivatives and total force

derivatives in sway and yaw with change of relative size
of rudder.

843

Norrbtn

moment derivatives at F, = 0.1 are increased by some 20 per cent
above their zero-speed values, an increase which is not fully
realized in model tests. A comparison of the results of this theory
with various experiments is presented by Newman, [41]. Newman
also points out that the free surface may give rise to a steady side
force as a thickness effect, and indicates a solution to that problem.

From an inspection of the experimental results for the drift
moment, which are the more consistent, a first approximation to
the speed dependence is given by

(NO) = (NE), + 2 Nie" (6.5)

uv'u uuy

where 3Ni,,* 1.3(NjV),- This suggests that the zero-speed values
will be some 20 per cent lower than those indicated by the mean line

of Fig. 11.

Viscous Frequency Effects and Small- Value Non-Linearities
in Lateral Forces

In dealing with the free-surface effects on added masses it
was concluded that so far the frequencies involved in manoeuvring
motions were to be regarded as low, but that frequency (or memory)
effects should be expected to appear in time histories were viscous
phenomena were of more concern.

The extreme exemplification is furnished by the pitching
submarine, the stern planes of which are operating in the downwash
behind the bow planes, but in case of submarines as well as normal
surface ships also the very stern portion of the hull is exposed to
velocities induced by vortices trailing from upstream hull and
appendages. Moreover, local separation within the three-dimensional
boundary layer flow over the stern directly affects the cross-flow
momentum and the impulsive pressures. The forces and moments
experienced by the hull in transient motions can then only be calcu-
lated by use of convolution integrals over the entire time history,
such as derived by Brard in case of a special descriptive model, [ 42].

For application to the mathematical model defined by ordinary
differential equations it is again still possible to use frequency
dependent coefficients, but unfortunately this frequency dependence
is likely to be subjected to scale effects. It is therefore advisable
to design experiments for Strouhal numbers or reduced frequencies,
which are low enough to produce steady-state values. From a sum-
mary of published data in Ref. [41] the limiting frequency will be
expected to be somewhere in the region 1<w'< 4. From a more
recent analysis of sinusoidal free-sailing tanker model data Nomoto
suggests that this limiting frequency is approached already at
w'= 0,5, [43]. This indicates that the high-frequency part of a normal

844

Shtp Maneuvering tn Deep and Conftned Waters

ship steering transfer function is obscured by the viscous frequency
dependence. (Cf. Section IX.)

The steady motion of a full form may also be accompanied
by a non-steady separation and shedding of vortices, which will
violate captive measurements, or it will modify the force field and
be a cause of unpredictable scale effects. In Ref. [44] Nomoto
drew the attention to an "unusual" kind of separation, which had
been observed not on the leeward but on the outer side of the after-
body of turning models. (Later on he reported the same phenomenon
taking place on full scale ships combining high block coefficients and
low length-to-beam ratios.) This separation may be responsible
for an almost constant increase in yaw damping moment — see
diagram in Fig. 14a — and so indirectly for the small-rate non-
linearity displayed in the yaw-rate-versus-helm diagram from spiral
tests with these hulls.

Unsymmetrical separation may also take place on a hull
moving along a straight line with a small angle of drift. If transverse
force and moment both are mainly linear functions of angle of drift
the centre of pressure will remain in a forward position, only
gradually moving aft with onset of viscous crossflow. A three-
dimensional separation, which suddenly develops on one side of the
hull, may explain the strange behaviour of the centre-of-pressure
curve of a tanker model tested by Bottomley [ 45], here reproduced
in Fig. 14b. New tests with modern hulls sometimes indicate
similar trends.

It is fully possible to approximate these effects by a small-
value non-linearity term in the mathematical model, which may
then be used, say, for the prediction of a ship behaviour which is
extremely sensitive to winds of varying directions [ 46]; if the sepa-
ration is peculiar to the model only this prediction is meaningless,
however.

Large-Value Non-Linearities in Lateral Forces

The predominant non-linearities present in the lateral forces
are due to viscous cross-flow resistances, and they can only be
established by experimental procedures. It will be assumed that
the empirical relationships may be expressed by finite polynomials,
derived by curve-fitting, and that these same relationships therefore
also may be fully defined by a finite number of terms in the Taylor
expansions. This convention motivates the use of appropriate
numerical factors in front of the derivatives within the hydrodynamic
coefficients.

From pure athwartship towing it is possible to define a Y-
force -Cp°* LT - v*, the sign of which is governed by ly|/v. Thus

X(v4, hives v2|v|/v, or, for convenience, 3Y vive

Yvvtivi/y) Ivlv

845

Norrbtin

Yaw rate r

Yaw damping due to
flow separation

Resultant yaw dampin

a) Nomoto’s explanation of effect of 3-dim. stern flow separation
( Above }

b) Lateral force centre of pressure acc. to measurements
by Baker and _ Bottomley ( Below )

F.P.

0.4 — ~ Cargo ship (normal curve )

— Smads

—_
= ——

0.3

0.2 Tanker

0.1

ne See ee Cs ee Se [ene
0° 2 Lc B 6°

Fig. 14. Smalli-value non-linearities in full form model

testing.

846

Shtp Maneuvering in Deep and Confined Waters

Note that the factor $ has been retained, which should not have been
the case if v and lv | had been treated as independent variables;
this, however, would only have been a formal artifice with no
physical significance.

In straight-line oblique motion the non-dimensional lateral
force is Y" (u", v", v"*, |v"|/v"), or, in accepted writing

Yu" vi") = ye uly" ie ave ve |v" (6.6)

where a =-Cp- ew (2. It is obvious that here two terms
are added, which each one corresponds to a certain flow field. In
the discussion of the "linear" term it was pointed out that the ideal
flow picture would remain valid over the bow portion of the hull,

and in view of the finite time required for the development of the
viscous cross-flow these conditions may still be true at larger
angles of drift. (Cf. non-linear theories for the lift of zero-aspect-
ratio wings. )

0.8

0.6

Cross - Flow Contribution

Viscous

0.4
Lee Ti
Zero - Aspect - Ratio
Wing Theory

0.2

0° 30° 60° 90°
B

Fig. 15. Calculated and measured lateral forces on a cargo liner
model in oblique towing.

847

Norrbin

An experimental evidence of the practical validity of the
superposition in Eq. (6.6) is illustrated in Fig. 15, based on force
measurements at SSPA on a 3.55 m model of a cargo liner with
rudder and bilge keels, [47]. In this diagram the quotient
Y¥/(p/2)V°LT =(2V/L°T)Y¥"/(u"'? + v"*) is plotted versus B = - arctg v/u,
and the viscous cross-flow component is seen to dominate the entire
range of 10°< B < 90°.

The variation of cross-flow drag coefficients with drifting
speed and hull geometry has also been discussed in several papers
by Thieme and by other authors, [48, 49, 50], In lack of experimental
results for a special case in the non-linear range it shall be possible
to use these results; a typical value of cross-flow drag of a tanker
formis C,)=0.7. The contribution of cross-flow drag to moment-
due-to-sway may then be ignored.

In a similar way it is possible to approximate the non-linear
rotary derivatives. If cp (> C)) is the mean section drag coefficient
the moment-due=to-yaw derivative is sNii =e (c,/32) + (LEST 72%
except for a three-dimensional correction factor. (For rough esti-
mates ZN], = 0.03 + 2Yiyy » which is verified from experiments.)
The force-yaw velocity derivative now is zero to this approximation.
Additional effects of skegs and screws contribute to non-zero values

1
of zNj\,, as well as Gare

In the general case the local cross-flow resistance is pro-
portional to lv + xr|(v + xr), and from symmetry relations the
coupling terms are seen to include the derivatives Yjyj,, and Yvypr),
etc. (In the cubic fits more often used these couplings are repre-
sented by terms in Yyyr and Yy-,, etc. — cf. Abkowitz, [51].)

The contribution to Y due to the combined sway and yaw may
be written Yj, |v|v(r/v) + Yiote |r|r(v/r), iee., Yjyj, may be looked
upon as the derivative of Yj,,, with respect to yaw velocity r per
unit v, etc.

Forward Speed and Resistance

The principal effects of viscous and free-surface phenomena
on the resistance to steady forward motion are well-known to naval
architects. The correlations of wavemaking and separation with ship
geometry are still less satisfactory. However, alternative methods
are available for full scale powering predictions from standard series
or project model data. As will be further discussed in next Section
the adequate synthesis should supply information not only on shaft
horse power and r.p.m. but also on hull resistance and wake
fraction. Speed trial data therefore require an analysis suchas
proposed and used by Lindgren; in case of very large and slow-
running ships it may be necessary to include scale effects also in
the open- water characteristics of the screw propeller, [52].

848

Ship Maneuvering tn Deep and Confined Waters

A simple guide to ship resistance values may be obtained
from the mean line of Fig. 16, which summarizes the results ofa
limited number of SSPA trial trip data in terms of the total specific
resistance R/A = - >See on basis of Froude number Fy or
u", (A similar plot of "total resistance in Ibs to displacement in
long tons" versus Taylor speed-length quotient, based on model data,
was published by Saunders, [| 53].) The mean line also reflects the
general trend of the resistance-speed-dependence for the individual
ships in the proximities of their design speeds.

8 !
Black circles relate to full speed, |
open circles to lower speeds
at same trial f

H

0 0.10 0.20 0.30 Fry

Fig. 16. Specific resistance figures as evaluated from ship trial
data at SSPA.

849

Norrbin

A close approximation to a resistance curve with typical
humps and hollows requires a multi-term polynomial inu. Estab-
lished practice in naval architecture makes use of a single exponen-
tial term R, (u/u, )° to characterize the curve in the vicinity of Uy.
For large slow-running tankers p* 2 over the entire speed range
of interest, which is associated with an almost constant advance ratio
for the screw. In confined waters it may be necessary to include a
higher-order term; see Section IX.

Forward Resistance Due to Lateral Motions

When the ship deviates from the true forward motion addi-
tional forces appear in axial direction. The main cause of speed
loss in a turning motion is due to the axial component of the centri-
petal mass force and the hydrodynamic contribution X,,° rv, of
second importance is rudder drag and finally the axial force due to
oblique-hull lift and wave-making shall be considered.

Ideal-flow hydrodynamics identifies X,;, with - Yy, i.e. the
mass effect is virtually almost doubled. (Cf. (5.6).) A recent
analysis of turning trial data indicates much lower values of X,,.

In a steady turn the ship proceeds with her bow pointing
inwards, so that (m + X,,)rv = - (m t+ ed igs) -B indicates a force
opposed to forward thrust. In running on a straight course the fre-
quency of the yawing motion normally is so low that yaw rate and
drift angle are in phase during most (but not all) of the time, and so
an average parasite resistance results.

: Let the response to a sinusoidal motion of the rudder be
b= q° sin (wt te,) and B = By°* sin (wt + €g). Averaging over a
number of complete periods gives

=B = Fao cos (e, - &) (6.7)

As the normal merchant ship will pivot round a point closely
aft of the bow at low frequencies a rough, estimate of the average
product is given by (rv)w.9* - (OP /2)uq.

A plane wing in a uniform flow will experience an induced
drag as given by Cpj = (1/mA)C,?. According to certain experiments
this simple relation may still be used with a correction factor for
the twisted flow over a rudder behind a screw. The calculation of
rudder lift will be shortly discussed in the next Section; using a
nominal aspect ratio equal to twice the geometrical one the correction
factor just mentioned will be of the order of 1.2 - 1.4.

850

Ship Maneuvering in Deep and Confined Waters

Typical estimates for tankers give as a guide value a relative
increase in forward resistance due,to a rudder deflection of 6
radians AX(5)/X(u) = 3.5 or 5° S For small sinusoidal helm
angles on a straight course the quasi-stationary application gives
AX(6)/X(u) = 1.75 or 26,°, which may be compared with the relation
given from propulsion tests with a Mariner ship model in Japan,
AT(5)/T = 2+ 6°, [54].

At propeller advance conditions removed from the steady
forward motion state the induced rudder drag will be given by
+ Xcess? |c|c6°, where c=c(u,n) is the effective flow velocity
past the rudder and where the coefficient a tens is proportional
to the control derivative +Y(,s and to the ratio a Vale In com-
puter applications a soft-type limiter will be used fo simulate the
conditions for a stalled flow.

The viscous lift experienced by a slender ship hull in oblique
translation is also accompanied by an induced drag, but the axial
component of the resultant force still is expected to be positive.
(According to the zero-aspect-ratio wing analogy the resultant force
will bisect the angle between the normal to the hull and the normal
to the flow. With increasing aspect ratios the resultants move
towards the normal to the flow.) The break-down of the ideal flow
over the stern causes a change of viscous pressure resistance,
however, and wave-making effects will cause a further increase of
forward resistance.

These effects are here illustrated in Fig. 17 by results of
axial force measurements on the surface ship model and the sub-
merged double-body form otherwise described in Ref. [18]. From
an inspection of these and other surface ship model experiments it
is suggested to use a term

lvf|ve (6.8)

uvvv

x(a, v) = 3 xX

to represent the axial force due to lateral drift. An approximate

value of the derivative is given by §X',, = - 200.

851

Norrbtn

Submerged Model

0.06
Surface Model
Y'= ? u
9 9 fe) > v2 LT
|
|
|
| ° 9
lee
| Fry =0.21 Zero - Aspect - Ratio
| Wing Theory
l 79 ‘
|
|
|
|

-0.020 : 0 -0.010 -0.005
e fy2_rT ov .
=3 vi Numbers at spots indicate
I drift angle A in degrees
-3dd-3
-0.02
-6
1g Pees lr (es

Change of longitudinal force with hull lift in oblique
towing of ship model and submerged double-body geosim

852

Shtp: Maneuvering tn Deep and Confined Waters

VII. SPEED AND STEERING CONTROL

In general the subject of steering and maneuvering may not
be separated from that of propulsive control, and this is specially
true in case of ship behaviour at slow speeds. Moreover, in model
testing the interactions between hull, propeller, and rudder are
likely to cause the main problems of model-to-ship-conversion,
including scale effects of a hydrodynamic nature as well as other
model effects due to the dynamics of the testing equipment.

Large seagoing ships are usually propelled by a single centre
line screw, or by wingward twin screws. In case of a tandem contra-
rotating propeller arrangement most of the characteristics discussed
below may be calculated for an equivalent single propeller. In case
of close-shafted twin screws of overlapping or interlocking types
the interaction with the rudder should be specially considered.

It has been repeatedly proven by handling experience that
twin screw ships should be fitted with twin rudders. Recent model
tests indicate that with a suitable design of the rudders, including
a certain neutral position toe-out, this arrangement may favourably
compete with the centre line rudder alternative also from a propulsive
performance point of view.

In the application of the first-order steering theory, first
introduced by Nomoto in 1956 and strictly valid only for inherently
stable ships, there appear only two constants: a (desired high)
"sain" K, which represents the ratio of rudder turning moment to
yaw damping, and a (desired low) "time constant" T, which
measures the sluggishness of the ship response, and which repre-
sents the ratio of ship inertia to yaw damping. As was subsequently
also shown by Nomoto [ 55] the non-dimensional quotient K'/T'
turns out to be proportional to the parameter LA,/V for ships with
similar stern arrangements. This quotient may therefore be looked
upon as a rudder-on-ship effectiveness factor, proportional to the
initial yaw acceleration imparted to the ship by a given helm.

Some ten years ago maneuvering trials were run with three

tankers of the Gotaverken 40 000 tdw series, all similar except for
the stern arrangements, [56]. The SSPA analysis of zig-zag tests
with respect to the rudder-on-ship effectiveness factor just mentioned
offers a unique illustration of the merits of these arrangements,
Fig. 18. In particular, note that the two alternatives with rudder
behind screw (screws) prove to be equivalent in case of same total
area of rudder, and that the use of the larger area of a twin alter-
native therefore is especially favourable.

A propeller or a rudder, or the combination of a propeller
and a rudder, acts as a stabilizing fin as well as a manoeuvring
device; the contributions to the fin effect from the propeller and from
the rudder-behind-propeller are of equal order. It should be

853

Norrbtin

100 A,/LT

Fig. 18. Results from first-order analysis of full-scale zig zag
tests with three 40 000 tdw tankers, similar except for
stern arrangements.

realized that a minor modification to a rudder does not appreciably
affect this fin effect or the size of a hysterisis loop in the yaw-
velocity-versus-steady-helm diagram of an unstable ship. However,
the higher control force per degree of helm then possibly achieved
will help in actual directional control, where the history of yaw
velocities and helm angles takes place well within the height of the
steady-state loop. (See also Section I.)

The general propulsion case will be represented by an arrange-
ment including one centre line screw and two wing screws, develop-
ing thrusts T,, Tg and Tp, respectively. Hull interference
generates axial forces t,T,, tT, and tpI,, in the opposite direc-
tions, as well as lateral or sideward forces s,° T and s,° T,-

In order to adhere to the thrust deduction concept the factors t —

which are not necessarily constants — will be taken as positive,
so that’the force in postitive’ x’ direction is -t* T. The facter
S, will be positive, and sp= - Sg. Roughly sg=tg* cot a, where

a is the effective waterline angle in front of the propeller.

Normally the lateral forces due to Tg and Tp are in balance,
but if Tp# Ts there is a resultant force applied some 0.4L behind
the C.G. of the ship. The turning moment thus obtained is much
larger than that produced by the axial forces along the shaft lines, [ 57]9

854

Ship Maneuvering in Deep and Confined Waters

Ya Bo

Splp Ss Ts
On >
tpTp 3 ts Ts
A I.
Normal twin - Starboard screw
screw propulsion idling

Fig. 19. Force fields on twin-screw tanker on straight steady
course.

The diagrams in Fig. 19 illustrate the symmetric force field
around a twin-screw tanker in normal straight course conditions,
and the steady state situation when running with starboard propeller
idling. The non-symmetric suction force on the port quarter is
balanced by the forces due to drift and checking rudders. The drift
angle is a fraction of a degree only, and some 90 per cent of the
compensation force is due to the rudders, set at some 5 to 7 degrees,
With the twin rudder arrangement it should be possible to maintain
75 per cent of the speed in this condition. The induced resistance
due to rudder lift would be larger in case of a single rudder between
the propellers, but the main cause of speed loss of a ship propelled
by one of its screws only is the additional drag from the idling

855

Norrbtin

propeller; again, that drag may well be increased by a factor of 3
if the propeller is locked.

The characteristics of a propeller in axial open-water flow
are usually given by tables or curves of well-known Ky and Kg
coefficients versus advance ratio J. In yawed flow the propeller
also experiences a lateral force and a (small) pitching moment, [ 58].

In behind conditions the effective angle of drift at the pro-
peller still is roughly 2/3 of the nominal local angle, high enough to
let the propeller contribute the fin effect already mentioned. (The
sidewash behind the propeller then has a further straightening effect
on the flow to the rudder.) The effective advance ratio is modified
by the effective wake in the factor 1-w; here w will be chosen as
for thrust identity. The effective wake, again, is modified by the
drift of the ship, being higher for a starboard drift angle than for a
port one and a right-handed propeller [59]; here that effect will be
taken as of second order.

Finally, the vertical asymmetry of the flow field is responsible
for the appearance of a lateral force on the propeller of a ship even
if drift or yaw are zero. In case of a single screw ship this latter
force may be put equal to 3 to 5 per cent of the thrust, [60]. A
right-handed screw tends to throw the stern of a loaded ship towards
starboard, thus requiring a small starboard helm to be carried on
straight course. Other free-running model tests prove that draught
conditions may change this picture, and that the ship on light draught
may have a tendency to turn to starboard, (oa).

The hydrodynamic'thrust T (T,; T,, T,) and torque: ©
(Q¢, Qs, Qp) — which is negative in case ofa right-handed screw on
a driving shaft — will be given as quasi-stationary functions of
instantaneous values of forward ship speed, u, and screwr.p.s.,
n (ng, ng, np). The thrust is a major factor governing the flow
velocity past the rudder, and this velocity likewise will be given in
terms of u and n. Rudder control derivatives usually are deter-
mined from model tests in one or two conditions of screw loading
only. In order to find an adequate prediction of full scale control
derivatives for the more general propulsion case it is necessary to
combine model results with a simple procedure for calculating the
total control force due to rudder deflection.

From the hydrodynamical point of view the typical all-
movable rudder in behind condition is equivalent to a twisted wing
on a pointed afterbody. There are a number of additional complica-
tions, however: The spanwise velocity distribution is highly non-
uniform, the flow along the chord is accelerating or decelerating,
the gap between wing and body is within a retarded boundary layer
flow and it also varies with the angle of deflection, the boundary
conditions at the free surface violate the vertical symmetry aspect
even if there is no suction-down of air, the shape of the body stern

856

Shtp Maneuvertng in Deep and Confined Waters

is far from say a simple axisymmetric cone. The modern half-spade
rudder on a fixed horn (the Mariner-type) is a hybrid of the alle
movable and the flapped types, and other common forms all have
their special characteristics. The procedure here adopted is not

a substitute for the detailed calculations necessary for a certain
project design, but it will furnish a good estimate of control forces
and make possible the extended use of model results referred to
above.

The Rudder or "Control" Derivatives

It will be assumed that for each rudder configuration may be
defined "equivalent" values of rudder area, rudder aspect ratio,
rudder angle and rudder advance velocity.

A detailed study of the velocity field in the slipstream ofa
propelled tanker model and of the pressure distribution over the
rectangular rudder fitted to this model was reported by Lotveit, [62].
The distorsion of the spanwise loading due to slip-stream rotation
was clearly demonstrated, but the diagrams did not indicate any
definite influence of the rudder image in the hull and free surface;
the gap distance from top of rudder to stern profile was some 12 per
cent of rudder height. Straightforward calculations of rudder lift
from known relations of lift curve slope versus geometric aspect
ratio and an average advance velocity based on the simple momentum
theory proved to give good agreement with the rudder forces measured
by a force balance or integrated from the pressure field.

Unfortunately in this case no simultaneous measurements
were made of the total hull-and-rudder forces, and there is stilla
lack of such data for normal surface ship forms. However, already
from the old experiments by Baker and Bottomley [ 63] it was seen
that the total force due to rudder deflection was increased by some
40 per cent in presence of a deep cruiser stern close above the
rudder, and that a third of the total force then was carried by the
hull.

Let b be the height of the rudder at the stock, or the higher
value forward of it, and let a be the depth to top of rudder at the
same station. With a projected area A, of the pudder the aspect
ratio of rudder + plane image is equal to A = 2b*/A,. The lift
curve slope aj is taken from the theoretical curve derived from the
Weissinger theory [64], or from empirical curves available.

The geometrical aspect ratio usually is of the order of 1.5,
i.e. the rudder is not a low-aspect-ratio fin, but it seems still to be
possible to make use of the results for wing-body interferences
applicable to such fins. In particular, the ratio of the liffon a rigid
combination of a wing and a cylindrical central body, Lae, to the
lift of the abridged wing alone, Lg, is simply given by

857

Norrbtin

(1 + a/atb)*, [65]. Next, for the calculation of the lift carried on
the axially oriented body and on the wing deflected to the flow, it is
observed that the exact t theory by Mirels [66] may be approximated
by 1 = yee 7 LNS= EN. (1 +a/atb). Except for a correction factor
the control derivative for the ship will be calculated as

a2 oo

A 1 Isgls
W = eM 4S ' = py nacre Are
where Y{ unlike Y. is defined also for zero forward speed.

The Pedeon nolt spade or Mariner type rudder has a fixed horn,
which divides the upper part of the rudder in ratio A,/(Ay - A,)-

The right-hand member of (7.1) may then be multiplied by a factor
i - (1/4) (ATR).

The effective rudder advance velocity c (squared) is calcu-
lated from the mean square velocity of the screw race and an esti-
mated mean square velocity past the rudder outside the race. If w
is the wake factor as integrated by the propeller (thrust identity) the
effective square velocity above ae race in a normal single screw
arrangement may be taken as u “(1 - 4 w)?, Inside the race, which in
average conditions has a diameter some 10 per cent smaller than
the propeller, the ultimate mean square velocity is given by

u*(1 - w )*(1 + (8/m) * (K,/J*)), where, for u>0,

J by
Ky = Ky, * ey M Ky, s Ky J eae Ky, J (7.2)

is to be approximated from the open water propeller diagram. Where-
as the thrust may be analytically defined for all combinations of u

and n — see below — the working conditions of the rudder are known
only for a positive thrust, in which case

=e 2 2 eve ites ee
COT 2Cyy BF Cy, UN F 2Ciniq |n|n F2Cqy

(7:3)

From an analysis of a large number of control derivative
measurements on models it appears that a correction factor of
0.7 - 0.8 shall be applied to Ae 1) when combined with (7.3) to give
the force Y(u,n,6) =pV/L> aie - &&. This correction factor is
understood to take care of gap ‘effects and non-ideal geometry of the
hull + rudder arrangement, etc.

The four constants in Eq. (7.3) depend on screw character-
istics and wake factors, and they are therefore unique for the model
scale. To facilitate a correction for this scale effect in the control
derivatives the diagram in Fig. 20 has been compiled, chiefly from
Ref. [67] and data available at SSPA. The slope of curves of wake
factors against ship or model lengths increases with hull fullness;
especially SSPA experience of full scale tanker trials rarely include

858

Ship Maneuvering in Deep and Confined Waters

0,4

.0;3

Fine Forms ee ae

0,1

Hull length L

1 2 5 10 20 50 100 200 m

Fig. 20. Scale effects on wake factor w as integrated by propeller
in model and full scale.

effective wake factors above 0.38.

In Fig. 21 the control moment derivative Ng for a 98 000
tdw tanker is presented as a function of forward speed u and shaft
speed n, for a 1:70 scale model as well as for the prototype.
(Extrapolation to slowly reversed propeller is shown dotted.) In
particular it is seen from the diagram that the turning moment from
the rudder at self propulsion point of ship is only some 60 per cent
of the model test value.

During a maneuver the effective change of angle of attack
of the rudder is a function of nominal helm deflection 6, drift v,

and yaw rate r, and change of screw loading. Again accepting this
quasi-stationary model it is

E quay
6e= S t+(ky> — +k, St )[8| (7.4)

859

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Mm RPM on
20 150
10
15
100
10
5
50

Dae oes <t
-5 ie Bas” Sate
en
-50 hes ee
-5 “= _9.2

0 0.05 0.10 0.15 0.20 0.25 u
SS ss Sess sss. sq

0 5 10 15 20 Vs
SSS. SSS SSS SSS SSS me.

0 0.25 0.50 0.75 1.00 1.25 Vm

Fig. 21. Relative change of rudder control force with change of
propeller advance conditions for 3.6 m model and 98 000
tdw tanker prototype. Diagram based on model tests at
VBD and SSPA, and on speed trials.

where typical values are ky=-0.5 and k,=0.5. (An alternative
but less explicit method to include the same phenomena is given by
Strom-Tejsen and Chislett [68], who make use of a number of
coupling derivatives such as Yeine etc.)

Helm Control

The manual or automatic pilot exerts the control through the
steering gear, which is supposed to have a time constant T,,
causing a small delay in the rudder angle 6 obtained. The value of
T, may vary say between 0.4 and 4s, the first figure being a good
catalogue value and the second one not seldom realized in shipboard
testing. The steering gear or telemotor system often has a back-
lash of about half a degree.

The function of an auto-pilot may be said to be essentially

860

Shtp Maneuvering in Deep and Confined Waters

of the "proportional + rate control" type, although an integrator
control shall be added to take care of stationary deviations. Com-
mercial type auto-pilots include special features, which shall be
includedin simulator applications.

With the simple ideal auto-pilot "calling for" the rudder
angle 5° = yi +o the transfer function of the feed-back loop is

y(4 qr 8)
Y, = Yan*? You, = ————-+— (7.5)
2 88 s"y 1 + Tes
Typical values for the gain or "rudder ratio" y and rate constant
o of a tanker auto pilot in deep water setting are y = 3 (degrees
helm per degree heading error) and o = 135 seconds (or 135 degrees
helm per degree per second of change of heading). (See Section XIII. )

Although the course keeping characteristics of say an inherently
unstable tanker may be studied in a Bode diagram by use of a total
system open loop transfer function Y,Y,, where the ship dynamics
open loop Y, is defined from the linearized equations, this method
is mostly avoided if an analogue computer is available. In case of
small value non-linearities — such as dead zones or lags — in gyro
compass and telemotors the equations~of-motion technique is un-
avoidable.

Much effort has been devoted to present the function of a
manual helmsman in terms of a transfer function. The helmsman
is a highly adaptive control system, which makes the task more
difficult, but which also makes it more important. In many cases it
is impossible to run real-time simulations because of lack of time,
in other cases it is impossible to run comparative simulations just
because of the learning ability of the operator.

Hooft tried to evaluate criteria for manual steering of large
tankers by use of a transfer function, in which the gain and time
constants were derived by extrapolation from high frequency pilot
dynamics, [69]. Undoubtedly new basic information is required.

Propeller Thrust and Shaft Torque

A majority of ocean-going ships are propelled by fixed-blade

screws driven by diesel engines or steam turbines, the normal

steady state outputs of which in principle are characterized by
constant torque Q5& proportional to fuel pump stroke — and constant
power — piaportional: to steam inlet pressure — respectively.
In running conditions the mechanical torque losses QF depend on

sign of r.p.m. but are more or less independent of its magnitude.
Shaft r.p.m. is governed by the simultaneous equations for longi-
tudinal resistance and for thrust and torque, for u>O here given as

861

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eT" =! - dr"? + Tun +L 37") [nin tL + at

eu D el -(/2 +1/2 EW
(Q, - Qin=Lg-Q etl 'g - OF

er ae me (7.6)

] -l
tls 2Qhiu +L Qy,un + 2Qihin [n|n + 20a

3
sy FeV «8 (i-2 wir 1K etc.

The steady-state hydrodynamic thrust and torque are given
as functions of forward speed, wu, and rate ae revolutions, n, based
on open water K. and K, characteristics; and K, are first
approximated by square functions of J = apie -w) or 4/J.
(Note that a linearization of these characteristics does not result
in a linearization of the (u,n)-dependence.) The Nordstrom data
[70] may be used when reversing or transient maneuvers are con-
sidered. In general it is then necessary to confine the analytical
functions to limited ranges of propeller advance coefficients, i.e.
to use alternative coefficients as in Eq. (7.2). Harvald has presented
useful information on the propulsive factors at arbitrary steady-state
advance conditions, [71]. The effects of separating boundary layer
flow along the stern of a retarding ship are still less predictable.

The added mass and moment of inertia involved in unsteady
maneuvering of the propeller are functions of the momentaneous
advance coefficients as well as of the rate of change of r.p.m. In
small changes from normal propulsive conditions the added inertia
is small as blade angles of attack are small. Naval architects often
use a value of 30 per cent of rigid screw inertia for the added inertia;
although this figure originates from model tests with screws oscil-
lating at zero advance coefficient it may still be used as an effective
average value during the short reversing stage of an engine maneuver.
In fact this stage is dominated by the large control torques and by the
way they are used.

When simulating maneuvers with diesel-powered ships it
shall be observed that normal r.p.m. control is not possible for n
less than some 35 - 40 per cent of design shaft speed ny. The torque
delivered is here rapidly reduced, mainly due to loss of charge air
pressure. (For high r.p.m. Q; is almost zero.) Slow speed
maneuvering must be performed by intermittent use of the propeller,
which requires repeated starting of the engine. Reversing maneuvers
must await drop of speed to some 60 per cent of the full speed value,
at which lower speed braking air may be applied. There is alsoa

862

Shtp Maneuvering tn Deep and Confined Waters

certain astern r.p.m. which must be attained before fuel may be
injected to start engine back. For a discussion of detailed features
of diesel maneuvering the reader is referred to a paper by Ritterhoff,
[ 72]%

The energy-converting efficiency of a turbine wheel has a
maximum of some 80 per cent at a certain ratio of blade velocity to
nozzle steam velocity, attainable at the design point. Assuming this
ratio equal to 0.5, and a parabolic curve of efficiency symmetric
to the design point, the following simple formula is obtained for
the torque output:

Q’ = 2xQk (1 - 2° n/M ) (7.7)

Here o- and n, refer to torque and shaft speed at design conditions
for full steam inlet xk = 1. The formula furnishes a good approximation
also for present multi-staged ship turbines. In practical applica-
tions to studies of slow-speed port approach maneuvering it must be
realized’ that steam production may then be limited to say kK=0.7.

VIII. MODELLING THE DEEP-WATER HORIZONTAL MANEUVER
The General Case

The ship will be regarded,as a rigid body moving under the_,
influence of the gravity force mg and the buoyancy force -p* Vo°g
— where Vo is the volume displacement at rest — as well as under
that of the external forces, including the control forces applied by
use of rudders and thrusters. Before reducing the problem to the
normal merchant ship case the more general form of the rigid body
dynamics will be included.

The centres of mass (G) and buoyancy (B) may be off-set
from the origin of the moving system (0), and it is then practical
to apply Newton's laws in a summation of the acceleration forces
on the mass elements (cf. (4.10) and (4. 4)):

dm 0 0 ay x m-pV, 0 0 0
0 dm 0 ay =a ly er 0 m-pV, 0 A JO
Only Or-idm. | ba Z 0 0 m-pVo g

Z jabs

863

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es abs

K : —VoraG Po Za) “YG. Plane y

M+] mzg-pVoZe 0 -(mxg- pV, Xp) A 0

N -(my,-PV yg) m™mx,-PV>Xg ) g
ey

Upon summation the coefficient matrices of the acceleration
terms, the mass and inertia tensors, expose as

me 0 0
m = 0 Myy 0 = dm
0 0 my
(8. 2)
Ax “Ly “Ly
Paepins tao aipeph ara % Sidm
“1. “I, L,
where the elements are defined by
)x.dm s,m x, > ty? + 22 dm = Iyy ), xy dm =1,y
Yydm=m+y, >) (2*+x)dm=ly Yyzdm=l, (8. 3)

yy z2dm=m-z, >) (x? + y*)dm = 1,, )) 2x dm = 1,

Many authors prefer to introduce the virtual masses and
moments of inertia into the equations given above. Here the "added"
masses will consistently be assigned to the hydrodynamic reaction
forces in the right-hand members; in Section Vit was seen that these
forces may include other inertia terms otherwise easily overlooked.

864

Ship Maneuvering in Deep and Confined Waters

In most practical applications the xz-plane is a plane of
symmetry, so that yg= yg=0 and I,yy=0. Except in a few special
cases, such as when dealing with hydrofoil crafts, etc. — the dis-
cussion of which is outside the scope of this paper — other terms
may be safely ignored in view of the smallness of the products of
inertia and the perturbation velocities involved.

The Merchant Type Displacement Ship

In what follows the discussion is restricted to displacement
ships, for which m= pV, and V*\V,. Forward speed is always
associated with a sinkage and change of trim, most obvious as
"squatting" in waters of finite depth, but the manoeuvring dynamics
will be sufficiently well described by the equations in four degrees
of freedom, i.e. the surge, sway, roll and yaw. Then

mfu - rv - ea + zorp{= x
my + ru +xgr - zp b= ¥ re

LP - lee . mz_(v + ru) = K - mg(Z, - Z5) sin >
ep mx,(v + ru) =N

Whereas the initial roll as well as the steady outward heel
may be appreciable in case of say a highspeed destroyer these
angles are also known to be quite insignificant in the tanker case.

In steady turning a heel, proportional to - (L/Re) * ie , may produce
an effective camber of the waterline flow around a fine hull, but this
hardly applies to merchant ship forms.

Leaving the roll equation the present deep-water model is
given as in Eq. (8.5). It shall be pointed out that the derivative Yy,
includes the potential-flow contribution Xj} and the derivative Nj,
the potential-flow contribution Y;' . In the forward speed equation
X\, is given a value that is smaller than its ideal value equal to - Yi

Ones, cae o eee ee 4
(4 - Xiu ie = Xqyt ae ee jot 55 Siu 2 = joy a LL | - t)
#(L+X™ yh L(x tox yb +L gt Ext ulylv?
vr G jams A Y & 6 uw
=| 4 (2
+L | Xess [el cd,

865

Norrbtn

F me ; “V2, <2. . 4 24
(i VAS Ey Styy st tes tap Ae Z Yuu tb
lon 3/2 -1/2 Ae Ss hee? =| 8S Seon
1 XN ay fide af a aN el = Ye eoltany
1 " rite [ey " ‘ W ‘
Ce rir Wie + YN, lv[ety vy
-| a 1 "l W
+L + Zig lelede + ket
i a : : 5 ‘ -3/2 -I2 :
(ko, FN )p = LU (NS - xt)v +L (NE - x8up+L og ee fe rat
-2 “8/2 -V2... A nn 12 ee |
PDN ay? ie 5g Nu 2S ay Ivlv
1 " Foulleg = A tt ‘ -I S u ‘
#5 Nie IEP PEO Nie tele EEO Ne fa
Fy 2 Nt. leleoue is tom"
2 lelcS e ys
(8.5)

Eq. (8.5) is to be combined with Eqs. (7.3), (7.4) and (7.6). In
case of twin-screw ships (7.6) is to be properly modified and terms
corresponding to sp°* Tp and s,° Ts are to be introduced in (32:5).

Some Elementary Concepts

So far as small motions are considered forward speed and
r.p.m. remain almost constant and the rudder force and moment
may be regarded as functions of nominal helm 6. The yaw-rate/helm
relation is given by the transfer function

eee | 1+ T3s

——— (8.6)
v5 et es | eee

and the open loop heading response by Y, = (1/s) - Yys which may
be used with Y, from Eq. (7.5) to study the closed-loop system
with transfer function F = Y,/(1 + Y,Y,).

The static gain and the three time constants in (8.6) are built
up from the coefficient of Eq. (8.5). T3 is always positive. The
two constants T, and T, are given by the roots of the characteristic
equation. If s, = - YT) 5 the root to the right on the real axis,
turns positive the ship is inherently unstable. The analytical criterion
for dynamic stability suggests the dynamic stability lever

866

Shtp Maneuvertng tn Deep and Confined Waters

qe sqws= xg - ue ee Nuv (8 7)
ai as ea cio mgs oh

uv

to be a suitable measure for the degree of stability. In particular it
provides a good illustration when studying the effects onto the stability
characteristics of changes in the stability derivatives.

Most modern large tankers are slightly unstable,or marginal
stable, i.e. 1, =1,. For such ships the pivoting point position is
given by the simple relation

OP 1-y"
ss ees. 7 ola eee)

uv

which may be approximated by OP/L = 0.45 + (1/3)(6,,(B/T) - 2).
For a typical tanker this corresponds to OP/L =0.5. (The formula
in fact indicates an acceptable value also for the destroyer, about
0.3.) Again, the pivoting point position — or the drift angle B —

is a critical parameter to study when entering shallow waters.

IX. CONFINED WATER FLOW PHENOMENA AND SOME RESULTS
FROM THEORY

Mostly on Resistance

In his notes for a third volume of "Hydrodynamics in Ship
Design" Saunders collected a number of citations, ranging from
Scott- Russel to Moody, which all illustrate the classical picture of
ship behaviour in confined waters as it has been derived from obser-
vations in full scale and in model tests, [| 73]. He also concluded that,
by 1960, the ventures and progresses made in analytical studies of
ship manoeuvring in shallow waters remained scarce. One exception
was offered by the papers by Brard, [74]. The problems of inter-
action between meeting or passing ships, or between ships travelling
abreast — closely related to the bank effect problem of the single
ship — had been dealt with by Weinblum [ 75], Havelock [76], and
Silverstein [ 77].

Undoubtedly much more effort had by then been devoted to
the changes of frictional and wave resistance of ships in axial motion
in confined waters, and an important survey and contribution had
been given by Schuster [78].

Ocean-going ships generally move at low speeds in shallow
or narrow waterways, and hence the deformation of the wave system
is small. According to Schuster the wave resistance is not notably
affected by a limited depth for speeds below Ean = 0.7, at which
speed the excentricity of the orbital ellipse corresponds to a diameter

867

Norrbin

difference of about 5 per cent. In case of a bottom depth of 15m
this again corresponds to a ship speed of 16 knots.

In Ref. [79] Weinblum demonstrated that the wavemaking
in a canal is a complicated function of speed, depth and width. In
general it is therefore not possible to define a single effective
length to characterize the canal dimensions in a speed number.
However, effective canal speed includes the back-flow, and just as
a critical speed in shallow water is defined by the speed of the
solitary wave, Vgh, experimental evidence advocates a critical
speed in a certain canal corresponding to a certain Boussinesque
number B= Fy,y(h/W) +1. (Here W is equal to half the mean
width of the section.) For a rectangular section Muller proved that
the maximum wave resistance occurred at Fp, = (2(h/W) + 1)-V2
[80]. In a canal 15 m deep and 120 m wide this corresponds to
Fy, = 0.81. Again, let it be assumed that a significant change of the
wave resistance due to the confinements will be found only at a speed
equal to or higher than 70 per cent of this critical speed: this now
gives a speed of about 13 knots, much to high to be experienced in
canal transits involving normal blockage ratios. It may be concluded
that the additional resistance terms to appear in the speed equation
normally need not to account for the oscillatory wave-making com-
ponents.

Reference shall here be given to recent studies of the un-
steady flow conditions existing within a critical speed range fora
ship in a canal; this range tends to zero when the width of the canal
tends to infinity, [81, 82].

At sub-critical speeds the wave-making itself may influence
the lateral force and moment on a ship moving along a bank, as
shown by Silverstein, [77]. In case of the low Froude numbers met
with in practice also these effects may probably be ignored, and the
water surface may thus be treated as a solid wall. At F,, = 0.078
or F,, = 0.32, realized for a 98 000 tdw tanker proceeding ata
speed of 14 km/h through the Suez canal, the longitudinal waves will
have. a,length.of some:,10.m,, i,.e.; only,4 per cent of the length ofthe
ship.

The back-flow producing an increase of frictional resistance
will also produce an increase of sinkage, and in case of small bed
clearances this will of course indirectly affect the lateral forces
sensitive to the clearance. These secondary effects must be born
in mind when comparing predictions from theory with results from
force measurements on models, which are free to heave and trim.
In the normal evaluation and presentation of such measurements,
however, it will be considered more practical always to use the
nominal under-keel clearance.

The viscous resistance, including frictional as well as
viscous pressure resistance, may be calculated accepting a plate

868

Ship Maneuvering tin Deep and Confined Waters

friction line and a form factor, characteristic for the super-
velocities arene the hull. This resistance now may be written

[ X"lwehza= 2 Xyyu*, where u is the forward speed of the ship. In
confined waters there are additional supervelocities, the effect of
which is equivalent to a back-flow along the hull and waterway
bottom, where another boundary layer is generated. The two bound-
ary layers will reduce the effective under-keel clearance, which
tend to increase the trim by stern. Separation and unsymmetrical
eddy-making within the boundary layers may initiate yawing ten-
dencies in straight running, or change the behaviour of the ship in
manoeuvres.

Graff has suggested to consider part of the mean back flow,
AUp, to be due to the lateral restriction, and the other part, AU,;
to be due to the finite depth, [83]. In normal applications AU is
small compared to u, so that

XY = Exyur(1 + SOUP ys + SOU) = xt + KY + Ky) (9.4)

The effects of a plane bottom at distance h below the ship
waterline and a pair of parallel vertical walls, each one at distance
W from the ship centreline, are those produced by an infinite array
of image bodies with spacings equal to 2h and 2W respectively.

At the double-body ship centreline the lateral perturbation velocities
cancel whereas the axial components add together. (This simple
concept is not valid for W or h small comparedto B or T, in
which case additional doublet distributions are required to prevent

a deformation of the body contour.) Graff choose to calculate an
approximate value of K, for an elliptic cylinder, extending from
the surface down to the bottom and having a beam given by the three-
dimensional form displacement. (Thus K, is dependent on canal
depth, although the final calculation is manely two-dimensional.)

For the calculation of K, he used an equivalent spheroid and results
for supervelocities Sanies published by Kirch, [84]. His final
results are given in graphs and compared with model measurements,
which confirm that this method offers acceptable values of resistance
allowances for moderate confinements. It is thereby also possible
to define a suitable form of resistance derivatives to be evaluated
from model experiments from case to case.

In particular, a limited re-analysis of some of the data
given by Graff indicates that the resistance increase in shallow
water will be proportional to the increase of an under-keel clearance
parameter € = T/(h-T). Further analysis of the results for sinkage
in shallow water according to Tuck's theory are likewise in favour
of the use of this parameter. (See below.)

In waterways severely restricted in width as well as in depth
the increase of resistance is a complex function of blockage conditions.

869

Norrbtin

From model tests with a Rhine vessel [85] it appears that the added
resistance at a given forward speed may be approximated by an
expression of the form AR=a-+ (BT/Wh) tb- (BT/Wh- ¢, or,
roughly,

AX(u;& n) = 5 Xyuts OF +. Xwurtar 67 (9.2)

where 1/2 = L/W is a bank spacing parameter defined from the
mean width 2W of the canal cross section. (See Section X.)

The higher resistance in confined waterways is associated
with a lower propeller efficiency, and the total propulsive efficiency
is further reduced by an increase of the thrust deduction. The
influence of flow restrictions on thrust deduction and wake factors
has also been considered in a paper by Graff, [| 86].In most simu-
lator applications this letter influence may be ignored. However,
the computed values of r.p.m. and speed attained at a given engine
setting should be compared with, say, diagrams compiled by
Sjostrom, [87].

Sinkage and Lateral Forces

Within the last decade the application of slender-body theory
has furnished new understanding and quantitative estimates to the old
experience on sinkage and lateral motions in confined waters.
Further developments of the theories and more accurate measure-
ments are required to bridge a gap still remaining in force pre-
dictions.

In an essentially forward motion of the ship in shallow water
the back-flow is increased all round the frame sections, and according
to the first-order theory of Tuck the dynamic pressure is largely
constant in the water around a cross section of the hull and over the
bottom bed close below it, [88]. Upon assumption of a water depth
of same order as the draught, the draught and beam being small
compared to the length of ship and waves, and by use of the new
technique of "matched asymptotic expansions" Tuck derived formulae
for the vertical forces and so also for the sinkage and trim at sub-
and super-critical speeds.

In case of ships with fore-and-aft symmetry the theory pre-
dicts zero trim for subcritical speeds, and zero sinkage for super-
critical speeds. For small to moderate Froude numbers based on
depth the sinkage varies as speed squared,and, using the under-keel
clearance parameter defined here, according to the upper curve of
Big 22).

870

Shtp Maneuvering in Deep and Confined Waters

Infinite width

0.05 04 0.2 0.5 1.0 2-0 5.0

2-0
(flow
af
1.5
A o2.4 6231
12 1+ 78 e-(1-F Ee)
1 Finite width
1-05
1-02
0.2 05 1-0 2-0
wee Af 2b pqs? 12
aw RAT = UF)

Fig. 22. Sinkage in shallow water of infinite and finite width,
recalculated from Tuck's results.

871

Norrbtin

In Ref. [89] Tuck has extended the theory to canals of finite
width, in which the ratio of sinkage into the water (or trim) in the
canal to the sinkage (or trim) in shallow water is given by a unique
curve on basis of a simple width-and-speed parameter. Replotting
this curve as in the lower diagram of Fig. 22 Tuck's results are
shown to yield a square dependence on the bank-spacing parameter
7 when FA, << 1.

In canals presenting higher blockage the total sinkage or
"squat" is dominated by the contribution from water level lowering
as a consequence of flow continuity. From the Bernoullie and conti-
nuity equations an approximate relation for the hydrostatic ship
sinkage in terms of ship lengths is given by

A 2
Tey x° Fa
Q A L Wp ; 2

Here g and aw are the prismatic and waterline-area coefficients
of the ship. Other methods of the practical calculation of squat are
discussed in Ref. [ 90].

At low speeds wave making is concentrated to bow and stern
of the ship, where changes of the local velocities do not influence the
blockage conditions, and it shall be possible to calculate the forces
on the ship without regard to wave making. The absolute speed still
is a parameter, as it is seen to affect the hydraulic as well as the
dynamic squat in a canal.

Kan and Hanaoka first presented low-aspect-ratio wing
results for the calculation of transverse forces and moments ona
ship in oblique or turning motions in shallow water , [91]. As the
theory predicts the same correction factor to be applied to all deep-
water values it seems to be essentially a two-dimensional theory as
it is in deep water. Newman studied the same problem by use of
the method of matched asymptotic expansions and by the assumption
of a three-dimensional flow, differently orientated close to the body
and close to the bottom (and upper image wall), [92]. His results
bear out the effects of finite length, most obvious in case of moments
due to yaw acceleration.

Newman considers the inner flow to be a two-dimensional
cross-flow of reduced velocity, at each section depending ona
blockage parameter in the velocity potential. The outer solution
assumes flow to take place in planes parallel to the bottom wall at
nominal transverse velocity as the body is reduced to a cut normal
to the flow, this being physically similar to the flow past a porous
plate. The results as applied to forces on a wing of low aspect
ratio (or to a ship) are given in a simple diagram in [92], and here

872

Shtp Maneuvering in Deep and Confined Waters

they are used for comparisons with ship model values in Figs. 33
and 34. (A limited comparison of sway force and moment derivatives
derived for the SSPA tanker model was included in [92]. A small
adjustment of model force derivative appears in the present compari-
son, due to modified assumptions for non-linear viscous cross-flow
contribution; cf. Section X.)

The lateral forces acting on a body of revolution in axial
motion in close presence of a vertical wall have also been studied
by Newman, [93]. The source distribution inside the body is
mirrored in the wall, and in addition the calculations require the
original distribution to be off-set towards the wall. This three-
dimensional source distribution defines the velocity potential and
so the forces may be found by use of the Lagally theorem. As
expected from experience and approximate image theories for
bodies not close to the wall there is an attraction towards the wall,
increasing monotonically up to a finite value of body-and-wall con-
tact. It is concluded that for geometrically related bodies with
same sectional-area distribution the suction force will be inversely
proportional to the length, whereas the yawing moment will be inde-
pendent of length variation. The results also indicate that there
will be a bow-away-from-wall moment for bodies with a stern, which
is blunt compared to the bow, and vice versa.

In Fig. 23 calculations by Newman's method are compared
with the results of force measurements on a tanker model towed
along the vertical wall of a ship model basin. (Cf. Section X,)
Basin depth was equal to 0.29 Lp», total basin width equal to
2.7+* Lj. The diagram is plotted on ratio of wall distance to
maximum radius of equivalent body of revolution, defined by length
and displacement of model hull t image. The better agreement is
obtained for that equivalent body, which also has the same sectional
area curve, but even then the experimental results are some 25 per
cent in excess of the prediction. At larger separations the differ -
ence is still larger. Comparative calculations using Silversteins
"not-too-near-wall" results for an equivalent ovoid [77], are
included in the diagram; in this case the prediction is better for
larger separations, but in all much too high.

As long as the body is not too close to wall contact the
Newman theory gives a linear dependence for the lateral force on
ratio of body radius to centre-line wall distance, i.e. it is propor-
tional to nN, or 1p, defined for starboard or port wall distances in
next Section. This linear dependence suggests that the lateral force
on the ship between two parallel vertical walls may be obtained by
adding the effects from each one, which idea may also be supported
by the new presentation of old DTMB data [94, 95] given in Fig. 24.
The diagram includes force and moment measurements on a twin-
screw tanker model in several canal sections of simple rectangular
form.

873

Norrbin

0.5
=
u"2 "' Equivalent '' ovoid
( Silverstein theory )
0.2
0.1
0.05
Tanker model
(Spots from tests
: at Fri= 0.078 )
Equivalent spheroid
(Newman theory )
0.02
e Body of revolution
with tanker hull + image
section area curve
(Newman theory )
0.01 FF
1 2 5 10 15
-Yp/ Tome

Fig. 23. Lateral force on a body moving parallel to a vertical wall.
Measurements on propelled tanker model and theoretical
results for bodies of revolution.

874

Shtp Maneuvering tn Deep and Confined Waters

6 uN \
< ‘ 1.0 se
~~ SS
S ‘

xX ~~
. L=720.6 feet ate & L=720-6 feet
\

a © 2W= 268 feet © 2W= 268 feet

rN 500

‘SY 4 500
ae N ia) 770

0 Fa eat. 10 0 5 nn, 10

Fig. 24. Asymmetric forces and-moments on a twin-screw tanker
model moving parallel to the vertical walls of canals of
differing widths and depths (From DTMB test data)

The theoretical results for bow-away-from-wall moments
are somewhat modified in practice, where bow wave and screw
action contribute to make the tendency felt in ships of all types.
Thus, in general a ship that moves off the centre-line of a canal
must use helm towards the near wall and it takes up a small bow-
from-wall equilibrium angle. Typical values derived from [ 94]
for the 721' tanker off-set 50' from centre-line of the 500' X 45'
section are 15° helm and 1° drift.

The motion in shallow port approaches may involve much
larger drift angles, and the behaviour of the ship is markedly
affected by the increase of lateral cross flow resistance due to
under-keel blockage. The diagram in Fig. 25 is compiled from
shallow water test data in Ref. [ 49], and from Japanese data in
Ref. [ 96], which also include measurements in presence of a wall.
Again the parameters ¢ and n are used for the presentation. For
moderate € cross-flow drag increases in proportion to €, just as
the linear force derivatives, butthe dependence on 1 is of higher
order, The cross-coupling between § and ) may probably be
ignored in practical applications.

875

Norrbtin

AC,
(Co)jan20

—O— Tujizal. ,/96/

——-=—= US Navy Tests, /49/

nle-j

Fig. 25. Ship model cross flow drag coefficients as influenced by
change of depth and presence of vertical wall

X. FORMAL REPRESENTATION OF CONFINEMENT EFFECTS

Waterway Description

The uniform straight canal with a rectangular section is the
most simple case of a waterway confined in depth and width, but
even there several parameters are required to characterize the
flow phenomena taking place. It was seen in the last Section that
the wall distance parameter n and the under-keel clearance param-
eter € both were useful tools for the description of certain effects.
Their first merit, of course, is due to the zero values defined in
unrestricted deep water.

Figs. 26 and 27 show a more general section of a canal.
Such a canal is usually described by its mean depth between the bed
lines, its widths at bed and beach lines, and its cross section area,
related to the midship section of a transiting ship by the blockage
ratio. The position of the ship in the canal is mostly given by the
off-centre distance, and by the angle to the canal centre-line. Here
approximate expressions involving the new parameters only will
be given for the main geometric characteristics.

876

Shtp Maneuvering in Deep and Confined Waters

/
Vap w
I
Esa
5
on &
tx
4
Wp.—yo Ws-yo

Yo

Fig. 26. Ship moving parallel to walls in a straight canal

The depth h is considered constant between the bed lines.
The mean width 2W is defined as the quotient between cross
section area A, and depth h. The ratio 2W/B is a better param-
eter for width-to-beam relations the more shallow is the canal.
For use with theoretical results for thin ships the width parameter
will here not be related to beam but to ship length L.

As seen from Fig. 27 the bank and ship positions may be
given by coordinates normal to a datum line essentially parallel to
the main direction of the canal. The orientation of the ship is given
by the heading angle J, measured from the same datum line. The
basic geometric parameters are defined as

Under-keel clearance parameter 6 = T/(h - T)
Port bank distance parameter Np = L/(Wp - Yo) (10.1)
St'bd bank distance parameter Ns = L/(W, - Yo)

877

Norrbin

—
Datum Line

Fig. 27. Ship moving in a canal of slowly changing form

Note that W. > : Pa Wes so that 7,>0 and TN) < 0. Itis also con-
venient to introduce

1 = Ns + Np dom

n= Ms - Np

878

Shtp Maneuvering tin Deep and Confined Waters

The mean width of the canal at the station considered is

= _ e n :
ZW = Wy - We = aap i (10.3)

where the parameter ratio in the right member is constant for all
lateral positions of the ship in the cross section. Expecially, when
ship on centre-line so that 1, = - Np = "/2, there is L/2W = 7/4.
In fact 7/4 alone is an acceptable approximation to the "ship
length-to-canal width" parameter ratio also at ship positions slightly
off-set from the centre-line: with yo = W/4 7/4 over-estimates the
ratio L/2W =7,/4 with less than 7 per cent. As a consequence }
and € may be used to define an approximate blockage ratio

BT 1 B 7 10
=—_- ant —_— _ °? L 4
2Wh Zan ae ae oa 4 (

For small ¢ the blockage ratio is proportional to 7,
for large € to nm alone.

Force Representation

The asymmetric forces appearing in presence of a single wall
or in a canal are highly increased by an increase of the under-keel
clearance parameter, and the general model will include complex
couplings. If a single canal depth is studied on basis of special
model tests it is of course possible to express the wall effect forces
in terms of Np and 1, only. Although the geometry of the inflow
to the propeller may be modified in confined water it is assumed that
the control derivatives remain unchanged and that changes of rudder
forces are due to changes of screw loading only.

When suitable theoretical and experimental information
becomes available it shall be possible to include the effects of ship
motions towards the wall and of the angular orientation along it. At
present solutions to the problem of motions oblique to a wall seem
to be known only for elementary singularities such as circular
cylinders and spheres, [97]. In particular, these results give a re-
pulsion by the wall on the body moving toward or away from it, but,
again of course, an attraction on the body moving parallel to it.

For the present investigation it shall be assumed that the
effects of the walls on a ship moving not to close to them will be
approximated by the quasi-steady asymmetry, and that the added
masses may be taken as those derived for low-frequency oscillations
in the centre of the confinement.

In the previous Section was shown that the attraction force

879

Norrbtin

on the ship in motion parallel to a wall is essentially inversely pro-
portional to the separating distance, i.e. Y(u,n,) = Zz uinee u"N">
and that the effects of two walls may be approximated by super-
position. Thus for Y(u,Ns»Np)

2
2 tuapel tT t 2 Yuunp Np = 2 Yyugh (10. 5)

where Yuyun= Yuuns= Yuunp*

As the ship moves closer to one of the walls, or as the walls
are closer, this expression shall be completed by terms in ng and
Tio | Tipe or alternatively, in nn.

The effect of a limited bottom depth is included by additional
terms in n& and nn&. The forces due to steady sway and yaw are
assumed to be increased in proportion to uvn and urn, andto uvnt
and urnG respectively. The dependence of, added inertias on the,
confinements are represented by terms in vt and vn rt and mt,
all so far evaluated from the results published by Fujino, [5].

XI. MODEL TESTS

Test Program and Model

Five years ago an experiment program was designed for a
tanker model with a view to put to test the analytical model set up
as well as to obtain basic simulator data for a first canal transit study.
Full scale measurements should subsequently be made with the
98 000 tdw prototype in the Suez Canal, but these plans could not be
fullfilled, of course.

In November 1965 a first series of three component force
measurements were ordered to be run witha 1:70 scale model at the
VBD Laboratories in Duisburg. The test program included straight-
line oblique towing of the propelled model in "deep" and shallow water
in the large Shallow Water Tank and rotating arm tests at same depths
in the Manoeuvring Tank. It also included straight-line oblique towing
of the same propelled model in two Suez-Canal-type sections witha
water depth equal to that of the shallow water tests. Most of these
tests were run at self-propulsion point of model, determined from
straight course speed runs in the waterways studied. All tests were
performed at maximum "Suez draught." Resistance and propulsion
tests had earlier been completed at SSPA with a 1:35 scale model on
several draughts, and ship speed trials were analyzed to support the
prediction of full scale screw loadings and control derivatives on
model test draught. (Cf. Section VII.)

880

Ship Maneuvering tn Deep and Confined Waters

Additional tests in the VBD Shallow Water Tank were ordered
in April 1966 to establish near-to-wall stiffness derivatives from
straight-line motion close to one of the vertical basin walls in "deep"
water. After a series of repeated tests with a modified recording
system the full captive model test program was completed in April
1967. The test data are included in reports [98] and tables from
VED.

The test program is condensed in Table IV. It shall be ob-
served that no acceleration derivatives could be obtained from these
tests. Most of the force measurements were made at a model speed
of 0.465 m/s corresponding to a ship speed of 7.6 knots or 14 km/h.

The ship prototype was a single-screw/single- rudder turbin
tanker of the Kockums 90 000 - 100 000 tdw series, delivered to the
owners in October 1965 for use in the crude oil trade through the
Suez Canal on a reduced draught. The main dimensions of ship and
1:70 scale model are given in Table V, and the body plan is shown in
Fig. 28. The prototype has a Mariner-type rudder, normal bow
and no bilge keels; a few tests were run to investigate the effects
of a bulbous bow and of bilge keels of common design.

Wl

Fig. 28. Model of 98 000 tdw tanker -- body plan and profiles.
Model tested on "Suez draught"

881

Norrbtn

DIZ

Il
peues

II-SD

ofc 00 oS¢F
o9F 00 o9F
G6°L s 0
: 7S-0F-9S 0- :
8t°¢ = Ee 30
z 62°0 5
Le Ge BESe LE SE
099} i 0086
O0EZ2 Ta 0086
02 OT = 0086
zi 00S =
FITZ PTC PZ
I I3}JeM IojeM
Teue) MOTTEYUS MAOTTIEUS
ie Si Sv MS

(WI G9T°O = L ‘Ut FI9°E = 44-7) [ePOW Teyxue], LOZ wrerzso01rg isa

OSOt

Ire“
0} Te3N

MN

00 oS¢F
00 o9F
: 0

¥S “07-9S. 0- =
= Le 0
62 °0
64 °O 67°O
= 0086
= 0086
5 0086
00St2 =
OSOt OSsOT
IayeM IoyeM
oo1q ay |
AVY MA

Q fo osuey
mh 20 ¢g jo osuey
|°u zo Su oxey

U/T = ,t fo oduey

SUOTJETISA IOJOWUIeIeYg

M2/T = $/%U
“/
(Z— O)A=

MZ ‘UIPTA Yue y
WPed An SUIPT Yue,

POU M ‘IPT Yue L,

Is ‘loJoWwe{Ip utseg

y ‘yydap rt903eM
:SUOTSUSTUIIP [Opopy

uolyeustsep
AVM IZIEM

SsoTias

“AI ATAVL

882

Shtp Maneuvering tn Deep and Confined Waters

TABLE V
Ship 1:70 Model
Length, Lop =e m 253,00 3.614
Beam, B m 38.94 0.556
Design draught, Tow. m he) Oatyc
Suez draught (38'), T m 14.558 0.165
Displacement, Suez draught, V m°> 91 933 0.2680
Slenderness ratio, Suez draught, - 5.606
"8
Midship coefficient, Suez draught, - 0.991
p
CB forward of L,,/2, x¢@/Lop +0.0185
Long. radius of gyration/Lpp - 0.23
Propeller diameter, D m fk Ge Oe HOal
Pitch ratio, P/D = 0.74
Area ratio, Ae ~ O65
Number of blades, z - 5
Rudder area, total, A, m* 64.8 0.01382
Horn area/A, - 0.182
Relative rudder area, A, json t Z 0.0221
Height at stock, b m ete 19) 0.140

883

Norrbin

Results for Force and Moment Coefficients

Figs. 29 - 32 show plot of force and moment coefficients
from tests in deep (free) and shallow water, and the analytical
approximation obtained by stepwise regression analysis. Results
from the near-to-wall experiments have been given in Fig. 23.

In the evaluation it was consistently assumed that the changes
of first and second order derivatives due to finite depth could be
approximated by terms proportional to ¢. As the tests did include
two values of & only, one of which very small, this does not effect
the derivatives derived for these two depth conditions, nor the "true-
deep-water" values.

Further, because of the scatter of experimental data it proved
suitable to perform the analysis with an assumed value for the deep-
water cross-flow resistance corresponding to Cp = 0.7; cf. Section VI.

In agreement with earlier findings the test results indicate a
very marked influence of shallow water on the non-linear force contri-
butions, and on the lateral force due to yaw in particular. It shall be
observed that the analysis involves a change of sign in the first-order
rotary force derivative as water depth is reduced.

The force and moment derivatives derived from shallow water
and canal tests will be presented in next Section.

Fig. 29. 98 000 tdw tanker — Force coefficient ¥"(B) Ja"? in deep
and shallow water.

884

Shtp Maneuvering tn Deep and Confined Waters

12

Fig. 30. 98 000 tdw tanker — Moment coefficient N"(B)/u in

deep and shallow water.

“ Note: Rigid mass included

Tests at Fry = 0-078 - 0.108

Fig. 31. 98 000 tdw tanker— Total force coefficient Y"(r') /u"®

in deep and shallow water.

885

Norrbtn

02 Tests at F nL = 9:078 - 0.108

Wigs 32. ye 000 tdw tanker — Moment coefficient N(x!) fa?
in deep and shallow water.

XII. RESULTS FOR CONFINEMENT DERIVATIVES

Figs. 33 - 38 present available empirical or semi-empirical
results for the main lateral hydrodynamic derivatives appearing in
the confinement terms of the completed mathematical model. The
derivatives Yyyy and Nyy, for the 98 000 tdw tanker have been
derived from the near-to-wall tests in deep water, and are not shown
here. (Cf. Fig. 23.)

In Fig. 33 the shallow water results obtained for the SSPA
tanker are compared with the experimental results published by
Fujino [5], and with calculations from Newman's theory,[93]. The
SSPA analysis is based on a linear dependence of the derivatives on
C and the results are given by plots on straight lines, also sug-
gested by the theory. Fujino's derivatives are evaluated separately
for each depth. In general the theory seems to underestimate the
influence of finite depth, especially for the stiffness moment.

Increase of added mass and added moment of inertia as

obtained from Fujino's experiments and Newman's theory is shown
in Fig. 34, again in poor agreement.

886

Ship Maneuvering in Deep and Confined Waters

5 § , 5 —O— SSPA/VBD Tanker
BY / OY ii
uv | uf as T
TYavij=0 | / Yor )fa0 ujino Tanker

7 & Fujino “Mariner”

(Black spots for higher
speeds)

Fig. 33. Stability derivatives as influenced by finite depth -- results
for SSPA tanker compared with Fujino tests and Newman
theory.

887

Norrbin

AYo
{ Yu)j=0

O . Fujino Tanker
& Fujino ‘Mariner’

( Black spots for higher
speeds)

<4

_
-—
=_—_
_
_
=—_—_—

c
_Newman_T heory

Fig. 34. Increase of linear and rotary acceleration derivatives with
increase of parameter € according to Fujino tests and
Newman theory.

Je)
7
8 vA
(Yur),
urs}
(urls $20 yy iat
/
7
7
7
red Fa = 0-0675
7
7,
ae A
pore a
a Z p= 2-00
ye ie PA !
Pig oO
Le
Say x ee "
~ : ae ee
ei Be ai Beer bain

‘
al

Fig. 35. Rotary force derivative for tanker as a function of waterway
depth and width, replotted from Fujino PMM data

888

Ship Maneuvering in Deep and Confined Waters

Fig. 36. Rotary moment derivative for tanker as a function of
waterway depth and width, replotted from Fujino PMM data.

Fig. 37. Rel. change of lateral acceleration force derivative fora
tanker as a function of waterway depth and width, replotted
from Fujino PMM data.

889

Norrbin

y, Fr = 0-0675

Fig. 38. Rel. change of rotary acceleration moment derivative
for a tanker as a function of waterway depth and width,
replotted from Fujino PMM data.

The diagrams in Fig. 35 - 38 are compiled from Fujino's
measurements of rotary and acceleration derivatives in shallow
waters and in canals. The dotted curves suggest a linear increase
of all these derivatives with ¢€ in unrestricted water, and a more
complex dependence of ¢ and y inacanal. (Cf. end of Section X.)

XIII. SOME ASPECTS OF SHIP BEHAVIOUR IN CONFINED WATERS

Here a few comments will be given on some of the results
obtained in a computer and simulator study performed for the
98 000 tdw tanker. The diagrams in Figs. 39 - 45 all include results
directly drawn on the analogue computer recorder.

The only full scale maneuvering trials with the prototype
ship so far available are a 20°/20° zig-zag test and a Dieudonné
spiral, both run at full speed on full draught. These results are
compared with the computer predictions — or hindcasts — for the
ship on Suez draught in Figs. 39 and 40. As the difference in draught
is not likely to have a significant influence the agreement is quite
good. It shall be observed that the derivatives with respect to

890

Shtp Maneuvering tn Deep and Confined Waters

Full scale /

(7 =13,45m) /

(T =11,58m)

0 1S min

10

Fig. 39. 98 000 tdw tanker zig-zag test in deep water. Comparison
of full scale trials and computer prediction.

v|r| and |v|r are not derived from measurements with this model
but taken from an analysis of rotating arm tests with another tanker
form, and the almost exact prediction of overswing angles might be
somewhat accidental.

The good correlation of speed loss in the zig-zag maneuver
is satisfying. The phase difference is likely to be due to the stern
position of the ship's pressure-type speed log.

The ship (and simulator model) is slightly unstable on
straight course in deep water; the total loop width is about 3.5°
at slow speed as well as at high. In Fig. 41 is also shown the
spiral prediction for shallow water (¢ = 3.37 or h/T =1.3). Here
the initial stability is further impaired, whereas the stability ina
turn is increased. A major factor governing the dependence of
initial stability on water depth is the change of Yy,-- From Fig. 33
was seen that Yure is negative for this model, so that the value of
lf = (xg - Ni - Nut c)/i - Yi, - ure 6) may diminish much faster
with increasing © than does 1} = (Ny + Nuvt Ae Yuve aa

891

Norrbin

Fig. 40. 98 000 tdw tanker — ,(5,)-diagram from spiral tests in
deep water, Comparison of full scale trials (x,o) and
computer prediction ( ).

892

Shtp Maneuvering in Deep and Confined Waters

Fig. 41. 98 000 tdw tanker — Low speed spiral diagram from
computer predictions for deep and shallow water

A similar trend is not unique, but it shall be observed that it may
be necessary to include higher order derivatives in ¢ to account
for a finite range of "dangerous depth" as defined by Fujino, [5].

Figure 42 shows predictions for 20° rudder step responses
in change of heading, yaw rate, and drift angle. The small drift
angle obtained in the shallow water case is associated with the large
increase of cross-flow drag. Similar results have earlier been
reported by Schmidt-Stiebitz, [99].

From simulator and full scale experience is known that the
helmsman may have some difficulties of controlling the ship in

maneuvers that involve a change of course in shallow waters.
Maneuvers by use of auto pilots are repeatable and well suited for

893

Norrbtin

Y
¥% ¥° ~*
Q3 3pi2
Y
0,2 20} 8
Pp
0) 104 qo

Fig. 42. 98 000 tdw tanker — Computer predictions of 20" rudder
step response in deep and shallow water. Approach speed

7.6 knots.

Approach Speed 16Knots

-5
Auto Pilot Constants:Rudder Ratio 3
Rate Constant135 s

Approach Speed 7,6 Knots

Fig. 43. 98 009 tdw tanker — Computer predictions of 10° course
change manoeuvres by use of auto pilot knob setting.
Two speeds in deep water.

894

Ship Maneuvering in Deep and Confined Waters

v% Approach Speed 7,6 Knots f 23,367
05
10
0,10
gost *
0 0
14min
-G@os -s5
6 Auto Pilot Constants: Rudder Ratio
_g Rate Constant 135s
Yi-8°
1S
Y $=0
Wy. 0 Approach Speed 7,6 Knots
IS
0,10
0,05 .
0 0
14 min
-0,

Fig. 44. 98 000 tdw tanker — Computer predictions of 10° course
change maneuvers by use of auto pilot knob setting.
Shallow and deep water.

es Re aati Vas
In Deep Water (} =0) ye

between Parallel Walls (7,=8,71)

Goin en ee

In Shallow Water ( | =3,37,7)=0)

In Deep and Wide
Water (}=0, 7] =0)

6%: 10¥+135¥* Q0175ye

Datum Line

S*210¥ + 135Y+0,0175ye

8°P10'¥ +135 40,0175 yo

Datum Line (€)

In Canal (} =3 37, 4=8,71)

Fig. 45. 98 000 tdw tanker — Computer predictions of on-track
control by auto pilots in deep and confined waters.
Approach speed 7.6 knots. Initial off-set 20 m to starboard.

895

Norrbin

comparative studies. The diagrams in Figs. 43 and 44 refer to 10°
course change maneuvers predicted for the tanker at two speeds in
deep water and at the lower speed in shallow water, all executed
using the same normal setting of auto pilot controls. There are
several overswings in shallow water, and checking helm is large.

The final diagrams in Fig. 45 furnish a condensed illustra-
tion to the changing problems of course control in shallow waters
and in canals. These problems are also dealt with by Eda and
Savitsky [100], and in considerable detail by Fujino, [101].

It is assumed that the ship is moving at low speed on a
straight course parallel to the required track (in a buoyed channel,
say) but off-set 20 m to the starboard side. A signal proportional
to this lateral error, calling for 1 degree rudder per m off track,
is fed into the auto pilot. The upper curves for the free water con-
ditions demonstrate that the rudder ratio setting must be increased
(from normal 3 to say 10) in order to stabilize the ship on the re-
quired track. This control works reasonably satisfactory also in
shallow water, but it tends to make the ship over-shoot the centre-
line in the alternative case between parallel walls in deep water.
Obviously the presence of the near wall accelerates the first swing
towards and beyond the centre-line.

The two lower curves of Fig. 45 relate to the ship ina
typical part of the Suez canal. In the shallow water the effect of the
near wall is even more pronounced, and the stern of the ship is in
danger of hitting the bank. However, by turning down the lateral
error knob to zero the auto pilot is made to behave like the ex-
perienced helmsman, already referred to in the introduction. Thus,
the ship first sheers bow-off the wall before the auto pilot applies a
counter-rudder in order to slowly press the ship laterally away
from the wall. The ship is seen to be almost steady on to the centre-
line within two ship lengths.

REFERENCES

1. Norrbin, N. H., and Goransson, S., "The SSPA Steering and
Maneuvering Simulator," (in Swedish), SSPA Allm. Rapport
No. 28, April 1969.

2. Mandel, Ph., "Ship Maneuvering and Control," in "Principles
of Naval Architecture" (revised), New York 1967.

3. van Berlekom, W., "Summary of a Simulator Study of Maneuver-
ing Large Tankers in the Approach to a Port at Brofjorden,"
SSPA PM No. 1643-10, Sept. 1969. (Contract Report)

896

fi.

12.

13.

14.

£5.

16,

a7.

Shtp Maneuvering tin Deep and Confined Waters

Bohlin, A., "Problems Arising from the Use of Very Large Ships
in Connection with the Alignment and Depth of Approach
Channels and of Maneuvering Areas," Paper S. II-3,

Proc. XXII©® Congrés International de Navigation, Paris 1969.

Fujino, M., "Experimental Studies on Ship Maneuverability in
Restricted Waters," Part I, Intern. Shipb. Progr., Vol.
15, No. 168, Aug. 1968.

Norrbin, N. H., and Simonsson, E. Y., "Observations onboard
a 60 000 t.dw. Tanker Maneuvering through the Suez Canal,"
unpublished papers, SSPA and Uddevalla Shipyard, Nov. 1964.

ITTC, "Recommended Symbols Classified by Subject," Appendix
II of Presentation Committee Report to the 10th ITTC,
London 1963.

SNAME, "Nomenclature for Treating the Motion of a Submerged
Body Through a Fluid," SNAME Technical and Research
Bulletin No. 1 - 5, New York 1950.

Norrbin, N. H., "A Study of Course Keeping and Maneuvering
Performance," SSPA Publ. No. 45, Goteborg 1960. (Also
Publ. in DTMB Report 1461, Washington D. C. 1960.)

Glauert, H., "A Non-Dimensional Form of the Stability Equations
of an Aeroplane," ARC R & M No. 1093, London 1927.

Kirchhoff, G., "Ueber die Bewegung eines Rotationskorpers ineiner
Flussigkeit, " Crelle, Vol. LXXI, Heft 237, 1869.

Lamb, H., "Hydrodynamics,"
Cambridge 1932 (6th ed.).

Cambridge University Press,

Milne-Thomson, L. M., "Theoretical Hydrodynamics,"
Mac Millan, London 1955 (3rd ed.).

Imlay, F. H., "The Complete Expressions for Added Mass of a
Rigid Body Moving in an Ideal Fluid," DTMB Report 1528,
July 1961.

Lamb, H., "The Inertia Coefficients of an Ellipsoid Moving in
Fluid," ARC R& MNo. 623, 1918.

Kotschin, N. J., Kibel, I. A., and Rose, N. W.,
"Theoretische Hydromechanik," Bd. I (from Russian),
Akademiee«Verlag, Berlin 1954.

Munk, M., "The Aerodynamic Forces on Airship Hulls,"
NACA Report No. 184, 1923.

897

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21.

22.

eS

24.

25.

26.

27.

28.

29.

30.

31.

Norrbtin

Norrbin, N. H., "Forces in Oblique Towing of a Modei of a
Cargo Liner and a Divided Double-Body Geosim," SSPA
Publ. No. 57, 1965.

Weinblum, G., "On Hydrodynamic Masses," DTMB Report
809, April 1952.

Grim, O., "Die Hydrodynamischen Krafte beim Rollversuch,"
Schiffstechnik, 3. Bd., 14/15 Heft, Febr. 1956.

Landweber, L., and de Macagno, M. C., "Added Mass of Two-
Dimensional Forms Oscillating in a Free Surface," J. Ship
Research, Vol. 1, No. 3, Nov. 1957.

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de Jong, B., "Berekening van de hydrodynamische coefficienten
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Shtp Maneuvertng in Deep and Confined Waters

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Havelock, T., "The Resistance of a Submerged Cylinder in
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Motora, S., "On the Measurement of Added Mass and Added
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Motora, S., Fujino, M., Sugiura, M., and Sugita, M.,
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Jones, R. T., "Properties of Low-Aspect-Ratio Pointed Wings
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Eda, H., and Crane, C. L. Jr., "Steering Characteristics
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Jacobs, W. R., "Estimation of Stability Derivatives and
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Newman, J. N., "Some Hydrodynamic Aspects of Ship Maneuver-
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Brard, R., "A Vortex Theory for the Maneuvering Ship with
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Nomoto, K., and Karaanog K., "A New Procedure of
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Nomoto, K., "Unusual Scale Effect on Maneuverabilities of
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Bottomley, G. H., "Maneuvering of Ships, Parts II-IV," Trans.
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Ambjorn, G.; "Sldpforsok med fartygsmodeller i sned och tvar
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Thieme, H., "Schleppversuche bei Queranstromung," Schiff
und Hafen, 6. Jahrg., Heft 6, 1954.

SNAME, "Notes on Ship Controllability "SNAME Technical and
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Crane, C. L., Jr., "Studies of Ship Maneuvering — Response
to Propeller and Rudder Actions," Proc. Ship Control
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Abkowitz, M. A., "Lectures on Ship Hydrodynamics — Steering
and Maneuverability," HyA Report No. Hy-5, May 1964.

Lindgren, H., "Ship Trial Analysis and Model Correlation
Factors," SSPA Publ. No. 54, 1963.

Saunders, H., "Hydrodynamics in Ship Design," Vol. II,
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Motora, S., and Koyama, T., "Some Aspects of Automatic
Steering of Ships," Japan Shipb. & Marine Engng., July
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Nomoto, K., "Analysis of Kempf's Standard Manoeuvre Test
and Proposed Steering Quality Indices," First Symposium
on Ship Maneuverability, Washington D.C., DTMB Report
1461, Oct. 1960.

Nordén, I., "Fartyg med tva roder," Gotaverken Journal
Skeppsbyggaren, No. 4, 1960.

Norrbin, N. H., "Steuern bei geringer Fahrt," Hansa, 101.
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Ribner, H. S., "Propellers in Yaw," NAGA Report No. 820,
1945.

Ditlev Jorgensen, H., and Prohaska, C. W., "Wind Resistance,"

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12

Ship Maneuvertng in Deep and Confined Waters

Wereldsma, R., "Experimental Determination of Thrust
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Mirels, H., "Lift Effectiveness of Slender Wings on Cylindrical
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Yazaki, A., and Yokoo, K., "On the Roughness Allowance and
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Str¢ém-Tejsen, J., and Chislett, M. S., "A Model Testing
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Nordstrom, H. F., "Screw Propeller Characteristics,"
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Harvald, S. A., "Wake and Thrust Deduction at Extreme Pro-
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Ue

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904

THE SECOND-ORDER THEORY FOR NONSINUSOIDAL
OSCILLATIONS OF A CYLINDER IN A FREE SURFACE

Choung Mook Lee
Naval Ship Research and Development Center
Washington,: D.C:

ABSTRACT

A nonlinear hydrodynamic response resulting from
vertical oscillation of a horizontal cylinder in a free
surface at the sum of two monochromatic frequencies
is investigated. The fluid surrounding the cylinder is
assumed incompressible, its motion irrotational and
its depth infinite.

It is shown for the case of a semi-submerged circular
cylinder that when the two frequencies are close to
each other the hydrodynamic force associated with the
difference of the two frequencies is greater than the
steady force. Inthe limit as the two frequencies
become equal the above two forces also become equal.
It therefore appears reasonable to include the differ-
ence-frequency force in the calculation of the maximum
steady force when the excitation of a body consists of
narrow-band frequencies.

I. INTRODUCTION

The hydrodynamic problem dealing with a horizontal cylinder
undergoing a vertical simple harmonic motion in a free surface has
been investigated by many authors. Ursell [ 1949] treated a semi-
circular cylinder using the method of multipole expansion and ob-
tained the pressure distribution, added mass, and damping of the
cylinder. Later Tasai [1959] and Porter [1960] extended Ursell's
work to cylinders of ship-like sections using conformal mapping.
Frank [ 1967] dealt with the foregoing problem by the Green's func-
tion which resulted in a distribution of singularities. Lee [ 1968] ,
following Porter's work, extended the potential solution to second-

905

Lee

order in a perturbation series in the ratio of motion amplitude to
half-beam. In the present work this cylinder-oscillation problem is
extended to the case where the cylinder is oscillated at the sum of
two monochromatic frequencies. In this case the second-order
forces acting upon the cylinder include the effects of interactions
between two frequencies, in particular, the sum and difference fre-
quencies of the basic spectrum. The magnitudes of these second-
order hydrodynamic quantities provide a measure of the non-
linearity of the frequency response of an inviscid incompressible
fluid to a periodic disturbance generated by an oscillating body in a
free surface.

Hydrodynamic quantities such as added mass and damping
obtained from the theory of oscillating cylinders in a free surface
with a monochromatic frequency are extensively used in the studies
of ship motions. Most of these studies are based on the assumption
of linear frequency response of ships to waves. Recently Tasai
[1969] and Grim [1969] emphasized the necessity of further investi-
gation on nonlinear ship responses to waves. The present investiga-
tion is an attempt to provide information on the nonlinear relation
between the motions of a body and surrounding fluid. This informa-
tion might lead to the study of nonlinear ship motions in waves,
perhaps by using the scheme suggested by Hasselmann [ 1966].

The problem to be investigated in this work is the following.
An infinitely long horizontal cylinder which is symmetric about its
vertical axis is semi-submerged and forced to oscillate vertically
at the sum of two monochromatic frequencies. The maximum dis-
placement of the cylinder from its mean position is assumed to be
small compared to the half-beam of the cylinder. The fluid in which
the cylinder oscillates is assumed inviscid, incompressible, and
infinitely deep. The motion is assumed to have existed for a period
significantly long that the initial transient phenomenon of the response
of the fluid has completely decayed. This problem can be formulated
as a boundary-value problem for a velocity potential. The kinematic
and dynamic conditions to be satisfied on the free surface are non-
linear and the position of the free surface is a priori unknown. An
exact solution of this problem in a closed form cannot be attained,
so an approximate solution based on a linearization of the problem
is pursued in this work. The linearization of the problem is carried
out by a perturbation expansion of the velocity potential in terms of
a small parameter formed by the ratio of the half-beam to a typical
displacement amplitude of the cylinder motion. The first-order
perturbation potential consists of two potentials, $,(x,y,t) and ,»
each of which involves only one of the two fundamental frequencies.
The second-order perturbation potential consists of five potentials.
Two of them are $, and 4, which are associated respectively with
frequencies of twice the fundamental frequencies. Two more are
@, and $, which are associated respectively with the sum and the
difference of the fundamental frequencies, and the last one, (x,y) 5
is independent of the frequencies and is a steady potential. The

906

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

solutions for the first-order potentials were given by Ursell [1949],
Tassai [1959], Porter [1960], and Frank [1967] among others.

The solutions for three of the second-order potentials, $3, $,, and

$,, were given by Parassis [1966] and Lee [1966]. Inthe present
investigation, the solutions for the remaining second-order potentials,
$5 and ¢¢, will be given.

These solutions are based on the method of multipole expan-
sions similar to that employed by Lee [1966]. An interesting prob-
lem arising from the present work is a surface-wave problem con-
cerning a non-decaying pressure distribution on the free surface.
The solution of this pressure-distribution problem is shown in detail
in Appendix C. The potential $,, associated with the difference fre-
quency, is of particular significance in practical problems. $¢¢ is
a potential which is slowly-varying in time if two fundamental fre-
quencies are close. In the case of bodies with insignificant restoring
forces, such as submersibles and floating platforms, any hydrody-
namic force, which is constant in time or varies slowly with time,
could cause large excursions from the mean positions of such bodies
if it acts for alongtime. 4, must be calculated in order to deter-
mine this slowly-varying hydrodynamic force.

In the present work numerical results obtained from the solu-
tion of ¢, are shown. These include the pressure-distribution about
a semi-submerged circular cylinder, the hydrodynamic force acting
on it, and the outgoing waves. These results are shown with other
first- and second-order quantities for comparison purposes.

II FORMULATION OF THE PROBLEM

A Cartesian coordinate system is used with origin at the
interaction of the undisturbed free surface and the vertical line of
symmetry of the cylinder. The x-axis is in the undisturbed free

surface and the y-axis is directed upward.

Any point in the space is described in complex notation by
z=xtiy= rel9, (1)

The region outside the cylinder and the cylinder boundary is mapped
from the region outside a circle inthe €-plane and its circumference
by the conformal transformation

00

Z, _ -(2n+1)

ae ae: (2)
n=O

t=Etin=re™, 21, (3)

907

Lee

where a and Bone are real constants.

Points on the surface of the cylinder at its mean position are
given by

= Ben Skene A)
x a} ro cos gee 3 y) Xene ,

(4)

és a sin (2n+1)@
Yor a}hgsin a - s eT Sa ?
n=O 8

where i, is the radius of the reference circle. When the cylinder
is at the rest position its half breadth and draft are given respectively

by
b = x,(A,, 0); (5)

aeelyiikly=n/2)| + (6)

The forced motion of the body is assumed to be the sum of
two vertical simple harmonic motions with different frequencies.
The motion of a point fixed in the body is expressed by

y(t) = h,(sin o,t + sin ot) (7)

where o, is greater than o, and h, represents the amplitude of
the individual simple harmonic motions.

The fluid is assumed to be incompressible and its motion to
be irrotational so the continuity of mass in terms of velocity potential,
@(x,y,t) is expressed by

2 92
(soe + gon) ® = V°B = 0. (8)

The boundaries of the fluid are the free surface which extends
to infinity along both the positive and negative x-axes, the fluid
bottom which is at infinity, and the immersed surface of the cylinder.

If we let the equation of the free surface be expressed by

y= Vixsth, |x| > b, (9)

908

Nonstnusotdal Oscillattons of a Cylinder in a Free Surface

the kinematic and dynamic boundary conditions on the free surface
can be given respectively by

(x, Y(x,t),t) ¥, (x,t) - a! + ¥Y, = 0, (10)
and
&, (x, ¥(x,t),t) + g¥ +3(O +B) = 0, (11)

where a constant atmospheric pressure and an absence of surface
tension on the free surface have been assumed. Taking the substantial
derivative of Eq. (11) and eliminating Y(x,t) by using Eq. (10),

we obtain

+ O25, + 20,6,6,, + O56,=0. (12)

Let the equation of the cylinder surface at its rest position
be given by

S(x9,Yo) = f(x) - yo = 0 (13)

where f(x,) represents an implicit functional relation between x,
and y, through the parameters X, and a. Then the equation of the
oscillating surface can be written as

S(x,,y, + y(t)) = £(x,) f h(sin ot + sin ot) - y= 0. (14)
The kinematic condition to be satisfied on the cylinder surface is

VO(x,,y, + y(t).t) > n= Vn=- st (15)

where n is the unit normal vector on the cylinder surface and points
into the fluid and Vn is the normal component of the cylinder-
surface velocity. Since

eae VS = CE ,-1)
2 Nei: - del S :

Eq. (15) becomes

909

® (xo,y, + y(t), t)f'(x,) - o, mir ho, cos ot + a, cos o,t). (16)

In terms of the stream function which is the harmonic conjugate of
@ the boundary condition on the body is

ay _ 8 _ dy(t) dx
op EL Ee

where s_ is the arc length of the cylinder contour in the counter-
clockwise direction. Thus we have

Weagsy, + y(t) t) = - x, SO |

To complete the specification of the boundary-value problem,
following conditions should be also given; the symmetry of the flow
about the y-axis implies that

@(x,y,t) = O(-x,y,t),

the zero normal component of the fluid velocity on a rigid surface
at the infinitely deep horizontal bottom is described by

@,(x,-0, t) = 0,
and the solution should represent outgoing plane waves as | x | 7160.

Ill. PERTURBATION EXPANSION

Assume a frequency-response system in which the relation
between the input X andthe output Y is given by

YaiAx+ Bx?
where the input X is given by
ae ee

and A and B are constants; it can then be shown that the frequency
components involved in the output Y are o,, 05, 20,, 205, 0, +O;
vg, - 0, anda '"d.c." shift. Therefore, we make a perturbation
expansion of the complex velocity potential

910

Wonstnusotdal Osetllations of a Cylinder in a Free Surface

H(z st) = @(x,y,t) i i(x,y,t)

in terms of a perturbation parameter € = h,/b in the following

fashion:

= eH! zh <*H'?) rs eH?) fice

= ¢(o'" ig”

y+ (Bl?) + jy + eH) +) 400,

(17)

Each velocity potential and stream function given above is further

expanded as

a" = $(x,y,t) + o(x,y,t)

t

ot -jo
+ P(xsy)e 2 ’

=)
9 (x,y)e

g!?)

-j2o,t -j2o4t
= 9,(x,y)e + ye +¢@

-jlo,-o,)t
ae ie

t % + g(x,y)

y= Ulxry.t) + bplx,y,t)

-jo,t -jo5t
W(xye | + W(x,y)e iu,

Y= Ys(x,y.t) +y(x,yot) tU(x,y.t) + olx,yst) + H(x,y)

-\lo,+o5)t
5°

-j2o,t -j2o,t -j( )t
= W,(x,y)e ee We ieee Vie de
-j(o,- o)t
+ Ye mete + W(x, y)

etc., where

P= Gegt Jeg?

v

k

Ves + jw

$(x,y.t) + o(x,y.t) + o(x,. yt) + o(x,y.t) + d(x, y,t)

ks’

(17a)

(1 7b)

(1 7c)

(1 7d)

for k=1,2,..-,6, and j=v-1. In these expressions, only the real

911

Lee

parts are needed, so whenever there appears an expression of the
product of two complex functions one of which is a time harmonic
involving with j = v-1, it should be understood that the real part of
the expression is to be taken.

The convergence of these perturbation expansions will not be
discussed. As usual it is hoped that the first few terms of the ex-
pression would yield an adequate approximation to the exact solution
of the complex potential H(x,y,t).

The expansion given in Eqs.(17) and the Bernoulli equation (11)
suggest that we also assume the expansion

e€

]

-j2o\t -j@o5t
+ e[ ¥,(x)e + Y,e

Y(x,t) = €[ ¥, Eyer)" a

-jlo,+o,)t -j(o,;-o.5)9
rye | at Yee ee

. + Y(x)] + O(e%) (18)

where
Tp Soe oR) Oke for fe 1G 23 Se oe

Substituting these expansions into the Laplace equation and the boun-
dary conditions and equating the terms of the same order in € as
well as of the same harmonic time dependence, we obtain a set of
linear boundary-value problems. In this linearization process the
instantaneous boundary of the fluid is expanded in Taylor's series
about the undisturbed position of the fluid boundary.

The linearized boundary conditions for the functions 9j
(j = 1,2,3,4,7) are shown in Appendix A where it is shown that in the
limiting case of o, =o, the relations 9, = 92, 93= 94 = p,/2 and
Yg= %7 can be established. These identities mean that when o, =o
the perturbation expansion given in Eqs.(17) reduces to that for the
case of a simple harmonic oscillation which was investigated by
Lee [ 1968].

The linearized boundary conditions for the functions ?. and
Yg are given next.

3.1 The Boundary-Value Problem for Pa(x,y)

gy, is harmonic in y <0 except in the portion occupied by
the cylinder at its mean position. On the free surface

Oey +0) - Kyo, = h(x) (19)

912

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

whe re
K,= (co, +o,)"/g:
h(x) = - jo, /(28)19,(*, 09,
~ jo5/(28) {99 x.0)(% yy

in which

2
jel On Bh
ae

On the cylinder surface,

- K,9))) - 2(9), P25 + Pry Pay)3

= Ki%y) a 2(9,, Pox + PiyPay)s

Pia,

! =-
5 4(XoVo)E(X,) - M5) = Mp(X,rY,)

where

rele.
m,(X9, Yo) == Jey {(Pixy (Xo Yo) + Poxy)£ (XQ) ~Piyy - Poyy) ’

or, in terms of stream functions,

Pals,
We 5s Vo) ea J 2 (Wy (xy, Yo) a Wy)

(20)

(21)

(22)

(23)

In the far field 9,,—~ 0 as y~-o and 9g, should represent out-

going plane waves as

flow condition which is expressed by $5 (%»y) = %5(-X,yY)+

3.2 The Boundary-Value Problem for 9,g(x;y)
yg, is harmonic in y< 0 except in the portion occupied by the

cylinder at its mean position.

On the free surface

where

K, = (o, - o,) /g;

913

|x| ~ oo. Furthermore there is a symmetric-

(24)

Lee

h(x) = - jo, /(2g){9,(x,0)(9,,, - K9,) - 219%, + 9) Py}
3 5%/(28) {Pl Pyy r Ky) y 2(P1, Pox + P1yPoy) (25)

and the bar signs mean the complex conjugates, i.e.

1 =%ic-ji?is for i=1,2.
On the cylinder surface
Pex(Xo2 Vo) f (Xp) = Pey = M,¢(Xo+ Vo) (26)
where
og ID ee = ,
M65 mae 2 {Poy y(Xo, Yo) ms Piyy ay (Poxy a Pixy)f (x,)h (27)

or, in terms of the stream functions,

We(Xo, Yo) = j 2 (Why = Wey). (28)

In the far field ggy~ 0 as y~-o and 9 should represent out-
going waves as |x| —~ oo. The symmetric-flow condition implies
that

9_(x+y) = 9,(-x,y)-

IV. SOLUTIONS FOR os AND 96

It will be assumed that solutions for the first-order potentials
gy, and g, are known. The method of multipole expansions for
finding g, and 9, is described in Appendix B.

The main difference between the first- and second-order
problems is in the free-surface conditions. A first-order problem
has a homogeneous differential equation for the free-surface condi-
tion (see e.g. (A-1) of Appendix A) whereas a second-order problem
has an inhomogeneous one (see e.g. (19))- When there exists a non-
constant pressure distribution on a free surface of negligible surface
tension the first-order free-surface: condition for an incompressible
irrotational flow is represented by an inhomogeneous differential

914

Nonstnuostdal Osctllattons of a Cylinder in a Free Surface

equation such as Eq. (19) or (24). The term on the right-hand side
of the equation of the first-order free-surface condition represents
the pressure distribution on the free surface. Thus the problems
for the second-order potentials Ps, and 9, presented in Sections 3.1
and 3.2 are the same type of boundary-value problems as those for
the first-order potentials except for the "non-constant pressure" on
the free surface. If we assume there is no body in the fluid, then
these problems can be treated as problems for surface waves arising
from variable free-surface pressure distributions. Solutions for
these problems are given in Wehausen and Laitone [1960]. If we
denote the velocity potential associated with the problem of variable
pressure distribution on the free surface by W and if we assume
that it is known,we can use it to find the potentials gs and gg.

This is done by introducing a new function G=g- W, where 9
could be either g, or 9.,s0 that the free-surface condition for G
is given by a homogeneous equation such as Gy(x,0) - KG=0 where
K is either Ks or Kg. The boundary-value problem for G is
then identical to those for the first-order potentials and the solutions
to these are well known. Once G is known the solution for @9 is
readily obtained from g=Gt+twW. This scheme was used by Lee
[ 1968] to find the second-order potentials gy, and g,. However
there are certain requirements on the "free-surface pressure
functions," h, and hg given by Eqs. (20) and (25) respectively,
to be satisfied before the known methods can be used to
find the potential W. These requirements are that the functions
hs(x) and h,(x) should be absolutely integrable in (-00,00) and
should satisfy the Hdélder condition. Although the proof of these
statements may not seem obvious from Eqs. (20) and (25), it can be
ee that both h, and hg satisfy the Holder condition and as

x] —*

hy = O(1/24 (29)
and

i€(K,-Kp)Ixt - B}
e

yea + O(1 /x?) (30)
where a, and 6 are given by
Z
ay = — Q,9,K,K (0, - 02) (30a)
and
oo di - do- 1/2. (30b)

Here the quantities Q, and q, for k=1,2 are associated with the

915

Lee

first-order potentials and can be best described by the expressions
of asymptotic behavior of the first-order potentials such as

Kyy j(K,Ixl-q, )
data xe “e K kK for Keak 2 as |x| — o.
It is apparent from Eq. (30) that hg is not absolutely integrable.
This implies the necessity of further consideration in deriving the
solution of W which is associated with hy.

4.1 Solution for a Case Where Free-Surface Pressure Distribution
is Specified

In this section we consider a potential-flow problem with a
given pressure distribution on the free surface. We restrict our
attention to the pressure distributions which have harmonic time
dependence and are even in x. Furthermore we consider the two
special cases: the one where the pressure distribution decays in
the manner of 1/x? as |x| — o and the one where the pressure
distribution behaves like that for outgoing plane waves as |x| ~ o.
Let w(x,y,t) be a harmonic function defined in y <0 and with its
time dependence of the form

-iwt

w(x,y,t) = W(x,y)e

where W=W,+jW, and w is an angular frequency. The free-
surface boundary condition is

Wy(x,0) - KW = h(x) (31)

where K = w*/g and h is a known pressure distribution and is even
in x. We expect that the solution of W should represent outgoing
plane waves as |x| — oo and furthermore that Wy (x, -00) = 02 "Wie
seek solutions to this problem in two cases.

Case 1: h(x) = O(1 /x?) as |x| — o. (32)

The solution for this case is given in Wehausen and Laitone [ 1960]
in the form of a complex potential

te
F(z) W(x, y) + iW (x, y)

oe) : Hs ;
1/1 Ae n(é)e'Kl2-8) & (-iK(z-€)) dé + 2i ( hleve iK(z-€) dé

00 :
+ (j-i) ‘a Helens 5 dé. (33)

916

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

Here E, is the exponential integral defined by

00 et
E,(z) - =~ at for larg (z)| <7.

Case 2: h(x) = Pe ase O(4 /x?) as |x| — c (34)

where K' (# K) is a wave number and A is a complex constant
with A = A, + jAg.

Before trying to solve for W, we introduce a harmonic
function W,(x,y) which is even in x and satisfies

W), (x0) - KW, = AelN™!

W(x» -00) =
and
W, ~ Be as |x| OS

where B is a complex constant with B=B,+jB,. The solution
for W;, is found in Appendix C as

Ky <y! iK'z -iK'z
W, (x,y) =o en - j 2 2 Re;[ A) + eel (35)

Now we let
Wo = W - W, in y = 0,
It ean then be shown that
jKIxl

Wz yl » 9) - KW, = h(x) - Ae = h,(x)

where
h(x) = O(1 /x*) as |x |— oc

and that

Wa y(x» 00) = 0 and Wo(x,y) = W,(-x,y).

911

Lee

The solution to this problem is given by Eq. (33) as

F(z) W,(x,y) ra iW, (x,y)

ise) , 00
aa hye “© & (-iK(2-€)) dé + 2i | nigje “2 ae

wT

iT]

ee ae ~iK(z-€)
jaa). Been Paes (36)
-00

Thus Eqs. (35) and (36) finally give

We Ww, it Wo»
“AES aK ial jAr Re, [© 7& (iK'z) man AG aS oe eal
PKK ae 2 aaa

+ Re, [z)" h{éye KE (-iK(z-6)) dé + 2S hye ae

+ (j-i) ie neve Fag |. (37)

If we seek a solution to the problem described in Section 3.1
except the body-boundary condition, given by Fq. (21), we can ob-
tain it from Eq. (33) of Case 1 of this section as

F(z) = W,(x,y) + iW, (x,y)

00 .
= tf née Mee (-iK,(z- 6). det 2 9 le -iK (2-€) at
00 ‘
+ (j - i) in hleje ies a

in which we let h,(x) =0 in - b<x<b since the velocity potentials
are undefined in this line interval. In the same way, if we seeka
solution which satisfies all the conditions except the body-boundary
condition given by Eq. (26) in Section 3.2, we can obtain it from

Eq. (37) of Case 2 with the constant A replaced by ae!” (compare
Eq. (30) with (34)). If we express the solution in the form of a com-

plex potential and let K'=K, - K,, we find that

918

Nonsinusoidal Osctllattons of a Cylinder in a Free Surface

*
F,(z) ca W, (x,y) a iW, (x,y)
K'y i(k 'Ixl-B) | .,_ i K'Ix-B)
= ones e + ije

sean K7E (iK'z) , e7 suena
K'+K,

‘s ~iK(2-€) ;
+t h,(&)e éz- E (-iK,(z-§)) d& + ail hale" Mee -€) =

-iK(z-€)

00
+ (j- vf h,(é)e dé, (39)
-00

where + signs correspondto x5 0 and h(x) =0 in -b<x<pb.

4.2 Solutions for gy, and 9,

We will now show how to use the solutions obtained in the pre-
ceeding section to find Ps and Pee We introduce a new harmonic
function G, defined by

G(x, y) = 9. = W, (40)

in the domain of y <0 except for the portion occupied by the cylinder.
Here the subscript k can be either 5 or 6 unless specified as one or
the other. The boundary-value problems posed in Sections 3.1 and
3.2 can be written in terms of G, as

G,(x,0) - KG, = 0, (41)

Gy lx yr ¥o)EK) - Gy = MylK yoo) — (Wy lXqrVq)f",) - Wyy)s (42)

ky

, . *
or in terms of the harmonic conjugates of G,, denote by G, , the
above boundary condition can be written as

* ._b x
G,(x,, Yo) a! | wa (W(x, Yo) Me v,,) co We = Bo(x,> Yo) (42a)
and

Gel(x,+y,) = is (W, (x52 V9) 35) We = B,(x,y,)> (42b)

Furthermore Gy, is evenin x, Gyy 0 as y~- oo, and G,

919

Lee

should represent outgoing waves as |x| — Os

These boundary-value problems are almost identical to those
for the first-order velocity potentials. Thus we find them by the
same method used to find the first-order potentials which is described
in Appendix B. It is often called the "multipole-expansion method"
since the potential is expanded in an infinite series of poles, located
at the origin, of increasing order with unknown strengths. Each
pole satisfies Laplacian equation everywhere except at the origin,
the linear free-surface condition of the type ® (x, 0) -\KO@ = 0 and
the infinite-depth condition, and is evenin x. However, since each
pole vanishes as | x | — o the radiation condition of outgoing plane
waves is not satisfied. To circumvent this a source singularity
which has all these properties plus the property of outgoing waves
at | x = oo is added to the multipole-expansion series. The unknown
source and multipole strengths are found by satisfying the remaining
condition which is the boundary condition on the body. Specifically
we assume the solution for G, to be

ee)
G (x,y) = S (Dem * 5S pan) My m(X(X52) sy(A,a))e cK, (43)
m=O

Here b,, = Q, = unknown strength of a source at the origin, c,,=0,
vee -J cre OsUpee Area sels conte (44)
0

= a source of unit strength at the origin,

ioe)
where f indicates that a Cauchy principal value is to be used,

M cos 2ma K sin (2m - i)ea
km em k (2m - 1)xem

(2m+2n+1)a

(2n ti) a,- 27 sin
~ 2m +2n ag yemeenel

for m= i1 (45)

= multipoles of unit strength at the origin,

by, and c,, form=i1 are unknown multipole strengths, and q
represents unknown phase relations between the forced motion of the
body and the pulsating singularities at the origin. The expression
for the harmonic conjugate of G, is

co
G* (x,y) = » (by mn + 5C pan) Man (0 9 @) ry(\,@)) (46)
m=O

920

Nonsinusotdal Oscillations of a Cylinder in a Free Surface

where
00 DYae-
* K
M. = f <a dp + 4re ete eee (47)
ko p-K k
(@) k
* sin 2ma {<22 (2m - 1)a@
M = - +K,a
Tes ae ra
km x m k (2m - 1) m
00
? (2n t1)as,4 cos (2m pennee \ he = 42 a8)
fy ee +2n #14 pemecns

Our task is now to find the unknown coefficients b,, and c,, and
the phase relationship q, from the boundary conditions on the body
surface given in Eq. (42). Since the boundary conditions on the body
are simpler when written in terms of the stream functions G, than
in terms of the potential functions G,, we use the Eqs. (42a) and
(42b) to obtain these unknown quantities. Thus we find that

00
p * -J4_
» (by 53 I<um) Mim (x5+yp)e = B(x 5+ Yo) : (49)
m=O
Since the q,;'s are independent of the points (x,,y,) on the body, we

see by choosing an arbitrary point on the body, say (x5 (A2'),
y' (X5.@')) » that

eK ___ Bylo yo) (50)

= *
np (Dim + IC Km) Mum (ho2%')
m=O

Substitution of Eqs. (50) into (49) yields

00

= * B (x ’ ) ne i]
ee M,..(X,,@) - Kt or¥o! na (yn, = 0,
i km * IC Km) (Mm (Xo 2) B bet oy) snlkase

and use of the earlier definition of Dig = Q, and c,,= 0 inthis
equation yields

oO
b * By(X_s Yo) * '
D, (abt Hate) [Mint hora) = Saale yg}

oo | «PYo ;
= Bxl%o Yo) (Xs Yo) (5 e__sin pxo px¢ dp + jre ”? sin Kx) -
B(x)» yo) fe) p - Ky

921

Lee

per ain px vole alt Ripe
- wre ae dp - jme sin K,x,]- (51)
gone h Sopk

Since this equation is valid for all values of @ inthe interval

(- r/2, 0) onthe circle of radius Xo, in principle we can take an

infinite number of @'s to set up an infinite system of a linear alge-

braic equation which can then be solved for by 2 and Chmn/ Qs

In practice, the infinite series is truncated, so only a finite number

of these unknowns is sought; this finite number is equal to the number

of chosen @'s. The proof for the convergence of such a truncation

scheme was given by Ursell [1949]. Once the values of anumber N

of b, Q, and c, Q, (i.e. m=1,2,..-N) is known the values of
re) m

q, an , are readily obtained from Eq. (50) by
-!f -Im; A
q, = tan ieee , (52a)
Q.= |Al, (52b)
where

yeep ee ies 5.1) 4
y (3 ‘ oe Mim (Xo @')

Thus we finnaly can express the solutions of the velocity potentials

g(x,y) = Gix.y) + W,(x,y) for k= 5,6. (53)

V. PRESSURE, FORCE, AND WAVE

5.1 Pressure on the Cylinder Surface

If we expand P which denotes the pressures on the cylinder
in the same way as ® was expanded in Eq. (17), we find that

P(xo,yo + y(t),t) = €{ Pr (x9r¥e + y(t))e + Pie

> -j2o,t -j2o,t -jlo,+o,)t
we {Pie + Piye + Pye
+P -j(o, -o,)t +P fs i} + ole3)
vie vir'*07% | Y :

922

Nonsinusotdal Oscillattons of a Cylinder in a Free Surface

We expand P, (i=I,II,...,VII) in Taylor series about the mean
position of the cylinder (x,,y,) and substitute

y(t) = h,(sin ot + sin ot)

€b(sin ot + sin oot)

in this expansion. We rearrange the terms in powers of € andin
time harmonics to obtain

-jo,t -jo,t
P= € {Pi (xorye)e + pe

ot -j2o,t
e
4

-j2 -j(o,+0,)t
+ e%{pe +p geuhil =

tp

-j(o, -o,)t
oper |e Peale) (54)

From the Bernoulli equation,

2 2
P = -~ pOj(xo, yo + y(t),t) - Palys + y(t) - $ (& + y),

we eliminate the static pressure pgy,, expand the right-hand side in
accordance with Eq. (17) and equate terms which are of the same
power of € and of the same time harmonics. We then find the ex-
pressions for pj (i=1,2,...,7) in terms of velocity potentials 9,
and their derivatives. The expressions for these P;'s are

Pj jp (o) 9) (X95 Vo) - gb) for i=1,2 (55)
ee er PER Gt tee for (1220 (56)
Pigg PF Ve dOU Mie Yo vive aie a iy Os aa?

- \- J(o, + o)e Axo ¥,) +5 (Px Pox t P1yPry)

ue)
a
N

+2 (0, 9,, a5 5% oy )\ ? (57)
Pe =~ PySoy ~ oppelx ) +5 (Po, 4 Oye
6 JAG, 2PeXorVol 7 F N\PixPoy PiyPoy)
z 2 (7% + T2Pey yt i in

2
b 1 — = F

923

Lee

where the bar sign means the complex conjugate e.g.

Pox Prex~ IPogy°
If we let

6; =tan (Re, pj/Im, pj) for i=1,2,...,6,

we can express these pressures in the form

p, = jlp ler?! for L=ilads ek 9.40
or, in association with the time harmonics,

-jwjt

pje = Ip; | sin (wt + 6.) for i7s4l 2 ,..<57,0

where

iT]
—_
w
ies)

oj for i

Wj = (61)

5.2.) Vertical Hydrodynamic Force on the Body

The vertical hydrodynamic force acting upon the body is given
by

L(t)
Erste i Pi cos (njy) dl.
£

0

Here £,(t) is the instantaneous position of the point of intersection
between the bottom of the body and the y-axis, 2(t) the instantaneous
position of the point of contact of the body with the free surface, and
cos (n,y) the direction cosine of the unit normal vector on the body
surface inthe y-direction. The positive direction of the unit normal
vector is into the fluid, and the integral is taken along the cylinder
contour. Eq. (54) enables us to show that for the family of cylinders

924

Nonsinusoidal Oscillations of a Cylinder in a Free Surface

mapped according to Eq. (2)

b A :
-jo ft -jJo.t
F = 2! dxo[e{P\(x-y0)¢ ad + p,e 52 \
O

-j2o,t -j2o,t -j(o,+o5)t
+ lp. + pye + pee

-j(o,-o,)t
£ Pee J o, os + p7t| + O(e*) c (62)
If we let
-jo,t -jo5t
F=e(fe"' +f, °°? )
2 -j2o,t - j2ont -jlo,+o,)t
te (f,¢ 2g f,e a: f,€
+ fg +i) +O), (63)

then we find that
b

a)
= 2 | P(X (A522) s¥Q(hy-%))T(2) dw = for f= 142,20%,7 ~ (64)
-17/2

where the expressions for p,; are given in Eqs. (55) through (58)
and

2
en¢
Xo

00
T(@) = - a4}, sina + » ase eriic sin (2n +1) a}.
n=0
If we let
y, = tan” (Re; £, /tm, £) for irs Dale ness
we can show that

fi.

Il

jlfife eM"

lf, | sin (wt + y;) for =) de seg 0 (66)

925

Lee

where the w,'s are defined in Eq. (61).

5.3 Outgoing Waves at |x| = 0

Equation (11) shows that
(x,t) = - - {B xb, ¥(x,t) +t) +5 (&; + a5).

If we substitute the expansions given in Eqs. (17) and (18) into this
equation and equate the terms of the same order and time harmonic,
we find that

¥j(x) = j + i (x,0) for i= 1,2, (67)

1. Kj Lp 2 2 :
Vis = 5 1520) ¢1,2%-0) - > PiPiy - | (Vix +o) for i=1,2, (68)

; Tt
Ys(x) = j A? og(x,0) - ZIP(o,o2y + P21y)

1
=e Pitan” Tage (69)

0, -Cv 0 | 0. = =
Y,(x) = j 1 le,(x.0) ur re (PP oy + PP iy)

1 _ ie
a 2g (91, Pox + PiyPay! » (70)

2
¥ Ax) = - a (9;,(% 10) Pj, + PiyPiy - 2Ki9;Piy)- (74)

If we let |x|— o (or X}—~o for @=0 or - 1) only the
pulsating sources contribute non-vanishing values (see the expression
for the first-order potentials in Appendix B, and for gz and g, in
Lee [1968]). Thus we find that

00
-jqj e?¥ cos px Kiy
(x,0) ~-Q.e lim ( dp + jre cos K;x}}]__
g (x, 0) sl y. ip Sa ee i Jly=o

j(Kjlx!-qj)
Sb MaGher ie Uhitemik inlet ha x3 ice (72)
J i

where the Q;'s are the source strengths, the qj's the phase re-

9126

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

lationship between the forced motion and the pulsating singularities
at the origin, and

os
Bie
2

Kig2 = ot F for es er Ae

We can also show from Eqs. (43) through (45) by letting |x| — oo
(or Xo for @=0 or -rn) that

00
-)q; py é
G,(x,0) ~ - Qie ' lim (5 ee dp ane cos K;x}| =0
|x |—oo re) P - Kj va
j(KiIxl-q;)
= jmQje for 1=55,.0 (73)
where
2
K, = Aumar)s ua >
g

2
K. = (21> 52)
6 &

?

and then Eqs. (38) and (39) can be used to show that

00 ; st)
We(x.0)~ 5) gee er® dé, (74)
i{(K,-K)IxI-B} oo -é)
Wales 0)~ S85 eas ti made at, (05)

where ao and B are defined in Eqs. (30). The far-field behavior
of the derivatives of the functions g; (i=1,2,3,4), Gj (i= 5,6),
Ws, and Weg can be shown to differ from those exhibited in Eqs. (72)
through (75) by factors of the appropriate wave numbers Kj. If
these results are applied to Eqs. (67) through (71) and some mani-
pulations are carried out,we can show that as A > 0

BLE,
i

¥i (x) ~ F Qie for i= 1,2, (76)

927

Lee

2
ol l4kjlxl- aj, 2) 4. (7Q;K))

j(2KjIxl- 2q;)
eon eee as

00 rau
F 4K i Ixl-@;
peut h, (6) cos 4K,& dé | aad
g b +2 I

Vier
¥; , {*) z Q

for i= 27

where h (x) and h (x) are defined in Eq. (A-2) of Appendix A and

s/ - Im, (Sf, digalé) cos 4K,& dé)

9i,2 = tan

- Re; (Sy risel6) cos 4K;§& dé)

Bae j(Kgx!-q5)
Y,(x) Z Q.(c, a v,)e

2 i{(K,+K )Ix!-q,-a.}
pe Q,Q,0,0,(K, + Ke | athe

00 2.

~ Std | h,(6) cos KE at| ears (78)
b

where

6, = tan"

Ps es Im j (J, gl) cos K,& dé)
- Re; pass cos K,& dé)

BIL! j(K dx! - gg)
¥,(x) : Q, fo, - o,)e

2 i{(K)-K>)IxI-(q,- a2}

00 ae
Si ie, = OP) ie Bey caster ag | qilKelx! +86) (79)

where

0. = tan” aaa) Cm h(E) cos Kg dé)
6- tan jayteNAD pod ow. tuo Doiaw

-Re, (f, he(é) cos Keé dé)

¥5(x) aug OP

928

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

VII. NUMERICAL RESULTS

A semi-circular cylinder of unit radius (b= 1) is chosen for
sample calculations of the pressure distributions, hydrodynamic
forces, and out-going waves. The inputs for the calculations are the
values of the fundamental frequencies o, and o,. Three values of
g, are chosen such that the corresponding length of gravity waves
in deep water, , = 2mg/c,*, are equal to 2b, 10b, and 20b. For
each value of o, the values of the eo cne sy ge o> are chosen
so that the wave lengths obtained by Ao omen lie in the interval
of 4,<A,< 24, which is equivalent to Pcs g)< bos/g < bo ?/g. The
reason euch a ae cow range of oo is chosen Stems. from our practical
interests. When the forcing frequencies are close the difference
between any two frequencies is very small. Thus any hydrodynamic
force response associated with the difference-frequency within this
band can often be treated as a d.c. force in practical situations. It
is of interest to examine how significant the magnitude of the hydro-
dynamic force of difference-frequency is compared with the pure d.c.
force:

Numerical results are obtained here for the d.c. component
of hydrodynamic pressure, p_,, hydrodynamic force, f,, and for
ae quantities which are associated with the difference frequency,
Pe f , and Y, which represent outgoing waves at | = oo. The
quantities associated with the sum-frequency, p,, f,, and y,, are
not computed because these quantities can often be approximated
from the known values associated with the frequencies 20, or 20
when o, *o5. For comparison purposes the quantities p;, fj, an
y,; for i= 2 and 4 which are borrowed from Lee [1968] are also
shown with the quantities presently calculated.

The deep-water gravity wave length based on the difference
frequency is given by

= 21g = x 2/1
6 c.- o,) 1 + (r,/X,) - 2Vr.,/d,

where

aut and ee
o | T>

X—/, as afunction of ,/X, is shown in Fig. 1. This figure shows
the wave length corresponding to the difference-frequency o, - 7,

compared to the fundamental wave lengths \; and ho.

In the rest of the figures the abscissas are \ which is defined
as X= o/dy- For the values of X,= 2b, 10b, and 20b, the corre-

929

Lee

Fig. 1 Difference-frequency wave length
vs. fundamental-frequency wave length

sponding frequencies are respectively
ov, (= V2mg/i,)=Vag/b , mg/5b, and Vng/10b.
o, is obtained from X= d/d, = Cre hay as
co = ¥ 298/008). -

Thus we have

ur
°
Le J
Pa
iT]
ia)
ion
q
ine)
tH
a
=
lon
a

930

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

for ,=20b, o,=¥mg/(10b)).

In Table 1, the values of 6) = osb/g and 6,= (0, - o,) b/g are given
for = 1.0 (0.1) 2.0 at each given 2X, = 2b, 10b, and Z0b. Thus
the results in the subsequent figures can be referred to appropriate
individual dimensionless frequencies suchas 62, 54= 40%b/g = 465,
and 6¢= 2mg/d,(1 +1/X- 2V1/X once X, and X are given.

TABLE 1

5, and 5¢ versus i, at three different values at h, and 6,

h, = 2b and 6, = wm |,=10b and 6,=m/5 |\,=20b and 6, = 7/10

0.365 x 10°
0.136 x 10°?
0.477 x 10°?
0.950 x 1072
15t X40"
Lethe alo:
.276.X 107
341 x 10°!
.406 X 107!
.474 x 107!
.540 x 10°!

0.680 x 1073
0.238 X 10%
0,475-x 10
0-753. x 10:4
0.106 x 10°!
0.138 x 10°!
O70 X10"
0.203 x 10°!
0.237 X 107
0x270 < 107!

238 x 107
.475 X 10°!
153 6-10"!
. 106
. 138
.170
. 203
eit
. 270

1
BA
3
4
5
6
it
8
)
0

eee Ee FE NY DY DN Dd DPD
Soo ©. O Go oO 6 Oo © 6 So
© ,O..0-.0170 SC 2 CO © 16 ©

1
1

i.
i.
si
i
i;
1

nla
Es

ea © —& & © Ce, oS ©

In Figs. 2, 3, and 4 the maximum hydrodynamic pressures
at three points on the cylinder, 0= - 900, - 450, and - 5°, are
shown as functions of \ for \,=10b. The maximum pressures
Ip. and lp. are obtained from Eqs. (58) and (59). The pressures
are non-dimensionalized by pgb and are denoted with bar signs
e.g. De= |p,|/pgb. The values of p, and_p3 which are not shown
in these figures are respectively equal to pp and pg at A=1.0.
The maximum hydrodynamic forces fp (= lt] /2pgb?) ; hgh? epcend
£7 which are obtained from Eq. (64) are shown as functions of X in
Figs. 5, 6, and 7 for \, = 2b, 10b, and 20b, respectively. Figure 8
shows the phase angles y,, y,, and y, which are defined by Eq. (65)

931

Lee

18 20

1.0 1.2 1.4 1.6
First- and second-order pressures vs. \ = h,/h,

Rig. 2
at 0 =- 90° for dy, = 10b

1.8 20 Vn

1.0 1.6
Fig. 3 First- and second-order pressures vs. Ge No/ dy
at 0 = - 45° for \,= 10b

1.2

932

Nonsinusotdal Oscillations of a Cylinder in a Free Surface

d, = 10b

: 6 =-5 DEG

Fig. 4 First- and second-order pressures vs. X= Xo/hy
at @=- 5° for X,= 10b

for 4, = 10b. The radiating-wave amplitudes at |x| = oo are shown
in Fig. 9 for 4; = 10b. In this figure Y, is defined by Y, = |Y,|/b =
7™Q,0,/(bg) and Ne and Y,, are obtained in the following way. We
can show from Eq. (79) that

-j(o,-o.)¢
Ye(x)e ASieee)

oa A, cos 1K |x| - (o, * o,)t a de |

+A, cos} K,|x| - (0, - o,)t + o,f

- Yepcos }(K, - Ky |x| - (0, - o,)t - (a, - at
where

A, ar: (0, = T)Q, ’

933

Lee

LY
are
if
pdf
fe i
fi
AWN E
ANY

Fig. 5 First- and second-order forces vs. \ for X= 2b

Oh) tte
So
oa

=

|)
ae)
aur

>
x

1.0 12 1.4 1.6 1.8 2.0 X

Fig. 6 First- and second-order forces vs. X for , = 10b

934

Nonsinusotdal Oscillations of a Cylinder tn a Free Surface

>|

Fig. 7 First- and second-order forces vs. \ for A, = 20b

2 ad
Ap= e (o, - o,) | \ h (5) cos K,§ dét ,

2
T

Furthermore we can reduce the above expression to
-i(o,-o,)t = !
Y,(x)e Ye cos 1K, |x| - (o, ~ o,)t + cm
= Lan cos }(K, - K,) |x| = (oc o,)t - (q, - q,)
where

(a 2 1/2
Yeo= LA, tA; +2A,A, cos (a,+,)]

Q' = tan’! Az sin 0¢- A, sin gg |
$ A, cos 0,+ A, cos qd,

935

Lee

Fig. 8 Phase angles of first- and second-order forces

vs. X for Y= 10b

Fig. 9 First- and second-order wave amplitudes at
lx| = co ve. \% for. X= 10b

936

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

7)
w
rT)
oc
9
wi
a)

Fig. 10 Phase angles of first- and second-order waves
at |x| =00 vs. X for d, = 10b

We then define

Y Y

In Fig. 10 the phase angles gq, (see Eq. (76)), 6,', and q, - q>
are shown as functions of X for i, = 10b.

VII. DISCUSSION

If the forcing motion on a floating body has a very narrow
frequency band,most of the second-order hydrodynamic responses
occur in a frequency range of about twice the forcing frequencies.
However, two components of the second-order force are exceptions
to this case. One is the steady-state force and the other is the
force with frequency equal to the difference-frequency between a
pair of the frequencies in the narrow-band spectrum. If the value
of the difference-frequency, 0, - oj, is very small, the force which

937

Lee

is associated with the difference-frequency changes very slowly in
time and often may be treated as a pseudo-steady force. In fact we
can show that in the limiting case of o, = og that the difference-
frequency force reduces to the steady-state force or in our notation,
f,=f, when o, =o 9. It is shown in Appendix A that for o, =o5

P= Po» P3=%4=%/2, and 96=9-

Equations (55) through (57) show immediately that p, = pp, P3= Py =
p,/2, and p,=p-7- Substitution of these relations into Eq. (64) leads
to f,=f;, for o, = aie 4p eae in Figs. 5, 6, and 7 that as

Mar 4.0, ie, ups oe £1 and in Fig. 8 that yg -0/2 as
Na 1.0. Since ee 265 eee sin yg from Eq. (66) and f7 is nega-
tive in this case, we'see ‘that Y6lo,2 05 = 7 m/2 in order to maintain

the relation f, = f. for 0, = >-

For sufficiently small values of o, - 63, we can show that the
expression of the forcing motion becomes

_y(t)

ho

sin o,t + sin Tot

R

2 sin opt cos sto t (80)

and the corresponding expression for the hydrodynamic force can be
derived from Eqs. (63) and (66) as

F = e2|f,| cos ee t sin (opt if Y>)

a efalt,| cos (co, - o,)t sin (20,t + y4)

+ lf, sin y,cos (o, - o4)t Dy £,{+ O(e*). (81)

This is a beat oscillation for small values of o, - o5. The response
of hydrodynamic forces to this beating motion is made of two kinds

of beat oscillations: a slowly-varying sinusoidal oscillation, anda
steady component. For comparison purposes the relative magnitudes
of the different components of the hydrodynamic force given in Eq.
(81) are shown in Table 2 for \,;=10b and do= 11b i.e. XSdeg4e
The values in Table 2 are obtained from Fig. 6.

938

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

TABLE. 2

The magnitudes of hydrodynamic forces for \, = 10b and = 1.1

2f, 1.58
4f, 1.00
f,| sin Yel 0.014
f, 0.04
¢, - 1éiL = for i= 2,;436,7)
2pgb

It is clear from Table 2 that the first-order force dominates
the second-order forces. For instance, if we assume € = 0.1 the
ratio of the first-order force to the largest second-order force is
2e|f,| /4e*|f,| + 16. It is also clear that the magnitudes of the
difference-frequency force and the steady force are much smaller
than the first-order force, so they appear unimportant. However,
when such forces act upon a body which has very small restoring
force for a sufficiently long period of time a considerable excursion
from its mean position can occur. One can see from Figs. 5 through
7 that the |f,| is largerthanthe |f,| in 1.0<X<2.0. This
means that for a sufficiently small value of the difference-frequency
an estimate of the maximum "steady" force acting on an oscillating
body should include the difference-frequency force.

If we assume that the motion of a wave maker is described

by Eq. (80), the expression for the free-surface elevation, Y(x),
for large x can be given in the form

Y¥(x) ~ €2C, cos hs 2 t cos (K,|x| =P host)

a <®}4c, cos (0, - o,)t cos (4K,|x| - B, - 20,t)
+ 4C, cos (o, - o,)t cos (2K, |x| - B, - 20,t)

+ ¥,(x) cos (o, - a,)t f + Ole?) (82)

where C,, C,, C3, B, ; B., and B, are quantities which can be ob-
tained from Eqs. (76) through (78) and Y,(x) is given by Eq. (79).
We can see from the above equation that the far-field outgoing waves
are made of four independent wave components. The terms other
than the one associated with Y, represent beating phenomena with

739

Lee

the beating frequencies of (c, - o,)/2 and g, - og. Although the
values of Cz and C,; are not shown in Fig. 10, they are found from
Lee [ 1968] to be about the same order of magnitude as Y,, and

XY; - This means that the dominant contribution to the free-surface
wave elevation comes from the first-order term whose beating fre-
quency is (0, - o,)/2.

We can conclude that the hydrodynamic force and the outgoing
waves associated with the difference-frequency of two nearly equal
frequencies are much smaller than the corresponding first-order
quantities and are of the same order of magnitude as the other second-
order quantities. An examination of the figures suggest that if the
difference-frequency is sufficiently small an estimate of the effective
"steady" force can be obtained by doubling the pure steady force.
However, since the magnitude of the difference-frequency force is
always larger than the steady component this estimate may be a
low one.

ACKNOWLEDGMENT

The author would like to express his gratitude to Professors
T. F. Ogilvie and J. N. Newman for their encouragement and valuable
discussions during the course of this work.

REFERENCES

Frank, W., "Oscillation of cylinders in or below the free surface
of deep fluids," Report 2375, Naval Ship Research and
Development Center, 1967, 46 pp.

Grim, O., "Non-linear phenomena in the behavior of a ship ina
seaway," presented at the 12th International Towing Tank
Conference, Rome, Italy, 1969.

Hasselmann, K., "On nonlinear ship motions in irregular waves,"
J. Ship Res., Vol. 10, No. 1, pp. 64-68, 1966.

Lee, C. M., "The second-order theory of heaving cylinders ina
free surface," J. Ship Res., Vol. 12, No. 4, pp. 313-327,
1968. Also published as Report No. NA-66-7, College of
Pnginee sine. University of California, Berkeley, 1966,
PP-

Lighthill, M J., "On waves generated in dispersive systems by
travelling forcing effects, with applications to the dynamics
of rotating fluids," J. Fluid Mech., Vol. 27, Part 4,
pp. 725-752, 1967.

940

Nonstnusotdal Osetllattons of a Cylinder in a Free Surface

Parissis, G. G., "Second order potentials and forces for oscillating
cylinders on a free surface," Report No. 66-10, Dept. of
Naval Arch. and Marine Engr., MIT, 1966, 141 pp.

Porter, W. R., "Pressure distributions, added mass, and damping
coefficients for cylinders oscillating in a free surface,"
Inst. Eng. Res., University of California, Berkeley,
Series 82, Issue No. 16, 1960, 181 pp.

Tasai, F., "On the damping force and added mass of ships heaving
and pitching," (in Japanese) J. Zosen Kiokai, Vol. 105, pp.
47-56, 1959. Translated in English in Series 82, Issue 15,
Inst. of Engr. Res. , University of California, Berkeley,
1960, 24 pp.

Tasai, F. and Takagi, M., "Theory and calculation of ship responses
in regular waves," (in Japanese) Symposium on Seaworthiness,
Society of Naval Architects of Japan, pp. 1-52, 1969.

Ursell, F., "On the heaving motion of a circular cylinder," Quart.
J. Mech. Appl. Math., Vol. 2, pp. 218-231, 1949.

Wehausen, J. V. and Laitone, E. V., "Surface waves," Encyclopedia
of Physics, Vol. IX, pp. 446-778, Springer-Verlag, Berlin, 1960.

APPENDIX A

Description of the Boundary- Value Problems for the
Potentials gy; for i= 12535-4245 and 7

The application of the perturbation expansions given by Eqs.
(17) and (18) to the exact boundary conditions given by Eqs. (12) and
(16) yields the following:

On the free surface:

Piy(X» 0) - Kj9, = 0 for b= iand 2, (A-1)
where K; = o?/g,
Piy(x,0) - 4Kj_29, = - je } i.2%+ 0G (i2yyy ~ Ki P(i-ady)
= (A-2)
= ar = h.
20% orn t Mticery) t = Bie
for i=3 and 4, and

941

Lee

2
94(X40) = » - me Rej [joj (+0) (Pky - KyPqy) | = box) (A-3)

where % = Me - JP xs°
On the body:

Pix (%o2Vo)£'(X,) Vit Fa bo; for i= and. 2, (A-4)

b |
Pix (Xs Vo)E'(X) - Biy ee a (Pi a)xy (Kor Yo) E(x) - Pi. ayy)

= m (x5>¥ 4) for i= 3 and 4, (A-5)
and
' b '
P2(XQr¥Q)E (x) - 92, = - x Im, [xy (X69 Vq) + Poy Jt (XQ) = Piyy - Poy |
= m{x),Y,)- (A-6)

In the far field
P jy(x»- 00) = 0 for La lj2y 35 45 and 7

and at | 3x | = oo the potentials g,; for i=1, 2, 3, and 4 should
represent outgoing plane waves. For the steady potential g_ the
condition at |x| = oo should be determined by the law of mass con-
servation (see Lee [ 1968]).

Symmetric flow condition:

yi(x,y) = 9 (-x,y) for le=) 2 ae ands

In the limiting case of o, = 0, the forcing motion given by
Eq. (7) reduced to

y(t) = 2hgsin ojt

and if we let € = 2h,/b, the perturbation expansion given by Eq. (17)
reduces to

-jot - j2o,t 2
B(x,y,t) = €9,(x,y)e + €g,(x,y)e + €°g_(x,y)

942

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

which is the same expansion as that assumed by Lee [ 1966]. We can

easily establish the identities gy, =, and g3= 9. It will now be

shown that for T= o, we also have 9, = 29, and g,=9,. Equation
: 2 5 3 6 G

(20) gives

e . Go
hg(x) = - § 5b} o,(*,0)(Pay - Kp,)

. F
= 201, ox + Pry Poy) t - 52] PalPiy ~ Kiry)

- 210), Pox + Py Poy) f >

LS ne 2 2\)
hex) = - jt }o,(x,0(g,, - Kye) - 2, +e, )f.
Comparison of this with Eq. (A-2) shows that
h,(x) — 2h{x).

Equation (22) gives

a
m(x,,y¥,) = - Jz | (Piny Coa Yg) 7 Pony)? Oy) Sig > Peyy t.

so for 6, =o,

Ng Ta F jb } P ixy (qr Yo)f'(X,) ~ Piyy t .

Comparison of this with Eq. (A-5) shows that m, = 2m, The far
field conditions and the symmetric-flow condition for both ?, and

9g, are essentially identical. The above results lead to the conclusion
that 9, = 29;- Q,_ can be shown to be equal to g_ by a similar proof
if h(x} (Eq. (25) for o, = ¢2) is compared with’ h{x) (Eq. (A-3))

and m,(x) (Eq. (27) for ov, = o,) is compared with m_(x) (Eq. (A-6)).

6

943

APPENDIX B

Evaluation of the First-Order Velocity Potential

There are two first-order potentials, g, and 9,, involved
in this work. Since their solutions are es sentially identical (they
differ only in the frequencies), g, will be chosen as the representa-
tive first-order potential. As shown in Appendix A, the boundary-
value problem for g, is

V9, = 0, (B-1)
(x0) - Kg, = 0, (B-2)
P(X or ¥)f(x,) - 9 = - boy, (B-3)
P(X» -00) = 0, (B-4)
9) (x,y) = 9 (-x,y), (B-5)

and the radiation condition can be explicitly written as

lim Rej(9), = jK,9,) = 0. (B-6)
x->+00

There are two methods for the solution of the above problem. One

of them is the method of multipole expansions (see Ursell [ 1949])
which is essentially an eigenfunction expansion of the unknown function.
The other is the method of source distribution (see Frank [ 1967] )

i.e. the method of Green's function. A brief description of the

method of multipole expansions will be given. First we consider

the problem without the boundary condition on the body given in Eq.
(B-3). If we transform the problem into the {-plane! we find that

7°M(X,e) = 0, (B-7)

(2n + i)a aM
Ka} - 2, Se Sarina | Min.0} - SE = 0, (B-8)

"Here, it should be recalled that the transformation given by Eq. (2)
maps the €- and y-axes into the x- and y-axes and maps the contour
of the semi-circle inthe (-plane onto the contour of the cylinder in
the z-plane.

944

Nonsinusotdal Oscillations of a Cylinder in a Free Surface

M(X,@) = M(\,7 - @), (B-9)
and in place of Eq. (B-4) and (B-6) we require
M— 0 as X — oO in -7w=as0. (B-10)

The solution of this problem is

a) = £28 2ma@ , jae (2m - i)a@
ide ea hem VU (2m - 1) 2m!
oo
- », (2n+1)agne, sin (2m+2n+1)a \ (B-11)
n=O
where m is a positive integer. M, is often called the multipole
of order m. Although this expression for Ma trivially satisfies
Eq. (B-6) in the C-plane,the expression above still does not repre-
sent the outgoing plane waves. To satisfy this radiation condition
we introduce a source function M,(x,y) which satisfies all the
required conditions except the boundary condition on the body. The
expression for the function M, is

M,(x,y) = ‘ a ea dk - jre” cos K,x (B-12)
.e)

0
where f means that a Cauchy principal value is to be used. There-

fore we represent our solution as
oo

9, = y (Dm + 5Cm) My (x(X,2) sy(X,@))e!4 (B-13)
m=0
where b,, and cm are the unknown strengths of the singularities,
q is the phase difference between the motions of the body and the
fluid, by = Q= source strength, and c,= 0. The unknown constants

bm» Cm, and q are to be determined from the boundary condition on
the body given by Eq. (B-3).

We introduce the stream function W, which is the harmonic
conjugate of the velocity potential g,. The Cauchy-Riemann relation
gives 89,/dn 2 OW /8s along the contour of the cylinder where s is

the arc length of the contour in the counter-clockwise direction. The
boundary condition for W, on the cylinder can be shown to be

Wi (x,,y,) =- bo,x,- (B-14)

The expression for W, in terms of the harmonic conjugates of My,

945

Lee

denoted by N, is easily found to be

00
<= » (bm + 5em) Nm(x(X@) sy (X,a)Jer*. (B-15)
m=O
where
N. = - Sin 2me + Kya {£28 (2m - 1)a
m 2m | (2 1)y2m-t
cy
(2n+i)a cos (2m+2nti)a
7 2 2m 1824 2 ment \ for m= 1, (B-16)
n=O
and
eX sin kx Kiy
No=- 4) apap dk + jme ' sin Kx. (B-17)
On alvala,

Substituting Eq. (B-15) into (B-14), we get

00
>. (bm zw jcm)Nm(x(5 2) ,y(h,a))e/4 i bo, x,. (B-18)
m=O

We choose any point on the contour of the cylinder between 6=0 and

== 7/2, say (x!,y5) in z-plane and (\,,@') in the ¢-plane, to show
that

00
Pe Ne ’
e 14 ie box! /{ > (5, r 3S on). Nin (Not 2) + QNo( x5, ye) } . (B-19)
m=!
Equation (B-19) can be substituted into (B-18) to give

00

) Am Ne(hor) = Nig(qo $4 = Nolxdsye) - Nol%orYe) (B-20)

Deantal€ ‘

Ag = —a 5m |

In principle we can choose an infinite number of points on the cylinder
(-1/2< @< 0) to set up an infinite number of simultaneous equations
from Eq. (B-20) for the unknown coefficients A,. However the

946

Wonsinusoidal Osetllattons of a Cylinder in a Free Surface

infinite series in Eq. (B-20) is truncated to a finite series to obtain
an approximate solution by a matrix inversion. After finding some
finite number® of b, and c, and using these coefficients in Eq.
(B-19) we find the values of Q and q.

APPENDIX C

Solution for the Problem of Sinusoidal Pressure
Distribution on a Free Surface

We seek a solution for the following boundary-value problem:
VW (x,y) = 0 in v <0,
W, (x,0) - KW, = Ae (Gan)

where K =w*/g, A is areal constant, and K'=w'*/g# K Further-
more we require that

Wiy (x , - 00) = 0;
Ww, ~ Bek Yeik'Ix! as |x| — oo
where B is a complex constant, and

W, (x,y) = W, (-x,y)-

If we let
Wy > Wig Aig
we can easily show that
Wi cy(*> 9) - KW, = A cos K'x,; (C-2)
W),y(x.0) - KW,, = A sin aa a (C-3)

?The exact number is determined in the sense of "an approximate in
the mean" for the function on the right-hand side of Eq. (B-20) by
the series on the left-hand side.

947

Lee

The value for the function W,, which has all the required properties
can be shown to be

AeKy

ie qe cos K'x. (C-4)

It takes little more effort to solve for Wj),. We find it by using a
transform. Let

00
e 3
Wis -{ e PAWis (x,y) dx.
- 00

The Laplace equation requires that

2u,* * =
- p Wis OW fey, 0
or
* Ipl
W,,(psy) = c(p)e’ (C-5)

If this is substituted into the Fourier transform of Eq. (C- 3),the left-
hand side yields

Wry = KW, = (|p| - K)c(p)

and the right-hand side yields

00 : co
al e”'P* sin K'|x| dx = 2A \_ e7!* sin K'x dx
on fe)
oo lie i(k" 2AK'

= (eilk'-Px _ Gi(K'ePe) Gy 5
1 fe) p--K"=

(C-6)

where the apparent improper integral above is interpreted as a
generalized function.> Thus we find that

- 2AK' - 2AK'

(|p| - K)c(p) a PLiae or c(p) = Ge K'2)(]p|] - K)

3 Another way of interpreting this is that of Lighthill [1967] who let
jK' jKolx! =
w'=wot+ je, € = 0 so that eK ixt LQ Mom! oo where K5 = (we - €*)/g
approaches K' when yp (= €2w,/g) — 0.

948

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

If this expression is substituted into (C-5) and the transform is
converted we find that

AKC ably
(p2 - K'2)(|p| - K)

|
W,, (x+y) P* dp

. _ 2AK' © eP%cos px aa
™ J, (p?- K%)(p - K)

A py { 1
-= e"’ cos px { ———_——>--—___
T Jo (K +K')(p +K')

Ges tl 2K! |
y dp. C-7
(K'- K)(p - K') (K'- aaa Pp ( )

Apparently there are poles at p=K' and p=K. However if the
inverse transform of the right-hand side of Eq. (C-6) is taken, it is
readily seen that the integral must be integrated as a principal-value
integral in order to recover the original function A sin K'|x|. This
means that the integral in Eq. (C-7) associated with the second and
third terms in the square bracket should be taken as P.V. integrals.

If we let
ward cos px en Re
124 S808 PX ap = Re, { —— dp,
0 pt+K 'Jo ptK

and make the change of variable t = i(p+K')x, we can show that

‘ 00 sy rage
I, = Re, eX? \ — dt = Re; | e'*7E, (iK'z) | :
iKz

Again the change of variable t = i(p-K')z enables us to show that

oo py 00 _-ipz
-5 e__COs8 px 4, Re; 5 & - dp
2 p- K' Y7O P- K

-iK'z

Re; [e E, (-iK'z) = ime“ 4

where + signs correspond to the case of x2 0. Similarly the change
00
of variable t = i(p-K)z in fs (e”! cos px) /(p- K) dp leads to

-iKz

-iK
I,= Re, [e""E,(-iKz) ¢ ine*7}

949

Lee

for x 20. This integral is the last term in Eq. (C-7), and we observe
that ae Iz= - meXY sin K|x|. This implies the existence of a sinu-
Xl» 00

soidal wave with wave number K in the far field. It obviously vio-
lates the radiation condition that the outgoing waves have wave number
K'. However a careful examination of the integral I, shows that it

is just one of the homogeneous solutions of the problém which can be
discarded, if desired, because of the radiation condition. Thus sub-
stituting the expressions obtained above for I, and [, into Eq. (C-7)
and discarding the last integral in that equation, we find that

A I if
Wty) = - 2 [eaber tebe |

7 LK+K'
iM tie pear t “1K Zi page K'y
--4Re([* EV (iK'z) ,e E | ( Re) +e sin eulaole
T K+K' K' 7K K"-K
(C-9)
We combine Eqs. (C-4) and (C-9) to finally obtain
: Aek Ye ik Ix!
Wilsall RRS RT
A eRe Kay , eke (-iK'2)
: jAre,| £ =k) + ———L | . (C-10)
T K + K' K-K

tiz
Since lim e E (+iz) = 0, we see that
|x — 00

as is required.

950

Nonstnusotdal Oscillations of a Cylinder in a Free Surface

DISCUSSION

Edwin C. James
Californta Institute of Technology
Pasadena, California

I would like to direct a question to Dr. Lee concerning the
pure steady force. Apparently this type of force can arise in free
surface problems and is attributed to a mean drift of mass in the
direction of wave propagation. The action of such a force applied
to an unrestrained body results in a sinkage or alift. The question
is then, how does one physically explain the steady force when the
symmetry of the problem dictates that the mass transport at the
station x = 0 should be zero?

REPLY TO DISCUSSION

Choung Mook Lee
Naval Shtp Research and Development Center
Washington, D.C.

A mass transport phenomenon arises in the higher-order
theory of surface waves (see, e.g. Wehausen and Laitone [1960,
pp. 660-661]). Since the present work deals with a second-order
problem of free-surface waves, it may be expected that mass-
transport will occur in the present problem also. Although I have
not touched upon this subject in the text, I discussed it in some
detail in my previous work (Lee [1968, pp. 317-318]).

As the discusser pointed out, there is no mass flux across
the y-axis. Then, the question arises as to the origin of the mass to
supply mass transport. I answered this question in this previous
work by showing that the role of the steady potential gy7(x,y) is to
counteract the mass transport phenomenon. This means that 9,
should behave like a steady sink whose strength is equal to the total
mass drift through two vertical control planes encompassing the
cylinder, divided by 217. The lowest-order contribution from ¢,
to the steady force is fourth order, as is proved by Bernoulli's
equation. Thus, the second-order steady force still exists while
the mass transport phenomenon is nullified by the pure steady
potential 97.

951

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vbeede stia oad yd be fit! i = WULF qj Gone ss BPKIT,, a
Li texeetat

HYDRODYNAMICS IN THE OCEAN ENVIRONMENT

Friday, August 28, 1970

Afternoon Session

Chairman: T. Y. Wu
California Institute of Technology

The Drifting Force on a Floating Body in Irregular Waves
J. H. G. Verhagen, Netherlands Ship Model Basin,
The Netherlands

Dynamics of Submerged Towed Cylinders
M. P. Paidoussis, McGill University

Hydrodynamic Analyses Applied to a Mooring and
Positioning of Vehicles and Systems in a Seaway
P. Kaplan, Oceanics

Wave Induced Forces and Motions of Tubular Structures
J. R. Paulling, University of California, Berkeley

Simulation of the Environment and of the Vehicle
Dynamics Associated with Submarine Rescue
H. G. Schreiber, Jr., J. Bentkowsky, and
K. P. Kerr, Lockheed Missiles and Space Company,
Sunnyvale

953

Page
955

981

1017

1083

4141

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olavy USE

.

eae

THE DRIFTING FORCE ON A FLOATING BODY
IN IRREGULAR WAVES

J. Hs Gs Verhagen
Netherlands Shtp Model Basin
The Netherlands

I. INTRODUCTION

A floating body in waves experiences a hydrodynamic pressure
force which is exerted by the surrounding fluid. Several factors
contribute to this wave pressure. One of them is undoubtedly the
conventional unsteady exciting force, which makes the body oscillate
at frequencies in the region comprising the bulk of the energy con-
taining waves.

Another factor originates from higher order forces due to
various non-linear effects. In general these non-linear effects are
too small to influence the high-frequency motions of the body. They
can, however, be of importance in that part of the frequency domain
in which the wave energy is very small, i.e. in the low-frequency
range, in particular if one of the natural frequencies of oscillation
of the body lies in that range.

In the limiting case -- zero frequency -- one arrives at the
well-known drifting force.

It will be obvious, to assume that the force on a floating body
in irregular waves comprises not only a steady part but also a slowly
varying part, slow in comparison with the mean period of the wave
spectrum. The steady as well as the slowly varying part of the wave
force, both of which are proportional to the square of the wave
height, are denoted by "drifting force." The present paper is con-
cerned with the slow drift oscillation of a moored vessel in irregular
waves. It is based on general observations revealed by an extensive
test program on the behavior of moored bodies in a seaway.

955

Verhagen

Il GENERAL OBSERVATIONS OF TEST RESULTS

A study of the test results on the behavior of moored floating
bodies in irregular seas revealed the following general observations:

14. The horizontal modes of motion -- surge, sway and yaw --
show two separate frequency regions. A low frequency
region corresponding to the low natural frequencies of the
moored system, and a frequency region corresponding to
those of the energy containing waves.

2. The long periodic motion is excited by waves or by a wave
group with amplitudes high compared to the mean wave
height. In the considered cases, where a linear stiffness
of the mooring system is employed, it appeared that the
amplitude of the long periodic motion for a given vessel
and mooring system is proportional to the square of the
significant wave height divided by the mean wave period
Coe for various long crested seaways coming from a
given direction.

3. For a given body and mooring system tested in various
seaways no clear relation could be discovered between the
time averaged excursion from the equilibrium position in
still water and the amplitude of the long periodic motion.

These observations are obtained from extensive model tests conducted
in the Seakeeping Laboratory of the N.S.M.B. at Wageningen.

The behavior is not unique to moored vessels. Also the towing
force of a vessel towed in irregular seas show the same tendency as
well as for instance, the slow oscillations in torque and thrust of the
propeller of a self-propelled model as observed in the seakeeping
model tests.

III. DISCUSSION OF THE RESULTS

The third point of the above mentioned observations deserves
particular attention. The drifting force on a floating body in regular
waves -- the time averaged position of the body is fixed in space --
is dependent on the joint action of waves and body motions. The
force is proportional to the square of the wave height and dependent
on the phase between wave and vessel motion.

If we consider an irregular wave as build up of a regular wave
whose amplitude is a slowly varying function of time (slow as com-
pared to the wave period) and a stochastic variable phase, the cor-
responding energy spectrum will be narrow. The drifting force on the
floating body in that case will show the same dependency of the time
as the square of the wave amplitude. The amplitude of modulation
will be the same order of magnitude as the time averaged drift force.

956

Floating Body in Irregular Waves

For a given moored system with approximately linear spring
stiffness and damping coefficient the same linear relationship must
be found between mean excursion and low frequency amplitude of
motion for various seaways. This appears not true. Especially in
heading waves large discrepancies can occur. From this observation
I am led to suppose that it will be not allowed to describe a practical
wave spectrum by a slowly modulated regular wave in order to
explain the obtained test results.

Based on the mentioned observations I am led to suggest the
following hypothesis.

Hypothests: The wave forces on a moored body in irregular
waves which are responsible for the excitation of the mass-spring
system in its resonance frequency are the second order low-frequency
wave forces on the body in fixed condition, i.e. the drift force due to
the reflection of waves.

The influence of the ship motions can be neglected.

One of the conclusions of this hypothesis is that the exciting
force for the long-periodic motion is a function of wave character-
istics and shape of the body alone, and not dependent on the mooring
system or on the weight distribution of the moored body..

The hypothesis is supported by numerical motion calculation
for comparison with experiments, which will be shown later on. It
is needed however to extend the number of comparisons in order to
obtain the restrictions of the proposition.

The proposition can be made acceptable in the following way:
Suppose the irregularity of the wave could be described by a more or
less regular wave pattern in which a few discrete steep waves are
present. Intuitively it can be stated that the occurrence of a few
high waves in an otherwise nearly regular wave pattern gives rise to
some violent ship motions. Through inertia effects these motions
occur mostly after the corresponding high waves have passed the
vessel.

Hence, interaction between the high waves and the resulting
motion on the pressure distribution around the ship's hull is drasti-
cally reduced, by the mentioned retardation between exciting force
and resulting motion. Hence, the effect of such a single high wave
on the floating body is consequently restricted to the instantaneous
effect, i.e. the effect of the wave reflection, on a fixed body. The
corresponding exciting force due to reflection is only dependent on
body form and wave characteristics.

Conclusion: The mean drifting force on a floating body in

irregular waves is dependent on the joint action between waves and
body motion.

957

Verhagen

The slow drifting oscillations of a moored vessel are caused
by forces due to the reflection of the waves against the fixed obstacle,

The remarkable observation that the amplitude of the long-
periodic motion for a given body and mooring system_in various
seaways of given direction is proportional to C¥,,, /T can now be
explained as follows: The force due to reflection of irregular waves
against a fixed obstacle is proportional to Cy ,3 if the body
dimensions (L) in the wave direction is not too small compared to
the mean wave length \ (L/\ >37). A "high" wave can be defined
more formally as a wave with amplitude a fixed number times the
significant wave height. The change of occurrence per unit time of
our socalled "high" wave is then proportional to 1/T provided all
of the considered random waves are Gaussian distributed, are at
least distributed in the same way. If the combination of floating
body and mooring system is considered as a mass-spring system
with linear stiffness and damping coefficients the resulting long-
periodic motions will be proportional to Cwy 3 /T. In consequence
of the many assumptions made in the above reasoning, one should
be careful to adapt the explanation without reservation. <A firm
foundation is needed.

IV. MODEL TEST RESULTS

An extensive test program has been carried out in the Sea-
keeping Laboratory of the N.S.M.B. ona number of models in order
to obtain systematic information on the station-keeping abilities of
moored vessels in a seaway. Some results of motion tests on one
of the vessels will be given in this paper.

The vessel is moored between four horizontal linear springs

and tested in various seaways approaching from ahead and from
abeam. A sketch of the mooring system is given below.

45° Head seas
ee

Beam seas

958

Floating Body tn Irregular Waves

The stiffness of each spring amounted to 5 tons/m. and is
independent of the elongation. The main particulars of the vessel
are:

Length - beam ratio = 5.3
Beam, - draft ratio = 3.65
Block coefficient = 0.750
Midship section coefficient = 0.997
Waterplane coefficient = 0.896

Wave spectra were produced similar in shape to those
analysed by Pierson and Moskowitz for fully developed seas. These
spectra can be described by:

Both the produced and the hypothetical spectra are given in the
Figs. 1 through 5.

The significant wave height is defined as:

The average wave period is

where
ay k
m, = { w S, (a) dw

The distribution of the wave elevations confirm well to the normal
probability distribution.

The motions of the vessel and the forces in the horizontal
springs were measured in three irregular head seas and three
irregular beam seas each with different characteristics. The
duration of each test run corresponded to approximately half an hour
for the full scale vessel. This time period is considered to be
sufficiently long for a reliable statistical treatment of the recorded
quantities. A typical recording of a horizontal motion of the moored
vessel is given in Fig. 6. A low-frequency motion in the natural
period of the ship-spring system is present upon which high-fre-
quency motions are superimposed. The spectral density of the,surge
and sway motions are given in Figs. 7 through 12. The distribution
of the forces and of the motions deviates from the normal distribution
though in many cases the deviation is not large.

959

Verhagen

WAVE ELEVATION IN m

©

Vv

c

©

{=

&

= |

1S)

Vv

°

oe

Cc

©

Vv

=

©

a
-1.0 -05 O ta 1.0
wave trough wave crest

Measured : Cwya= 1.20 m a =» 9.8 sec.
Scenes > Cwiyse 1.25m 729.0 sec.

0.3

SU (W) In msec.

eee ss Bi dal
FE 0 dV
Ba Fh FP SB DS
Fa ee a | i i PG
Ra a a nia A NSS

1.5

W in alot 108

Fig. 1. Wave distribution and spectrum

960

Floating Bodies in Irregular Waves

WAVE ELEVATION IN m

percent occurrence

Measured Sway 2.00 m, f=96 sec.
------ Theoretical : Gupae 2.00 m, T= 9.0 sec.

TTT TLL
pol SEO Se a
Paeee eee

W in ect

Fig. 2. Wave distribution and spectrum

961

Sc(w) in m? sec.

Verhagen

WAVE ELEVATION IN m

20

CTT A
CCCCaTe
CC

-1 Oo 1 2
wave trough wave crest

percent occurrence

Measured Cwyys" 236m, Ts 6.6 sec.
Theoretical : Cw" 250m, T= 6.0 sec.

a Se
Aerie
Sa A ae

a PTALN Agi
eee teee

Eee

W in ae

Fig. 3. Wave distribution and spectrum

962

Floating Bodtes tn Irregular Waves

Wave elevation in m.

§ 40

: -

WwW

i 4

[4

=)

oO

3 Se
20

- — |

o hi

y ee ai

" ia —
° = nm

2 1 ° 1 2

WAVE TROUGH WAVE CREST

MEASURED , fi1/3= 2.52m., #s 95 sec.

———— —— THEORETICAL (Pierson- Moskowitz), Ai 2.501m:,, qt s 9.0 sec.

a ed se a

pa thf

aa Pe Ed
aaah

1.5

cay smal

W in rad.sec.

|

Fig. 4. Wave distribution and spectrum

963

Verhagen

Wave elevation in m.

G
max.es 46 m.
max.- s 44 m.

= zs
MEASURED, Hy32 4.80 m., Tz 10.1 sec.

THEORETICAL (Pierson - Moskowitz), Fits: 5.00 m., ¥. 9.0sec.

fnh (W) in m® sec.

Rl
nn es eS SS

W in rad. sec”!

Fig. 5. Wave distribution and spectrum

964

Floating Body in Irregular Waves

SWAY
WAVE

mean
ie]

Fig. 6. Irregular wave and sway motion

965

TIME

Verhagen

SURGE DEVIATION IN m

© G = 0.87 m
e 49 max.+= 2.4 m
t max.-= 3.0 m
=)

U

Yy

° 20

Ee

p

©

a2 0 aii | be

-4 -2 ie) 2 4
—=> ship forward

Head sea : =2.30m, fT. 6 6sec.

wr
Gwi/3

W inrad.sec”

Fig. 7. Distribution and spectrum of surge

966

Floating Body in Irregular Waves

SURGE DEVIATION IN m

percent occurrence

Sy (W) in msec.

W inrad.sec>

Fig. 8, Distribution and spectrum of surge

967

3
3)

Sy (W) in m? sec.

Verhagen

SURGE DEVIATION IN m

by of

c¢ 40-—max.+

7)

L max.- 8.9 m

5

Vv

13)

° 20

E

©

5 ii i

rT)

a oO ape fi Sia
-8 -4 oO 4 8

—=~ship forward

SURGE SPECTRUM

7]
Head sea : =4.883m, T=101 sec.

w
Gw13

W inrad.sec”

Fig. 9. Distribution and spectrum of surge

968

Sy (w) in m?sec.

Floating Body in Irregular Waves

c-o7m > i.
mMax.«2z 3.2 m

maxcz_ 23m [Nmeen | |
a
eee Saez
-4 -2 re) 4

ship to starboard ship ta port

SWAY SPECTRUM

= 120m, #098 sec.

40

percent occurrence
~
re)

oO

Beam sea : Goris

a ea ee
LaRES ZS SERRA
LESERPSESASe See

BEReeeo NS2ae
BE GR Z aa aise 2 cies
A 1.

W inrad. sec-'

Fig. 10. Distribution and spectrum of sway

969

Sy (W) in m? sec.

Verhagen

SWAY DEVIATION IN m

percent occurrence

-10 0 5 10

W inrad.sec~'

Fig. 11. Distribution and spectrum of sway

970

Sy (w) in m?sec.

Floating Body tn Irregular Waves

SWAY DEVIATION IN m

“ a
Beam sea Cwiy/3" 2.36 m , T=66 sec.

W in rad.sec.~'

Fig. 12. Distribution and spectrum of sway

Ar ge

Verhagen

V. COMPARISON BETWEEN EXPERIMENTAL RESULTS AND
CALCULATIONS

_ Some tentative calculations have been carried out amplifying
and illustrating the aforementioned suppositions.

Mean Drifting Force

The mean drifting force on the moored vessel in the long-
created irregular waves has been determined by a linear superposi-
tion of mean drifting forces in the regular wave components of the
known wave spectrum. As has been shown by Maruo [1] a.o. the
principle of superposition can also be applied to determine the non-
linear mean drifting force. For head seas the mean longitudinal
drifting force is

2nr®
F. = he Fl) _ S.(w) dw (4)

© petiB/L §

In beam seas the mean lateral drifting force is

)
peie

@
F = 2pet | mre S(u) do (2)
O 2pgo L

Formulas for the mean drifting force in regular waves on freely
floating, completely restrained or elastic moored bodies are
obtained by Maruo[2] and Newman[3]. Numerical calculations
are usually carried out on specific formulas for the drifting force
on a slender body. [3] As is well known, the agreement of these
calculations with experiment is on the whole not very satisfactory.
The slender body approximation results f.i. in a vanishing mean
lateral drifting force. Therefore an engineering approach is pre-
ferred using available experimental data on a similar ship form.
The estimated curves of the longitudinal drifting force in regular
head waves at zero forward speed and the transverse force in beam
waves are given in Figs. 13 and 14. Using these data the expressions
(1) and (2) for the mean drifting force in irregular waves can be
solved. The results compared with the experimental results are
shown in Table I. The agreement is reasonable.

972

Floating Body in Irregular Waves

06

Oo
0.5 1.0
W in rad.sec~!

Longitudinal drift force for head seas

Pig. 13.
20
15
>
| Xin 640
re ee
fo)
a
0.5
oO
Oo 0.5 1.0 1.5
W inradsec>!
Fig. 14. Lateral drift force for beam seas

973

Verhagen

TABEE: 1

Wave Mean drifting forces
characteristics in tons

*® in sec

force measured

force calculated
force measured
force calculated

height Cy,, in m
Average

period

Wave direction
Longitudinal
Longitudinal
Transverse
Transverse

»
G
@?2
0

ot

MS

ot
G
ter)

ran

YW)

The low frequency drifting force

This part of the drifting force has been estimated as follows:

The measured wave height record {€,(t) is squared c(t). This

squared wave height function can again be analys ed by a spectral
analysis. The spectral density function of 7¢°(t) is determined.
For an example see Fig. 15. As can be seen from this figure it
contains the sum and difference frequencies of the original wave
height spectrum. The difference frequencies are now of special
interest. They are related to the envelope of the original wave

height record.

The r.m.s. value of the low-frequency energy variation is:

V2

Cedi | f Si p2(w) ao

diff. freq.
Te att is proportional to the mean square value of the fluctuating
part of the wave height envelope. The energy in the fluctuating part
of the envelope curve depends largely on the occurrence of "high"
waves. As discussed earlier the occurrence of "high" waves is pro-
portional to 1/T per unit time. So the relation between the r.m.s.

974

Floating Body in Irregular Waves

0.3

Vv
©
rr
ae
Eoce
£
ra)
ww)
x
WY
0.1

oO 05 1.0 1.5 2.0 2.5
W inrad.sec.~

Fig. 15. Spectral density function of a half times the
square of the wave height

value of the low-frequency energy variation and the characteristics
of the waves becomes

=n

ows
= constant °

TE ditt

Aa

Figure 17 shows that the produced wave spectra fit this relation quite
well.

Now the exciting force must be determined keeping the vessel
in a fixed position. In that case the exciting force is due to wave
reflection alone. In regular head waves this force is

F, = $pga*B sin* a

x

975

Verhagen

ord
SE ate in m:

w2 2 3
Swiys /# in m2. sec.-1

Fig. 16. Relation between the low-frequency energy
variation and wave characteristics

when a is the wave amplitude and sin*@ is the mean square of

the angle between the tangent at the ship's waterline and the longi-
tudinal axis.

Now the re-m.s. value of the low-frequency exciting force
becomes

2
Op, = PEF aitt B sin” a

and the r.m.s. value of the low-frequency motion is

a
x By * Woy

The surge damping B, is obtained from an extinction curve. The
numerical values are B,* w,,= 1.50 ton/m, sin*a@=0.24, The

results compared with the experimental results are shown in Table II.
The agreement is good.

976

Floating Body in Irregular Waves

TABLE II

o, inm o, inm
calculated measured

0.96 0.87
0.86 0.77
2.87 2.54

where y is a coefficient depending on the beam-wave length ratio.
For the wave lengths under consideration (\/B = 5) the mean value
of y is obtained from experimental data on Series 60 models is
about 0.5. This value increases up to one for shorter waves and
decreases for longer wave lengths.

Now the r.m.s. value of the low-frequency exciting force in
irregular beam seas becomes

Oy = YPB% aieg £

and the r.m.s. value of the low-frequency sway motion is

The sway damping is again obtained from an extinction curve. The
numerical value is Byw., = 4 ton/m,. The results compared with
experimental results are shown in Table III.

TABLE III

~ x o, inm co, inm

sec
1.19 9.8 0.72 0.67

1.96 9.6 2012 2.05
2.30 6.6 Se? 4.56

a

Verhagen

Taking in mind that in the last case the value of y must be

higher than 0.5, due to the shorter wave length, the agreement is
very good again,

© Gy SURGE

uw2
Swis /# in m2 sec.-'

Fig. 17. Relation between surge, sway and wave
characteristics

ACKNOW LEDGEMENT

The author would like to thank the staff of the Seakeeping
Laboratory of the N.S.M.B. for the many fruitful discussions. He is

indebted to Mr. Tan Seng Gie who carried out the experimental
program.

978

2.

3

4,

Floating Body in Irregular Waves

REFERENCES
Maruo, H., "The excess resistance of a ship in rough seas,"
Int. Shipbuilding Progress, 1957.

Maruo, H., "The drift of a body floating on waves," Journal
of Ship Research, 1960.

Newman, J. M., "The drift force and moment on ships in waves ,"
Journal of Ship Research, 1967.

Verhagen, J. H. G. and van Sluijs, M. F., "The low-frequency

drifting force on a floating body in waves," Int. Shipbuilding
Progress, 1970.

979 :

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Bee ik mn $4 | bo op Oye Shel Ghar laa vel Ul) yee
Dg nak meet ‘erry tLe oP wy Gee: ie ip ee st
pe a4 Op? 7

®ageadgs0% ri olde 6 te sankielsad eaeoxe odT" , Tee

4 : BEA a aenthar el galb lt aga le 7
a farane! * ~ovaw fe gaitach ypod 6 io yh vot 4 cil
fe ty td

C07 Se | { aly boxes de ne
" ry a4

Arye vaew al bqlde nm doerrets bos Satoh Mish say a ae a

oy) l steel dozaoeoH que 40 cas ab

Weutrnus.-wolets” g.% aoa ,a(icte cav bos .O. oH ic a

ives ...rat -" dovaw! Ab ybod ys Pools ao so7ok yale
- | OTe! ,sastgon |

; 7 '

. -
‘ i, ‘ ; al i
i ae 1a
7 a Bs
i
' - , uv = { ry s
a
rs
.
J =e -e hy oAaenty ‘
- '
a } s |
.
cas
. 7a 4 ew i vyav¢
t 7 >)
ACN IS Lal Ss
. » > «¢ , = F
28 @ "ine Y= 1% a H > Covi Hs A ced ire amie ap
acc paewry e1 Ahe N, £) it. Be. for thy mary Leu fal Gis rea Brin.
te, Letitod) & if Tun ig Cale ws sPYisd oft the levi” af
: Biya Tair.

DYNAMICS OF SUBMERGED TOWED CYLINDERS

M. P. Paidoussis
MeGtll University
Montreal, P.Q.; Canada

I. INTRODUCTION

Interest in the dynamic stability of towed ships dates back to
the halcyon-days when solutions to engineering problems could still
be obtained by experience, without the aid of sophisticated analysis.
Certainly, operators of horse-drawn barges in canals must have
been aware of possible instabilities and remedial actions. Never-
theless, to the author's knowledge, the first substantive paper on
the subject, by Strandhagen, Schoenherr and Kobayashi [ 1], did not
appear until 1950. This is also surprising, if one considers that
both the analytical techniques and physical concepts were understood
long before that; indeed much earlier work does exist on the closely
related topic of stability of airships moored to a mast and kite bal-
loons, starting with the work of Bairstow, Relf and Jones [ 2] in 1915,
and followed by the work of Munk [ 3], Glauert [4], and Bryant,
Brown and Sweeting [5], for instance.

Strandhagen et al., and the discussors of their paper, firmly
established the following important criteria for stability of a towed
ship: (i) the point of attachment of the tow-rope should be ahead of
both the center of mass and the center of pressure of the (static)
lateral hydrodynamic forces acting on the ship; (ii):the ship should be
stable when moving untowed; (iii) in cases where (ii) is not satisfied,
then the system could be rendered stable by either short enough or
long enough tow-ropes. It is noteworthy that the criteria for stability,
at least for the linearized theory of small departures from course,
apply to all towing speeds, so that for a given configuration a (rigid)
towed ship is either stable or unstable irrespective of how fast it is
being towed. Instabilities were found to be of two distinct types:

(a) yawing, i.e. azero-frequency, amplified motion which in aero-
elasticity would be referred to as 'divergence', and (b) oscillatory
instability, where the system, when disturbed, oscillates about its
position of rest with increasing amplitude.

More recently, interest in the instability of submerged towed
bodies has arisen mainly in connection with sonar applications. Here

981

Patdoussis

the body housing the sonar device is towed deeply submerged by sur-
face craft, for the purpose of hydrographic survey, submarine
detection, or location of schools of fish. We must refer to the work
of Strandhagen and Thomas [6], Richardson[7], Laitinen [8],
Patton and Schram [9], Jeffrey [10], Schram and Reyle [11], and
Whicker [12]. The stability problem for the sonar-type towed bodies
is of course quite similar to that of a towed glider [5]. The work
referred to here deals with the dynamics of the towed body system

as a whole; the geometry of the towed body for these applications
tends to be fairly complex, and the analysis quite elaborate.

A considerable amount of work also exists on the equilibrium
configuration and thedynamics of towing cables, starting with McLeod's
[13] and Relf and Powell's [14] work, to more recent work by
Landweber and Protter [15], Pode [16], [17], O'Hara[18], Kochin
[19], Eames [20], and Albasiny and Day [ 21]; this represents a by
no means exhaustive list of references.

The author's interest in this field comes from work associ-
ated with yet another application: that of the Dracone flexible barge,
which is a flexible sausage-like container towed behind a small craft,
and used for the transportation of oil and other lighter-than-water
cargoes, including the sea transport of fresh water to arid lands
(e.g. to some of the Aegean islands from the mainland). The new
element that enters the problem in this case is that of elastic forces,
making this a problem in the general area of fluidelasticity (cf. 2a) \e
The first analysis of stability of the Dracone was by Hawthorne [ 2a »
Later, the author studied systematically the dynamics of flexible
slender cylindrical bodies immersed in axial flow, for various con-
ditions of end-constraint [ 24] , [ 25], including the case of a towed
slender cylinder [26]. In the latter case, both rigid-body type in-
stabilities and flexural instabilities were shown to exist; stability
was highly dependent on the towing speed.

It was suggested [ 27] that cylindrical or quasi-cylindrical
containers towed underwater by a small submarine could be used to
transport liquid cargoes to and from arctic ports, avoiding the
hazards of surface transportation in ice-covered seas. The con-
tainers could be either flexible or, more likely, rigid; there could
of course be a string of such containers towed by the same submarine.
This idea has taken added poignancy since the oil discoveries in the
Arctic.

In this paper we shall re-examine the problem of stability of

a submerged cylindrical body, both flexible and rigid, towed by a
submarine craft.

982

Dynamtes of Submerged Towed Cylinders

Il THE EQUATION OF SMALL LATERAL MOTIONS OF A
FLEXIBLE SLENDER BODY IN AXIAL FLOW

We shall derive the equation of small lateral motions of a
slender body of revolution of the type shown in Fig. 1(a); the body
is supposed to be supported somehow so that it is not washed away
downstream. The fluid is incompressible and of density p; it is
flowing with velocity U parallel to the x-axis, which coincides with
the undisturbed longitudinal axis of symmetry of the body. The body
is of mass pér unit length m(x), cross-sectional area S(x), and
flexural rigidity ElI(x).

Fig. 1(a) Diagram of a flexible, slender body of
revolution in axial flow

We consider small motions y(x,t) and assume that y,
dy/ax, 8*y/ax* to be all small, so that no separation occurs in cross-
flow. Moreover, we assume that dS/dx is small everywhere,
except perhaps at the ends of the body, so that no separation occurs
in the axial flow (except perhaps at the rear end), and so that
slender-body theory may be used. Also d(EI)/dx is assumed to be
small, which, together with the restrictions on the displacement
function, allows us to use the simple Euler-beam approximation to
describe the flexural forces. The body is further assumed to be
of null buoyancy and uniform density, so that no constraining force
in the y-direction nor a moment is necessary to keep it lying along
the x-axis, at least at zero flow velocity. Furthermore, the motions
are considered to take place within the (x,y)-plane, which for the
sake of simplicity is assumed to be horizontal, Finally, we neglect
internal dissipation in the material of the body.

We now consider an element 6x of the body. The forces and
moments acting on it are shown in Fig. 1i(b). Q is the transverse
shear force, ™ is the bending moment, T is the axial tension,

F, and F, are the normal and longitudinal components of frictional
forces per unit length, and Fy, is the lateral inviscid force per unit
length.

983

Patdoussis

—
we
ce
ais

[ove nev,
~*~ |e
ao
tad

x

Fig. 1(b) Forces and moments acting on an element
a(x) of the body

Taking force balances in the x- and y-directions and a moment
balance we obtain

aT dy _

Sx0 Fu En Bam (1)
aQ 8 in by dy a*y _

da 7 Fut q(T By) + FL GR - Fa mogee = 0 ®
am

Q= a3. 9 (3)

where the inertia forces in the x-direction have been neglected.

We next consider the functional form of the forces. The
lateral inviscid force F,5x represents the reaction on the body of
the force required to accelerate the fluid around it, and may be
written as

F, = [(2/at) + U(8/dx)] (Mv), (4)

as discussed by Lighthill [ 28], [29], where v is the lateral relative
velocity between the body and the fluid flowing past it, and M is the
virtual mass of the fluid. Here the effects of sideslip have been
neglected, effectively assuming that each cross section of the body is

984

Dynamites of Submerged Towed Cylinders

part of an infinite cylinder; boundary layer effects have also been
neglected. The virtual mass M(x) = pS(x), and v(x,t) = [ (8/8t)
+ U(8/8x)][ y(x,t)] , which substituted into (4) yield

F, = pS[(8/at) + U(8/ax)]*y + pU[(dy/at) + U(dy/8x)] (dS/dx). (5)

The frictional forces, as proposed by Taylor [ 30], and elabo-
rated by Paidoussis [ 24], [25] are taken to be

tr}
"

2¢,(PS/D)U* sini and F, = 3¢,(pS/D)U* cos i,

where i is the instantaneous angle of incidence on the cross-section
and is given by i= sin! (v/U), and D= D(x) is the diameter.
Accordingly, Fy, and F, are given by

Fy = 2¢y(PS/D)U[(ay/at) + U(ay/ax)] and F, = 2¢,(PS/D)U?. (6)

Finally, we note that the bending moment is related to the flexural
rigidity by

M = El(0*y /ax?). (7)

Now, substituting (6) into (1), neglecting terms of second order
of magnitude, and integrating from x to L, we obtain

L
T(x) = T(L) + e,pu*{ [ S(x) /D(x)] dx,
x

where T(L) is the value of T at the downstream end. We consider
that T(L) is non-zero and that it arises from possible form drag
at the end. We accordingly write

e
2
T(x) = 2copS(L)U nF 2c, purl [ S(x) /D(x)] dx, (8)
x
where cy, is the form-drag coefficient.

Substituting now (3), (5), (6), (7) and (8) into (2), making use
of (1), and neglecting terms of second order of magnitude, we obtain

985

Patdoussts

2 2 os
&, (er 24) + 05(2 tuhyy +eu(R+u¥) ss

+ 5 uC) + Us

i 2
-5 ous [¢,S(L) + J oo ax |33 tm2y =0, (9)

xX

which is the equation of small lateral motions. For a uniform cylin-
der this equation becomes

eroy +u(2 ru zyy +5 onl (ge + Use

2 2
1 2 L-x] 9 8 4
-3 MU [cg +c D |< +m Be = 0, (10)

where the diameter, D, and M=pS are now constant.

We note that in the absence of frictional forces, (10) becomes
the governing equation for small motions of a cylindrical beam con-
taining flowing fluid [31], where we interpret M as the mass of the
contained fluid per unit length. The physical similarity between the
internal and external flow cases is striking, albeit that in the former
case fluid friction does not enter the problem. We shall refer to this
later.

We finally note that Eqs. (9) and (10) also hold to describe the
motions of a towed flexible body, if we identify U as the towing
speed, provided the tow-rope forces are taken into account as part
of the boundary conditions.

III. BOUNDARY CONDITIONS

Clearly the boundary conditions will depend on the mode of end
constraint. Let us consider the case of a towed flexible cylindrical
body shown in Fig. 2. The body consists of a uniform cylinder ter-
minated by a rounded 'nose' and a streamlined, tapering 'tail', incor-
porated to provide reasonable axial flow conditions over the body.

We assume that the towing craft moves horizontally in a straight
course with uniform velocity U, so that the tow-rope in its undis-
turbed state lies along the x-axis; we also consider the assumptions
made at the beginning of §2 to hold.

We may use Eq. (9) to analyze the system, together with
boundary conditions stating that (a) at the downstream end, x=L,

986

Dynamtes of Submerged Towed Cylinders

Fig. 2 Diagram of a towed flexible, slender cylinder
with streamlined "nose" and "tail"

the bending moment and shear force are zero, and (b) at the upstream
end, x= 0, the bending moment is zero, but the shear force is equal
to the normal component of the tow-rope pull. It is obvious, however,
that the very form of Eq. (9) will depend on the shape of the nose and
the tail. As we are only looking for the general characteristics of the
dynamical problem, this is not convenient. We shall instead proceed
as follows: (i) we shall use Eq. (10) which satisfactorily applies

over the uniform, cylindrical part of the body; (ii) the forces acting

on the non-cylindrical ends will be lumped and incorporated in the
boundary conditions. For this process to be meaningful we must have
4, <<L and £,<<L, where £, and £, are the lengths of the nose

and tail, respectively; yet £, and Ly are considered to remain great
enough to permit the use of slender hody. approximations [ 23], [ 26].
Since £, and £) are small, compared to L, we may further simplify
the problem by considering y and the lateral velocity v to be approxi-
mately constant over OS xf, and L- 4, =x=L, and by neglecting
the skin frictional forces over the same intervals. Hence, integrating
Eq. (2) and using (4), and incorporating the forces arising from the
tow-rope pull, P, we obtain

f iad ne
wer ag 19" (ZF +uZ) bls a wean,

and

8Q 8 8 3 sta
{ 29 ax-£,{ (& + vB) (esv ax -f m 34 dx = 0;
Lt mt Llp

the parameters f, and f,, which are equal to unity according to
slender-body inviscid-flow theory, were introduced to account for
the theoretical lateral force at the nose and the tail, respectively,
not being fully realized because of (a)-the lateral flow not being truly

987

Patdoussis

two-dimensional, (b) boundary-layer effects, and (c) the use of fins.
The tow-rope pull P is equal to the tension in the cylinder at x=0,
plus the form drag at the nose, i.e.

P= }MU%(c, +c, +c,L/D).

Substituting P into the above equations and assuming y and v to
be constant over the intervals of integration, we obtain

9° 8 F)
[er Sy +emu(3 + ue

2
i
+3 MU (cy areca. dace! < + (m + £,M)x, eI = 0. «(f4)
; hs 8 8 9 3
[- EI5 - fpMU ( 5¥ + U5.) + (m +eM)x, 24] =, Os ffeil 42)

where
1 f, ict
x,=5 i S(x) dx and x,= 45 S(x) dx.

Here M=pS, and S and D are quantities pertaining to the cylindri-
cal part of the body.

In the above the forces arising from form drag were taken to
be inthe x-direction. If they are taken to be along the cylinder,
then terms }c,MU“(8y/8x) and $c,MU%dy/8x) should be added to
Eqs. (11) and (12), respectively.

The other two boundary conditions are obtained by making the
reasonable assumption that there are no bending moments at x = 0
and x=L, or

[ yo [31 Os (13)

The advantages in this method of analysis, in which the shape
characteristics of nose and tail were absorbed in the two parameters
f, and f3, are obvious. The disadvantages are equally obvious: al-
though we can estimate f, and f,, we cannot easily calculate them.

988

Dynamtes of Submerged Towed Cylinders

The range of f, and f, will be taken to be between zero and unity,
the latter limit representing well-streamlined, gradually tapering
nose or tail, and no flow separation; it is obviously much more likely
for f, to approach bared than for f,. In the case of a blunt tail, on
the other hand, fp —>

IV. EQUATION OF MOTION AND BOUNDARY CONDITIONS OF [ 26]

The equation of motion given by Eq. (10) is not identical to
that previously derived by Paidoussis [ 26]. The difference is in the
frictional terms, because of the different manner in which frictional
forces were resolved in [26]. The boundary conditions are identical.

As we shall make use of the results obtained in [ 26] , we give
the equation of motion below, for reference.

4

erge+m(2+uZy)y+3e(MZ)(¥
bc, ( MU ates = 0. (10a)

The equation of motion and boundary conditions used in the
‘new theory' presented in this paper are believed to be more self-
:onsistent than those of the 'old theory' of [ 26].

989

Patdoussis

Ve. DYNAMICS OF TOWED FLEXIBLE CYLINDERS

5.1 Method of Analysis

Upon expressing the equation of motion and the boundary con-
ditions in dimensionless form, the dynamics of the system may be
found to depend on the following dimensionless parameters:

(i) £, and f,, which were defined in §3;

(ii) €cy, €c,, c, and c,, where € = L/D;

(iii) A= s/L; the ratio of tow-rope length to body length;

(iv) x, =x,/L and x, =x,/L, where x, and x, were
defined in §3;

(v) u=(M/El)'”UL, the dimensionless towing speed.

It is noted that according to the assumptions made in the theory,
m=M.

We shall not present the analysis here, as it is adequately
documented elsewhere [26], [27]. Suffice it to say that solutions
were obtained of the type

where Y is a function of x/L, T is a dimensionless time and w
is the dimensionless frequency given by

Q being the circular frequency of motion. In general, w will be
complex. Clearly, we have an infinite set of frequencies, W;, as
the system has an infinite number of degrees of freedom. If the
imaginary components of the frequencies, Im/(wj;), are all positive,
then the system will be stable. If, on the other hand, for the jth
mode we have Im (w;) < 0, then the system will be unstable in that
mode; now if the corresponding real component of the frequency,

Re (w, ), is zero this will represent a divergent motion without oscil-
jailons: which we shall call yawing; if Re (wj) # 0, then the insta-
bility will be oscillatory.

990

Dynamics of Submerged Towed Cylinders

The calculation procedure was as follows: (a) a set of values

GE tp fp» EC.» ECL, Cy» Co» Xj» X_ and A were selected; (b) the
complex frequencies of a few of the lowest modes of the system were
traced as functions of u, starting with u = 0.

MERGES WITH FIRST MODE

3.6

3.5

ZEROETH

Fig. 3

WwW
a
a
%
2
> )

MODE

(A x
Re (w)=0 Re(w)

The dimensionless complex frequencies of the zeroth and
first modes of a flexible cylinder with ecy=ec,= 1,
f,=f,=1,¢, = co = 0, A= 1 Ne = 2 = 0.01, as a function
of the dimensionless towing speed u. (Theory of [ 26]).

991

Patdousstis

SECOND MODE

40 50 60

Re (w)

WwW
a
re
”
z
5

-0.2

fe) 20

Fig. 4 The dimensionless complex frequencies of the second and
third modes of a flexible cylinder with ec,=ec,=1,
f,=f,=1, c,=c,=0, Aas Ge Xj =e = 0.01. (Theory

5.2 Results Based on the Theory of | 26

Typical results are shown in Figs. 3, 4 and 5, obtained by
using Eqs. (10a), (11), (12) and (13).

We first consider Figs. 3 and 4 applying to bodies with well
streamlined nose and tail, and A=1, Figure 3 shows the behavior
(with increasing towing speed) of the two modes which at zero towing
speed have frequencies w)=w,=0; these are the so-called zeroth
and first modes and, at low towing speeds, are associated with quas!-
rigid body motions -- a matter to be further discussed in §6. Figure
4 shows the loci of the so-called second and third modes of the system
as functions of towing speed. The frequencies of these modes at zero
towing speed correspond to the second- and third-mode frequencies
of the flexible body treated as a free-free beam; accordingly, these
(and all higher modes) are flexural in character.

992

Dynamics of Submerged Towed Cylinders

We observe in Fig. 3 that both the zeroth and first modes
lead to instabilities for small, finite u. The instability associated
with the zeroth mode is a yawing one, while that associated with the
first mode is oscillatory. We see that for u> 3.05 the oscillatory
instability ceases in the first mode, re-appearing at u* 3.65.
However, at much lower towing speed (u* 2.3) the system loses
stability in its second (flexural) mode, as shown in Fig. 4, and at
u* 4in its third mode. In short, this particular system is subject
to several types of instabilities; at low towing speeds it is subject to
quasi-rigid body instabilities, and at higher towing speeds to flexural
oscillatory instabilities as well.

Figure 5 shows the zeroth and first mode of a system with a
well streamlined nose and a very blunt tail. We see that it is not
subject to yawing instability, and the first mode is only unstable in
the range 0<u<0.9. It is, however, subject to flexural oscillatory
instability (not shown) in its second mode for u> 5.29. Accordingly,
a blunt tail stabilizes the system considerably. Also shown in Fig. 5
is the first mode of a system with a less than perfectly streamlined
nose; we see that the range of first-mode oscillatory instability in
this case is larger, i.ewg O<u< 1.75.

FIRST MODE (f, =!)

0.4 4 FIRST MODE (f, = 0.8):
DS 03
Im(w)
: 02 Me CROSSES TO STABLE REGION
: AT u=0.9
0.5 0.1 ae
ey

ZEROETH MODE (f, =!)

UNSTABLE

=lh
Re(w)=0

3 4
Re (w)

Fig. 5 The dimensionless complex frequencies of the zeroth and
first modes of a flexible cylinder with cole €cr=i,
f,=1, c= 0, f,=0, co=1, A> 4d, = 0, Ol. Also
the first mode with f, = 0.8. (Theory e [ 261).

993

Patdoussts

Based on such complex-frequency calculations it was possible
to construct stability daigrams illustrating the effect of various
parameters on the stability of the system. Examples are given in
Figs. 6 and 7 showing the effect of stability of f, and A, respectively,
Other similar stability diagrams may be found in[ 26]. The following
general conclusions may be drawn:

(a) for optimal stability the tail should be blunt (f, small,
cy large), the nose should be well-streamlined Oe ae)
and the tow- rope length should be short (A small);

(b) asystem that is.unstable by yawing, within a range of
towing speeds, can be stabilized by blunting the tail, but
not by manipulating the length of the tow-rope;

(c) in some cases it is possible to stabilize a system which
is unstable at low towing speeds, by towing it faster,
within a specified range of towing speeds.

Conclusions (a) above are not contrary to reported experience
with rigid bodies. On the other hand, (b) may sound surprising. The
fact is that the onset of yawing is not a function of A, nor is its
cessation (§6.2). This is also true with f,, Finally, conclusion (c)
is characteristic of the dynamical behavior of towed flexible cylinders.

6

SECOND MODE

OSCILLATORY INSTABILITY FIRST MODE

5 OSCILLATORY INSTABILITY

fy,

‘

Oy

Ne
SS

4 ae

u STABLE REGION ZEROETH MODES aus

FIRST MODE

Za

YAWING

YAWING AND OSCILLATORY
INSTABILITY
FIRST MODE
OSCILLATORY INSTABILITY

0 0.1 0.2 0.3 0.4 O15 0.6 0.7 0.8 0.9 1.0

BEUNT TAIL ———— f, ——-» ELONGATED STREAMLINED TAIL

Fig. 6 Stability map showing the effect of the tail shape fora

flexible cylinder with ecy= ec;=1, f,=1, c= 0; A=
X, =X> =0001 and c,=1-£, (Theory of 126i fe

994

Dynamies of Submerged Towed Cylinders

/
/

/
THIRD - MODE
OSCl LLATORY INSTABILITY

FIRST- MODE
OSCILLATORY INSTABILITY

SECOND - MODE
OSCILLATORY INSTABILITY

SECOND MODE

STABLE REGION

~
wor
YAWING A/S YAWING AND FIRST- MODE
FT OSCILLATORY INSTABILITY
| va
fo)
0.1 0.2 0:5 o:5 | 2 5 5 10

A

Fig. 7 Stability map showing the effect of A for a flexible cylinder
with €cy=ecy=1, f,= 1, c,=0, f,=0.6, cy = 0.4,
X, =X, = 0.01, (Theory of [ 26]).

Some experiments were performed designed to test the theory
[26]. Rubber cylinders of neutral buoyancy were held in vertical
water flow by a nylon 'tow-rope'. Provided the tail was streamlined
and the tow-rope not too short, 'criss-crossing', essentially non-
flexural oscillations developed at very low flow; these were inter-
preted as corresponding to first-mode oscillatory instability. At
higher flow velocities, flexural oscillations developed with a modal
shape corresponding to that of the second mode; sometimes, at yet
higher flow velocities, oscillations with a third-mode modal shape
developed. These flexural oscillations were interpreted to cor-
respond to second- and third-mode oscillatory instabilities, Se-
quences of ciné-film frames depicting these oscillations are shown
in Figs. 8 and 9. Finally, it was observed that for sufficiently blunt
tail and short tow-rope, the system was completely stable. Thus
the experimental results were in generally good qualitative agree-
ment with theory.

995

Patdoussts

(a) (b)

Fig. 8 Photographs in consecutive frames showing a cylinde.
11.1 in.long and 0.54 in.diameter with streamlined nose
and tail executing (a) criss-crossing, essentially rigid-
body, oscillation (8 frames/sec), and (b) second-mode
flexural oscillation (24 frames /sec)

Quantitative agreement in the various instability thresholds
and stable zones, based on estimated values of some of the theoreti-
cal parameters, was also fairly good.

5.3 Results Based on the New Theory

Typical results based on the new theory and obtained by using
Eqs. (10) - (13) are shown in Figs. 10 to 13.

996

Dynamtes of Submerged Towed Cylinders

(a) (>) (c)

Fig. 9 Photographs in consecutive frames showing a cylinder
15.8 in. long and 0.68 in. diameter executing (a) criss-
crossing, essentially rigid-body, oscillation (8 frames/sec),
(b) second-mode flexural oscillation (24 frames/sec),
and (c) third-mode flexural oscillation (24 frames/sec)

Figures 10 and 11 show the dynamical behavior, with in-
creasing towing speed, of the zeroth, first, second and third modes
of a system with well streamlined nose and tail and A = 1; this is
the identical system, the dynamical behavior of which, according to
the 'old' theory, is shown in Figs. 3 and 4. We observe that, accord-
ing to the new theory, the system is considerably more stable than
predicted by the old theory. Thus, the first mode is unstable only
for u<0.74 (not discernible in the scale of Fig. 10); moreover,
the unstable locus originating from merging of branches of the
zeroth and first modes regains stability at u = 6.3. Similarly, the
system loses stability in its second and third modes at respectively
higher towing speeds than predicted by the old theory.

Further calculations were conducted for the same system as
above but with other values of f,, always taking c,:= 1\- foe It was
found that the first mode is not uniformly stabilized with decreasing
f5 as was the case with the old theory (cf. Fig. 6). The ranges of
instability of the first mode, for various values of fo, were found
to be as follows: 0<u<0.74 for f,=1; 0<u<41.66 for f,= 0.8;
O<u< 1.65 :for f,= 0:6; and '0 <u < 0.70 for f, = 0.4. Thus the

POF

Patdoussts

UNSTABLE—~—|—~ STABLE

36,

3:5

Re@)=0O Re@)

Fig. 10 The dimensionless complex frequencies of the zeroth and
first modes. of,a flexible cylinder with 4, =%4 - cy =f
fo=1-cpg=1, Ecy=ecy=1, A=1, x, =x, =0.01, as
a function of the dimensionless towing speed u. (New
theory).

curve corresponding to that relating to the first mode in Fig. 6 will
now exhibit a maximum at f,< 1; i.e. the system is least stable

in its first mode, not for a perfectly streamlined tail as predicted
by the old theory, but for a somewhat less perfectly streamlined
tail. The second and third modes, on the other hand, are both un-
conditionally stabilized as the tail is made blunter; thus, the
threshold of instability of the second mode is at u = 2.83 for f,=1,
at u= 3.85 for f2= 0.6, and at u= 4.38 for f,= 0.4.

Figures 12 and 13 show the dynamical behavior of a system
with €cy=ec,= 41, A= 1, XX. = 0.08, £, = 007, ve, = 0) and
fo=1-cp)=0.7. We note that the effect of a less than perfectly
streamlined nose is to destabilize the system in all its ocillatorv
modes. Thus the first mode is unstable for 0 <u< 2.80, and the
second and third modes lose stability at, respectively u = 3.21 and
u = 5.40 (cf. values given above).

Also shown in Figs. 12 and 13 (dashed line) is the behavior
of a system with €cy= €cy= 0.5 and all other parameters the same.

998

Dynamites of Submerged Towed Cylinders

O 10 20 40 50 60 70
Re(@)

Fig. 11 The dimensionless complex frequencies of the second and
third modes of a flexible cylinder with f,=1- c, = 1,
fg=1-cp= 1, ecy=ecz=1,A=1, xy, = xX, =0.01. (New
theory).

This system is unstable in its first mode for 0 <u < 3.43 and loses
stability in its second and third modes at u= 3.04 and u= 5.14,
respectively. As we may regard the smaller values of ec, and

€c,; to represent a smaller € = L/D, we may conclude that reducing
the slenderness of the system renders it less stable.

Other similar complex frequency calculations establish that
the general conclusions regarding optimal stability are essentially
identical to those given by the old theory, in spite of quantitative
differences in the thresholds of instability. The main difference
appears to be that the first-mode instability is less extensive, in
terms of the range of parameters over which it is possible, than
predicted by the old theory. (Here it should be mentioned that these
calculations are still in progress and that stability maps of the type
of Figs. 6 and 7 are not yet available.)

The question now remains on how well this theory is capable
of predicting the experimentally observed dynamical behavior of the

ae

Patdoussts

FIRST MODE +

uW

=|

x

4 a
0 |

uJ

=)

4 a

5

3, 2 Zz
=)

4 10
Re(@)= O Re(@)

Fig. 12 The dimensionless complex frequencies of the zeroth and
first modes of a flexible cylinder with f,= 0.7, c¢, = 0,
foe P= cp = 0.7, AH; ‘ecy= ecp= 1, %, = = 0.01 (——).
Also shown (---), portions of the zeroth and nee modes
with €c,= €c,;=0.5. (New theory).

system. The observed behavior of flexible cylinders with increasing
towing speed [ 26] can be summarized as follows: (a) at low towing
speeds a 'criss-crossing' oscillation developed in which the cylinder
inclination was of opposite sign to that of the tow-rope; (b) at
slightly higher towing speed, sometimes a narrow region of stability,
or a region of stationary buckling, was observed; (c) at higher
towing speeds, second-mode, and at yet higher towing speeds,
third-mode flexural oscillation developed. The above are typical
observations provided that the tail is not blunt and the tow-rope not
too short; if they are, then the system remains stable for apparently
all towing speeds.

We first note that, in terms of qualitative agreement, the
results depicted in Figs. 12 and 13, for instance, agree with the
experimental observations. Thus, at very low towing speeds the
system is subject to first-mode oscillatory instability and yawing,
the former ceasing at slightly higher towing speeds, while yawing
persists (presumably corresponding to the observed buckling). At
yet higher towing speeds, the second mode loses stability, followed
by the third mode at even higher towing speeds.

1000

Dynamics of Submerged Towed Cylinders

0.4

Im@)

- 0.2

Re(@)

Fig. 13 The dimensionless complex frequencies of the second and
third modes of a flexible cylinder with f, =0.7, c,=0,
fo-=1-c,=0.7, A=1, x, =x, = 0.01; ——- ec, = €c,= 1;

’ T
--- €c,=€c,= 0.5. (New theory).

We next consider quantitative agreement for one specific
case, the details of which are given in[ 26]: a cylinder with quite
well streamlined nose and tail, € = 20.4 and A=1 (cf. Table 2
of [ 26]). Theory is compared with experiment in Table 1. The
rationale for the choice of parameters used to obtain the theoretical
values has been discussed in [ 26] and will not be unduly elaborated
here. The parameters usedare e€cy=e€c;=1, A=i, £,=0.8,
c,=0, fp=1-cp9= 0.7, X; =X. = 0.01. It is noted that, although
the tail is quite well streamlined, f,< 1 and cy# O were taken
(cf. | 26]), as the tail cannot be considered to be perfect in the sense
described in §3, i.e. with regard to two-dimensionality of the lateral
flow and lack of separation in the axial flow. On the other hand, the
nose, although of identical shape to the tail, must have a value of f,
nearer unity, as no separation takes place over the nose. Accordingly,
f,= 0.8 and c, =0 were-taken in the new theory} (the calculated
values of the old theory, also given in Table 1, were obtained with
f,= 1, which is considered to be unrealistic, as the lateral flow over
the nose is no more truly two-dimensional than over the tail).

1001

Patdoussts

TABLE 1
THEORY COMPARED WITH EXPERIMENT

Description

Criss-crossing oscillation

(first-mode osc. instability)

Stationary yawing (zeroth-
3% 3-3 e 8

mode instability alone)

Second-mode oscillation

threshold

Second-mode with first-mode
4,.4-7.2 4.4-6.8
oscillation superposed

Third-mode oscillation

threshold

fe
€cy= €c,=4, £/=\1-¢c, = 1, f2= i-ep = 0.7, Ali=4y.X, =X,

same as above, but f,= 0.8, c,=0.

Psame as for Tf, but f,= 1-c,= 0.8.

We see that agreement between the new theory and experiment
is comparable to that between the old theory and experiment. No
more can be said at this stage, until a more extensive comparison of
the new theory with experiment has been undertaken, and also because
of the several incompletely tested assumptions involved in the deter-
mination of the values of the system parameters. Agreement between
the new theory and experiment is least satisfactory in predicting the

1002

Dynamites of Submerged Towed Cylinders

point of cessation of criss-crossing oscillation. Here it might be
argued that f,= 0.7, c,=0.3 may be too severe for low towing
speeds; comparison with theoretical values calculated with f,=1-c)=
0.8 {shown in parentheses in Table 1) yields better agreement, as
anticipated.

VI. DYNAMICS OF TOWED RIGID CYLINDERS

6.1 The Equations of Motion

We consider exactly the same configuration as in Fig. 2, but
impose the restriction that the body be rigid. In this case the system
is reduced to one of two degrees of freedom. The generalized co-
ordinates may be taken to be the lateral displacement of the center
of mass, y,, and the angle that the body makes with the x-axis, 4,
Accordingly, the displacement at any point is given by

yY=Y_o +t xo (14)

For the sake of simplicity, we assume that the center of mass coin-
cides with the geometric center of the body, x, and x, being small.
For convenience we now measure x from the center of mass, so
that the body extends from x= - L/2 to x=L/2.

Instead of deriving the equations of force and moment balance
independently of the previous work, we shall proceed as follows.
We shall integrate Eq. (10) or (10a) formally to obtain an equation
of force balance, and similarly integrate the product of the forces
in (10) or (10a) by x, to obtain the equation of moment balance. The
boundary conditions are incorporated through the integral of the
first term of these equations; alternatively, the shear forces at the
ends may be viewed as forces replacing the effect of nose and tail on
the main part of the body.

Thus using Eqs. (10) to (14) in the manner described above,
we obtain the following two equations:

[M(L +x f, +x,f,) +m(L +x, +x] ye

+[5¢MUL/D +£, - £] y, +35 MUc_y

c 2s c
1 ss fetes
- 5[m(x, - x,) + M(x,f, - x,f,)] L¢+ MUL [2 - fff] )
ae ae o 1 L
+MU [sey tf - fa- GF cr d= 0, (15)

and

1003

Patdoussts

zs
2

+ oat +P) + mal AN] 3

1 ee e
- sl ML(x,£, =—!%f5) + mL(x, - x,) | Yo > MULE CT ty.

L 2
fxg MU Srey

erf.-f,, 1. L7°
+ MUL [p< +saayp]¢

11 f, +f,
@MUL 3 erp 5 Je=o, aL

where Crp= roe By (1B +c, tCo. The same equations could have been
obtained from first principles. It is noted that here L is the length
of the cylindrical portion of the body which is smaller than the over-
all length, as used in §5, by L, +45, the difference never exceeding
a few per cent.

We non-dimensionalize these equations by introducing n= y,/L,
A=s/L, €=L/D, x, = x*,/L, Xp = x,/L and 7 = Ut/L, and consider
solutions of the form

1=He”* and d= @e

where w is a dimensionless frequency defined as w= QL/U,
being the complex circular frequency of oscillation. Substituting 7
and $ into the non-dimensionalized equations, and noting that our
assumptions require m = M, we obtain

{[2 +x, (1 +4) +xe(t t£)] (-04) +[ 5 ecy +4, -£,] wi) +[ 5c, /A]{ H
+ {- Fix, +8) - x, H6)] (-04 +L 2-5 (E, +6] (oi)
+[5 Ecytf,- f,-4 (1/Mcql | ® = 0, (17)

and

1 + Xi (a +2) - 4 Xp(4 + £,)(-o*) -[ Sf, 7£,) | (wi) - [ 5 Cr p/4\) 1)
+ \[Z +4 x) (te, Sxl! + £,)] (-w*) +S, - f,) +3 €cy] (wi)

+ [5 (1/Mcyp- 5 (ie sito] i = is (18)

1004

Dynamics of Submerged Towed Cylinders

Similar equations were obtained when using the theory of [ 26],
i.e. Eqs. (10a), (11), (12) and (13), namely

}[2+x, +4) +xo(1 +£,)] (- ) +L Ecytf, - fal (wi) +15 cy)/A]{ H
at }- 4 [x,( +f) - X2(4 + f5)] (-w*) 7 [ 2 - Fig, +£,)] (wi)

+5 ele, he) +£,-£,-4cqp/A] 1 = 0, (17a)

}- [5 x14 +) - Sxolt +e] (-04 -[ 4 +2)] (od) - [4 c,,/Al tH
+} taxi (+8) +4 xo Hey) (-04) +14 (e- 6) + A ecy] (wi)

tlgcp/A-F(, +4)]1 6 = 0, (18a)

For non-trivial solution, the determinant of the coefficients of
H and ® in (17) and (18), or in (17a) and (18a), must vanish, yielding
a quartic in w,

i PAG Bao eco =o, (19)

6.2 Calculations Based on the Theory of | 26]

The aim here was to compare the dynamical behavior of the
rigid body to that of a flexible body; as the rigid body may be regarded
as a flexible one of very large flexural rigidity, it would be reasonable
to expect correspondence of the dynamical behavior of the rigid body
to the 'rigid-body' modes of the flexural one, i.e. the zeroth and
first modes. Recalling that the dimensionless flow velocity in the
case of a flexible body was defined as u = (M/EI)'/* UL, the dynamical
behavior of the rigid body should approach that of the flexible one as
uO. Two sets of calculations were conducted, as described below.

The four rigid-body frequencies, given by (19), were computed
for a number of cases and the values compared with the existing com-
plex frequencies of the flexible body. As an example, let us compare
the case corresponding to Fig. 3. The four frequencies are w, = 1.956,
we= -0.761, w,,= + 0.582 - 0.3571. These compare well with the
four frequencies associated with the flexible body for u=0.7,

namely w, = 1.934, O5.=9 0. 734, = + 0.580 - 0.350i; the first

ek

1005

Patdoussits

two are associated with the zeroth mode, and the other two with the
first mode and its mirror image about the [ Im (w)] -axis.

Surprisingly, the correspondence of the rigid-body frequencies
to those of the flexible body, for the apparently arbitrary value of
u = 0.7 persists for other values of f,, as shown in Table 2. This
value of u= 0.7 can be explained as follows. We have defined the
dimensionless frequency of the rigid body by w,p= QL/U. On the
other hand, the dimensionless frequency of the flexible body was
defined as w,,= [ (M+m)/EI]’“QL*, which may be rewritten as
o.,=[(M +m)/M]/2u2L/U, where u is the dimensionless flow
velocity (§5.1). The assumptions made in the theory require that
m-=M, sothat wy,= V2uQL/U. Now, if the dimensional frequency,
22, of the rigid body and of the flexible body are identical, we may re-
write this as Wey = ANE and we can see that identity of the dimen-
sionless frequencies will occur when u = 1//2* 0.707,

Calculations were also conducted to pin-point the thresholds of
yawing and oscillatory instability in terms of f,, A etc., and to
compare with the existing stability diagrams, e.g. Figs. 6 and 7.

TABLE 2

RIGID-BODY AND FLEXIBLE-BODY FREQUENCIES COMPARED
Other parameters: A=1;€cyaé€cps1, f) = t-c, = 1, 1¢, = 1h
X; =X2 = 0.01

Rigid body Flexible body (u = 0.7)

0. 58-0. 36i 0.58-0.35i
0. 84-0. 39i if 0. 83-0. 38i

4.28-0.04i 1. 28-0.004i

In the case of a rigid body with parameters corresponding to
those of Fig. 6, it was found that oscillatory instability exists for
0=f,=1 and that yawing occurs for f,>0.5. Correspondingly, for
the case of Fig. 7 it was found that yawing persists throughout, and

"Upon examination, the st: »le branch of the zeroth mode as given in[26]
was found to be in error; the locus moves away from the [Re (w)] -
axis much faster than shown in Fig. 3, | 26]. The corrected value
for u= 027 is given here.

1006

Dynamtes of Submerged Towed Cylinders

that oscillatory instability occurs for A>0.20. Once again agree-
ment in behavior of the rigid body and the flexible body (for u = 1/2)
is good. Similar calculations confirmed agreement with the other
stability maps of [ 26].

Two stability diagrams were constructed (Figs. 14 and 15)

showing the effect of ECy, €Cr, fp and A on stability, for comparison
with those to be obtained using the new theory.

a

YAWING
a
Pg

Fig. 14. The effect of €cy, €c; and f, on stability of a
rigid cylinder with A= 1, f, = 1-c,=1,
Co= 1-f2 and x, oa 01, —— ec, = 0.1;
=-— €¢,= 0,5; --- ec,=1, (Theory of [26]).

In Fig. 14 we observe that unless ec, is considerably less
than €cy, the region of oscillations practically covers the whole
plane; moreover, oscillations persist to lower values of f, than
yawing does.

In Fig. 15 we see that a sufficiently short tow-rope has a very
definite stabilizing effect on the system, as far as oscillatory insta-
bility is concerned. Very long tow-ropes, on the other hand, evi-
dently have a very weak stabilizing effect.

The foregoing clearly establish that the dynamical behavior
of the rigid body is represented by the behavior of the zeroth and
first modes of the flexible body at small u.

One noteworthy aspect of the analysis is that the existence of
yawing instability cannot be affected by varying A, i.e. by altering

1007

Patdoussts

OSCILLATIONS

Fig. 15. The effect of A on stability of a rigid cylinder
with f,=1-c,=1, ecy=ec,=0.5, Cy = 1-f

2
and xX, =X, = 0.01. (Theory of [26]).

the tow-rope length. In the case of the rigid body this becomes
obvious upon considering equation (19). Since the threshold for
yawing instability implies w= 0, this threshold is established by
the equation E=0. Now E is found to be

E = (c4p/2A)[5 €(cy + Cy) - 2£,] é

Clearly we see that the threshold is not dependent on A. This seems
to be in contradiction with Strandhagen's et al. [1] criterion (iii) for
the stability of towed ships (as given in §1); on closer examination

of their own work, however, we see that the equivalent of term E,

in their case also, contains A as a common factor. Accordingly,
we must conclude that the only form of instability the existence or
non-existence of which may be controlled by the tow-rope length is
oscillatory.

6.3 Calculations Based on the New Theo

Calculations were also conducted with the new theory. It was
found that, in this case also, the dynamical behavior of the rigid
body corresponds to that of the zeroth and first modes of the flexible
one at low towing speeds -- quantitative correspondence of frequencies
occurring at u=1/72 as before.

Stability plots were also constructed (Figs. 16and17). These

are markedly different to those given by the old theory (Figs. 14 and
15), the main difference being in that oscillatory instability according

1008

Dynamtes of Submerged Towed Cylinders

OSCILLATIONS (ec =1)

fe

OSCILLATIONS (€c,=0-5)

OSCILLATIONS (€ c;= eae
se

OSCILLATIONS(€c,=1)_ —
ees

OSCILLATIONS =
(€c,=1) {-- <=

Fig. 16 The effect of. €c,, €c, and f, on stability of
rigid cylinders with A=1, c,=,1-f,, c¢) =0
and xX, = Xo= Oe Oty a ee f, = 0.8;
--- f, = 0.7. (New theory).

OSCILLATIONS

Fig. 17 The effect of A on stability of a rigid cylinder;
Co= 1-fp, ECy= €Cp7= 0-5, X,; = Xo = 0,01;
— f, = 1-c, = 1; ---f,=0.8,c,=0. (New
theory).

1009

Patdoussts

to the new theory occurs over a much more limited range of system
parameters, while yawing is more prevalent.

Comparing Fig. 16to Fig. 14 we note the following essential
differences: (i) yawing, being independent of c, according to the
new theory, is represented by a single line; (ii) according to the
new theory oscillations persist to progressively lower values of fp,
as €cy, is reduced, while the opposite trend was predicted by the
old theory; (iii) according to the new theory, for f, = 1, there
are large regions in the (€cy, f,) parameter space where yawing
occurs alone, but not where oscillations occur alone; on the other
hand, according to the old theory the opposite is true. However,
this last point applies only for f, = 1. It may be seen that for f,= 0.8
and 0.7, the results of the new theory become much more like those
of Fig. 14 in this respect.

We note that the onset of yawing is independent of f, as well
as Cy, so that the line shown in Fig. 16 applies to all cases examined
therein. Once again considering term E of Eq. (19), which in this
case is given by E = (CAN €c,- 2f,), we see that £,, Cy, Cos Cy
and A are all parameters that cannot affect the onset of yawing.

We next compare Fig. 17 to Fig. 15. The results are quite
similar, except that (when f, = 1) oscillatory instability occurs over
a more limited range according to the new theory than predicted by
the old theory. However, the results of the new theory for f, = 0.8
when compared with those of the old one for f, = 1 are quite similar.
The results for f, = 0.7. not shown in Fig. 17, are of interest in
that oscillatory instability, in that case, occurs practically over the
whole plane, i.e. for fg>0.013 for A=0.1 and for f,> 0.008 for
N= 02.

VII. CONCLUSION

In this paper we have reviewed an existing theory for the dy-
namics of flexible cylindrical bodies towed underwater, and developed
a parallel theory for rigid cylinders. It was shown that, whereas the
dynamical problem in the case of rigid cylinders is independent of
towing speed, in the case of flexible cylinders the dynamical behavior
(and stability) of the system is highly dependent upon towing speed.

It was found that, in general, flexible towed cylinders are subject

to both flexural and 'rigid-body' instabilities, the latter occurring

at relatively low towing speeds. It was also established that at low
towing speeds, the dynamical behavior of the flexible cylinders in
their two lowest modes (the so-called zeroth and first) correspond to
that of rigid cylinders, which of course have but two degrees of free-
dom. Thus the study of the dynamics of towed flexible cylinders
yields sufficient information to establish the dynamical behavior of
the corresponding rigid bodies.

1010

Dynamics of Submerged Towed Cylinders

A new theory was also presented (for both flexible and rigid
cylinders) which, it is believed, represents the physical system
more closely. The main difference in the results obtained by the
old and new theories are associated with the behavior of the rigid-
body modes of the system; specifically, the new theory predicts the
system to be more stable in its first (oscillatory) mode and less stable
in its zeroth (yawing) mode than does the old theory.

The new theory is in general qualitative agreement with ex-
periment. Quantitative agreement cannot be assessed definitively
until a means is found for accurately determining the values of some
of the dimensionless system parameters, particularly f, and fp.
Nevertheless, it is possible to make intelligent estimates of these
parameters based on experience from other experiments [25]. On
that basis quantitative agreement between theory and experiment, for
one particular experiment (Table 1), is seen to be fair, although
clearly leaving a good deal to be desired.

In all the above discussion, as in [ 26] » the observed criss-
crossing instability was identified with the theoretically predicted
first-mode oscillatory instability, despite the fact that in most cases
theory predicts that the system is also subject to yawing instability
over the same range of towing speeds. This is supported by the
observed frequency characteristics of the oscillation and the ob-
served effect of varying A, for instance, being essentially as
theoretically indicated for the behavior of the first mode. It has
thus been presumed that oscillatory instability is the prevalent form
of instability. There is, however, an alternative interpretation of
the observed behavior, namely that criss-crossing oscillation is a
nonlinear manifestation of yawing. This may be postulated, but can-
not be proven by the present linear theory.

In fact a number of questions remain. More careful and ex-
tensive experiments, including experiments with rigid cylinders, and
more extensive theoretical calculations are necessary to resolve
these questions.

We next consider briefly the mechanism underlying the onset
of instabilities, to the extent of identifying the physical forces at
work.

We first consider the mechanism involved in yawing. The
first thing to recognize is that yawing must involve angular motion as
opposed to pure translation. This is evident upon considering the
cylinder momentarily displaced parallel to the x-axis; in this case
the forces acting on the cylinder are exactly as in the equilibrium
configuration, except that the tow-rope exerts a restoring force on
the body. We next imagine the cylinder momentarily displaced such
that the y-displacement of the nose is positive and that of the tail
negative. Then, considering boundary conditions (11) and (12), we
note that the inviscid hydrodynamic force at the nose is £ MU(dy/dx)

1011

Patdoussts

while at the tailitis - f MU (ay /8x) » producing a moment tending to
exaggerate the original inclination. However, there are Coriolis
forces proportional to MU(@y/8x 8t) which always oppose rotation
[ cf. Eqs a10)].. ;

To understand this action of the Coriolis forces we consider
the related physical system of a hinged-free tube containing flowing
fluid (as mentioned in §2), depicted in Fig. 18(a), which was first
considered by Benjamin [32]. We see that if the system rotates
about A without bending, the fluid suffers a Coriolis acceleration
which has a reaction on the tube always opposing the motion. This
is clearly a stabilizing effect, as energy has to be expended by the
tube to keep the motion going; as further elaborated by Benjamin,
this represents the action of a pump from the energy-transfer point
of view.

Fig. 18 Rudimentary representation of a pump and a
radial-flow turbine

We next consider flexural instabilities. Clearly everything
mentioned so far applies here also. But we also have another force
coming into play. Once again we consider the hinged-free tube con-
taining flowing fluid, as shown in Fig. 18(b), where the tube is
momentarily 'frozen' in the bent shape shown. The centrifugal force
of the fluid acts to increase the curvature further. This is clearly
a destabilizing force, energy flowing from the fluid to the tube; it is
the action of a radial-flow turbine. In flexural oscillations we have
a play between these 'centrifugal' forces and the Coriolis forces;
«hen the former prevail, then instabilities may develop.

1012

Dynamites of Submerged Towed Cylinders

More formally, we may consider the work done, AW, on the
cylinder over one period of oscillation, ti» in much the same way as
was done in[ 24]. We find that

t, \ e °
AW = (1- ans by + Uyy'] veg ade etd fgmul (ees Uyy'],., dt
0

Seg iG: MU) (5? + Uyy') dx dt. (20)

If AW< 0, oscillations will be damped, while if AW >0O oscillations
will be amplified, i.e. the system will be unstable. We first note
that if f, = f,= 1, then instability can only arise from viscous effects
(cf. [24]). We next consider the first two terms of (20). We note
that for ice ae sien i stability will be governed by whether

(1-f 3 yé dt - (1-f OI i yi dt is positive or negative; it is clear,

therefore, that a well streamlined nose (f, ~ 1) and a blunt tail (f2< 1),
both tending to make AW <0, will promote stability. For higher U,
however, the situation becomes more complex, as Ulyy' >y may
now obtain, and yy! may be either positive or negative, the bar
representing the mean value over one period of oscillation. (It is
noted that from Figs. 8 and 9 it may be found that for oscillatory
instabilities we generally have (yy')o being strongly negative, and

(yy vy"), also negative but with smaller absolute value.) Stability will
depend on the magnitude of f,, f,, Vos val etc., and no simple general
rules can be formulated beyond the statement of Eq. (20).

It was found that the most effective way of stabilizing a towed
system is by making it blunt at the tail, which has the disadvantage
of increasing the towing drag. Clearly, what is needed is a blunt
tail without separated flow! The present work and that of [ 26] indi-
cate that small f, and large cy» (both associated with a blunt tail)
have individually stabilizing effects on the system. Clearly then
what we need is a sufficiently small f2 for stability, and a small
Cg for moderate form drag. From the boundary conditions we note
that a small f, has the effect of reducing the lateral shear exerted
by the tail on the cylinder. Accordingly, if the tail is made very
flexible with the rest of the body essentially rigid, the full shear
force might not be transmitted to the cylinder, simulating the effect
of a small f,; yet insofar as axial flow conditions are concerned,
they would be fairly good. Of course, this particular solution might
give rise to other problems, e.g. whiplash-type behavior of the tail
may be envisaged.

Another point of possible practical interest hinges on the
fact that a towed flexible body, which is unstable at low towing speeds,
may be stable at an intermediate range of towing speeds. (On the
other hand, a rigid towed body of the same shape would be unstable

1013

Patdoussts

at all towing speeds.) Accordingly, in the case of a flexible towed
system this suggests the possibility of removable stabilizers; these
would be operative only at low towing speeds, and would be removed
at the operating speed to reduce drag. Incidentally, the above
would generally also apply to articulated towed systems, made up of
a number of rigid tubular sections flexibly connected [ 32], [33].

ACKNOWLEDGMENTS

The author is grateful to his student, Mr. Jean J. Baribeau
for assistance in the preparation of this paper during the summer of
1970. The author wishes to thank the National Research Council
(Grant No. A4366) and the Defense Research Board (Grant No.
9550-47) for financial support making this research program possible.

REFERENCES

[1] A. G. Strandhagen, K. E. Schoenherr, P. M. Fobayashi,
"The Dynamic Stability on Course of Towed Ships,"

Trans. Soc. Naval Arch. and Marine Eng., Vol. 58,
pp» 32-66,°1950.

[2] L. "Bairstow, E. F.-Relf, R. Jones, "The Stability of Kite?
Balloons: Mathematical Investigation," A.R.C. R& M
ZOOS), 2o158

[3] M. M. Munk, "The Aerodynamic Forces on Airship Hulls,"
NACA Rept. No. 184, 1924,

[4] H.Glauert, "The Stability of a Body Towed by a Light Wire,"
ASRZG SOR Gin £st2 74930).

5 L. W. Bryant, W. S. Brown, N. E. Sweeting, "Collected
y g
Research on Stability of Kites and Towed Gliders,"
A.R.G. R°& M 2303; 1942.

[6] A. G. Strandhagen, C. F. Thomas, "Dynamics of Towed
Underwater Vehicles," Rept. No. 219, U.S. Navy Mine
Defense Lab, Panama City, Florida, 1963.

[7] J. R. Richardson, "The Dynamics of Towed Underwater
Systems," Eng. Res. Associates Rept. No. 56-1,
Toronto, Canada, 1965.

[8] P.O. Laitinen, "Cable-towed Underwater Body Design,"

Rept. No. 1452, U. S. Navy Electronics Lab. , San Diego,
California, 1967.

1014

[12]

[13]

[14]

[16]

[17]

[18]

[19]

[ 20]

Dynamites of Submerged Towed Cylinders

T. Patton, J. W. Schram, "Equations of Motion of a Towed
Body Moving in a Vertical Plane," ReptsJNo. 750, U.S.
Navy Underwater Sound Lab., Fort Turnbull, Conn., 1966.

E. Jeffrey, "Influence of Design Features on Underwater
Towed System Stability," J. Hydronautics, Vol. 2,
pp. 205-13, 1968.

W. Schram, S. P. Reyle, "A Three-dimensional Dynamic
Analysis of a Towed System," J. Hydronautics, Vol. 2,
pp. 213-20, 1968.

F. Whicker, "Oscillatory Motion of Cable-Towed Bodies,"
Ph.D. Thesis, Univ. of Calif., Berkeley, California, 1957.

R. McLeod, "On the Action of Wind on Flexible Cables with
Application to Cables below Aeroplanes and Balloon Cables,"
Wake G. R& M554, 1918.

F. Relf, C. H. Powell, "Tests on Smooth and Stranded
Wires Inclined to the Wind Direction and a Comparison of
Results on Stranded Wires in Air and Water," Adv. Comm.
Aero., R & M 307, 1917.

Landweber, M. H. Protter, "The Shape and Tension of a
Light Flexible Cable in a Uniform Current," Rept. No.
533, David Taylor Model Basin, Navy Department,
Washington, D. C., 1944,

Pode, "An Experimental Investigation of the Hydrodynamic
Forces on Stranded Cables," Rept. No. 713, David Taylor
Model Basin, Navy Department, Washington, D. C., 1950.

Pode, "Tables for Computing the Equilibrium Configuration
of a Flexible Cable in a Uniform Stream," Rept. No. 687,
David Taylor Model Basin, Navy Department, Washington,
DeGeg 19515

O'Hara, "Extension of Cylinder Tow Cable Theory to
Elastic Cables Subject to Air Forces of a Generalized
Form," A.R.C. R& M 2334, 1945,

E. Kochin, "Form Taken by the Cable of a Fixed Barrage
Balloon under the Action of Wind," Appl. Math. Mech,
Vol. 10, pp. 152-64, 1946.

C. Eames, "Steady-state Theory of Towing Cables,"

Quart. Trans. Roy. Instn. Naval Arch., Vol, 110,
pp. 185-206, 1968.

1015

[ 21]

[ 22]

[ 23]

[ 24]

[ 25]

[ 26]

[ 27]

[ 28]

[ 29]

[ 30]

[31]

[ 32]

[ 33]

5

s

Patdoussts

L. Albasiny, W. A. Day, "The Forced Motion of an Ex-
tensible Mooring Cable," J. Inst. Maths. Applics, Vol. 5,
pp. o>-71; 19609.

H. Toebes, "Flow-induced Structural Vibrations," J. Eng.
Mech. Div., Proc. ASCE, Vol. 91, No. EM6,-pp. 39-66, 1965.

R. Hawthorne, "The Early Development of the Dracone
Flexible Barge," Proc. Instn. Mech. Engrs., Vol. 175,
pp. 52-83, 1961.

P. Paidoussis, "Dynamics of Flexible Slender Cylinders
in Axial Flow -- Part 1. Theory," J. Fluid Mech., Vol. 26,
pps (17-30; 1966;

P., Paidoussis, "Dynamics of Flexible Slender Cylinders
in Axial Flow -- Part 2. Experiments," J. Fluid Mech.,
Vol. 26,5 PPe 731-51, 1966.

P. Paidoussis, "Stability of Towed, Totally Submerged
Flexible Cylinders," J. Fluid Mech., Vol, 34, pp. 273-97;
1968.

P,. Paidoussis, "Stability of Towed, Totally Submerged
Flexible Cylinders," Rept. Eng. R-5, Atomic Energy of
Canada, Chalk River, Ontario, 1967.

J. Lighthill, "Mathematics and Aeronautics," J. Roy. Aero.
SOG si> Vol. 64, PPpe 375-94, 1960,

J. Lighthill, "Note on the Swimming of Slender Fish,"
Tie Fluid Mech., Vol. 9; PPe 305-17, 1960.

I. Taylor, "Analysis of the Swimming of Long and Narrow
Animals," Proc. Roy. Soc. (A), Vol. ‘214, pp. 158-83, 19522

W. Gregory, M. P. Paidoussis, "Unstable Oscillation of
Tubular Cantilevers Conveying Fluid -- I. Theory,'
Proc.” Roy. Soc. (A); Vol. 293, pp. 51 2-27 P1966?

B. Benjamin, "Dynamics of a System of Articulated Pipes
Conveying Fluid -- I. denies " Proc. Roy. Soc. (A);

P. Paidoussis and E. B. Deksnis, "Articulated Models of
Cantilevers Conveying Fluid: The Study of a Paradox,"
J. Mech. Eng. Sc., Vol. 12, pp. 288-300, 1970.

1016

HYDRODYNAMIC ANALYSES APPLIED TO A
MOORING AND POSITIONING OF VEHICLES
AND SYSTEMS IN A SEAWAY

Paul Kaplan
Oceantes, Ine.
Platnvtew, New York

I. INTRODUCTION

At present, increasing interest is being devoted to the prob-
lems of deep sea operations of vessels that must remain on station
for an extended period of time in order to accomplish their intended
mission. This concern was given its initial impetus by the success-
fully conducted preliminary operation of drilling through the ocean
bottom from a surface ship in the operation known as the "Mohole
Project," as well as the increase in oil exploration in deeper water
depths. On the other hand, from the point of view of military
operations, there is need for placing instrumentation packages and
other military systems on the ocean floor for various purposes of
National Defense. These operations require a definite degree of
precision, safety during the course of the operation, and the capa
bility of returning to a particular locale and retrieving information
and/or the equipment itself for further study of data or for emplace-
ment in another location.

As a result of this emphasis on deep-sea operations, it is
necessary to determine the response of representative moored ships
in the open sea, and also to determine the characteristics of the
important parameters associated with lowering loads from sucha
vessel to the ocean floor and returning them to the ship. The
parameters that are of interest to the personnel aboard the ship are
the forces in the mooring cables, the displacements and tensions in
the lowering lines, the degree of precision in placing the loads, the
accelerations acting on the loads, and the magnitudes of impact on
the ocean bottom. In order to arrive at some appropriate engineering
estimates of the capabilities of carrying out such operations, appli-
cation of available theoretical hydrodynamic studies can be made to
deal with problems of this nature.

The study of motions of ships at sea is a general problem of

1017

Kaplan

naval concern, and has received increasing emphasis during the last
fifteen years or so by virtue of the advance of statistical methods
which describe the effects with greater realism than in previous
studies based on simplified wave representations. Major concern
has been devoted primarily to the problems of an advancing ship in
head seas, with the prime variables of concern being the heave and
pitch motions. Recent studies, however, have been concerned with
motions in oblique waves, wherein lateral motions (sway, yaw, and
roll) are also important. All of these studies involved large ships
advancing in waves, and only limited theoretical studies have been
developed to predict adequately the motions in all six degrees of
freedom under these operating conditions. A treatment of the motion
of a free ship with six degrees of freedom in waves is a formidable
problem that has not achieved a complete solution at the present
time, and when the influences of moorings are also included, the
problem is further compounded. Nevertheless, there exists a need
for some means of preliminary estimation of the expected motions of
a moored vessel, and there is sufficient hydrodynamic information
available to allow a study that will indicate the expected range of
amplitudes of motion so that the results obtained can be used as
guide-lines for operating personnel.

Another related problem that is assuming more significance
recently is that of a moored buoy system. These smaller payloads
are planned for use over large ocean regions to provide a network
of environmental reporting stations that will yield continuous data on
the important properties of the ocean and atmosphere for use in
weather forecasting and other technologies dependent on air-ocean
interaction. The effective design and engineering development of
such systems requires an ability to predict the buoy (and hence the
transmitting antenna) oscillatory motions and structural acceleration
loadings in various seaways; the determination of the tensions along
the cable under various operating conditions; etc. Knowledge of such
results will greatly enhance the design of handling equipment for
both launching and retrieving of buoys at sea, and will also provide
basic information on system survivability under extreme environ-
mental conditions.

A tool that can provide engineering estimates of such informa-
tion is a mathematical model that describes the essential mechanical-
dynamic characteristics of a moored buoy system. This mathemati-
cal model will be a system of equations and relationships that allows
the calculation of the spatial configuration, dynamic motion and
internal tensions of a specified moored buoy in a given excitation
environment. The hydrodynamic force acting on the buoy hull and
the forces acting on thecable system (hydrodynamic, inertial and
elastic) are coupled so that each affects the other, especially when
considering dynamic effects and rapidly varying motions. Certain
similarities exist between this problem and that of a moored ship,
together with definite differences as well. The applicability of basic
techniques of analysis from one problem to another provides useful

1018

Mooring and Postttoning of Vehicles tn a Seaway

insight and extends the utility of basic "tools" used in hydrodynamic
and dynamic investigations.

When considering the problem of maintaining a ship on station
for a long time period, various concepts for achieving a minimum
deviation from a derired operating point are possible, with the two
main methods being that of fixed mooring or by use of a dynamic
positioning system. In certain situations where mobility is required,
as well as due to the high capital cost of a mooring system for very
deep water operations, the associated high cost of emplacement and
the dangers of damage due to large storm conditions, a mooring
system does not appear to be attractive. Dynamic positioning is a
more recent development, which has only received limited applica-
bility to date.

In order to provide the information ncessary to determine
the possibility of an application of dynamic positioning, it is neces-
sary to carry out particular analyses to determine the environmental
conditions appropriate to possible operating areas; the resulting
forces and moments acting on the ship; the arrangement and type of
control effectors; the possible signal systems that provide the error
and command signals for actuation of controls; possible control
system concept designs; etc.

The important quantities that must be determined for proper
design of the positioning system are the disturbing forces that act
on the ship. The major forces and moments that affect the ship
stationkeeping ability in this case are the more-or-less steady type
of "drifting" forces imposed by the environment, and these quantities
are amenable to computation by means of hydrodynamic analyses
using available theory.

In all of the foregoing situations the importance of hydrodynamic
force evaluation and its applicability to obtain desired engineering
performance data is paramount. Many publications are available in
the literature on ship motion theoretical studies that can be applied
to the above problem areas, with reasonable expectation of validity
for the results. The central theme of this Symposium, "Hydrody-
namics in the Ocean Environment," is certainly appropriate to the
present International Decade of Ocean Exploration which will em-
phasize the technology that will yield benefits to Mankind. The
application of the basic developments in hydrodynamics of ship motion
to the applied engineering problems associate with maintaining
vessel operations at fixed positions in the ocean, which will be
required as part of this extensive international effort, is a vital
element in achieving improved system performance. It is also a
good illustration of the direct application of many years of basic
research toward the solution of problemsthat are anticipated as
further and deeper ventures into the sea are made. The present
paper is aimed at providing a limited description of the use of
hydrodynamic analysis when applied to some of these problem areas.

1019

Kaplan

II. SCOPE OF INVESTIGATION

It is easily seen that there are a host of problems associated
with the subjects considered in this paper. As a result, some limi-
tations are imposed so that only certain aspects are considered in
detail. The region of application of the results in this paper is in
deep water, so that no shallow water effects are considered. This
limitation thereby excludes problems of ship oscillation when
moored at docks in harbors, which is an important problem that can
be treated in a similar fashion to those herein by proper inclusion
of shallow water effects. The main emphasis within this paper is
on the seaway and its effects, and in some cases the influence ofa
current will not be considered. However it is known that currents
are often present together with sea waves, and their combined effect
is often very important. In addition the presence of a current is
often necessary to establish certain static equilibrium conditions
for a vehicle about which the seaway disturbances are imposed,
and in that case certain assumptions are made as to the existence
of such initial conditions for purposes of simplifying the analysis.
Similarly, the presence of any wind effects is also not considered
in detail within this paper.

When considering the problem of the motions of moored
systems, it is known that the effects of drift forces are also present
and that they produce an important influence on the resulting motions
and cable forces. However, in an effort to obtain tractable solutions
and to provide information on the characteristics due to different
force mechanisms, these effects will be considered separately.
Illustrations of the different influences that act on vehicles and
systems in a seaway will be presented separately, with some dis-
cussion given to the expectations with combined effects in a realistic
situation when more than one mechanism is acting on a system.

The discussions of results are devoted to the more important
phenomena influencing performance of a system in the sea, and they
will be given throughout the paper for each case treated.

III TECHNIQUES USED FOR MOORED SHIP ANALYSIS

In order to determine the motions of a moored ship in irregular
waves, it is necessary to determine the response in regular sinu-
soidal waves. The aim is to predict these motions, and the technique
to be utilized is that of spectral analysis [1] wherein the statistical
definition of the seaway in the form of its energy spectrum is used
as the initial data. The energy spectrum of the time history of each
motion of the vessel in response to irregular waves is evaluated for
the corresponding degrees of freedom to the energy spectrum of the
seaway. These operators are obtained from the solutions for the
motions in sinusoidal waves, and in accordance with the basic
premise of this technique of analysis, a linear theory of ship motions
is a prerequisite.

1020

Mooring and Postitioning of Vehicles tn a Seaway

The equations of motion in regular waves, for six degrees of
freedom, are formulated according to linear theory by the balance
of inertial, damping, restoring, exciting, and coupling forces and
moments. Both hydrodynamic and hydrostatic effects due to the body-
fluid interaction are included in the analysis, together with the
influences of the mooring system. The longitudinal motions (heave,
pitch, and surge) are coupled to each other, and similarly, the
lateral motions (sway, yaw, and roll) are also coupled. There is
no coupling between the two planes of motions, in accordance with
linear theory.

The hydrodynamic forces and moments such as damping,
exciting effects due to waves, etc., are determined by application
of the methods of slender-body theory. Essentially, this theory
makes the assumption that, for an elongated body where a transverse
dimension is small compared to its length, the flow at any cross
section is independent of the flow at any other section; therefore,
the flow problem is reduced to a two-dimensional problem in the
transverse plane. The forces at each section are found by this
method, and the total force is found by integrating over the length
of the body. A description of the application of slender-body theory
to calculate the forces acting on submerged bodies and surface ships
in waves is presented in [2], where simplified interpretations of
force evaluation in terms of fluid momentum are also given. The
hydrostatic and mooring forces and moments are combined with the
hydrodynamic terms, resulting in linear combinations of terms that
are proportional to acceleration, velocity and displacement in the
various degrees of freedom. All of these expressions, when related
to the appropriate ship inertial reactions by Newton's law, lead to the
set of six linear coupled differential equations of motion.

Solutions of the equations are found for regular sinusoidal seas
with varying wave length and heading relative to the barge. The
response amplitude operators are found from these solutions together
with the phases of the motions relative to the system of regular waves.
Assuming a knowledge of the oncoming irregular sea conditions (e.g.
in terms of sea state, as specified by an associated surface-elevation
energy spectrom from information in[3]), the set of energy spectra
for the ship motions are determined. Information on average values
and probabilities of relatively high values of the amplitudes of oscil-
lations in the ship-motion time histories for the different degrees of
freedom are found from the ship-motion energy spectra in accordance
with the methods of [1]. Cross-spectra are also used to determine
the energy spectra and hence the various average values and the pro-
babilities for the remaining quantities of interest, such as load-
displacement time histories and other quantities which are linear
combinations of the ship motions and their time rates of change (the
presence of lowering lines for placing loads on the ocean floor is
considered inthis analysis). These energy spectra may also be
obtained from the solutions of the differential equations by linear
superposition, and explicit use of cross-spectra here is necessary

1021

Kaplan

only for obtaining phase information.

The ship is assumed to be placed in a currentless seaway,
with no wind effects being considered. This may be somewhat un-
realistic from the practical point of view, but since concern here is
devoted only to the motions induced by the seaway, this neglect is
reasonable (as discussed previously). The ship is assumed to be
moored with bow and stern moorings of conventional line and anchor
type. The line and anchor mooring system utilized for this study is
a particular system especially suited to deep-sea operations [4],
and utilizes a taut line. Other types of mooring lines can be con-
sidered as well, but separate analyses to determine the static
orientation, restoring force variations, etc. must be carried out.
The extent of linearity for these different mooring arrangements
must be determined for use in the present type of analysis. The
effects of the moorings will be to provide restoring effects in the
particular displacements of surge, sway and yaw, thereby providing
"spring-like" terms in the equations for these degrees of freedom.
As a result, there are certain natural frequencies associated with
these motions, which do not ordinarily occur in case of free (un-
moored) ships. The moorings are assumed to have a negligible
influence on the motions of heave, pitch, and roll, which have large
hydrostatic restoring effects.

Following the evaluation of the various motions of the moored
ship, equations are formulated to determine the forces in the moor-
ing cables, and the displacement of and tension in the lowering line,
as a function of the different degrees of freedom of the oscillating
platform moored inthe seaway. The lowering line displacement and
tension, which are functions of the ship motions are then related to
the seaway and all of the resulting spectra determined. Operations
on these quantities provide information on expected amplitudes for
particular sea states, and in addition the vertical accelerations of
the loads are determined and similarly expressed, where this infor-
mation is useful for study of impact ofthe loads on the ocean bottom.

IV. EQUATIONS OF SHIP MOTION

The equations of motion of the moored ship are derived on the
basis of linear theory, with the body allowed to have six degrees of
freedom. A right-hand cartesian coordinate system is chosen with
the axes fixed in the body, and with the origin at the center of gravity
of the body. The x-axis is chosen positive toward the bow, the y-axis
is positive to port, and the z-axis is positive upward. These axes
are defined to have a fixed orientation, i.e. they do not rotate with
the body, but they can translate with the body. The body angular
motions can be considered to be small oscillations about a mean
position given by the axes. The dynamic variables are the linear
displacements x, y, and z along the respective axes, and the
angular displacements $, 8 and yw which are defined as positive in

1022

Mooring and Postttoning of Vehteles tn a Seaway

a direction of positive rotation about the x, y, and z axes,
respectively, (i.e. port upward, bow downward and bow portward).
The positive directions of the forces and moments acting on the
body are similarly defined.

The force (or moment) acting on the body is composed of the
inertial force due to dynamic body motions (denoted as Fj), the
force due to damping (denoted as Fy), the force due to hydrostatic
restoring action (denoted as F)), the force due to the moorings
(denoted as Fm), and the force due to waves (denoted as Fy). The
equations of motion are then established as

me = F au Dept (1)

for rectilinear motions (with s representing any rectilinear dis-
placement, and m the mass ofthe ship), with similar representa-
tions for the angular motions. A discussion of these different types
of forces is given below, together with some results obta ned, for
purposes of illustration.

The hydrodynamic forces and moments due to dynamic body
motions are of inertial nature, and do not contain any terms of dissi-
pative nature. The effect of the free surface is accounted for by
different frequency-dependent factors that modify the added masses
of each section. All couplings of inertial nature are exhibited in the
results of the analysis. In the case of dynamic body motions, the
simplified results of slender-body theory states that the local force
on any section is equal to the negative time rate of change of fluid
momentum [ 2]. For the vertical force (z-force), this is expressed
by

D
ae = at | A535 > (2)

where A is the added mass of the cross-section and Wy is the
body vertical velocity, given by

w, = yy (2 - £8) = % - 60. (3)

In the above equations, the coordinate € is a "dummy”™ variable
along the longitudinal coordinate x (and coincident with it), and the
time derivative D/Dt is just the partial derivative 0/8t, since there
is no forward speed. The quantity A33 is the added mass of the
cross section, including free-surface effects, which is obtained from
the work of Grim [5] for the class of sections known as Lewis forms.
The total vertical inertial force is then found to be

1023

Kaplan

£p ! eo &b 1 ee
zi =~ | aggadt- 2 +) Pape at - 6 (4)
& bs

where § and & are the bow and stern €-coordinates respectively.

In a similar manner, the lateral force (along y-direction) may
also be expressed by use of this same procedure, but certain addi-
tional factors enter in that case. These factors are the necessity
of including roll effects which influence the lateral velocity, and
also the fact that the representation of the lateral force is based upon
added mass terms that are evaluated for motions relative to the free
surface level, rather than the body center of gravity position. Cor-
rections to refer the final forces to the center of gravity position are
made after finding the forces referred to the free-surface position.
The detailed procedures for determining these inertial force (and
moment) results, as well as all other forces of hydrodynamic,
hydrostatic, etc. nature are described in [ 6] , which is the basic
report on which the present section of this paper is based. In view
of this, only limited discussion of the remaining forces and moments
will be presented.

The damping forces and moments are dissipative in nature, and
are primarily due to the generation of waves by the ship motions on
the surface, which continually transfer energy by propagating outward
to infinity. In accordance with the two-dimensional treatment used
for the analysis of inertial forces due to body motions, the same
concept is used in evaluating the local forces at a section of the ship
due to wave generation. With the ratio of the amplitude of the heave-
generated two-dimensional waves to the amplitude of heaving motion
of the ship section denoted by A,, the vertical damping force per
unit vertical velocity of the ship section is expressed as

o—2
1 _ pg Az _ rR =2
N= nae po (>) A,2 (5)

where A, for Lewis-form sections are available as a function of
w°B* /2g = mB '/i, for different beam-draft ratios and section coef-
ficients, where B- is the local beam and 2 the wave length.

The vertical damping force at each section is
dz 1s .
“ze == N,,(z - E80), (6)

and this is integrated over the ship length to determine the total
vertical damping force, given by

1024

Mooring and Postttoning of Vehtecles in a Seaway

Zaz - Nz +NiO, (7)
where
2 ee
N, = po(%) VA? at, (8)
fs
and
é
28>
Nzg = po(>) \. Aré dé. (9)
Ss

Similar treatments yield the lateral damping force, pitch damping
moment, etc.

In the initial discussion of damping, emphasis was placed upon
energy dissipation due to wave generation. Actually, viscous effects
also manifest themselves and contribute to damping. The contribu-
tion of the viscous damping term is quite negligible’for most motions,
with the possible exception of roll. Roll damping due to wave genera-
tion is often small for most normal ships and viscous effects (or
other drag mechanisms, such as eddy-making) assume greater
importance, especially if the ship is fitted with bilge keels. In that
case, the roll damping is often of nonlinear form, and an approxi-
mation is used to determine some equivalent linear representation.
Knowledge obtained from model experiments [7] was used to deter-
mine the value of roll damping used in treating the illustrative ship
case in this paper.

The hydrostatic restoring forces and moments are, as the
name implies, due to buoyancy effects arising from static displace-
ments. The only displacements that will result in hydrostatic
restoring effects are heave, pitch and roll. On the basis of linear
theory, the local hydrostatic vertical force change due to vertical
displacements is

ae =- pgB*(z = 50), (10)

where the ship is assumed to be almost wall-sided near the inter-
section with the free surface, and the effective buoyancy change
comes from the total immersion. Similarly, the hydrostatic
restoring pitch moment is

dMp _ dZp
“xe fe ome

1025

Kaplan

leading to total hydrostatic restoring vertical force and pitch moment

given by
b x bx
Z, = - PS B d&«z + pe B & de = 65 (42)
s és
and
fb fb
M, = ee) BE ab 2- pe) B*t’ dé +e. (13)
Ss Ss

In the case of roll motion, the hydrostatic restoring effect is given
by

K, = - pgV|GM|¢= - W/GM|4, (14)

where VY is the displaced volume, |GM| is the metacentric height,
and W = pgV is the ship displacement.

The exciting forces and moments due to waves are obtained as
the sum of terms due to buoyancy alterations as the waves progress
past the ship hull, together with hydrodynamic terms of inertial
and damping. The buoyancy effect for the vertical force is repre-
sented by

pgBn(é xt) (15)

at each section, and these contributions are combined to determine
the total forces and moments due to waves. The analysis includes

an allowance for the waves to be propagating at an oblique heading
with respect to the ship, and a further allowance for the influence of
the non-slenderness of the ship is also included. A correction factor,
relating the beam to the wave length and the heading, is included for
this purpose since the shipforms considered for mooring application
are often not very slender. Details of the evaluation of wave forces
and moments by these methods are presented in [6].

Before discussing the mooring forces and moments, informa-
tion on the characteristics of the vessel studied in this investigation
is given below. The particular vessel for which the equations are
formulated and solutions carried out is the CUSS I, which was the
vessel used in the preliminary Mohole drilling operation. This ship
is considered representative of the class of construction type barges
which will be utilized for deep-sea construction operations. A
diagram of the barge, together with its mooring and load-lowering
lines, is shown in Fig. 1. A summary of the numerical values of the
parameters characterizing the moored-barge system is presented
in Table 1.

1026

Mooring and Postttoning of Vehicles in a Seaway

Draft: 10

Center of gravity

12,000- foot mooring cable

Fig. 1. Schematic diagram of moored barge
(Profile and plan views)

1027

Kaplan

260 ft

48 ft

LO ft

9.8 £

ot Nae 8 =

15.2 2

8.16 ft

2823.2 long tons
6.324x10® lbs

Table 1
Numerical Values of Moored-Barge System

Length =
Beam =
Draft

Vertical distance from CB to CG =
Vertical distance from free surface to CG =
Vertical distance from CG to keel =
Metacentric height =
Displacement

Weight =
Mass =

|
Pitch moment of inertia =
Yaw moment of inertia: =
Roll moment of inertia” =

Total roll moment of inertia (including
added inertia due to fluid) =

Surge period® =
Sway period =
Heave period =
Pitch period =
Roll period =

Effective spring constant for mooring
cable =

Effective mooring system spring constants:
Surge =

Sway =
Yaw =

Depth of barge

' Assuming longitudinal gyradius = 0.25 L.

T surge
T sway

Theave
T pitch

Trott

197.624x10° slugs
706. 7x10°® slug-ft*
706. 7X10° slug-ft?
49x10° slug-ft®

6
78.69X10 slug-ft-

79 seconds

= 64.5 seconds

4.6 seconds
4 seconds

7.75 seconds
4250 lbs/ft

1250 lbs /ft

3750 lbs /ft

633. 75X10° lb-ft /rad
15 ft

2Without added fluid inertia; it is assumed that transverse gyradius =

Bys.

3For all motions these are uncoupled periods determined in terms of
effective spring constants and values of total masses or inertias. The
effects of coupling will change these somewhat, but for first app roxi-
mations andinterpretation of critical conditions, this. will suffice.

4F rom model tests [ 7].

SBridge strand wire rope, of cross section 0.595 in,

1028

Moortng and Postttioning of Vehicles in a Seaway

In analyzing the mooring forces and moments, the barge is
assumed to be moored by a conventional line and anchor system,
with both bow and stern moorings. However, for application to
deep-sea conditions with depths of the order of 1000 fathoms, a
certain particular mooring scheme is utilized. This scheme
utilized a long-wire rope for each mooring leg assembly (12,000 ft
in length), which is supported in the water by a series of submerged
spherical buoys. The buoyancy of these buoys keeps the rope taut
along its entire length, thereby not allowing it to assume the usual
catenary shape. Withthis arrangement, an initial tension is applied
along each mooring let, and any changes in mooring forces on the
ship (and therefore also in the cables) occur as a result of elastic
forces resulting from ship displacements. A layout drawing of such
a system is shown in [4], which has direct applicability to ships of
the same general displacement as the construction barge presently
studied.

The displacements having greatest influence on the moorings
are in the horizontal plane, and these are surge, sway and yaw.
Since the mooring lines are fairly taut and are under an initial
tension, the elastic restoring effects may be taken to be fairly
linear, i.e. the restoring force is proportional to the displacement.
The proportionality factor for an effective displacement along a
single mooring cable is found from a knowledge of the modulus of
elasticity of the cable material. For the present case of 1-inch
diameter bridge strand wire rope, which is 12,000 ft long, has a
cross section area of 0.595 in®, and an assumed modulus of
25 X 106lb/in®, the effective spring constant for a single wire rope
is found to be C = 1250 1b/ft. This linear result only holds below
the yield point of 60,000 lb of static force (in a single cable), but it
is anticipated that the maximum deflection necessary for attaining
this force (viz. 48 ft) will not be experienced in the present case.

For the purposes of analysis, the barge is assumed to be
moored in an arrangement similar to that shown in the following
sketch of the mooring plan. A longitudinal displacement of the barge

along x, denoted as Ax, leads to an effective displacement along a
single cable given by Axcos @, where @ is defined in the sketch
above. The force in a single cable is then C Ax cos a, The longi-

1029

Kaplan

tudinal force component at one end of the ship is represented by
(CAx\cos @)cos @ + (CAx cos*e)cos,o = .2CAx cos* Q,

and since an extension of the cable at one end of the ship requires a
contraction at the other end, a similar force occurs. These forces
are restoring forces and the net result is a longitudinal force in the
barge due to the moorings, given by

X, = 4C cos‘ a@*x=- k,x, (16)

where x is the surge displacement variable.

In the case of sway displacement, the effective displacement
along the cable is y sine, and combining components for net Y-
force on the barge, accounting for all the cables, leads to a net
mooring lateral force given by

Xm etc gin oy = — bys (17)

For yaw displacements, Y* L/2 where L is the ship length. The
lateral force at one end of the ship is then

2C sin®a- Sy = Cl sin?a- y, (18)

and the contribution to the yaw moment is
2 1 4 2 we
CL sin’ @* ¥ (5) =5 CL sin a-w (19)

at each end. Since the forces at each end are equal and opposite
(approximately, since the origin is not exactly at the ship center),
the net yawing moment acting on the barge is given by

Nm=- CL’ sin?a* w= - kyl (20)

The variations in the force in the mooring cables due to the
motions of the barge can easily be found, since they are related
kinematically to the motions. It is seen that the longitudinal dis-
placement, x, andthe net lateral displacements, y +(L/2)W at
the bow and y - (L/2)\ at the stern, can be combined to determine
the net variation in elongation of each mooring cable. The cable
displacements due to surging motion on the barge are x cosa, while
the cable displacement due to the motions of sway and yaw are‘
[y+(L/2)W] sin a, according as the cable is at the bow or the stern.

1030

Mooring and Posittonting of Vehicles in a Seaway

Different effects as to the cable displacement directions occur for
the cables, at either the bow or the stern, for the influence of the
lateral motions, while the same direction of displacement (at either
bow or stern) occurs for the surge motion. The general expression
for the fluctuating cable force may be written as

Fe = c[x cos a+ (y+ 5v) sin a|

where C is the effective spring constant for a single wire rope,
and particular values for each of the four cables are given in the
following, where a positive cable force is defined as that which pulls
on the restraining anchor support on the ocean floor.

The expressions for the individual cable forces (c.f. sketch
of mooring-line system) are listed below:

Bow
F, =- c[x cos @ +(y +>) sin a | port
(21)
_ zi ‘ b
Fa — - Ci cos a-- (y >) sin a | starboard
Stern
F, = c [x cos @ - (y - 1) sin a| port
(22)
F, = c [x cos @ + ( - >) sin a | starboard
4> y "2

For the present case where the barge is moored with a = 60°,
L = 260 ft, the mooring system restoring constants are

k, = 1250 lb/ft
ky = 3750 lb /ft (23)
ky = 633.75 X 10° 1b-ft/rad

These values are the effective spring constants for surge, sway and
yaw, and as a result there also exist natural periods for these
motions in the case of moored ships. There still exist natural
periods of heave, pitch and roll, as in the case of free ships, and
these natural periods are relatively unaffected in the present case.
The introduction of the existence of natural periods in surge, sway
and yaw (with possible large motions associated with resonances in

1031

Kaplan

these degrees of freedom) is the main characteristic of moorings
applied to ships that distinguishes the resulting motions from those
of free ships in waves.

V. SOLUTION OF EQUATIONS

The equations of motion result from combining all of the
constituent terms discussed above, and solutions can be obtained
by converting them to a simpler form for sinusoidal waves. Since
the exciting forces and moments are sinusoidal functions, the
motions will also be sinusoidal with the same frequency. Defining

— iwt — iwt — iwt = iwt = _lwt
c= xe ; y=ye ; VA AS he Xy = Xe , Yw.= YC rq s, sete.

the equations of motion are then converted to (complex) algebraic
linear equations. In matrix form the equations may be represented
by

a I 0 as ]f x x
0 Aon aos Z = 4. (Zi (24)
a ass as 6 M

for the longitudinal motions, where the coefficient matrix is sym-
metric, i.e. aj3 = a3,, a,,= a3,- The matrix elements are defined
by:

a, = (- mo* + ioN, + k,) (25)
aj3 = a3, = m|BG|o (26)
aon= = (m + ( As, at) ot ioc No + ral Bae (27)
és €s
2 bot 4 b *
Ags = ago = w (; A336 d& - iwNzg - af B Gude (28)
Ss Ss

{p i ee 2 fb 2
an =| I +( A.W dé )'@ + iwCgNo + pe | BtE dé (20)
33 GF f 33 ) es

The lateral equations are represented by

1032

Mooring and Postitioning of Vehicles in a Seaway

by be bsry ng
bo, bop = Bag ff P| =| N
be Bean bs i) K + (OG)Y

(30)

where the matrix here is also symmetric, i.e. bia = bo, by3 = bas

by, = b3, The elements are defined by

€b
by =-(m¢t \ Ago d&) w + iwCyNy + ky
és
2 Sb ;
b,, = b,, = - w ( Ae dé + iwNyy
s
bit ss
be bs, Jf (Ago + (OG) Age) dé + iwCyNy | BG |
s

E
b= - (1, + ( ” Ast dé) w° + iwCyNy + ky
és
€b
by, = Dap = - wf (Ago + (OG) Ax.) & dé + ioNyy| BG |
s

bene ol, + iwNg + W|GM|

The presence of symmetric matrices helps in effecting an

(31)

(32)

(33)

(34)

(35)

(36)

easier solution of the equations, obtained by matrix inversion on a
large digital computer. The solutions are then available for each
degree of freedom and also for any linear combination of degrees of
freedom. The real form of the final solutions is obtained by taking
the real part of the complex function, which was the original defini-
tion implied in the complex representation of the solution variables.

The lowering line displacements are related kinematically to
the body motions, and hence they are relatively simple to determine
once the different methods of lowering loads are specified in this
study, viz. center-lowered loads and boom-lowered loads. Center-
lowered loads, as the name indicates, are lowered through some
sort of opening through the ship's keel, and it is assumed that this
is done at just about amidships. The instantaneous displacement

vector components of the load and lowering line are s,, s

and are then given by simple geometry as ;

1033

and s,

Kaplan

S, =x
sy=y + |KG|¢ (37)
S$, = 2

where x, y, z and @¢ are the instantaneous ship motions of surge,
sway, heave, and roll, respectively, and KQ@ is the vertical distance
between the center of gravity and the keel.

The tension, T, in the lowering line is given by the relation

T-W

=—s =

; (38)

Wigs FW pat
—- Zz
g
where Wg is the weight of the load, and only vertical effects are
considered to affect the tension. At rest,

so that upon representing the tension as
- = !
Lots t.t aM, gee

where T' is the tension change due to dynamic effects, one obtains

Thus the tension variation due to the dynamics of the ship motion is
directly related to the vertical acceleration of the load, and it is also
proportional to the weight of the load.

In the derivation of the formulas given above, it is assumed
that the trajectory of the load attached to the line is such that at each
instant it is on the vertical line through the point of attachment of
the lowering line to the barge. It is also assumed that the elastic
effects of the lowering lines may be neglected; the only dynamic
influences considered being those due to the ship motions. The
neglect of elastic effects in the lowering line appears to be a fairly
safe assumption, since the major influence would occur only if the
wave frequencies excited the natural frequency of wave propagation
in the lowering line. In view of the lack of specification of the line's
physical characteristics, as well as the expectation of wave-propa-

1034

Mooring and Posittontng of Vehicles in a Seaway

gation frequencies out of the range of interest in the present problem,
the tensions and accelerations are considered adequately represented

by Eq. (39).

For boom-lowered loads, the situation may be visualized by
reference to the accompanying sketch, where the boom length £ is

stern

elevated at an angle a'. An appropriate value for the relevant
horizontal projection of the boom length (£ cos @') is considered,
for computational purposes in the present case of a 260 ft length
barge, to be 150 ft. As shown below, the boom is also oriented
horizontally at an azimuth angle y, measured from the bow, positive
in the counterclockwise sense, viewed from above. The load and

stern bow

lowering-line displacements about their respective equilibrium
positions are given by

s? =x - (f cos @') sin y> w
slay + (2 cos a') cos y> w (40)
az = it cos a') cos y° 6 + (£ cos @a') siny >

and the line tension (fluctuating part), Tt!” and vertical acceleration
are represented by

1035

Kaplan

rT” 2 it os ee oo
Moe Eh S cos a'(siny > 6 - cos y= @)] (41)

where 4, 8, and W™ are the rotational barge motions, roll, pitch
and yaw, respectively, and the superscript y denotes the boom
azimuth angle. These quantites are derived on the same basis as
those for center-lowered loads, it being assumed that the boom
pivots about the ship CG. The instantaneous magnitudes of these
quantities thus appear as linear combinations of the instantaneous
ship-motion solutions.

Each motion of the barge in response to a regular sinusoidal
wave having a given frequency and propagating in a given direction
will also be sinusoidal, of the same frequency, but will, in general,
possess a different phase. In addition, the amplitude of each motion
will, in general, differ from that of the wave, the ratio of the former
to the latter being a function of the wave frequency and the heading
of the wave relative to the heading of the barge, and this amplitude-
ratio function is known as the response amplitude operator for the
particular motion of interest. In order to arrive at an effective
characterization of the barge motions in a random sea, in which
case these motions themselves have a random nature, the function
known as the spectral energy density, or the energy spectrum, of
each motion must be found. This spectrum is a measure of the vari-
ation of the squares of the amplitudes of the sinusoidal components
of the motion, as a function of frequency and wave direction. The
total area under the spectral-energy density curve contains much of
the statistical information on average amplitudes, near-maximum
amplitudes, etc., for the particular motion considered. For an
arbitrary motion, represented by the i-subscript, the energy
spectrum of that motion, due to the effects of irregular waves, is
given by
(i,i)

@"' (w) = | Tig(w) | A (o) (42)

for a particular fixed barge heading in a unidirectional irregular
sea, where A (w) is the wave spectrum and (Tia is the response
amplitude operator for that heading.

For computational purposes in the present study, the Neumann
Pierson spectral-energy description of the seaway has been adopted,
and calculations made for these particular sea states, corresponding
to three particular wind speeds. The following table illustrates the
conditions.

1036

Moortng and Posittoning of Vehicles in a Seaway

Table 2
Sea Wind Speed Sig. Wave Surface Elevation (Time
State Vw (knots) Ht. Hyy3 History and Energy Spectrum)
(ft)
rin... Value,, Lt. energy,
o , (ft) o%, (ft)? 20% = E
3 14 S05 0.81 0.66 pW Ye
= 9 6.9 iit 3.05 6.10
5 Ze 10.0 2.50 6.25 12.46

The Newmann wave spectrum for a unidirectional fully-
developed sea represented by

2
6 -29°/(wVy)
iw tS

A*(w) = C (43)

where C is an empirical constant having the value 51.5 ft*/sec,
vw is the wind speed in units of ft/sec, and A*(w) has the units
ft?-sec. The wave spectrum for a non-unidirectional sea, allowing
for angular variation (a two-dimensional spectrum), is represented
by

2 -6 -29°/(wVy)* T T

— Cw e “ cos® By, for -353<By< ts, 0<w< to

2 T 2 2

vs (w, By) =
0, otherwise, (44)

where Py is an angle measured from the direction toward which the
wind is blowing (the predominant wave direction). In this case,

the motion spectrum occurring for a particular barge heading f,,
measured relative to the wind direction is

+1 /2

2 2

BUN) (a) = A2(e) a ABy cos® By| Tin (w.) [2 (45)
-17/2

where £6 = By- Bg, and this energy spectrum will depend upon the

angle B,.

From the spectral density function, of? (w), for a particular
motion, there may be obtained, in principle, all the statistical or
probabilistic properties possessed by the random process. The
total area, Ej;, under the spectral density function curve, as defined
above,

1037

Kaplan

© (i,i)
Ej -( dw ® (cw) (46)
(@)

is equal to 2a i.e. twice the variance of the ordinates on the cor-
responding time-history curve. Under the assumption that the
seaway is a Gaussian or normal stochastic process which is exciting
a linear system (in this case, the barge), the set of responses of

the system will in turn represent a Gaussian stochastic process.

The probability of an ordinate of a particular response lying between
two values is given by the definite integral of the Gaussian proba-
bility density between those two limits, and will be a function of the
variance oj. Thus, Ej or oj may be used to estimate the proba-
bility of the occurrence of instantaneous values in any range of
interest, for any given barge motion, including infrequently-occurring
large or near-maximum values. Characteristics of the motion time
history may be obtained in terms of the quantity Ej; by relating the
behavior of the envelope of the record (interpreted as the instantane-
ous amplitude of the time history curve) to this quantity. Such
relations are based on assumed narrow-band behavior of the energy
spectrum, and yield expressions for the mean amplitude of oscilla-
tion (half the distance between the trough and crest of an oscillation),
the mean of the highest 1/3 of such amplitudes (known as the signifi-
cant amplitude), and other related statistical parameters of interest
for a specified sea condition. In particular the relations for average
pitch amplitude and significant pitch amplitude are

Oa, = 0.88 (Ea
(47)
8 5ig. = 1.41 VEg

VI. DISCUSSION OF RESULTS

Computations of the amplitudes and phases of the six separate
motions of the moored barge for the complete range of possible
headings were carried out for wave lengths varying from 100 feet to
800 feet, which covers the range of periods significant for ship
motion in an operational environment up to Sea State 5. Solutions to
the equations were obtained for the complex response operators
(both amplitude and phase) of the various motions relative to the wave.
Representative solutions for a particular wave length, for both the
longitudinal and lateral motion amplitudes, as functions of the heading
angle B are shown in Figs. 2 and 3. From this data the response
amplitude operators, as functions of frequency (since w=V2tg/d),
are obtained and representative curves are presented in Figs. 4 and
5. Application of the techniyues of spectral superposition theory [1]
results in spectral energy density values for particular barge motions
in Sea State 5 (as an example), and these values are indicated in
Figs. 6 and 7, as illustrations of some of the results.

1038

Moortng and Posittoning of Vehicles in a Seaway

X (feet)
1.0

180 150 120 90 60 30° => 0 + So 60 90 120 150 180

Z (feet)
1.0

180 150 120 90 60 30 - O + 30 60 90 120 150 180

@ (radians)
025

Eee
180 150 120 90 60 30 - O + 30 60 90 120 150 180
A (degrees)

Fig. 2. Amplitude of response for unit-amplitude wave as a function
of direction of wave relative to barge. Longitudinal
motion; X= 300.

1039

Kaplan

Y (feet)
10

0.8

180 150 120 90 40 30 - 0 + 30 60 90 120 180 180

¢ (radians)
.08

.06

180 180 120 90 60 30 - 0 + 30 60 90 120 150 180

y (radians)
.010

.008

.006

180 180 120 90 6Q 30 - 0 + 30 60 90 120 150 1860
43 (degrees)

Fig. 3. Amplitude of response for unit-amplitude wave as a function
of direction of wave relative to barge. Lateral motion;
X= 300°.

1040

Mooring and Posittoning of Vehicles in a Seaway

Taq |? (dimensioniess)

w (rad/sec)

2
(Response amplitude operator)~ for surge, Tis I

[Tz 9 (dmenssoniess)

w (rad/sec)

2
(Response amplitude pparataris for heave, IT2 9

=
= .0004
~
i]
a
°
a
me
+ 0002
§
=

w (rad/sec)

(Response amplitude operator) for pitch, Teel”

Fig. 4. Response amplitude operators for longitudinal

motions.

1041

Kaplan

Fy
2 10
0.8 90°
v_ 0.6
=
- 04 45°
0.2
°e
0 (e}
10) 05 10 Le j
w (rad /sec)
(Response amplitude operator)@ for sway, [Tval”
006
N
= .004
x 902
nN
wv
ec
«002
AS 45°
19) 0.5 10 1S

w (rad/sec)

2
(Response amplitude operator) for roll, IT gal

.00006
45°
.00004
.00002
o°, 90°
te)
0 0.5 1.0 us

w (rad/sec)

IT py I?, (rad? ft?)

(Response amplitude operator)* for yaw, [Toel*

Fig. 5. Response amplitude operators for lateral
motions.

1042

Mooring and Postttoning of Vehicles in a Seaway

3 SEA STATE 5

WwW (rad/sec)

- SEA STATE 5

ENERGY DENSITY (ft*sec)

{@) 0.5 10
Ww (rod /sec)

Fig. 6. Spectral energy density for translational barge motions
for indicated barge heading Bg

1043

Kaplan

8 SEA STATE 5

Ww (rad/ sec)

.005

coe SEA STATE 5

.003 @

=0°
002 Py =O
001

ENERGY DENSITY ( rad@-sec)

0 l ee) eee ee ee ee

ia] 05 1.0 15
W (rad/sec)
nOn09 SEA STATE 5
.0003
38 = 135°
0002
0001
My
15
w (rad/sec)

Fig. 7. Spectral energy density for rotational barge motions for
indicated barge heading Bp

1044

Moortng and Posittoning of Vehicles in a Seaway

In the present study the angle for the predominant wind direc-
tion was taken to be By = 0, and a variable barge heading angle, Bp,
introduced to allow for the relative heading of barge to wind. The
relationships of the wind direction, the wave heading, and the barge
heading are shown in Fig. 8, together with the difference angle
Bw- Bg representing the wave heading relative to the barge heading.
Also shown in this figure are the conventions made use of later for
the designation of the forces in the mooring cables and the azimuth
angle for the boom used to lower loads from the barge.

VECTOR WAVE
PROPAGATION DIRECTION

PREDOMINANT
WIND DIRECTION

L,

MOORING LINES ?
DIRECTION OF

BARGE HEADING
La (RELATIVE To
PREDOMINANT WIND)

Fig. 8. Orientation and relations between barge, wind and waves

1045

Kaplan

Figures 9 and 10 show, for each of the six barge motions,

the variation of total spectral energy with barge heading, for each
of the three sea states considered. The ordinate plotted for each of
the curves is the r.m.s. value, oj, for the time history of the barge

otion represented, and it will be convenient to refer to the variance
gj of the function as the total energy. The r.m.s. value of any
time-history function is therefore the square root of its total energy
(assuming, here, as always, the mean value of any time-history
function to be zero.

Representative examples of calculated spectral energy density
functions for the case of the center-lowered load are presented for
two sea states and two barge headings relative to the predominant
wind direction in Fig. 11. The spectral energy density functions
shown for the load were calculated from those of the fundamental set
of cross-spectral energy density functions, i.e. those of the six
barge motions. Since the time histories of the load and amplitude
operators for the former may be obtained by forming appropriate
linear combinations of the complex response operators, Tjn, for
the barge motions, and calculating their squared absolute values.

The r.m.s. values (as defined here) were obtained for all
quantities of interest for the load lowering operation such as dis-
placements, accelerations, tensions, etc. as well as the forces in
the mooring cables, for each sea state and barge heading relative to
the waves. Similarly variations of these quantities as a function of
the boom azimuth angle were found, from which an optimum boom
angle (which minimizes the r.m.s. values of any one of the time
histories of interest) may be determined. As an example of results
obtained for a 200 ton load lowered in a State 5 sea with a crosswind
barge heading and with the optimum boom azimuth angle (here 180°,
i.e. boom over the stern), the r.m.s. value of the added-dynamic
line tension given by Fig. 12 is (2.38)(200)/32.2 = 14.8 tons. From
data on the normal probability curve for this r.m.s. value, it can
be shown that the downward force of impact on the bottom would
exceed 25 tons approximately 2.3% of the time, if the instant of im-
pact were allowed to occur at random. For a center-lowered load
under the same conditions, the r.m.s. value of its acceleration is
1.17(200)/32.2 = 7.27 tons, and the downward impact force on the
bottom would exceed 14.3 tons approximately 2.3% of the time.

The r.m.s. value of the fluctuating component of the force in
each of the four mooring cables is shown in Fig. 13 as a function of
barge heading for each sea state. The four are seen to have nearly
the same r.m.s. value for any particular barge heading in a Sea
State 3, with the actual values varying between 300 and 600 lb. For
a Sea State 4 the range is from 1100 to 1750 lb, with the differences
between r.m.s. values for the four fluctuating cable forces being as
much as 150 1b. For Sea State 5, the range is from 1800 to 2700 lbs
with differences in r.m.s. values between cables of 250 1b. In all

1046

Mooring and Postttoning of Vehicles itn a Seaway

SEA STATE 3 RMS VALUE
Mee

180 150 120 90 60 30 - O + 30 60 90 120 180 180
Ag. BARGE HEADING (degrees)

SEA STATE 4 RMS VALUE
(feet)
1.2
1.0
Z
0.8
Y¥
06
x
0.2
180 150 120 90 60 30 - O + 30 60 90 120 150 180

$y, BARGE HEADING (degrees)

SEA STATE 5 RMS VALUE
eae

1.6

0.6
0.4
0.2
180 150 120 oe 60 307° — 3 O}) + 50 60 90 120 150 180

Ag. BARGE HEADING (degrees)

Fig. 9. RMS values of the translational barge motions as a function
of barge heading at indicated sea state.

1047

Kaplan

SEA STATE 3 RMS VALUE

(radians)
.018

016

O14 p

O12

002

180 180 120 90 60 30) =" “0? + - 30 60 90 120 180 180
A, , BARGE HEADING (degrees)

SEA STATE 4 RMS VALUE
eet ba

180 150 120 90 60 30° =~ '0)'+°'30 60 90 120 150 180
4,, BARGE HEADING (degrees)

SEA STATE 5 RMS VALUE

(radians)
.09

.08

01

180 150 120 90 60 207 — Ors 30 60 30 120 150 180
4, . BARGE HEADING (degrees)

Fig. 10. RMS values of the rotational barge motions as a function
of barge heading at indicated sea state.

1048

Mooring and Postttoning of Vehicles in a Seaway

(TT)
$(w) FoR Bg = 90°

—e
"e T' = Added dynamic tension in
lowering line I f load
Bi wering line per slug o SEA STATE 5
aan
SEA STATE 3
Oo
(¢) 0.5 1.0 1.5
Ww (rad/sec)
16

(Sy,8y)
@(w) FOR SEA STATE 5

Sy = Port-starboard displacement
of center-lowered load

SPECTRAL ENERGY DENSITY (ft2-sec)

fo) 0.5 1.0 1.5
w (rad/sec)

Fig. 11. Spectral energy density functions for added-dynamic
tension in lowering line and lateral displacement for
center-lowered load, for indicated barge heading and
sea state.

1049

Kaplan

MINIMUM
RMS VALUE
SEA STATE 5 3.0

iE
i*
2.4

7 =+180° Y=+165 345 7 =-165~ 7 =-180°

7=180% 722
SEA STATE 4

1.8
° ° 1. 6
7 =+180 7= +165 7=-165° 7 =-180°

7 =180Y + 1.4
1.2
1.0
SEA STATE 3 wine

7 =+180

0.4

0.2

180 150 120 90 60 P 30 - O + 30 60 90 120 150 180
Bs, BARGE HEADING (degrees)

Fig. 12. Minimum r.m.s. values of added-dynamic line tension
(pounds /slug) and vertical load acceleration (feet/second )
for boom-lowered load, as a function of barge heading at
indicated sea state

cases the cable force r.m.s. values are greatest near crosswind,
and least for upwind and downwind barge headings.

All of the results obtained in this study provide useful infor-
mation for application to many operations that can be performed at
sea, using a moored ship as the base. The major questions con-
cerning these results are their degree of validity, as well as the
capability of extending the results of related situations such as shallow
water operation, different mooring systems, the effects of nonlinearity,
etc. Some extensions and/or applications of the present theory have

1050

Mooring and Postttoning of Vehicles in a Seaway

RMS VALUE
(pounds)
3000

SEA STATE 5 aie ae

p crams (

Fa Fe
SEA STATE 3
iw ~ 500 mat
Sap Re RI 2055. idea hn RRs ay
180 150 120 #90 60 30 s— 10) 14,30 60 90 120. 150 +180

Ag, BARGE HEADING (degrees)

Fig. 13. RMS values of mooring cable forces as a function of barge
heading at indicated sea state.

1051

Kaplan

been carried out, where comparisons between theory, model experi-
ment, and (in some cases) prototype behavior were made (see [8],
[9]). The conclusions of those studies were that linear theory
produced good predictions of motion response operators; shallow
water effects may be easily incorporated; the effects of other mooring
arrangements (such as the usual catenary form of weighted chain
cables) can be represented in linear form and produce results agree-
ing with theory, within the range of lower sea states.

While agreement between theory and experiment was generally
obtained for almost all conditions, some degrees of freedom of the
vessels considered in [8] and [9] were not properly predicted for
irregular sea conditions. The motions of surge, sway, and yaw
exhibited large spectral response characteristics at very low fre-
quencies where little (if any) wave energy was present, but close to
the natural frequencies of those motions (due to the mooring "spring™
forces). Since these motions are very lightly damped, a very small
amount of input excitation can still produce relatively large motions
at these low frequencies. There are a number of possible explana-
tions for this behavior, but the most plausible one is related to the
influence of the nonlinear "drift" forces and moments, which will be
discussed in a later section when considering dynamic positioning.

The theory described here supplements what may be known
qualitatively for moored vessel behavior by furnishing quantitative
estimates for the motions and their inter-relationships. While the
validity of these analytical results for any particular vessel is sub-
ject to test, the results for other moored vessels using this same
analytical procedure give support to the reliability of the predictions.
Thus, it is feasible to treat all six degrees of freedom of a moored
ship in a realistic seaway and obtain results for response character-
istics of various motions of the ship and any associated load.

Considering the results and examples concerning the load-
lowering operation, there are two main conclusions. First, motions
having amplitudes of oscillation or giving rise to forces and accelera-
tions sufficiently high to influence construction operations may occur
under certain of the environmental conditions considered in this
study, particularly when loads are lowered by means of a boom ina
high sea state. Secondly, the violence of these motions, forces, or
accelerations may be significantly reduced by the proper choice of
vessel heading relative to the wind, and boom azimuth angle. The
latter factor regardless of the sea state, has by far the greater effect
in minimizing the energy of the fluctuating tension in the lowering
line, the vertical acceleration of the load, and its three displacement
components. These results provide useful information for conducting
operations from a moored ship platform, and hence the capability of
obtaining guidelines for operating vessels and performing engineering
work at sea is available with the tools of theoretical hydrodynamic
analysis presented here.

1052

Mooring and Postittoning of Vehicles itn a Seaway

VIL. MOORED BUOY ANALYSIS

A moored buoy system is similar in many respects to the
moored ship case, and simplifications are made in order to treat a
representative problem. The buoy system is assumed to be a single
point mooring, with a surface floating buoy hull connected by a
flexible line to the ocean bottom. Both slack and taut types of
moorings are included in the analysis, and the surface buoy form
can be either a ship-like form, a spar shape, or an axisymmetric
discus shape. The analysis is restricted to motion in a single plane
and the current direction and wave direction thereby lie in this plane,
making a two-dimensional problem. Allowance for current magni-
tude variation with depth is considered, with its main influence being
in the static equilibrium problem (which will not be treated in detail
here).

Considering the static equilibrium problem, a free-body
diagram of a differential element of the cable in the plane of interest
is shown in Fig. 14. The cable bends and the tension varies along
its length so as to keep all the indicated forces in equilibrium. The
cable weight acts vertically and the tension forces are directed
along the cable axis. The hydrodynamic forces due to the current
are resolved into components normal and tangential to the cable
direction. These unit forces are represented as follows:

F($) = Cys 5 pc(Ve sin 4)° (48)

G($) = C_+ 5 pc(V, cos 4)” (49)

1
where

p = mass density of fluid

cable chord length (in current direction)

ie)
Ul

and C,, C, are appropriate drag coefficients. These coefficients
depend on the cable cross section and surface geometry.

The summation of forces along the direction of the cable
axis yields:

T + G(o)(1 + €) ds - We, ds sin > - (T - dT)cos (dd) = 0

For a differential element, d¢—~ 0, so that cos (d¢) ~ 1.0. This
gives the differential equation for cable tension in terms of the inde-
pendent variable s, as follows:

£053

Kaplan

(p)(I+e) ds

F(¢) (1+ «)ds

Ve = current velocity

JT = cable tension
g = distance along relaxed cable
« = cable strain

@ = angle of cable from horizontal
W,= unit submerged weight of cable

G¢) = tangential unit force componet due to current

a
=
Hl

normal unit force component due to current

Fig. 14. Cable Free-Body Diagram

1054

Moortng and Posittoning of Vehicles in a Seaway

dT =[- G(¢)(1 + €) + W, sin ¢] ds. (50)
The summation of forces normal to the cable axis yields:
F(¢)(1 te) ds + W, ds cos $ - (T - dT) sin (dé) = 0

and with d@é—~ 0, we can approximate sin (dd) by dq. Neglecting
higher order terms involving products of differentials, results in an
equation for the differential angle:

do = = [F(t +e) + We. cos 4] ds. (51)

The strain, or cable elongation, is obtained from the following simple
relationship for an elastic cable material:

€= a (52)

where

cable (load-bearing) cross section area

>
uN

E. = effective static elastic modulus of cable material.

Associated with these equations is the representation of the
forces and moments acting on the buoy due to the wind and the cur-
rent (not considered here, but discussed in[10] from which the
present analysis is abstracted). All of these effects are considered
to be in equilibrium with the weight, buoyancy, and cable forces.

All the forces and moments acting on the buoy due to current
and wind are considered to be in equilibrium with the weight, buoyancy
and cable forces. At the surface buoy we then have:

D, + Dp = T cos 6
L, + B(6,h) = W+T sin $ (53)
2 2,1/2 lize
M, +M,= T(4, +24) sin (@ + © - tan 5 me + B(6,h)GZ(0,h)
c

where

1055

Kaplan

D, = drag due to wind acting on a buoy

Do» Les Me = current-induced drag, lift, and moment acting

on buoy
T = cable tension (at buoy attachment point)
@ = cable angle from horizontal (at same point)

B(@,h) = buoyancy force
W = total weight of buoy

4,, Z, = horizontal and vertical distances respectively
from cable attachment to CG

Z(8,h) = hydrostatic righting arm

Thus, for a given buoy'configuration in a particular condition of sub-
mergence in a given current, Eq. (53) can be solved for 0, T and
$; that is, the buoy trim equilibrium and the cable tension and angle
at the buoy. This result then becomes the initial condition for the
static equilibrium cable geometry calculation.

When considering the problem of the dynamics of the complete
moored buoy system, separate considerations in the analysis are
given initially to the buoy and to the mooring system, with ultimate
combination (i.e. coupling) exhibited later. The motion of a buoy
in waves considers the buoy to be equivalent to some type of hull
form, and the restriction to planar motion results in analyzing only
three degrees of freedom which may be considered to be surge,
heave and pitch. The equations of motion of the buoy are formulated
in the same general way as for a surface ship, described previously.
The only possible additional influence in the present case of the buoy
is to allow for the effect of a uniform surface current, which can be
included in the equations by interpreting the current as an equivalent
forward speed of the buoy hull through the water. However, for
simplicity here, this effect is deleted when analyzing the buoy wave
responses.

For the case of a ship-form buoy hull the hydrodynamic
force derivation is similar to that shown previously for the moored
ship. The general equations of motion in the vertical plane for the
coupled motions of surge, heave and pitch can be represented in
a more specific form as

ax tasx tia 6 = et x, (54)

1056

Moortng and Postttontng of Vehteles in a Seaway

ee oe
Zz a,

me zt ang a a579 + a5,0 + A909 = Lan F Ly (55)

a 5

a,x tayz ta,z ta,z ta,0 ta,0 ta,0 = Mn t My (56)

6

where the mooring forces are represented in general form and the
wave forces can be represented as sinusoidal functions of time for
different wave frequencies. The mooring forces will depend upon

_ the mooring arrangement (i.e. number of cables, attachment point,
etc.), whereas the functional form and degrees of freedom in the
force representation depend upon the geometric arrangement.

For the surge degree of freedom, the coupling with the pitch
equation, and vice versa, occurs as a result of hydrodynamic
inertial coupling (potential flow theory) and hence the symmetry
relation a,7= a3, is attained. This result is due to the equivalence
of the off-diagonal terms of the added mass tensor representation
of inertial forces. With the longitudinal force mx assumed to act
through the center of buoyancy (CB) of the hull, a pitch moment

m|BG|x occurs, i.e. a,,=m|BG| = a,7z where LBG| is the
vertical distance between the CB andthe CG (center of gravity).
The remaining terms in the surge equation are a,, = m and some

estimate for surge damping Ajos
The surge damping can be represented in a number of ways,
either linearly with allowance for the current by means of perturba-
tion theory, or in a nonlinear form as a drag coefficient representa-
tion, etc. (see [10] for more details). Ordinarily, this surge
damping term is not very important in its influence on the resulting
ship or buoy motions since there is no natural resonant response
in surge. However, in the present case of a moored buoy there isa
restraining surge force from the mooring cable and there may be
some resonant surge motion. Thus, the proper inclusion of the
surge damping force on the buoy hull can be important for dynamic
behavior calculations.

The mooring cable forces acting on the buoy hull are con-
sidered separately further ahead in this study. They are important
since such forces affect the buoy motions, and the buoy motions in
turn determine the boundary conditions as well as the input excitation
for the cable dynamics. The techniques for inclusion of these effects
in the overall mathematical model are considered later in this inves-
tigation.

A spar buoy hull form is axisymmetric about the vertical
axis and hence motion analysis can be carried out for the three
degrees of freedom with slender body theory techniques used in the
analysis of the hydrodynamic action on a long slender spar form.

mi = = pgs, aihiy Pa ee (57)

1057

Kaplan

where S, is the spar cross section area at the waterline intersection.
The mooring force will depend upon the mooring arrangement and
geometry, and is deleted temporarily from consideration. The wave
exciting force can be evaluated and the wave generation damping is
determined from work in[11].

The velocity potential and the pressure on a slender axisym-
metric body in waves are found using the results of [12] for the case
of a vertically rising body in waves, at zero forward speed and
evaluated for the condition where only the submerged portion from
the waterline down is of interest. The fluid pressure on the body in
regular sinusoidal waves is (from [ 12])

P,=.pea ES a sR cos 8' cos wt - sin wt) (58)

where € and @! are the longitudinal and angular coordinates of the
spar hull and R is the local hull radius. The local vertical force
on a section of the spar buoy is then

wv

a =-2R tana p dae’ (59)
e 0

with tan a = dR/d€ (the slope of the body contour), the local vertical
force is

2nrt/ ds!
SEs = pga sinwts+ e m™ ae (60)

where S' is the local hull cross section area, leading to

0
ké ds!
Zw = pga sin ut | Oar dé

(61)
where k= 2n/N= w/z, which can be simplified further by integration
oy parts (with S'(L) =0).

The surge equation for the spar hull form is represented as

0 eo
mx = - of. st(é)[x + (€ - E,)8] d& + Xqt Xy t+ Xm (62)

where &, is the C.G. location along the €-axis, and the wave force
is obtained as

1058

Mooring and Postttoning of Vehicles tn a Seaway

O
Xy -f i p cos 6'R dé dé = pau | Bete Nae ocoaai Co)

The surge damping force expression due to wave generation (pro-
portional to x) in [11] and to this should be added the surge damping
due to real fluid drag effects, which is nonlinear. This drag term
is represented in the form

0
ESP ay) [e+e - eg ag (64)

where Ap is the lateral projected submerged area of the buoy and
Cy is a drag coefficient whose value is = 1.2, the value for long
slender bodies and sections in an oncoming normal flow.

The pitch equation for the spar hull form is represented by
@e 0
Ho =-) suenlx +6 - 9616 - &) a
Bt

fe)
=: oa (€ - &)S'(&) dé > 0 +Mg+tMy+Mm (65)

and the wave induced exciting moment is given by

fe)
dX
oe he - &,) Se ag

2paw aa - Eg)eMs! (E) d& * cos ut (66)

M

The pitch damping moment coefficient due to wave generation is
given by [11], and as with the surge damping above, the pitch
moment equation has additional nonlinear damping given by

O
S04, ) Pea DOl(E-&) 48 (67)

which must also be induced. By considerations of symmetry, a
cross-coupling damping term due to pitch angular velocity, 0, will
appear in the surge equation, which is given by

6) 0 °
x,=- 289 Moye ag Je - toielsuey asd (6

1059

Kaplan

and a similar term will occur in the pitch equation, proportional to
x, given by

nee ° ;
Me = - oan este) ag - Mn ( - Eg)e“*s'(é) d&- x. (69)

All of the above expressions can be combined to produce the coupled
surge and pitch motion equations of the spar buoy, together with the
heave motion equation. The effects of the mooring are included in
terms of the appropriate degrees of freedom to allow computation of
the complete system response, which will be the end product of the
program.

The disc-shaped buoy hull is analyzed as a case of a shallow
draft vessel. The section in the water is a circular cylinder with a
small draft compared to the cylinder diameter, and the form is
axisymmetric. The hydrodynamic and hydrostatic forces are found
using the shallow draft approximation, as in [13], together with other
simplified representations for the wave-induced forces. Because of
symmetry relations, where the disc-shaped buoy is assumed to be
circular shape, some of the coefficients in the basic equations of
motion, Eqs. (54) - (56), are immediately evident:

(70)
aeg9 = az6 = 0.
Specific values of certain other coefficients are readily evaluated
for a circular discus shape, and they are given below. Assuming

that the discus buoy is a cylinder of radius R, and draft d', the
following heave restoration coefficient value is found:

2
aog= pgmR. (71)

For the case of the pitch restoring moment, the basic term (cor-
responding to the hydrostatic portion) of the coefficient az, is
obtained from the expression

M, = - W|GM|é (72)

where

=
Mr

weight of buoy

|GM| = metacentric height

1060

Mooring and Posittoning of Vehtcles in a Seaway

|GM | is the difference between the distance between the CG and the
CB of the submerged portion of the buoy, and the metacentric radius
between the CB and the intersection of the displaced buoyancy vector
with the vertical axis.

The metacentric radius is determined from the value of the
lateral displacement of the CB of the submerged section of the disc
cylinder. This is determined as the ratio of the moment of inertia
of the waterplane area about the buoy vertical centerline plane, to
the displaced volume. With this moment of inertia found to be
mR*/4, and with the valug of the buoy volume given by mR*d', the
metacentric radius is R /4d', leading to a metacentric height |GM|:
given’ by

ne
|GM | Sore |BG| (73)

where |BG| is the vertical distance between the CB and CG of
the buoy.

Similarly, values for added mass and added inertia for the
disc-shaped buoy can be found from the work of [13], based on con-
sidering this hull as a shallow draft vessel. In that case the total
added mass in the vertical direction is given by

‘ Al, dx = pR°My (74)

where the value of My is given in[13]. Similarly, the added pitch
inertia term is represented by

{ Al.x’ dx = pR°I! (75)

where the value of I} is given in[13]. These values are weakly
frequency dependent for the range of significant wave lengths of
concern in the buoy problem, and an appropriate approximate
constant value can be used. The damping coefficients for heave and
pitch are represented, respectively, by

a

es Ni dx = pR*wN, (76)

and

; 2 5
asg= | Nyx" dx = pR'wH, (77)

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Kaplan

where the quantities N, and H, are indicated as frequency-dependent
parameters in (77).

For the surge degree of freedom the same expressions as for
the ship hull form for the coupling terms with pitch, and vice versa
for pitch with surge, are valid for the case of the disc-shaped buoy.
The damping due to surge also has the same expression, and can be
carried over to the present case of a disc-shaped buoy, with appro-
priate values of drag coefficient and reference area for the disc.

The wave forces acting on the disc-shaped buoy are primarily

due to hydrostatic action and are evaluated on that basis. The verti-
cal wave force is expressed as

R
Zy = 2pg io £(x)n(x,t) dx (78)

where

f(x) = /R* - x? (79)

is the lateral offset of the buoy circular section and (x,t) is the
wave height representation, as follows:

n(x,t) =a sin (x cosB - cyt)

a sin(=™ - wt) (80)

where the dependence on the heading angle B is deleted due to sym-
metry. The resulting expression for the vertical wave force is

Zy = - ApgRa sin at | (1 - o* cos Yo) do (81)
10)

where o =x/R and y = 2mR/\, leading to

Zw=- 2mpgRea , 240) sin wt (82)

where J,( ) is a Bessel function. The pitch moment term due to
waves is given by

R
M,,= - 20g | xf(x)n(x,t) dt (83)

1062

Mooring and Posittoning of Vehicles in a Seaway

leading to

_ 7

My => pgR-alJ,(y) - Jgly)| cos ut . (84)

For the surge force due to waves the pressure component in
the axial direction is required, and this is found in terms of the
axial gradient of the wave amplitude record along the disc. Fora
hull of draft d', the surge force due to waves is given by

R
X, = - 2pgd' \ f(x) a n(x,t) dx (85)

which leads to

X, = - 2egmd'RaJ\(y) cos at. (86)

Thus the above expressions complete the representation of the terms
required for treating the motion of a disc-shaped buoy in regular
waves.

VIII. MOORING DYNAMICS

The initial treatment of mooring cable dynamics will be based
upon a complete formulation of equations of motion for a continuous
line that is assumed to be completely flexible and extensible. The
analysis is restricted to two-dimensional motion in a single plane,
which is coplanar with the oncoming current, and the velocities,
etc. are converted from directions along x- and y-axes (fixed in
space) to those along the normal to the cable, which leads to con-
sideration of the velocities U (normal) and V (tangential) relative
to the cable as basic variables.

The basic equations of motion inthe x- and y-directions are

du _ 9 Ox
Kae =Ry (r dette) + G(ite) cos $+ F(ite) sin > (87)

av 8 3
fegy = s(t sais) + G(ite) sin @- F(ite) cos ¢- We (88)

where p is the sum of the cable mass and added fluid mass per unit
length, when considering an elongated element of the cable, of length
(1{+e) ds. From geometric considerations

1063

Kaplan

ox = (ite) cos 4, oy = (ite) sin $ (89)

and with the definitions

ney veer (90)
U=usin @-vcos ¢ (91)
V=ucos ¢+vsin $ (92)
it can be shown that
U, - Vo, = - (1+ 94, (93)
and
V, +t Ud, = & (94)

where the s and t subscripts represent partial derivative operations.

By considering effects normal and tangential to the cable, the
basic dynamic equations can be expressed in the form

pl U, - Vo,] = - To, + F(ite) + W, cos # (95)
pl V, + Ud] = T, + Glite) - W, sin 4. (96)

In addition the relation

E yA (97)

is also necessary, so that the basic equations governing the cable
dynamics are Eqs. (93) - (97), where the first two relations are
basically kinematic. For the steady state case, i.e. neglecting
time derivatives, Eqs. (95) and (96) reduce to the same expressions
as given in the static equilibrium case, i.e. Eqs. (50) and (51).

To solve the quasilinear partial differential equations given

in Eqs. (93) and (97), a linearization procedure can be applied.
Defining the expressions

1064

Mooring and Posittonting ‘of Vehicles in a Seaway

U = Us) + U'(s,t) (98)
V = Vols) + V'(s,t) (99)
T = T,(s) + T(s,t) (100)
= $(s) + $'(s,t) (101)
€ = €,(s) + €'(s,t) (102)

where the '-symbol quantities are perturbations about the equilibrium
positions (o-subscript terms), and expanding various constituent
terms up to first order terms alone, leads to

sin @= sin ($9t+ $') = sin 6, + $' cos $5 (103)

cos $= cos ($,t+9') = cos $, - $' sin $9 (104)

The hydrodynamic loading terms are expanded in the form

Fak, + Fu" + FyV't+ Fy?" (105)
G=G,+GyuU' + Gw' + Gyo! (106)

where the partial derivatives of the loading functions with respect
to particular velocities are indicated. This can be accomplished
when considering the steady state velocity solutions (from Eqs. (93)
and (94) when time derivatives equal zero) which, when combined
with Eqs. (114), are

Up, = 0 (107)

Vior= 0 (108)

0

The linear perturbation equations are then given by
-pU' = Tod; +790 (ite, )[F,U'+FV'+ Fo] +F,e'+ Ww, ¢'sin d, (109)

-pV' = Tj +(1te,)[ GU'+GWV' +Gyo'] +G,e'- Wed! cos $o (110)

1065

Kaplan

(1 + €,)o, = V'bo, + Vos - Us (141)
€} =e vert U" do, (11:2)

ut
oe oer (113)

which are a set of first order linear partial differential equations.
The boundary conditions for this set of equations is the next task of
importance. For a moored buoy in a combined current and seaway,
the current is felt acting on the buoy and also on the cable. How-
ever, the wave effects attenuate rapidly with depth, and hence the
wave forces act on the buoy along with no influence assumed on the
cable. In that case the buoy motions due to a regular sinusoidal
seaway (assumed for analytical simplification) are transmitted to
the cable at its attachment point, and the cable motions are then
sinusoidal in time at that point.

The boundary conditions at the anchor point at the sea bottom
are given by

U=V=0, s=0 (414)

and at the buoy attachment point for the cable, s =£, the boundary
conditions are much more complicated. The velocities at the buoy
attachment point are given by

U= x = 2°6 (115)

c
v=z- 4,0 (116)

where x; z and @ are the wave-induced surge velocity, heave
velocity, and pitch angular velocity, respectively, and Ze and f¢
are the vertical and horizontal distances from the buoy CG to the
cable attachment point. The normal and tangential velocities are
defined by Eqs. (91) and (92), and considering the wave-induced
motions to be of the same order as linearized perturbation terms,
the boundary condition relations for the perturbations are

U! = (x - z,0)sin $, - (2 - 4,8) cos 9%, (417)
V' = (x - z,0)cos &+ (z - £,0) sin $, (118)
at s=2, where x; z and @ canbe represented in the et form

for a sinusoidal wave input. The boundary conditions at s=0 are

1066

Mooring and Positioning of Vehicles in a Seaway

U'=v'=0 (119)

where only four boundary conditions are necessary since e' can be
eliminated as a variable by use of Eq. (113).

The representation of the boundary conditions at the upper
end of the cable (s = 2), at the attachment with the buoy, shows how
the cable motions are influenced by the buoy motions. However,
the buoy motion is also influenced by the cable system dynamics
since a mooring force acts on the buoy as well. The mooring force
that affects the buoy motion is due to the component of tension at the
attachment point, which leads to

Xm=- T'(L) cos off) + To(L) sin o(£)'(£) (120)
Zm=- T'(L) sin $,(£) - TolZ) cos (2) $'(£) (124)
MS Se ee eZ (122)

where these expressions are component terms on the right-hand
sides of the respective equations, e.g. Eqs. (54) - (56). With
T'(£,t) and #'(£,t) represented as f(s)e®t forms the total system
of buoy and cable can be solved using linear equations of motion for
sinusoidal wave inputs at different frequencies (assuming the non-
linear damping terms in the buoy motion equations are linearized).
The "feedback" nature of the equations governing the buoy motion and
the cable motion is illustrated by the above discussion, where the
cable tension force influences the buoy motion directly and the buoy
motions determine the cable upper point boundary condition.

As mentioned earlier, the study of a moored buoy system is
closely related to other mechanical cable system problem areas.
The case of a moored ship is, of course, very similar to a moored
buoy but the distinguishing difference is the relative masses that
are involved. For a moored ship case, the ship is so large (rela-
tively) that it can be realistically assumed that only the quasi-static
forces applied to it by the mooring cable are significant, and that the
cable dynamics do not influence the ship's response; that is, mooring
cable dynamic forces can be assumed small with respect to other
excitation forces. Thus, the dynamic problem of the ship and the
cable can be treated separately. Similar reasoning applies to the
surface condition of a cable-towed body system. However, the
analysis of the component forces involved in such systems is appli-
cable to the present case of a moored buoy system, keeping in mind
the required coupling in the mathematical model, as shown above.

The equations developed here for a moored buoy system have
to be solved in order to determine the necessary information on

1067

Kaplan

system performance. The methods to be applied should recognize
the problem as involving complicated two-point boundary value prob-
lems, or alternatively another technique that replaces the equations
by a set of difference equations or differential-difference equations
similar to the case of a beam vibration problem can be applied (see
[10] for a discussion of different computational techniques for this
problem). The solution of the class of partial differential equations
given above is a specialized simulation problem that is the subject
of presently on-going research so no further detailed discussions can
be given. The development of these equations is another illustration
of the application of knowledge of hydrodynamics of ship motion
toward other related problems of engineering significance.

IX. DRIFT FORCES DUE TO WAVES

When a floating vessel is acted upon by waves it experiences
forces and moments that are predominantly oscillatory-like in
nature, with the frequency characteristics similar to that in the
spectrum of the oncoming wave system. These forces are also
linear with regard to wave amplitude. In addition there are also
nonlinear force contributions that arise from the presence of the
vessel hull modifying the incident waves by virtue of its function
as an obstruction, as well as the effect of interaction between the
vessel motion and the incident waves. These nonlinear forces are
much smaller than the linear wave forces, but nevertheless exert a
significant effect on certain degrees of freedom of the vessel.

The major nonlinear drift forces of importance to the problem
of maintaining a desired position in a seaway are in the longitudinal
and lateral directions relative to the vessel, as well as the yawing
moment that tends to rotate the vessel in heading. Some theoretical
studies of these quantities have been made, but only for determining
average values in regular waves, with the work of Havelock [14],
Maruo [15], Hu and Eng [16] and Newman [17] serving as typical
examples.

Havelock [14] treats only the drift force in head seas. His

formula is based on the heave and pitch motions and their relative

hase to the incident wave. The theoretical approach of Hu and Eng
Pt 6], which follows that of Maruo [15], yields expressions only for
the lateral drift force and draft yaw moment in waves. (Maruo's
results only considered the lateral drift force.) While their results
are quite general, they have only been reduced to workable formulas
under the restrictive assumptions of a thin ship, with small draft,
in long waves. These results indicate infinite (practically unrealis-
tic) forces and moment as the wave length goes to zero. The maxi-~
mum lateral drift force occurs in beam seas (varying as sin°§),
and the maximum moment occurs at an angle of 45° (varying as
sin B).

1068

Mooring and Postitioning of Vehteles in a Seaway

Newman's method [17] , based on slender body theory, does
show some comparison with very limited experimental work for
lateral drift force and yaw moment which indicates rough agreement.
His results for longitudinal force, for which no experimental com-
parison is given, indicate that this force in head seas generally
exceeds the lateral forces for a given wave length condition. Further-
more, these results indicate ‘that the maximum lateral force occurs
in bow waves (B = 45°), with the force going to zero in both head and
beam waves.

The results of Hu and Eng [16] include the effects of sway,
yaw and roll motions, with no influence of heave and pitch included
(as to be expected for thin ship analysis) while Newman [17] only
accounts for heave and pitch motion effects without any influence of
the three lateral degrees of freedom. Thus there is a question as
to the proper representation of the drift forces that would reflect
the influence of the important dynamic motions that produce these
forces. The analysis by Maruo [15] presents a final expression for
the average lateral drift force in beam seas that depends upon the
reflected wave amplitude, which in turn is defined in terms of the
relative motion between the incident wave and the resulting heave
motion. The presence of sway motion has no effect on the lateral
drift force since the body acts like a wave particle in beam seas
and no relative motion occurs (to that order). A similar result is
indicated for submerged cylinders in the work of Ogilvie [18],
where the average lateral force identically vanishes.

In all of these hydrodynamic studies, the force is found to be
proportional to the square of the incident wave amplitude, since the
nonlinear pressures are represented in terms of squares of generated
wave amplitudes, squares of fluid velocities, and products of first
order oscillatory displacements with derivatives of fluid velocities +>
While the previous hydrodynamic analyses have been concerned with
the average drift force for a regular sinusoidal wave, it is important
also to determine the actual time histories of these forces, especially
for the case of an irregular incident wave system. In that case it is
expected that the drift force will be a slowly varying function, in
terms of the frequencies contained in the defining incident wave band-
width, and it is of interest to determine the basic representation of
the forces, the response of floating vessels to such forces, the
statistical properties, etc.

In order to illustrate the basic characteristics of these forces,
particular attention will be given to the case of the lateral drift force
acting on a vessel in beam seas. Using the results of Maruo (15);
the later drift force acting on a cylinder (in the two-dimensional cas e)
is given by

FE 1 —2 2 1 2——2 2
> = 5 pgA,|2-71| = 5 Pga A,|=-1| (123)

1069

Kaplan

where A, is the ratio of the amplitude of the heave-generated two-
dimensional wave to the amplitude of heaving motion of the ship
section, and |z- n| is the absolute value of the relative heave mo-
tion. It is seen that this expression is thus proportional to the square
of the incident wave amplitude, with the force only occurring due to
the relative motion between the heave and the incident wave. It can
be shown that the mean value of this force in an irregular sea
characterized by a wave spectrum such as the Neumann spectrum is
given by

Qo ——

: ~—
mee SG) See (124)
o a

where A*(w) is the Neumann spectrum representation given in
Eq. (43).

The results obtained in the basic derivations for determina-
tion of the drift force have been carried out for the case of a single
sine wave at a fixed frequency. Inthe real case, the waves are
composed of many frequencies in a band, and for the purposes of
simplification of the following analysis it will be assumed that this
bandwidth is relatively narrow. If a combination of two different
frequencies is present in a wave, and hence in the relative heave
motion represented by

Z-N= b, sin wt +b, sin (wet + ¢) (125)
the square of this term is given by

(z - n)°= be sin’ wt + bs sin® (wot + 9) +2b,b, sin w,t sin (wot + >)

1.) 2 2
= she +b. + 2b, b, cos [(w, - w, )t + 6}

2 2
- 54) cos 2w,t + b, cos 2(wt +) - 2b,b, cos [ (wy + o,)t ral}.

(126)

It can be seen that this expression is made up of a group of terms
that are essentially constants and slowly varying terms (due to the
narrow band assumption), and another group of terms representing
higher frequency oscillations, i.e. at higher frequencies than the
wave terms. If a time average of this quantity is made, it will be
seen that the combination of the constants and the slowly varying
term remains and the higher frequency terms drop out, and that this
first grouping of terms can be represented as the square of the
envelope of the combined signal given in Eq. (125) (see Er9}).

1070

Mooring and Postttoning of Vehicles in a Seaway

If the wave system, and the resulting relative heave motion,
are represented by the sum of a larger number of terms of different
frequencies, with these frequencies having only small increments
relative to a single reference frequency (as a result of the narrow
band assumption), the contribution to the drift force value in that
case can be shown to be given by an expression that is also identified
as the square of the envelope of the total signal. This identification
and interpretation of this type of expression for the drift force can
generally be extended to the case of an arbitrary input, including a
random input which is assumed to be made up of a combination of
different frequencies within a narrow band. Thus a simulation
technique in the time domain for this term requires determination
of the envelope of the input signal (relative heave motion), squaring
this quantity, and applying the appropriate constants to produce the
required time history signal.

The drift force has been shown to be a nonlinear function of
the wave amplitude, and in a random sea it is a slowly varying
function of time, where this slow variation is considered relative to
the wave frequencies and the linear wave-induced forces. Since
the wave surface elevation and all linear terms derived from it are
assumed to be Gaussian random processes, the drift force is known
to be non-Gaussian in regard to its probability density. In order to
obtain further characterization of the properties of the suction force,
it would be useful to determine the probability distribution and
spectral properties of this force. The accomplishment of this task
will be aided by the simplified interpretation of the drift force that
was presented above.

Considering the drift force as the square of the envelope of a
Gaussian random process, certain information is available concern-
ing the probability density of this type of function. A square-law
detector produces an output proportional to the square of the envelope
of the input, if the input is a narrow band Gaussian random process
[ 20] which is the assumption used in the present analysis. Fora
particular input into such a square-law detector, the probability
density function of the output (denoted as w) is given by

1 -w/ E(w) s
p(w) - E(w) © ’ w= 0 (27)

where E(w) is the mean value of the square-law detector output.
On that basis, the probability density for the drift force in a random
sea can be represented by

-F//F
p(Fy) = se de seke nabie ee (128)
F
y

where Fy, is the mean drift force, and hence the probability distri-

1074

Kaplan

bution is given by
P(= <x) =1-e" (129)
F

In addition to information on the probability density, additional
statistical properties of the drift force are provided in terms of the
autocorrelation and power spectral density functions for that force.
The relationship between the statistical characteristics of an input
to the nonlinear form of drift force representation, to the output
characteristics, as defined by the autocorrelation and spectral
density, is a useful description which can be applied in further
analyses and for simulation studies. The problem of a square-law
detector has been treated in the available literature, e.g. [20] and
the results can be applied to the present case. If a general narrow
band Gaussian random process, represented by the variable x(t),
is the input to a square-law device, and the output is definedas r,
the autocorrelation function R,2(7) is given by

Z 2
R.2(7) = 407 + 4R,(7)
BAG aod ae
=(r°) + 4R,(7) (130)
where
x = 202 (134)

is the mean value of the square of the envelope, with of the mean
square value of the input function x(t). The relations between the
autocorrelation and power spectral densities of the input function
are given by

00 @
S,(w) = zt R,(t)e tat, —-R, (7) = a Salen’ da (rae)
-©

where
{ 0
R,(0) = o4 = a“ Sy(w) dw (133)
0

thus being in conformity with the relations described by the Neumann
wave spectrum and all linear functions derived from that spectral
formulation.

1072

Mooring and Posittoning of Vehicles in a Seaway

The power spectral density of the square-law device output
is then

Oo
2 -i
S 2(w) = ai R, (T)e te dt
J a 00

=4(r*) §(w) +3 Rilrje de (134)

where 6(w) is the delta function. The last term onthe right in
Eq. (134) can be evaluated by using the definitions in Eq. (132), so
that

ro) re -
2 u; ; res
s( R,x(T)e Lae ae = aN S,(w') au" f R,(T)e i(w-w )T oe
T JLo T J-00 -00
©
-{ S,(0!)Sx(w - w!) dos! (135)
-00
which leads to the final result

aN oo
S 2(w) = 4(r?) 6(w) +{ S,(w') Sy(w - w') dw. (136)
-00

The power spectral density of the square-law device output, whichis
proportional to the drift force, contains the delta function term (that
represents the non-zero mean value of the force) and a convolution
integral of the input spectral density whose value will depend on the
nature of the particular input. The square law detector output is
obtained by operating on the output of the square law device with an
ideal low pass zonal filter that filters out completely the high fre-
quency part of its input, thereby leaving only the low frequency part
representing the envelope.

A particular application to illustrate the results of applying
this analysis is given for the case of a vessel moored in a seaway
such that an irregular beam sea is present. The vessel chosen for
this illustration is the same CUSSI moored barge treated previously,
and the sea condition is represented by a Newmann spectrum cor-
responding to a 24 kt. wind (upper Sea State 5). A mathematical
representation of a filter circuit whose amplitude characteristics
are approximately the same as the square root of the Newmann
spectrum formula (Eq. (43)) was derived, programmed on an
analog computer with a white noise generator input, and produced
an output that represented a continuous time history of the surface
waves with that desired spectrum, r.m.s. value etc. A simplified
constant coefficient second order differential equation for ship heave

1073

Kaplan

motion in beam seas was set up on the analog computer, with the
input wave force excitation assumed to be proportional to the wave
record, and solutions obtained for z(t), the heave motion as a
function of time, and also for [z(t) - n(t)], the relative heave
motion.

At the same time another equation was programmed on this
analog computer representing the uncoupled sway motion of the
moored barge, viz.

(m + Agdy + Nyy + kyy = Yox(t) (137)

where Yo,(t) is the wave exciting force. This exciting force was
simulated to represent the linear wave-induced force and then the
nonlinear drift force, with separate solutions obtained for each
excitation above in order to illustrate the different output results.
The linear wave excitation force was represented as proportional
to n(t), after a 90° phase shift over the pertinent wave bandwidth,
which is an adequate approximation. The nonlinear drift force was
represented by

= 2 2
Y aritt = pgLA, {z - n} ’ (138)

where the { } symbol represents the envelope operation, and a
constant value is assumed for A,. The envelope of a time-varying
function is obtained by rectifying the signal (i.e. an absolute value
circuit), followed by a low pass filter.

The results of this simulation study are shown in Fig, 15
for the case of linear wave force excitation and in Fig. 16 for the
drift force input. The time histories of the surface wave motion,
sway motion output, and input exciting force are shown in each
figure. The linear wave force response is seen in Fig. 15 to be
generally oscillatory, of the same general frequency content as the
wave input, and with an amplitude of the same order as the wave.
The input excitation force has somewhat higher, frequency content
(since it is proportional to n) and it reaches amplitudes of 4X 10° lb.

The sway motion in Fig. 16, due to the nonlinear drift force
input, has an entirely different character than the surface wave
motion or the wave-induced sway motion shown in Fig. 15. Itisa
long period, almost regular response at the natural period of sway
for the moored ship, viz. 64 sec. (see Table 1). The input force
that caused this response is also shown in Fig. 16, as derived
according to Eq. (138) and it can be seen to be a slowly varying
function of time, reaching a maximum value of about 50,000 lb and
causing a response reading up to 15 ft in amplitude. Thus the
characteristics of the slowly varying nonlinear force, of much

1074

Mooring and Positioning of Vehicles ina Seaway

Fig. 15. Sway motion response to linear wave force in irregular
beam seas.

15

=3
ae 10" Yarift

lb.

Fig. 16. Sway motion response to nonlinear drift force in irregular
beam seas,

1075

Kaplan

smaller magnitude than the linear wave force, produce large motions
of moored vessels by causing resonant responses with the low fre-
quency modes introduced by the moorings.

If the two responses given in Fig. 15 and 16 were linearly
combined, as in the realistic case at sea when both forces are
generated simultaneously, the total output would represent the actual
motion of the moored vessel. The resulting large motions would
cause significant stretching of the mooring cables, leading to larger
forces in the cables than indicated by the linear wave effects above
(e.g. as shown in Fig. 13). Thus proper consideration of the non-
linear wave forces and their influence must be included in any
analytical estimation of expected motions and forces of moored
vessels, thereby requtring further effort at understanding and simu-
lating these effects.

As an aid in obtaining further insight into the characteristics
of the lateral drift force in this case, an evaluation was made of the
power spectrum of this force using the expressions given in Eq. (136)
for the part that represents the random variations of this force
about its mean value (the convolution integral term). The result
of this evaluation is shown in Fig. 17, and when considering the
effect of the low pass filter, all values for w> 1.0 will be eliminated.
Thus it can be seen that the drift force itself is concentrated at low
frequencies, and that the response of a dynamic system with a very
low natural frequency (i.e. the ship sway motion) will result ina
response spectrum concentrated at an even smaller low frequency
band. This is what is usually found in the results of model tests and
full-scale experience, and thus an explanation is provided by the
preceding analysis.

X. APPLICATION TO DYNAMIC POSITIONING

When considering the case of dynamic positioning, various
forces act on a free vessel in the open sea that cause it to move from
its required position. These forces are the relatively steady forces
due to wind and due to current, the oscillatory-type forces due to
waves and the drift forces. The wind generates forces and moments
because of its impingement upon the abovewater surfaces of the hull
and superstructure, while the current forces act on the underwater
hull (and any submerged drilling equipment, if that is the purpose for
the vessel). These steady forces can be overcome by the generation
of steady forces by some type of thruster mechanism that will act to
maintain the ship more-or-less in its desired location.

The oscillatory-type forces due to waves are very large, and
no force-generating system installed on a vessel is expected to be
able to overcome such effects. The ship will therefore oscillate
"back-and-forth" in response to these large wave forces with
essentially no net deviation of significance from its average position.

1076

Mooring and Postttoning of Vehicles in a Seaway

Sy (w)
drift 0.04

2(pauk)

ft.*-sec.

0.02

0 0.5 0 1.5 230 2.5 3.0

w, rad/sec.

Fig. 17. Representative power spectrum of lateral drift force.

It is the drift forces and moments whose mean values tend to move
and/or rotate the ship off position, with their level of fluctuation
causing dynamic responses that produce ship motion. Thus some
means of control must be applied to "modulate" the forces developed
by the thruster system in order to minimize the ship's average
motion relative to its desired position.

The ship will experience drift-like forces in the lateral and
longitudinal directions, as well as a yaw moment, and the philosophy
of applying control forces to the ship will be aimed at countering
these forces by orienting the ship in the proper direction so that the
resultant force is acting along the ship's longitudinal axis and there
will be no significant moment. It would then be possible to use the
main propulsive thrust of the ship, assuming controllable pitch
propellers, to counter this resultant force. Lateral forces that are
developed by particular thrusters, or other force systems, will be
used to overcome any tendency of the ship to rotate out of its pre-
ferred direction due to any resulting yaw moments. The force
magnitudes in regard to average values can be estimated from the
results of some of the cited references given in this paper, and
some idea of the time history variations and maximum magnitudes
expected relative to the average values can also be inferred from the
work presented here.

1077

Kaplan

In a complete simulation study of the resultant motion of a
ship in which a dynamic positioning system is to be installed, all
three degrees of freedom (surge, sway and yaw) will be coupled and
the forces and moments will be dependent upon the relative orienta-
ticn with respect to the incident wave system. This will be a some-
what complicated analysis, but the tools are generally available for
determining the various hydrodynamic parameters entering inte such
a study. It will also be necessary to consider the type of signal
system that would inciate the position errors of the ship, together
with a signal processing operation (i.e. control system design) that
will be necessary in order to achieve the desired type of operation.
Similarly, some estimate of the response time of the thruster force
development must be included in determining ship response so that
a measure of positioning accuracy can be obtained as a result of the
analysis.

In view of the complexity of this problem, a discussion of a
simple application will be given for the case of sway motion alone in
beam seas. In that case the equation of motion will be similar to
that given in Eq. (137), without the presence of the linear spring
term (that was due to the mooring in the previous case). The
response due to the linear wave forces will be generally the same for
this case as in the case of the moored ship, as shown in Fig. 15.
However, the effect of the drift forces will cause the ship to continu-
ally deviate in position within a very short time. The deviation will
be almost a quadratic growth with time since the response is similar
to that of a constant force acting on a system primarily represented
as a pure second derivative dynamic response. Thus a control force
is necessary, and the control rule should include terms proportional
to sway displacement and velocity, i.e. the control force will be of
the form

Y, = <iCplys=cyolh= Coy (139)

where y represents the lateral error displacement relative to the
desired position, y,, and this control force is included in the basic
equation

(m + Age)y + Nyy = Ywaves * Yarits * Ye ° (140)

The lateral position error, which can be obtained from an
acoustic reference system placed on the ocean bottom, will contain
the influence of the higher frequency response due to the linear wave
forces and in addition the control signal that includes the lateral
velocity error will contain more "noise" in the resulting control
signal. This can be overcome by the inclusion of appropriate filter
circuits associated with the control signal processing, which involves
the use of standard servomechanism techniques within the state-of-
the-art of control design. A closed loop feedback system using the

1078

Mooring and Posittoning of Vehteles in a Seaway

appropriate measured inputs can then be evaluated in detail by com-
puter system simulation to determine the optimum gains to be~used
in the control rule in Eq. (139). The only guidance that can be given
for this selection, based on simple dynamic principles, is to select
the value of the gain C, that will produce a resultant frequency that
lies between the frequency associated with the maximum spectral
energy of the predominant wave system and the very low (near zero)
frequency for large responses to drift forces. The value of the gain
Cy should be such that, when added to the normal ship damping,

the resulting response of the system will be relatively "flat" through-
out the major band of disturbing frequencies for the drift force.

All of these characteristics can be refined in the course of control
system analysis and design, as well as from the simulation results,
and further discussion lies beyond the scope of the present paper.

XI. CONCLUDING REMARKS

All of the preceding problem areas discussed in this paper
have illustrated the application of a specific area of Naval Hydrody-
namics, viz. hydrodynamics of ship motion in waves. The utility
of presently existing techniques of analysis for solution of practical
problems in ocean engineering, with emphasis on mooring and posi-
tioning of vessels and other systems in a seaway, has been shown
within the limits of the present state of development of this field.
Greater emphasis toward consideration of certain nonlinear hydro-

dynamic forces for application to these problems has been indicated,
especially in view of their predominant effect in certain modes of
motion. Possible directions for future research and development
activities in this field will involve consideration of better techniques
of representing the form of these forces in terms of body geometric
parameters, more concentration of basic model measurements for
comparison with theory, and techniques of simulatiorm in dynamic
analyses of motion behavior. This information will be fundamental
in establishing computer models for determining many aspects of
system performance at sea prior to actual construction, thereby
providing insight as to expected problem areas and methods of solu-
tion. The methods of applied hydrodynamics for these purposes are
generally available now, and it remains for the ocean engineering
profession to determine the utility or applicability of these tools

to the particular practical problems that they face in their own
operations.

1079

Te

9.

10.

11.

Kaplan

REFERENCES

St. Denis, M. and Pierson, W. J.: "On the Motions of Ships in
Confused Seas," Trans. SNAME, 1953.

Kaplan, P.: "Application of Slender Body Theory to the Forces
Acting on Submerged Bodies and Surface Ships in Regular
Waves," Journal of Ship Research, November 1957.

Pierson, W. J., Neumann, G. and James, R. W.: "Practical
Methods for Observing and Forecasting Ocean Waves by
Means of Wave Spectra and Statistics," U.S. Navy Hydro-
graphic Office Pub. No. 603, 1954.

Deep sea mooring plan and typical mooring legs, and deep sea
mooring details, U.S. Navy, Bureau of Yards and Docks,
Dwegs. Nos. 896131, 896132, June 1961.

Grim, O.: "Die Schwingungen von schwimmenden, zweidimen-
sionalen Korpern," Hamburgische Schiffbau-Versuchsanstalt
Gesselschaft Rpt. No. 1172, September 1959.

Kaplan, P. and Putz, R. R.: "The Motions of a Moored Con-
struction Type Barge in Irregular Waves and Their Influence
on Construction Operation," Report for U.S. Naval Civil
Engr. Lab. under Contract NBy-32206, August 1962.

Technical Memorandum containing information concerning
"CUSS I," supplied by U.S. Naval Civil Eng. Lab.,
Port Hueneme, California, August 1961.

Muga, B. J.: "Experimental and Theoretical Study of Motion of
a Barge as Moored in Ocean Waves," Hydraulic Engineering
Series No. 13 of the U. of Dlinois, January 1967.

Hsieh, T.,.Hsu, C., C., Roseman, D, P. and Webster, /Wa:Ga:
"Rough Water Mating of Roll-On/Roll-Off Ships with Beach
Discharge Lighters," Hydronautics, Inc., Tech. Rpt. 631-1,
July 1967.

Kaplan, P. and Raff, Alfred I.: "Development of a Mathematical
Model for a Moored Buoy System," Oceanics, Inc. Rpt. No.
69-61, April 1969.

Newman, J. N.: "The Motions of a Spar Buoy in Regular Waves,"
DTMB Report 1499, May 1963.

1080

12.

13.

14,

15

16.

17.

18.

19.

20,

Mooring and Posittoning of Vehicles itn a Seaway

Breslin, John P, and Kaplan, P.: "Theoretical Analysis of
Hydrodynamic Effects on Missiles Approaching the Free
Surface, Including the Influence of Waves," Proceedings
BOHAC Hydroballistics Symposium, September 1957.

Kim, W. D.: "On the Forced Oscillations of Shallow-Draft
Ships," Journal of Ship Research, Vol. 7, No. 2, October
1963.

Havelock, T. H.: "The Drifting Force on a Ship Among Waves,"
Philo. Mag., Vol. 33, 1942.

Maruo, Hajime: "The Drift of a Body Floating on Waves,"
Journal of Ship Research, Vol. 4, December 1960.

Hu, Pung Nien, and Eng, King: "Drifting Force and Moment on
Ships in Oblique Waves," Journal of Ship Research,
Vol. 10, March 1966.

Newman, J. N.: "The Drift Force and Moment on Ships in
Waves," Journal of Ship Research, Vol. 11, March 1967.

Ogilvie, T. Francis: "First- and Second-Order Forces ona

Cylinder Submerged Under a Free Surface," Journal of
Fluid Mechanics, Vol. 16, 1963.

Terman, Frederick E.: Radio Engineering, McGraw-Hill
Book Co., Inc., Second Ed. , CELE

Davenport, Wilbur B., Jr., and Root, William L.: An Intro-

duction to the Theory of Random Signals and Noise, McGraw-
Hill Book GCo., Inc., 1958.

1081

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rao .

WAVE INDUCED FORCES AND MOTIONS OF
TUBULAR STRUCTURES

J. R. Paulling
University of Caltfornta
Berkeley, Caltfornta

ABSTRACT

Many types of stable ocean platforms consist of space-
frame assemblages of tubular structural and buoyancy
members. An approximate method of predicting the
hydrodynamic forces and resulting motions of such
structures is described. In this procedure, the force
on each member is’computed by assuming that the
member is long and slender and all other members are
absent. Such forces for all members are summed and
introduced into the linear equations of motion of the
entire structure, which may then be solved for the re-
sulting platform motions. Reasonably good agreement
is obtained between the results of such analysis and
model experiments with several different platform
configurations.

I. INTRODUCTION

Many of the stable floating platforms which have been pro-
posed or constructed for deep water drilling, mining, or emplace-
ment and recovery of heavy objects can be described as space-
frame assemblages of tubular members. R. H. Macy [1969]
describes and illustrates several oil-drilling platforms of this type.
One of them, BLUE WATER II, consists of a square base configura-
tion approximately 200 feet square, made up of cylindrical members
14,5 feet in diameter, with four vertical corner caissons 24.7 feet
in diameter supporting the main deck. This platform normally
operates at a draft of about 40 feet. A second platform, the SEDCO
135, consists of three main vertical caissons located approximately
at the vertices of an equilateral triangle, several diagonal tubular
truss members, and, at the bottom of the main caissons, elongated
pontoons of oval planform. These platforms are moored by a spread
array of anchors, and operate in water depths of up to 600-1,000 feet.

1083

Paullting

McClure [ 1965] described a platform which was designed for the
MOHOLE deep sea drilling project. This platform was to consist of
two submerged main horizontal pontoons 35 feet in diameter, and 390
feet long, with a centerline separation of 215 feet. Three vertical
caissons extended from each horizontal pontoon through the free
water surface to support the main working deck. This platform was
intended to operate in a water depth of 14,000 feet, and was to be dy-
namically positioned by means of trainable propulsion units controlled
through a central computer system. The foregoing types of platforms
are referred to as "column stabilized," which implies that pitch and
roll static stability are obtained primarily from the waterplane ma-
ment of inertia of the surface piercing vertical column.

A third type of platform for which a tubular space frame con-
figuration has been proposed is the tension leg platform, an example
of which is shown by Macy [1969] and described by Paulling and
Horton [1970]. This is a moored stable platform for which the
buoyancy exceeds the platform weight, and the net equilibrating
vertical force is supplied by vertical tension mooring cables secured
by deadweight or drilled-in anchors. As a final example of tubular
stable platform structure, we mention the spar-type platforms, the
prime example of which is FLIP, described by Fisher and Spiess
[1963]. The platform consists of a single cylindrical member of
tapering cross section arranged to float vertically, with a small
portion of its length projecting above the surface of the sea.

All of these platforms share a common characteristic in that
their configuration consists of a space frame assemblage of relatively
long, slender, cylindrical members, with the addition in some cases
of small buoyancy chambers or pontoons. All share a common ob-
jective of producing a working platform having minimum wave-
induced motions, even under relatively severe sea conditions, i.e.,
a platform which is "transparent" to the waves. The positioning
methods used differ greatly in each case, ranging from essentially
no positioning, in the case of FLIP, dynamic positioning with no
physical connection to the sea bottom, in the case of the MOHOLE
platform, to various types of anchoring systems exemplified by the
tension leg and the column stabilized platforms. The sea environ-
ment and resultant platform responses are similar in each case,
i.e., all are intended to operate in relatively deep water under
severe environmental conditions, with platform motions which are
small compared to the overall dimensions of the platform and to the
length of waves involved. Our objective here is to describe a pro-
cedure for analyzing the forces and motions which can be applied
equally to all of these platforms if suitable account is taken of the
type of anchoring or positioning restraint involved.

Such an analysis of wave-induced forces and motions forms
an essential part of the process of designing a platform to perform
a specific mission. At least three such functions are envisioned.
First, for a platform of given geometry, a range of sea conditions
may be investigated to determine what limitations may be imposed

1084

Wave Induced Forces and Mottons of Tubular Structures

on the platform's performance by the resulting wave motions.
Second, given a set of sea conditions and platform requirements, we
may investigate a family of platform configurations to determine
those members of the family which will be able to perform the
specified mission under the stated sea conditions. This kind of
analysis, in turn, might form a part of a more extensive system
study aimed at determining the most cost effective platform system.
As a third function, the force distributions on structural members
which are obtained during the force and motion analysis may be

used in connection with the detailed structural design of the platform.

The general procedure followed in analyzing the dynamic
behavior of such a platform is to assume that it behaves as a rigid
body having six degrees of freedom. The external forces which
excite the motion of the structure are associated with the fluid motion
relative to the structure, and with the structure's mooring or position-
ing system. Two alternative methods are available for the computa-
tion of the fluid forces. In the first, the fluid is assumed inviscid
and its motion irrotational, and we proceed on the basis of classical
hydrodynamic theory to seek a solution to Laplace's equation in the
fluid region subject to certain boundary conditions. These include
kinematic boundary conditions on the free water surface and on the
wetted surface of the structure itself, a constant pressure dynamic
boundary condition on the free surface, a dynamic boundary condition
on the wetted surface of the body, which is derived from the rigid
body equations of motion, and other conditions far from the body
which are necessary for uniqueness of the solution. This approach
yields great insight into the fundamental nature of the fluid phenomena,
and is exact within the limits of the necessary fluid idealization and
motion linearization. Its implementation, however, is beset with
almost insurmountable difficulties unless the geometry of the body is
extremely simple.

The second method is less exact in principle, but provides
approximate means of including real fluid effects and of dealing with
geometrically complex, realistic configurations. This procedure,
which is employed in the present analysis, is termed "hydrodynamic
synthesis." Here we consider the complex structure to be assembled
from a group of simpler bodies whose individual hydrodynamic pro-
perties are known, perhaps as a result of an analysis of the first type
above. A fundamental assumption is then made that the hydrodynamic
force on the assembled structure may be computed by taking the sum
of the forces of all of the component members. In the simplest:case,
these forces are computed as though.each member were completely
remote and independent of the rest of the structure, but subject to the
same pattern of body and fluid motions. The forces computed in this
way might be refined by introducing modifications to the fluid flow to
account for the hydrodynamic interaction between adjacent members.

The result of this hydrodynamic synthesis is a system of

hydrodynamic forces acting upon the assembled structure, containing
terms dependent upon the incident wave system and upon the motion

1085

Paullting

of the structure itself. Additional restraint forces are introduced

to account for the effects of the mooring or dynamic position system.
This total system of external forces is then equated to the mass times
acceleration of the body by Newton's second law, yielding a system
of coupled differential equations of motion. These equations are then
solved to obtain the time-dependent motion of the structure.

In the present analysis, a linear relationship will be assumed
to exist between all forces and the appropriate motion parameters.
Two important consequences follow as a result of such an assumption:
(1) The hydrodynamic forces acting on the structure may be divided
into two independent parts, one depending only on the incident wave
motion, and the second depending only on the platform motion.

(2) A prediction of the platform response to a realistic random sea-
way may be obtained by superimposing the responses to the seaway's
regular wave components.

The validity of such a linearization may be tested either
empirically or by comparing its results with results of an "exact"
analysis. An exact analysis is normally possible only for sucha
simplified class of geometries that the validity of the comparison
for the realistic case is subject to question. We are, therefore,
forced to an experimental test. For the present, we have considerable
evidence on the usefulness of linear techniques in predicting ship
motions as in Gerritsma [1960], and motions of platforms of the
present type, Burke [1969], Paulling and Horton [1970]. Some
further experimental comparisons are given in the present paper.

II. THE EQUATIONS OF MOTION

The motion of the platform will be expressed as a small
deviation from a mean position, and for this purpose it is convenient
to define two coordinate systems. The first, OXYZ, is fixed rela-
tive to the structure suchthat O is located at the structur's center
of gravity, Y is directed vertically upward, and OXZ is parallel
to the mean waterplane. In many cases, we may take advantage of
symmetry to arrange these axes so that one or more of them is a
principal axis of inertia. Also, in some cases, a designer's co-
ordinate system may be used for drafting or other design purposes,
which is parallel to OXYZ but whose origin is located elsewhere.
Quantities defined in this latter system may always be transformed
to the OXYZ system by simple coordinate transforms, and it is
assumed that this is done.

A second coordinate system, oxyz, is fixed in space such
that it occupies the mean position of OXYZ as the platform moves
in waves. In general, it is found most convenient to express the
inertial properties and the forces acting on the structure in OXYZ
since the geometry of the structure is fixed in this system. On the
other hand, it is more convenient to express the equations of linear
motion in the space system, oxyz, since this is an inertial system,
and because we ultimately wish to obtain the motion of the platform

1086

Wave Induced Forces and Mottons of Tubular Structures

in terms of time dependent deviations from this mean position. The
linear displacements of the center of gravity of the platform from its
mean position may then be expressed by the small quantities x(t), y(t),
z(t), measured in the oxyz system. We next express the rotational
motion of the platform in terms of the Eulerian angles e(t), B(t), y(t).
These angles are so defined that the angular displacement between the
two coordinate systems, oxyz and OXYZ, may be created by imagin-
ing the platform as first oriented such that the two coordinate systems
coincide. It is then rotated about OX through the angle a, then about
the new position of OY through the angle 6, and finally about the new
position of OZ through the angle y to bring the platform to the final
position of angular displacement. For small values of a, B, y, the
two coordinate systems will now be related by:

x 4 -Y B Xx
y = y 1 -a@ . Y e (1)
Zz -B a 1 Z

The equations of motion may now be written. It is convenient to
first write the equation for translatory motion in oxyz, thus it is
assumed that all forces acting on the body have been expressed

in this system, giving

f, = mx, , = 4i,. Zed s (2)
where the x; = x(t), y(t), and z(t), respectively.

The equations of angular motion may be most easily written
in the body coordinates, OXYZ, since the moments and products
of inertia of the structure are constant in this system. If the exter-
nal moments are also expressed in OXYZ, the Euler equations for
rotational motion in rotating coordinates are obtained:
se
ial

8. = components of the angular velocity vector in OXYZ,

moment of inertia about i-axis if i= j,

i

(-) product of inertia if i# j.

*
Note that the transformation expressed by Eq. (1) can be applied to
forces and velocities as well as to coordinates.

1087

Paulltng

Now, M; may be transformed by (1) into components, m, i?
expressed in the space ue system. We note, further, that if
the angular velocities, , are small quantities, the moments, m,,
causing the motions eee small of the same order. The transfor-
mation of moments, therefore, after dropping products of small
quantities, yields

mj = M;.- (4)

Similarly, the components of angular velocity, Q,, in OXYZ
may be transformed into the space coordinate system. wae small
angular velocities, this results, approximately, in

or oe (5)

Here, the @, are the components in oxyz of a small rotation
of the structure about an instantaneously fixed axis in space. In
other words, for the small angular motions to which we limit the
present analysis, the Euler angles are approximately equal to the
components of the body rotation during a small time interval about
the fixed space axes.

We may now write the translational and rotational equations
of motion, in the fixed coordinate system, in the combined form

6
y mj; X; — 1, as a ae 6\. (6)
jzl

Here f. »>x.; i=i1, 2, 3, are the forces and displacements in the
X,Y, Z- directions ; fi» x.3 i= 4, 5, 6, are the moments and rota-
tions about the x, y, z-axes,

m,,* M,5» m,,= m= mass of structure,

33

m,,° M,., M,, = moments of inertia about XYZ-axes,

respectively.

M4. = ™M,4= ih XY dm

Mgg= Meg= - we YZ dm | Products of inertia about

| the OXY Z-axes.
Myg = Mgq= - seat XZ dm

1088

Wave Induced Forees and Mottons of Tubular Structures

The force system acting on the structure comprises hydrodynamic
forces resulting from wave and platform motion, hydrostatic forces
from the changes in displaced volume associated with static displace-
ments of the platform and the restraining forces exerted by the
positioning or anchoring system.

Tl. COMPUTATION OF THE FORCES

The total hydrodynamic force exerted on the body is assumed
to be composed of three parts:

(1) The force resultant of the pressure exerted by the un-
disturbed incident wave train on the stationary body in
its mean position.

(2) The force resulting from the disturbance of the incident
waves by the body occupying its mean position.

(3) The force resulting from the motion of the body computed
as though it undergoes the same motion in calm water.

In a formalized linear analysis, these terms would be associ-
ated with velocity potentials representing the incident wave train,
a diffracted wave train, and the waves generated by the motion of
the body. The force represented by (1) above, i.e., that part ob-
tained by neglecting the effect of diffracted waves and body motions,
is termed the Froude-Krylov force.

Procedures based on assumptions similar to these have, as
previously noted, proven successful in predicting the motions of ships
in waves. The present situation is somewhat simpler than the ship
case because the structure has no forward speed.

As noted in the Introduction, we shall obtain the total force
on the structure by computing the force separately on each member
of which the structure is composed. Two principal assumptions will
be made in computing these forces. First, each member is assumed
‘to be either a cylinder whose cross sectional dimensions are small
compared to the lengths of both the cylinder and the incident waves,
or the member is a small pontoon (point volume) all of whose di-
mensions are small compared to the incident wave lengths. Second,
all hydrodynamic interaction effects between adjacent members will
be neglected, thus the force on an individual member will be com-
puted as though the member occupies its mean position in the field
with all other members absent.

In Fig. 1, we show a cylindrical member located in the flow
field corresponding to a train of waves and undergoing motions cor-
responding to the rigid-body motions of the entire structure. The
force per unit length at a given point along the length of the member

1089

Paulling

,Y — NOTE: €€-PLANE IS MEAN

WATER SURFACE

LINE IN €C-PLANE PARALLEL TO X-AXIS
8 WAVE DIRECTION

ORIGIN
OF ent

Fig. 1. Coordinate systems and nomenclature for member

is assumed to be expressed as
Foe are pn ds + C,\ we = Uap) af: Cuil ara anp)- (7)

The first term is the resultant force obtained by integrating
over the surface of the body the pressure, p, which would exist
at that location if the body were not present and is seen to be the
Froude-Krylov force. Note that n is the unit normal vector
directed out of the body into the fluid.

The second term is called the drag force and is assumed
proportional to the relative normal velocity between the member and
the fluid. The third term is called the added mass force and is
assumed proportional to the relative normal acceleration between
the member and the fluid.

The first term and the first parts of the second and third
terms are seen to be dependent on the wave motion while the remain-
ing parts of the last two terms depend on the structure's rigid-body
motions. In order to evaluate these forces for the cylinder, we must
define two additional coordinate systems. The first, ox yz, is
associated with the member such that the x-axis lies along the
centerline of the member, the y-axis is directed generally upwards,
the z is defined to form a right-handed system. The second co-
ordinate system, o§nf, is used to express the waves, and is so
defined that the €-axis lies in the mean water surface and is positive

1090

Wave Indueed Forces and Mottons of Tubular Stretures

in the direction of propagation of the waves, 7 is directed upward,
and € is defined so as to form a right-handed coordinate system.

Wave Forces
In order to be consistent with our assumption of small plat-
form motions, an assumption of small incident waves must also be

made. The velocity potential for regular infinitesimal gravity waves
moving in the direction of the positive §€-axis is

= h k(n+d)
9(&.,0,t) = $2 SOS AT sin (k§ - ut). (8)

The pressure is given by
C)
P=- pen- par, (9)

and the linearized velocities and accelerations in the &- and n-
directions by

_ 99 . 99

Se v= 1"
(10)

. . OU See BS

U= 3p = °

We shall first compute the Froude-Krylov force which is
given by the first term in (7). This integral, which is to be evaluated
over the entire immersed surface of the body in order to obtain the
total force, may be replaced by the following volume integral through
the application of Gauss' theorem

cee ae se

The corresponding integral expressed for the moment referred to the
Ent-axes is

ah p(r Xn) ds
SSS (cB ng. SSE 3 oe - & ge) av. (1 1b)

Note that in defining these integrals for a member which projects

ti

—
Th

iN

1091

Paulling

through the free water surface, the pressure and its derivatives
vanish for that part of the member's surface or volume above the
free suriace.

The evaluation of both of these integrals is a straightforward
but rather tedious process, yielding the components of force in the
Eng-directions, and the moments about these axes. We first per-
form the indicated differentiation of the two terms in the pressure
equation with respect to €, n, and {, then substitute for €, n,
and {, their values transformed to the oxyz-coordinates. The
element of volume is given in oxyz by Adx where A is the
constant cross sectional area of the cylinder. Since the cross
sectional dimensions are assumed small compared to the wave
length, the integrand may be assumed constant over A and, conse-
quently, the volume integrals are reduced to one-dimensional
integrals in x to be evaluated over the length of the cylindrical
member.

If the member is completely immersed, the evaluation of
this integral yields two terms corresponding to the two terms in
Eq. (9) for the pressure. The first term is the static buoyancy
force or moment and the second is a time-dependent "variable
buoyancy" corresponding to the variation in effective weight density
of the fluid as a result of the wave motion.

If the member projects through the free surface, the region
of integration must be dealt with in two parts. The first part is the
constant volume of the member below the mean water surface, and
the second is the time varying part of the volume of the member lying
between this mean waterline and the instantaneous water surface.
Evaluation of the integrals over the first part of the volume, i.e.,
that part below the mean waterline, yields a result identical with that
for a completely submerged member. In evaluating the integral over
the second part of the volume, we note that this volume is of the same
small order of magnitude as the wave amplitude. Since the velocity
potential, (8), and therefore the second term in the pressure, (9),
is of this same small order of magnitude, only the first term in the
pressure, i.e., the hydrostatic part, has a linear contribution to
this part of the integral.

In evaluating these integrals, (11), for the pontoon or point
volume, we note that the assumed small dimensions of the pontoon
imply that the integrands are constant over its volume. Therefore,
the integration reduces to taking the product of this volume with the
appropriate derivatives of the pressure expression.

The first parts of the last two terms in (7), i.e., the drag
and added mass forces associated with wave induced water motion,
are each computed in a similar manner. Let u; and uj; represent

the components of fluid velocity and acceleration in off, with u;
and u, the corresponding components in oxyz. A coordinate

1092

Wave Induced Forces and Mottons of Tubular Structures

rotation may be expressed by ai such that

The components of force on an element of length, dx, inthe oxyz-
directions are given by

dF, = @,u, +%,,u,) &, j=1,2,3 (13)

where p; and }; are added mass & and linear drag coefficients. For
the cylindrical member, only p55 Xoo, and X33 are non-
zero, while for the point volanie FAT hee the diagonal terms in p» and

X are nonzero.

The forces may now be expressed in o€n{ by the inverse of
the above transformation

dF = aj, dF;

it
R
rom
hy

= (piu, + u,) dx. (14)

Here, the added mass and drag coefficients in of G are seen to be
related to those in oxyz by the following expressions.

al amenicl a as

dij =a Rigi (15)
_Corresponding expressions for the elementary moments

about oxyz and o§nG may be written. The total forces and

moment may be obtained by integrating these expressions over the

length of the member, noting that uj and uj, contain terms with

the same trigonometric functions of time which appear in the Froude-

Krylov buoyancy integrals, (11), and may, in fact, be combined with

them in carrying out the integration over the member length.

Evaluation of these integrals yields a set of forces and
moments in o§nf. A coordinate translation and rotation may now be
applied to express these forces and moments in the space coordinate
system oxyz.

1093

Paulling

Motion-Dependent Hydrodynamic Forces

We now consider a velocity vector, us whose components
are the velocities of the center of gravity of the structure in oxyz
and are given by x a‘ ip 1 x, in Eq. (6). We may similarly define
an angular velocity. vectors QR, whose components are the three
rotations about oxyz, and were denoted X4» X53» Xe in (6). The
resultant velocity of a point P(x,y,z) is given by

ee

—> —> —
us UTM Xr,

where r is the radius vector drawn from the origin of oxyz to the
point P. Similarly, the resultant acceleration is given by

ea
—_
=u

+
a

rs)

+Yxr.
(Note that this neglects the radial or "centripetal" acceleration which
is of the order of the square of the (small) angular velocity.) Point
P may be thought of as lying on the x-axis of a member of the
structure, We therefore may obtain the velocity and acceleration
vectors u,p and app, appearing in Eq. (7) by a transformation similar
to (12)... THe a, however, relate the member coordinates OxyzZ

to the space coordinates oxyz inthis case. Also note that the
velocities and accelerations are now unknown quantities.

Applying these transformations, we obtain expressions similar
to (12) where the Uj» u; are replaced by the components of up and
ape The elementary forces are given by an expres: sion which is the
negative of (13) but containing the same p's and X's. These are then
transformed back to the oxyz-directions by the inverse of the above
transformation. Integration of these elementary forces and their
moments over the length of the member is somewhat simpler now
since the velocities and accelerations vary in a somewhat simpler
manner than in the case of the wave-induced velocities and accelera-
tions.

The Added Mass and Drag Coefficients

The idealization made in arriving at the subdivision of forces
illustrated in Eq. (7) is a computational expedient at best. In reality,
the total force experienced by a member of the structure will be the
resultant of distributed normal pressure forces and tangential shear-
ing forces arising from viscosity. The subdivision into a Froude-
Krylov force, a force proportional to relative fluid velocity, anda
force proportional to relative fluid acceleration, is done partly on an
intuitive basis and partly on the basis of our knowledge of simpler
problems. Such a subdivision has the great advantage of leading to
solutions in which there is a linear relationship between platform
motion and the exciting wave motion. Let us review some of the

1094

Wave Induced Forces and Mottons of Tubular Structures

justification for this simplification.

Maruo [ 1954] and Havelock [| 1954] have discussed the forces
on submerged bodies which are caused by waves of small amplitude
in an ideal fluid. Maruo deals with the two-dimensional problem of
a horizontal cylinder completely submerged below the surface of an
inviscid fluid, and gives some results which can be compared directly
with Eq. (7). In particular, he shows that, for the case of a deeply
submerged cylinder, the "exact" total force is equal to twice the
Froude-Krylov force, Eq. (7), and that this corresponds to a value of
the added mass coefficient equal to that of the cylinder in an infinite
fluid combined with the wave motion at the centerline of the cylinder.
If the cylinder is near the free surface, the force differs from this
deeply submerged value by an amount dependent upon the depth of
submergence and the wave length. The error, however, is small
if both the depth of submergence and the wave length are greater than
several cylinder diameters. Similarly, C. M. Lee [1970] has
analyzed the problem of an oscillating cylinder submerged beneath
the free surface, and has shown that the added mass coefficient for
forced motion approaches the infinite fluid value within a small error
if the depth of submergence is more than two times the cylinder
diameter and the length of generated waves is more than about five
times the cylinder diameter. For our present purposes, these
results imply that we may assume a constant value of the added
mass coefficient in Eq. (7), since in the majority of practical situ-
ations the cylindrical members will be sufficiently deeply submerged
and of sufficiently small diameter compared to wave lengths of
interest to fulfill the above conditions. Thus the first and last terms
in Eq. (7) can be expected to give a good approximation to the non-
dissipative parts of the force on the individual member considered
here.

The drag or velocity dependent force acting on an oscillating
body under a train of waves is associated with two phenomena: (1) the
dissipation of energy in surface waves which are generated as a
result of motion of the body, and (2) the viscous effects which are
felt both as tangential forces on the surface of the body, and as a
deviation of the pressure distribution from its ideal fluid value. This
latter effect, which is associated with the formation of a wake and
vortices downstream of the body, will cause the added mass coefficient
to differ from its ideal fluid value as well. The drag force associated
with free surface wave effects decays to zero with increasing depth
of submergence at the same rate that the added mass coefficient
approaches the infinite fluid value. Therefore wave damping is of
little significance to the configurations being considered here. The
drag forces associated with viscosity are generally of much greater
importance, and also less clearly defined. The usual method of
approximating these forces, Wiegel [1964], is to assume that they
behave in a manner similar to the drag on a body immersed in a flow
of constant velocity. In such case, the drag force is expressed asa
quadratic function of velocity and the drag coefficient is found to be a

1095

Paulltng

function of Reynolds number. In applying this concept to the present
situation in which we have a periodic fluid motion resulting from the
superposition of wave and body motions, we might compute a Reynolds
number using the mean absolute velocity, and choose the drag coef-
ficient accordingly. The drag term in Eq. (7) should then be a
quadratic function of velocity. This, however, destroys the linearity
of our analysis, and in view of the crude approximation involved, it

is not worth the added complication. In order to preserve linearity,
an equivalent linear drag coefficient is therefore defined, as described
in Blagoveshchensky [1962], such that the linear drag force dissipates
the same energy per cycle of periodic motion as the nonlinear drag
force which is being approximated.

The derivation of the equivalent linear drag coefficient is as
follows. Assume a sinusoidal variation of the relative fluid velocity
given by

Vv =v, sin wt. (16)

The linear drag force is given by

Dy FEY
and the nonlinear drag by
- n
D, = Cyav -

The energy dissipated per quarter cycle of motion is given in the
linear case by

~W/2w
Cc v? dt;
OL 0

and in the nonlinear case by
T/2w
nl
Gi vo pdt.
ie)
Equating the two energies enables us to solve for the equiva-

lent linear drag coefficient in terms of the assumed nonlinear coef-
ficient. In the case of n= 2 (quadratic drag) the result is

+, Bo neass (17)

ot ae oe

1096

Wave Induced Forees and Mottons of Tubular Structures

Thus, it is seen that the use of such equivalent linearization
requires a prior knowledge of the amplitude of the motion. No
difficulty is introduced by this in the case of the wave force ona
stationary member. However, the amplitude is unknown for the
absolute motion of the member. This leads to the necessity for an
iterative solution in which we first assume an amplitude of motion,
compute the equivalent linear coefficient, and then solve the equa-
tions of motion using this value. This solution then is used to com-
pute a refined value of the linear drag coefficient, which is used for
the second solution of the equations of motion, and so on. It is
questionable whether the approximations involved warrant more than
two iterations, as noted by Burke [1969].

Hydro static Forces

A floating body which is displaced in heave, pitch, or roll
from its equilibrium position experiences hydrostatic forces pro-
portional to these displacements as a result of the changes induced
in the immersed volume. There will be no forces in surge, sway,
or yaw since these displacements, which are parallel to the free
surface, cause no change in the immersed volume,

These forces, including coupling terms, are computed by

standard naval architectural formulas. Thus, the vertical force
resulting from a small heave displacement, x,, is given by

F = - pgA,x,, (18)

where A, is the waterplane area.

Similarly, the moments of this force about the x- and z-
axes (static coupling terms) are given by

M

= pgAy zx
x wow 2 (19)

M

ae PEAY XX, »

The roll and pitch moments resulting from small angular dis-
placements are given by

= - pgVGMa, (20)

where GM is the appropriate metacentric height, V is the volume
displaced by the structure, and a is the small roll or pitch angle,
either x, or x, in the notation of Eq. (6).

Finally, the force in the y-direction resulting from a small

1097

Paulling

roll displacement, X4, is
Fy = PR AWZyX4\s (21)
and for a small pitch displacement, Xe

Fy = - pgAwXw%¢ » (22)

Xw, Zw are the coordinates of the center of gravity of A,. These
forces are included in the equations of motion as static restoring
or coupling force terms.

The Restoring Forces

Three types of restraints have been described in the Intro-
duction, dynamic positioning, spread array mooring, and vertical
tension leg mooring.

A dynamic positioning system incorporates two principal
components: sensors for detecting deviations from the desired
position, and thrustors which may be activated automatically or
manually to exert a force tending to restore the structure to the
desired position. In the simplest system, the thrustors are actuated
without time lag to exert a force proportional to the displacement,
This would be termed a pure proportional controller. Real systems
seldom operate this simply but incorporate time lags, back lash,
and other non-ideal characteristics. Increased sensitivity and
response may be built into the system by having it sense velocity (rate
control) and acceleration. If the system can be approximated by
linear features, i.e. , if the applied thrust can be linearly related to
displacements, velocity, and acceleration of the structure, then the
control system constants may merely be introduced in the force terms
of the equations of motion, (6), as additions to the already defined
hydrodynamic and hydrostatic terms.

In a spread mooring system, several pretensioned anchor
lines are arrayed around the structure to hold it in the desired
location. If the structure moves from its mean position, the tensions
in the anchor lines change and these changes may be related to the
geometry (catenary), elasticity, and hydrodynamic properties of the
anchor lines. It is usually permissible to neglect the hydrodynamic
forces on the anchor lines and to approximate the force by a linear
relationship between force and displacements in the plane of the
anchor line. The displacements at the point of attachment of the
anchor line may be determined in terms of the coordinates of this
point for given displacements of the structure. These are resolved
into horizontal and vertical displacements, xy, yg, in the plane of
the anchor line by a transformation similar to (12). The horizontal
and vertical forces exerted by the anchor line may then be expressed

1098

Wave Induced Forces and Motions of Tubular Structures

as

Bay “a Its - Kye (23)

hy
NW

Ly Ks + KO be

These forces may then be transformed back to the oxyz coordinates
for inclusion in the equations of motion. The anchor spring constants,
ky eee ky are computed from a knowledge of the aforementioned
elasticity and weight-shape characteristics of the anchor line.

In a tension leg mooring, the mooring lines are vertical and
provide essentially total restraint against vertical movement of their
upper ends. Figure 2 illustrates the horizontal force which results
when the upper end of such a mooring line is displaced horizontally
as a result of surge, sway, and yaw. The restoring force in the
direction opposite the displacement, x,, of the end of mooring line
n is given by

(24)

The displacement, xX,» may be expressed in terms of its components
in the x- and z-directions, in which case the corresponding com-
ponents of F, in these directions will be given by (24).

Xr“ HORIZONTAL COMPONENT

_ _XnTn
ft La

Th

MOORING LEG "n"

Fig. 2. Restoring force in tension mooring legs

1099

Paullting

IV. SOLUTION OF THE EQUATIONS OF MOTION

The total system of forces described in the preceeding
sections are now introduced into the equations of motion, Eq. (6).
Our linearization of the problem has resulted in the subdivision of
these forces into two categories: those forces resulting from the
wave motion in the presence of the stationary structure, and those
resulting from the motion of the structure in a stationary fluid. The
former category contains the first term and the first parts of the
second and third term on the RHS of Eq. (7). The platform motion
dependent forces are contained in the second parts of terms two and
three of the RHS of Eq. (7), plus the hydrostatic and restraint forces.
The wave motion depending forces are seen, as a result of the velocity
potential assumed to represent the wave motion, Eq. (8), to be sinu-
soidal functions of time. If we rearrange the equations of motion into
the standard form, placing the motion-dependent terms on the left-hand
side and the time dependent forcing terms on the right-hand side the
result is a set of six simultaneous second-order differential equations
of the form

6
Ds [ (mj; a aij) x; + bij Xj 28 ci, x;] = F,j sin (wt + Ej), (25)
j=l

where the exciting force amplitude, F,;, is proportional to the wave

amplitude, a. The solution of these equations may be expressed in
the form

x; =X; sin (wt + 6),

where x,,; is proportional to Fj, therefore to the wave amplitude.
The quantity ws or the amplitude of response to unit waves

varies with wave frequency, since the exciting force Fo is a function
of frequency and because the coefficients a, b, c the LHS of (25)

may also vary with frequency. The square of this unit response is
then the response amplitude operator, which may be combined with
the wave spectral density function to obtain the platform response to

a random seaway.

V. MODEL EXPERIMENTS

A number of model experiments have been conducted in the
University of California Towing Tank in order to test several parts
of the procedures described in the previous sections. The initial
objective of the study was to evaluate the tension leg platform, and
all experiments deal with this configuration. Initial experiments
were made on single cylinder members to test some of the hydro-
dynamic force predictions and the linearity of the resultant motions
in regular waves. Next, experiments were conducted in regular

1100

Wave Induced Forces and Mottons of Tubular Structures

waves using a platform model of triangular plan form to investigate
the predictability and linearity of motions and mooring tensions of a
composite structure consisting of a number of cylindrical members.
Finally, experiments were made in random seas to test the applica-
bility of linear superposition.

The arrangement of the model and experimental apparatus
is shown in Fig. 3. The model configuration shown is typical of a
number of those tested, consisting of three base cylinders arranged
to form an equilateral triangle with three or more vertical legs
supporting the deck. An important geometrical parameter studied
in these tests was the relative proportion of buoyant volume contained
in the vertical and horizontal legs.

PULLEY
“

0000

LEADS TO

INTEGRATOR

WAVE
SURFACE

X- AXIS

PLATFORM
MODEL INSTRUMENT

TENSION LEADS
TO RECORDER
ad

METERS

DEAD-WEIGHT
ANCHORS

V
TANK BOTTOM

Fig. 3. Arrangement of experimental apparatus

11014

Paulling

From Fig. 3 it is seen that instrumentation was provided
for measuring model motions, tension variations in the mooring legs,
and incident wave amplitude. The surge motion was sensed and con-
verted to an electrical signal by a miniature low torque potentiometer
driven by the model through a string and pulley arrangement. Yaw
was sensed by a rate gyroscope mounted on the model, the output
of which was integrated electronically to give the yaw displacement.
Tension meters were installed in each mooring leg. These consisted
of small proving rings fabricated from seamless stainless steel
tubing and mounted with etched foil strain gages. Four gages on
each ring were connected to form a four-arm Weatstone bridge, the
output of which is proportional to the applied force. The bridge was
balanced initially to bias out the initial static tension. Therefore
only the time dependent variations are recorded.

The outputs of these force and motion transducers, as well
as the output of a resistance wire wave meter, were recorded, using
a multichannel oscillograph. During experiments in random waves,
a simultaneous recording was made of the same quantities in digital
form on magnetic tape for processing by electronic computer.

Single Cylinder Experiments

The first group of experiments were conducted using a single
circular cylindrical model having hemispherical ends and moored
by two legs, one at each end. Only the incident regular waves and
tension variations were recorded. The model dimensions and test
conditions were:

Length 3.44 ft

Diameter 0.282 ft

Depth of model 0.792 fit = 2.8 X dia.
Weight 6.5 1bs

Water depth 4.17 ft

For this configuration, the computed tension variations in the
mooring legs will be equal to the hydrodynamic forces expressed in
Eq. (7), i-e., there will be no linear coupling between the tension
variations and the motions of the model. These results, for regular
waves of 1.45 second period, and several different amplitudes striking
the model at 0, 45, and 90 degrees, are shown in Fig. 4. Experimen-
tal points show the amplitudes of force variations which were measured
in the two mooring legs. The theoretical lines have been determined
by the method described here and by Havelock [1954]. Havelock's
procedure gives the wave force and moment on a spheroid having its
long axis horizontal, moving beneath a train of regular waves. For
the present computation the approximating spheroid was assumed to
have the same length and diameter as the cylinder model. The
dashed curve labelled "present work" was computed by Eq. (7)
assuming the infinite fluid value of unity for the added mass coefficient

1102

Wave Induced Forees and Mottons of Tubular Structures

, LBS

FORCE AMPLITUDE

°6 Ol G2" 03804. 05). 06 OF;
WAVE HEIGHT , FT

Fig. 4. Tension teg single cylinder

of the circular cylinder and a quadratic drag coefficient of unity.

Within the limits of experimental accuracy, the forces are
seen to vary linearly with wave height for a range of heights tested.
The two theoretical procedures are seen to give results of about
equal degree of conformity with experiments.

Triangular Platform 1 in Regular Waves

Experiments similar to those with a single cylinder were
next conducted with a complete platform model. This model was
similar to the one shown schematically in Fig. 3 except that the main
horizontal pontoons were of oval cross section and the above-water
deck was supported by a space frame arrangement of very small
vertical and inclined tubes. The dimensions of this model, referred
to as Model 1, are given below:

1103

Paulling

Weight 40.13 lbs
Buoyancy 56.40 lbs
Length of side of

equilateral triangle 3292 ft

Main pontoon cross section,
vertical and horizontal

semi-axes 0.175 X0.109 ft
Vertical member - radius 0.031 ft

Draft to centerline of main

pontoon 0.77 ft

Water depth 4.18 ft

This model was subjected to a large number of tests in regular and
irregular waves. Only a few of the regular wave tests are reported
here,

-

[NS

WAVE PERIOD = 1.15 SEC
WAVE DIRECTION = 0°
Beal EXP THEORY
LEGS IAND3 ©
LEG 2
WAVE PERIOD = 1.44 SEC

AN

Nai

NE

NE
NLL,
ERASE

WAVE PERIOD = 1.74 SEC

\

mt |
LAAT

PEAK-TO-PEAK TENSION VARIATION, LBS
Co © ©, © Ss | ~ © i; © cs ¢* - ©

N I
:

Paz]
er
ise 3
isi]
Ps

wie HEIGHT, FT

4 0.5

Fig. 5. Tension variations in legs -- Model 1

1104

Wave Induced Forees and Mottons of Tubular Structures

0.5

WAVE DIRECTION = 0°

THEORY:

DRAG COEF = 1.0
DRAG COEF = 1.5

°
rs

SURGE MOTION PEAK-TO- PEAK, FT

) 0.1 0.2 0.3 04 0.5
WAVE HEIGHT, FT

Fig. 6. Surge motion -- Model 1

Figure 5 contains a comparison of measured and computed
tension variations in the mooring legs for three different wave
periods and Fig. 6 contains the platform surge motion. Again, the
agreement between computed and measured values is about as good
as in the case of the single cylinder. Two features should be noted
here. First, a mean curve drawn through the experimental tension
variations appears to curve slightly concave downward with increas-
ing wave height, thus indicating a measurable nonlinearity in this
quantity. Second, better agreement between the computed and
measured values is obtained for motions than for forces. Note that
the motions are shown for two different values of the assumed
quadratic drag coefficient. The effect of a substantial change in
this quantity is seen to be slight.

Triangular Platform 2 in Random Waves

A second triangular platform was tested in both regular and
irregular waves. This platform, designated Model 2; was similar in
arrangement to the platform depicted in Fig. 3 and had the following
characteristics:

Weight 28.49 lbs
Buoyancy 32.30 lbs
Length of side 3212 ft

Main horizontal pontoon dia. 0.187 ft
Vertical cylinder dia. 0.381 ft

1105

Paulling

Draft to centerline,
horizontal pontoon 0.78 ft

Water depth 4.70 ft
This model was tested in six different random sea conditions,
representing two families of wave spectra. The first family of
spectra, designated "A," have their peak ordinate at a wave period
of about 0.9 second. The second family, or "B" spectra, have their
peak at about 1.55 seconds. Both families for several different
significant wave heights are shown in Fig. 7. These two spectra

were used in order to adequately excite the model over the range of
wave periods of interest.

"B " SPECTRA

SEA

CENCE
mn
CORINA
peel MET AL

BPS aa
oncgh SiN

0 0.4 08 12 16 20 oe
PERIOD IN SECONDS

Bakre

A SPECTRA x 104
B SPECTRA x !0°

WAVE AMPLITUDE HALF - SPECTRUM

Fig. 7. Experimental tank wave spectra

1106

Wave Induced Forees and Mottons of Tubular Structures

The model response is shown in Figs. 8 - 10 for waves mov-
ing parallel to the X-axis. The first two of these figures show the
mooring tension variations, and the last shows the surge motion
versus wave period. In each case the ordinate is the double ampli-
tude of the force or motion in question divided by wave double ampli-
tude, thus the amplitude of the transfer function obtained from a time
series analysis of the random wave tests. Points on the figures dis-
play these random wave results for three significant wave heights
within the applicable range of periods for the "A" and "B" spectra.
Also shown on these figures are results from experiments in regular
waves, and theoretical predictions.

It is interesting to note that the random sea tension variation
results display an apparent amplitude dependence in the range of
longer periods. This is the range in which drag forces would be
most strongly felt and no doubt points to a possible deficiency in
the process of linearizing the drag force.

As before, the surge motion shows very good agreement
between experiment and theory.

0.2 0.6 1.0 1.4 1.8 2.2 26
WAVE PERIOD IN SECONDS

Fig. 8. Tension variations in anchor leg 2

P2LO-7

Paulling

5)
DIRECTION OF
WAVE ADVANCE

sre B-SPECTRA a

EXPERIMENTS IN REGULAR WAVES
THEORY

5a

Ne

0.2 0.6 1.0 1.4 1.8 22
T IN SECONDS

Fig. 9. Tension variations in anchor legs 1, 3

1.6
B
2 12 A-SPECTRA| B-SPECTRA
: y
> O08 P
ae A

(2)
BSS

a be oft

0.2 06 10 1.4 1.8 22 2.6
T IN SECONDS

Fig. 10. Surge motion

1108

Wave Induced Forcees and Mottons of Tubular Structures

VI. CONCLUSIONS

In the previous section, a comparison is shown of experi-
mental measurements and theoretical predictions of platform motions
and mooring leg tensions, using the procedure developed here. The
theoretical results were obtained by the simplest form of the pro-
cedure in which constant infinite fluid values were assumed for added
mass coefficients. Constant linear drag coefficients were used, and
no attempt was made to account for the hydrodynamic interference
between members of the structure. The first two groups of experi-
mental results show that the observed performance of single members
and assemblages does, indeed, follow a nearly linear pattern in
regular waves, and this pattern is well predicted by the present pro-
cedure. The last group of experimental results show nearly equally
good results in both regular and irregular waves. There is, however,
some consistent nonlinear amplitude variation in longer waves, as
may be seen in Fig. 9.

It is probable that the good agreement is obtained because
the structures tested consisted of assemblages which satisfied the
initial assumption reasonably well, i.e.,

(1) All members were long, slender cylinders relatively
sparsely distributed throughout the structure.

(2) The bulk of the members were submerged sufficiently
deeply below the free surface.

(3) The cross sectional dimensions of all members were
small compared to the waves used in the experiments.

(4) The motions of the models were small compared to the
model dimensions and to the wave lengths.

The aforementioned nonlinear behavior probably illustrates
the failure of a single value of the linear drag coefficient to ade-
quately represent this component of the hydrodynamic force over the
entire range of frequencies.

ACKNOWLEDGMENTS

This work was conducted under the sponsorship of Deep Oil
Technology, Inc. and the author expresses his appreciation for
their permission to publish the foregoing results. The assistance
of a number of individuals in conducting experiments and performing
calculations is also acknowledged. Special thanks in this respect are
due to Mr. O. J. Sibul, and graduate students Kwang June Bai and
Nabil Daoud of the University of California, Department of Archi-
tecture, and Mr. Paul Gillon of Deep Oil Technology.

1109

Paulling

REFERENCES

Blagoveshchensky, S. N., Theory of Ship Motions, Dover, 1962,
p. 142.

Burke, Ben G., "The Analysis of Motions of Semisubmersible
Drilling Vessels in Waves," Paper No. OTC 1024, Offshore
Technology Conference, Houston, 1969.

Fisher, F. D. and Spiess, F. N., "FLIP -- Floating Instrumental
Platform," J. Acoust. Soc. Am., v. 35, no. 10, 1963,
pp. 1633-44,

Gerritsma, J., "Ship Motions in Longitudinal Waves," Netherlands
Research Center, TNO Report 35S, Feb. 1960.

Havelock, T. H., "The Forces on a Submerged Body Moving Under
Waves, Trans., RINA, 1954, pp. 1-7. Also "Collected
Papers of ...," pub. by ONR, Dept. of the Navy, ONR/ACR-
L036

Lee, C. M., private communication, 1970.

Macy, R. H., "Drilling Rigs,"Ch. XVI of "Ship Design and Con-
struction," A. M. D'Archangelo, Ed., SNAME, 1969.

Maruo, H., "Force of Waves on an Obstacle," J. Soc. Naval Arch.
Japan, v. 95, 1954, pp. 11-16.

McClure, Alan C., "Development of the Project Mohole Drilling
Platform," Trans. SNAME, v. 73, 1965, pp. 50-99.

McClure, Alan C., "Delos: An Application of Oil Field Marine
Technology to Space Programs," Marine Technology, v. 6,
no. 2, 1969, pp. 156-170.

McDermott, J. Ray, Inc., "Feasibility Study of a Floating Ocean
Research and Development Station (FORDS)," Final Report
to Dept. of the Navy, Bureau of Yards and Docks, under
Contract NBy-37640, April 1966.

Paulling, J. R. and Horton, Edward E., "Analysis of the Tension

Leg Stable Platform," Paper No. OTC 1263, Offshore
Technology Conference, Houston, 1970.

Wiegel, R. L., Oceanographical Engineering, Prentice-Hall,
1964, p. 248ff,

1240

SIMULATION OF THE ENVIRONMENT AND
OF THE VEHICLE DYNAMICS ASSOCIATED
WITH SUBMARINE RESCUE

H. G. Schreiber, Jr., J. Bentkowsky, and K. P. Kerr
Lockheed Misstles and Space Company
Sunnyvale, Caltforntia

I. INTRODUCTION

The U.S. Navy's first Deep Submergence Rescue Vehicle
(DSRV) was launched at San Diego, California on January 24, 1970.
This vehicle was designed and built by Lockheed Missiles & Space
Company (LMSC) under contract to the U.S. Navy's Deep Submergence
System Program Office (DSSPO) to provide the capability to rescue
the crew of a submarine immobilized on the ocean floor. The DSRV
is 50 feet long, 8 feet in diameter, has a fiberglas external hull and
an inner (pressure) hull made of three interconnected HY140 steel
spheres, Propulsion and control of the vehicle are provided by a
stern propeller in a movable shroud, horizontal and vertical ducted
thrusters located in pairs fore and aft, and a mercury trim and list
system. An Integrated Control And Display (ICAD) system developed
at the Massachusetts Institute of Technology Instrumentation Labora-
tory enables the DSRV operators to correlate information from
sonars, Closed circuit television, and advanced navigation devices,
in order to perform this intricate rescue mission. The mission
scenario of the DSRV is as follows. Word and position of a dis-
tressed submarine is received and the DSRV and its support equip-
ment are flown by three C141 aircraft to a nearby port. The DSRV
is then loaded on to a mother submarine, by being attached to the
after escape trunk, and transported to the area of the downed sub-
marine. The DSRV then detaches itself from the mother submarine
and descends to the disabled submarine, and mates to one of the
escape trunks of the distressed vessel as shown in Fig. 1. The
rescuees are then transferred into the aft two spheres of the DSRV
and returned to the mother submarine, 24 at atime. Because of the
possibility that the distressed submarine may be at an unusual atti-
tude, and there may be bottom currents, the DSRV must be able to
perform this hovering and mating maneuver in a one knot current
and at attitudes up to 45 degrees in pitch and roll.

se Fk

Bentkowsky and Kerr

Sehretber,

Iojsuerly, sonosoy

“ty

"81a

1Vt2

Vehtele Dynamics Assoctated with Submarine Rescue

This hovering and mating operation puts the DSRV in a new
and growing class of submersibles which because of their missions,
are required to hover, work, search, and otherwise maneuver at
low speeds. This requirement for low speed, high angle of attack
maneuverability is far outside the range of operation of the conven-
tional fleet type submarine and consequently analysis designed to
predict the dynamic behavior of conventional submarines is not com-
pletely applicable to the prediction of motions of the DSRV and other
submersibles of the same class. The adequate prediction of the
DSRV dynamics requires six degrees of freedom and a simulation
capable of predicting the forces and moments at high angles of attack
wherein the vehicle will experience lateral forces equal in magnitude
to the axial forces. To be useful, the simulation must be precise
enough for use in the design of the automatic control system. The
operational environment is also quite different from that normally
simulated in that the vehicle must hover and maneuver in currents
at near zero forward speed and in the presence of the disabled sub-
marine which causes considerable disturbances to the flow field.
This paper, which is divided into three general parts, presents one
approach to the simulation of the dynamics of a highly maneuverable
submersible. The first part describes the simulation of the free-
stream vehicle dynamics or thé dynamics outside the influence of
the distressed submarine. The second section deals with the inter-
action forces and moments caused by the presence of the distressed
submarine and includes a discussion of a test program conducted to
measure these forces and moments. The third section describes
the application of the resulting equations of motion in conducting a
man-in-the-loop simulation of the DSRV motions during the mating
maneuver.

The equations of motion were developed at LMSC and pro-
grammed on a Remington Rand 1103A computer. They were used
to determine the preformance characteristics of the vehicle to be
used in design studies and to provide equations of motion for use in
the control system development. The interaction forces were
measured in the 12-foot variable pressure wind tunnel at the Ames
Research Center in Mountain View, California. Tests of this nature
were necessary due to the lack of data on interaction forces and the
possibility that these forces would provide a significant influence on
the vehicle and control system design. The manned simulation was
performed at the Marine Systems Division of the Sperry Rand
Corporation to provide demonstration of the ability to manually con-
trol the DSRV within the limits necessary for mating and to deter-
mine operational limits for this mode of operation.

1113

Schretber, Bentkowsky and Kerr

Il. FREE STREAM DSRV DYNAMICS SIMULATION
EQUATIONS OF MOTION

The development of a dynamic simulation of the Deep Sub-
mergence Rescue Vehicle (DSRV) follows a different approach than
the methods used in most submarine studies. This deviation from
the standard approach is necessary because of the basic differences
in the mode of operation of the DSRV compared to that of conventional
submarines. While the analysis of a submarine is generally con-
fined to prediction of the vehicle dynamics at speed in an infinite
fluid, the DSRV dynamics must also be simulated while hovering and
docking in the presence of a downed submarine. The conventional
method used to simulate the dynamics of a submarine is to calculate
the position of the center of gravity of the vehicle using linear force
and moment coefficients for the complete vehicle which are refer-
enced to its center of gravity. The basic equations of motion for
the DSRV differ in two ways from this conventional method, first in
the choice of an axis system and secondly in the manner of handling
the forces on the vehicles and appendages.

Axis System

Since the DSRV is required to assume angles of 45° to the
horizontal in pitch and roll (very unrealistic for a conventional sub-
marine) a mercury trim and list system is incorporated which moves
the vehicle's center of gravity (c.g.) to accomplish these attitudes.
The fact that the vehicle's c.g. moves with respect to the vehicle
during maneuvers makes it a poor choice as a reference point for
describing force and moment coefficients since they would have to be
changed as a function of c.g. position. Using the c.g. as a refer-
ence axis system would also lead to complications in describing the
vehicle's motion with respect to the distressed submarine since the
motion of the axis system with respect to the vehicle would be in-
cluded in the velocity of the axis system. Therefore, an axis system
fixed to the body was used as a reference point. Since the axis sys-
tem is not always at the center of gravity and terms to account for
this shift must be included in the equations of motion there is no
advantage in choosing the nominal vehicle c.g. as the center of the
axis system. There are, however, advantages to having the x-axis
lie along the vehicle centerline since the basic DSRV shape is a body
of revolution. This axial symmetry provided by having one axis of
the system lie along the vehicle centerline greatly reduces the number
of cross coupling coefficients required to describe the forces and
moments on the body. The positive direction of this axis is forward
so that positive vehicle velocities are associated with vehicle forward
motion. Similarly with the z-axis through the centerline of the transfer
skirt (280.8 inches aft of the forward perpendicular) the number of
cross coupling coefficients are reduced and the direct reference to
the centerline of the transfer skirt simplifies the description of
relationships between transfer skirt and the hatch during mating

1114

Vehtele Dynamics Assoctated with Submarine Rescue

maneuvers. The positive direction of this axis is downward com-:
mensurate with standard submarine dynamics analysis. The y-axis
is through the x-axis z-axis intersection with positive direction to
the starboard to provide a right handed orthogonal system. This

xX, y, 2 body axis frame is related to an inertial axis system,

X, Y, Z, through the ordered rotations WV, @, and @ about the

Zz, y and x axis. The origin of inertial axis system is located at
the origin of the vehicle axis system at the start of a computation
and the X Y plane is parallel to the water surface with the vehicle
axis in the X Z plane.

The Euler angles are formed in the following manner. With
the two systems initially coincident, a first rotation, (ZW), is
performed giving the system (x,, y,, Z). Next a rotation, (y,®),
is performed about the y, axis resulting in the system (x, y,, Zo).
A third rotation, (x4), about the x axis brings the body axis
system, (x, y, z) to the final position. The transformation matrix
relating the (X, Y, Z) system to the (x, y, z) system through the
above ordered rotations is then

x cos@ cosw cos®6 sin sind x
y |=| sin® sin@cos-cos@ cosy sin@ sindsinbtcosdcosi sinécosé fl y

Zz sin8 cos@coswtsiny sing sind cosdsiny-sinécosW cosb cos@}} Z

It remains to relate the Euler angle rates to the roll, pitch and yaw
rates, (p,q,r) respectively. The relation is

11/15

Schretber, Bentkowsky and Kerr

4 p t+ tan 0(q sin #6+r cos @)
@ |= qcos $- r sin $
w (r cos 6 + q sin $)/cos 0

High Angle of Attack Considerations

The second major difference between the DSRV simulation
and the conventional method is required because of the high angles
of attack experienced during the hovering and docking maneuvers.
This high angle of attack problem becomes accentuated by the vehicle
which, by using its thrusters, is capable of turning without forward
way, a maneuver entirely outside the scope of those covered by
conventional analysis. These shortcomings of the conventional
analys is were overcome by a technique used during development of
LMSC's DEEP QUEST research submersible where the hydrodynamic
forces on the body and appendages (in this case the shroud ring) are
considered separately. This allows for adequate representation of
stall characteristics of the shroud ring as a function of the local
angle of attack at the shroud ring which is essentially impossible to
account for when a total coefficient for the body-ring combination is
used in the simulation. The coefficients of the body itself are handled
through addition of a normal drag components to the standard small
angle of attack representation of forces. These high angle of attack
considerations will be discussed further in the following sections.

The equations of linear motion are derived from the funda-
mental equation

e = 2 (mY) (1)

external

stating that the sum of the external forces acting on a rigid body of
mass m, equals the time rate of change of the momentum of the
body. The momentum, mVg,, is a vector quantity and V, is the
inertial velocity of the center of mass. Expressing V. in terms of
the velocities and rates about the vehicle fixed axis system described
earlier

u+aqZg- rY¢ +t Xe

wtpY,- 4X_tZ

1116

Vehtele Dynamics Associated with Submarine Rescue

where Xg, Yg, Z,g are the coordinates of the vehicle center in the
body axis system. Neglecting the velocity and acceleration of the
ceg.e with respect to the body (Xg, Yg, Ze¢, X_q Yo, Zg= 0) because
they are small and performing the operations

= d pee ne =
Fexternal = ar (™Ve) =mV, body pmo aN
we obtain
u tqw - rv - X(q° + 1°) + X(pq - r) + Z,(pr + q)

Pexternol =myfu tru - pw - Yg(r" + p*) + Z,(ar - p) 7 X(ap + r)

w+ pv - qu- Z¢(p? +g?) + X,(rp - q) + ¥,(rq +)

The equations of angular motion are derived from

Mexternal = £- (le) (2)

which states that the sum of the external moments acting on a body
equals the time rate of change of the angular momentum of the body
with both the moments and angular momentum expressed about the
same point. Since the hydrodynamic moments are described about
an off c.g. axis system the development of the right-hand side of
the equations consists of expressing the time rate of change of the
angular momentum of the body about the center of the axis system
with the rotational vector expressed in the directions of the body
axis, The results of this operation [1] yield:

Tp + (i - i )qr +m ¥,(w + pv - qu) - Z,(v tru - pw)|
Mexternal = fea AU ESE) yeep es! | Z,(u + qw- rv) - X,(w + pv - qu)}

lLrt+(I, -I,pqtm X.(v + ru - pw) - Y,(u + qw - rv)
z y x G G

The moments of inertia I,, I, and I, are the moments of
inertia about the center of the body axis system and not about the
vehicle cénter of gravity. Since there is near symmetry in weight
distribution about this body axis system, cross products of inertia
have been dropped from the equations of motion.

The external forces and moments, Foyterngi ANd Meyternai »
on the vehicle during free stream operations come from three

1217

Schreiber, Bentkowsky and Kerr

sources; the body, the shroud ring, and the thrusters.

Fexternal = F body + Fghroud * Fthruster -

The following sections will discuss the simulation of these three
classes of forces.

BODY FORCES

The forces on the body can be divided into several types:
the static forces, Fhoqy, static » associated with buoyancy and weight
and the dynamic forces, Fhoqgy dynamic? which depend on vehicle
motions with respect to the water.

Static Forces

The weight or gravitational force (W) acts inthe Z direc-
tion and the buoyancy, A, acts inthe -Z direction. The total
vehicle weight, W, can be best expressed as the vehicle weight
when in neutral trim W, =A plus the change in variable ballast
from initial conditions. The resulting static forces on the body are:

- sin 6
F pody, static = (Wo +My, - S)| sin @cos 0
cos $cos 8

The DSRV contains the following ballast and trim systems which will
affect the static forces on the vehicle:

Main Ballast

Variable Ballast

Transfer Ballast

Rescuee Ballast

Mercury Trim

Mercury List and BG Control

In operation, the main ballast tanks are full when submerged.
The Transfer Ballast tanks are empty except when they are being
used in the dewatering process, and rescuee ballast is exchanged for
rescuees providing a constant value during submerged operations.
These systems do not vary during normal submerged operations and
therefore are not included in the simulation.

The tanks whose contents vary during submerged, unmated
operation are the variable ballast (T, and T 3); trim (T, and Ts),

1118

a

ee oe

Vehtele Dynamics Associated with Submarine Rescue

and list (T,, T,, and T;) system tanks, Fig. 2.

The variable ballast tanks (6, 7) are hard tanks each witha
capacity of 500 pounds of seawater. Water can be transferred to and
from the sea from either or both tanks at a rate of 2 gallons per
minute.

The mercury trim system is a set of two tanks (4,5) con-
taining 225 pounds of mercury and 15 pounds of oil. A pumping rate
of 3 gallons per minute provides a net weight change of 5.3 pounds
per second between the two tanks. With the list system reservoir
full (Z, maximum) a 23 degree trim angle is attainable.

T4815
16817
12 13
TI
Tl LIST SYSTEM RESERVOIR
T2 STARBOARD LIST TANK ene
13 PORT LIST TANK x 4g + X5Ws
T4 FORWARD TRIM TANK CG = —W=Ww
T5 AFT TRIM TANK .
T6 FORWARD VARIABLE BALLAST TANK
17 AFT VARIABLE BALLAST TANK Y5W> «YW
Y ~ =
G + Z
ti
z(w__w_)
7. =< Z nf n n no
G = “co* —Wl+8W,

Fig. 2. DSRV Trim and List System Tanks

The list system consists of three spherical tanks each witha
capacity of 2780 pounds of mercury. The two list tanks (2, 3) are
located 2.2 feet above the vehicle centerline and are separated by
4.4 feet. The reservoir (1) is located 3 feet below the centerline.
The configuration of the list system piping and valving is shown in
Fig. 3. It is noted that transfer can be effected between list tanks,
between each list tank and reservoir and between the two list tanks
tied together and the reservoir. Pumping rate can be varied between

1119

Schreiber, Bentkowsky and Kerr

1/2" TUBE

N = NEUTRAL (AS SHOWN)
E = ENERGIZED

Fig. 3. List System Schematic

0 and 28 GPM (51.3 pounds per second net change). During roll
damping operation the rate is controlled proportionally. In all other
modes of operation, maximum rate is used.

Variation of BG is accomplished by transferring from the
reservoir to the two list tanks. Roll damping is accomplished by
transferring between list tanks for list angles less than 22.5 degrees,
and between reservoir and the appropriate list tank for list angles
greater than 22.5 degrees.

n = net weight in tank n - pounds

rate of change of weight in tank n - pounds /second

commanded rate of change

228%
iT]

net weight in tank n for vertical buoyancy, neutral
trim and maximum z,- DSRV on surface

D = Depth, feet
Baseline operation is represented by 200 pounds plus a cor-

rection for depth in each of the variable ballast tanks, 93 pounds in
each list tank, 2404 pounds in the list system reservoir, and equal

1120

Vehicle Dynamics Assoctated with Submarine Rescue

distribution of mercury between the two trim tanks. Note that all
mercury system weights listed represent the difference in weight
between mercury and an equal volume of oil.

Table 1 summarizes the location of all tanks and the value of

Wg:

TABLE 1

Physical Parameters of Ballast and Trim Systems

W max

=
i

140,369 pounds
ZG = 0.1335 feet

The effects of variations in the weight of water in the variable
ballast tank on. zg have been included in the basic vehicle equations
and the z_, variations computed in this section are due only to vari-
ations in the list and trim systems.

Changes in depth effect both the density of the water and the
compressibility of the hull. The net effect on buoyancy, using tem-
peratures and salinities corresponding to sub-tropical waters cor-
responds to a gradient of 0.1 pounds per foot of depth. Under normal

conditions the required ballast change is divided equally between the
two variable ballast tanks. Thus, for neutral conditions

W, = Wy + 0.05D

All operations are written with t = 0 corresponding to neutral

1121

Sehretber, Bentkowsky and Kerr

buoyancy at the initial operating depth of the problem.
Thus

Wn = War, * Wr dt aap Maree eS Pe arc
0

t
Wy = Wyo +9-05D +f) W, dt n= 6,7

The term Wy which is used in the vehicle equations matrix is
given by

Wii = We t Wr - Weg - Wao>

Location of the center of gravity is given by

wee xqWa te x5W5
G
Wo + ie

Yo= W> + y3W

gz ee
Wo + DW

D2 Wr aq] Wao)

Zg= Cs)
Wo " DW;

Dynamic Forces

The hydrodynamic forces arise because of the motion of the
body with respect to the water and are defined in terms of hydro-
dynamic force coefficients. The hydrodynamic force and moment
coefficients used in this report are not the standard non-dimensional
coefficients used in most studies. It has been found that a set of
dimensional coefficients provides a much easier nomenclature. The
force and moment coefficients are represented by subscripted capital
letters of the form Xgpce._. The letter denotes the direction of the
force (X for forces along the x-body axis, Y along the y-body axis,
and Z along the z-body axis) or the axis about which the moment
results (L, Mand N for moment coefficients describing the mo-
ments about the x, y, and z body axes respectively).

The subscripts vary in number and form and denote the
variable quantities that the coefficient must be multiplied by to
obtain a force or moment on the body. For example, Xyjy is the
dimensional axial drag coefficient since when multiplied by u |u|
the square of the axial velocity, u, it results in an axial force

Pi22

Vehicle Dynamites Associated wtth Submarine Rescue

Xywulul = Fx drage Similarly, Muwwlu| is the pitching moment
used by the normal velocity, w. The use of absolute values in these
coefficients provide for the proper signs on the force and moments,
and because of most of the near fore-aft symmetry of the DSRV

less the shroud, the coefficients are independent of the direction of
various velocity components. The direction of the normal force
Zwiui W}u} is dependent only on the direction of the normal velocity
regardless of whether the vehicle is going forward (u > 0) or back-
ward (u<0). Since the sign of Zwiy is negative the normal force
due to normal velocity is always in the opposite direction of the
normal velocity. A brief description of the development of the
representation of hydrodynamic forces on the body follows. First
consider the forces on the axisymmetric bare body of the DSRV and
then add forces resulting from asymmetries, such as the transfer
skirt and splitter plate. The representation of lift, acceleration
and axial drag forces on an axisymmetric bare body is relatively
well known and can be obtained from slender body theory, Ref. 2,
other potential flow analysis, Ref. 3, or test data and is of the form

Xyu t Xyy ula

F Yyv + ¥pr + Yow rlul + Yuu v1 ul

EXT lift, acceleration, axial drag

Zyw + Zaq + Za a/ul + Zwiyiwlu|

These lift, accelerating and axial drag forces are those normally
used to simulate the dynamics of submarines and provide a very
adequate representation of the forces and moments at low angles-of-
attack (a < 15°), At high angles-of-attack, however, they become
inadequate. For example Zwyjw|u], the only force resulting from
normal velocity, w, goes to0as u goes to 0, while a vehicle
normal to the flow experiences a significant normal force. This
normal force is due primarily to flow separation and is, excluding
Reynolds number effects proportional to the normal velocity squared
w*, Wind tunnel and water tunnel tests on the Polaris, Poseidon,
DEEP QUEST, and other vehicles have shown that a reasonably
good representation of the forces and moments due to normal
velocity can be obtained by using the two terms Zwyy wlu| i Zwiww | w |
where Zww is measured force at u=0 (g = 909) divided by the
normal velocity squared and Zyjw is the slope of the force versus
angle of attack curve. Pitching (yawing) of the vessel will cause a
variation in normal velocity along the vessel and therefore a variation
in this normal drag over the body. It remains then to develop a
method to account for the distribution of this local normal drag
caused by pitching and yawing. The use of a strip theory method
provides that the normal force due to normal drag can be expressed
as Znormal drag= Ziawiw'|w'| dx where Zi‘, is the local value of
the normal force coefficient and w' is the local normal velocity and
can be expressed as w'=w+tqxX where X is the distance between

1123

Sehretber, Bentkowsky and Kerr

the point in question and the center of the axis (- forward). This
integral can be evaluated at each step in the integration of a simula-
tion when the distribution of Z\,y is known but it proves both
cumbersome and time consuming. On the other hand, for a nearly
cylindrical body such as a missile or the DSRV, test data has shown
that a fair representation of both the force and moment are obtained
when a constant value is used for Zw, from the nose of the vehicle
to the forward edge of the shroud, a distance L, from the nose.
This then allows the value Z'wiw to be removed from the integral
and sets the equality Zwiw= Zwiwi/Ls. Replacing the local normal
velocity w' by its equivalent w + qX, the integration
Z ata ‘Ioody (w + qX)+ |(w-qX)| dX still poses some problems because
of the absolute value signs. To accommodate these two integrals
are formed depending whether the center of rotation, the point where
w' =0, is on or off of the body. Expressing the ratio of the distance
the center of the axis system is off of the nose, Lj), and the length
L, as K,= L/L, the center of rotation is forward of the nose when
w/qLs < Ks and aft of the body when w/aLs >K, - 1 and the value
w'|w'| can be replaced by (|w|/w)(w')"=(|w|/w)(w? + 2wxq + x?q?)
and the integration

ae LeLy

Ziel w' |w'i|*dse= Z! ll { (w* + 2wxq + xq?) dx
E w -L

wlwl

results in three terms

wi 2
Cywlw] + Ga lw] + Cyl g

where
2 mt
C, = Lg, Zwiwi
3 !
4 2 '

When the center of rotation is on the body K,= w/ql, = K,- 1 the
integral must be divided into two parts to account for the sign change
in w'|w'| at the center of rotation and

Lg ly -w/q Lg ly

2
Zit ww? |SdX*="= Zs Jal] ¢ w' ax sh w! ax|
-Ly -w/q

where the lal/a is used to denote the direction of force since the
local normal velocity forward of the center of rotation depends only

1124

Vehicle Dynamites Associated wtth Submarine Rescue

on q. This when expanded to

- Zwiwi lal hie (w? + 2qxw + (qx)*) dX ey)

q Ly

Le L,
(w? + 2qxw + (qx)*) ax|

Ww

and integrated yields a four term expression for the normal drag
with the center of rotation on the body

C4a/la| w°/a + Cya/lal w? + wlal +c alal

where
Caw = 21/3 Ziel

Cs Baele (1 - 2K )z'

wiwil

Cogn (i) Ke Ze

wiwl

Grit Kk, + kG - 2K) Zz

Tw wiwl

In a similar manner the lateral drag terms for the sway or
equation are developed and result in C, abs us ie at lv | + GC, e/a

for the center of rotation off oe the bod “vy /-rL or v/-rL, < K,-1
and Cqar/|r|v3/r + Cove /|t - my A # eta for the center of
rotation on the body K, = ia = - 1 with

2
Ciy = Ls Y vivi
3
Co = fs (2K, - 1)Y'!

vivl

EN at a 3K et 3K. )yt

Us
<
i}

viv

Cay = 23/3 Yvivi

2
C.. =, (2Ko- 2) iui
> 2
Ce, = Le Lt -~K y ee 3 mg be
4 2
Cae, /3 = Kort tee aK, 4) y au

This then completes the simulation of forces on the axisymmetric
bare body of the DSRV. This representation has been developed

1225

Sehretber, Bentkowsky and Kerr

keeping in mind the fact that it will be programmed on a digital
and/or analog computer and several simplifying assumptions were
made to allow for mechanization of the resultant equations. As in
any simulation a trade-off must be made between the accuracy of
representation of forces and moments, the ability to mechanize
certain types of expressions, and the availability of data.

The presences of the transfer skirt and splitter plate add
additional coefficients to the body because of the asymmetries they
provide. A complete set of these terms due to asymmetries were
developed, their numerical values determined, and their relative
importance established, with the reduction in the number of terms
in the simulation as an end goal. It was doin that additional
terms of the form Xquw > Xryrv, X, r’, Koad’: Xrprp» YpPp,» YpwPw,
Yap QP » Yuiwiv] ws z ppP* » Zprpr; aa » Zryrv, Zeer , and Zyjyju u|
would be required foe adequate simulation of the dynamics of the
DSRV during the hovering maneuvers.

qs, The addition of forces due to the shroud, Fyproug,s propeller,
Forop» thrusters, Frthr» and the interaction forces during mating,
Tae , complete the force equations.

A similar line of development was used for determination of
a set of moment equations. Values for most of the coefficients were
obtained from model tests conducted at NSRDC (Ref. 4) and Hydro-
nautics (Ref. 5), and theoretical values were computed for the re-
maining coefficients. A complete set of equations of motion used for
the DSRV Model for Analysis (Ref. 6) follow along with numerical
values of the hydrodynamic coefficients, Table 2.

Surge:
m[u + qw - rv - xg(q? + r*) + yelpq - r) + zg(pr + q)]
= Xu + Xyaw + Xyrvit+ Xywulul + Xr + Xa + Xprp

- (Wo +) We - 4) sin © + Xghroua + Xprop + Xthr + Xdist

Sway:
m[v + ru - pw - ye(r* + p*) + agar - p) + x,(qp + r) |
SY 7 Ypert Ypp + Yriu r|u| + YpwPw + Yqpqp

| wi ch Monomer LY yi, V Pa act

1126

Vehiele Dynamtes Assoctated wtth Submarine Rescue

*
PhGeviy| + Conv + oul r*]

# Cyst — + Caer y +Gevir| + Car|r(|]

+ (Wo + > wy, - A)cos 8 sin @ + Ygnroud + Yprop + Ythr + Yaist

* Vv Vv
Terms are cancelled when “1, >Kg, or ie <K,- i
+ Vv
Herms are cancelvedi when Ky = = Ke 1
s~ -rL, s
1 2
Cy =z Pls Yyivi
4 3 ’
Coy iss pl, (2K, - 1)¥ viv
1 4 é !
C3, = 5 pl, (ol eI oe) ae,
1D 7211
Cay = = Pz Yui
1 2
Coy =z PL (2Kg - 1)¥ viv
{ 2 2
Cey = = pl, iE Key Re xii
Cres ol Bl eK ek ee
i7ve= No Peg “Ns s gee vivi

Heave:

; 2, 2 ; °
m[w +pv - qu-z(p +q ) +x,(rp - q) + yg(rq + p)]
; ; 2 2 2
= Zyw t+ Zaq aR ZopP + ZyyPV + Z5-Pr + Za sp Sra ice Lopt
+ Zyiyu |u| + Zui wal + Za all

+ (Wo +) wy - A) cos 8 cos $6 +

27

Schretber, Bentkowsky and Kerr

aK
+ Cywlw] + Cyalw] +c, bl g ]

3
Bigs

+ [ Cawrap qt Swat ©” + Cou i a +C,alal]”

+ Zone + Z gist

A Z shroud i Z prop

*
Terms are cancelled when w/ql,>K, or w/ql,< kK, - 1

**
Terms are cancelled when K, 2 w/ql, = K, - 1

Cw = pl, Z wlwil
3
Cow= > PLS (1 - 2K)Zyiwy
4 2 1
Caw pl, /3 (4 ~ 3K, + 3K, )Z iwi
p2L,/3 Zalwi

ie)
a
=
1

2
el, (i 5 2K)2

wliwl

OQ

o

=
N

3 2 2
pls [« - K,) + a) a

2)
N

Q
N By
= =
I Ne
Ne ne aK NK Ne NK Ne

4 2
pL, /a(i - K+ K, 1 - 2K)Z 4.

Yaw:
Lr + (I, - I,Jap + ml xg(v + ru - pv) - ye(t. + qw - rv)]
= Nex B Niv + Npged + Nuvi lulv + Netut r{u| + NwpwP
+ Nyqvd + Nopwip |u| + Niwiv | wlv
+ (WoxX, a > Wy x4) cos 8 sin ¢ + Woycesin 8
+ Yehroud*shrd * Nope + Npropt Naist

4 *
+ Cher] el + Copy A Corie Te? + Ca,|r|v]

1128

Vehtele Dynamtes Assoctated with Submarine Rescue

**
Cs,|vir + Cy viv + epi r?]

* Vv Vi
Terms are cancelled when - FL. >Ks or - ai < K, - 1
A Vv
Terms are cancelled when K,= - ns K,- 1
s
4 5 4
eee = pL, /4[K, + (1 - Ky Yui
{ 2 2
Cor = > pies /2lK, + (1 - Ke) Win
4
C. ae 5 pL, /6 Yyivi
en 4 3 4 3 y1
‘ates wire 2L,/3{ Kg - (1 = K,) Tein
4 4 ) 3
Cz, = o p 2125/3 Ket (i= Ks) aii
iM 3 2 2
Ce, = 2 pl, [2 [K, ia (4 ~ Ks) 1 Yviw
4 5 4 4 xy:
Cr, = > pl, /4[Kg - (1 - K,) ger

Roll:

Lp PL. = qr + ml Yg(w + pv - qu) = Zg(Vv + ru - pw)]
= Kp + Ky + Kar + Ky, v|w| +K, rw + K, va + K,pw
+Kyy viel +Kyy viv] +Kyyplul +K,jy rlul + Kyi P P|
+ Woyg cos 8 cos $ - (Woz, +) Wy, Z,)cos 8 sin >

+ Korop + K gist

1129

Schretber, Bentkowsky and Kerr

Pitch:
Iq + (I, - L)pr + m|[ zg(u + qw - rv) - X_(W + pv - qu)]
= Mga + Myw + M,pr + Mutwi | ul w * Maui a| | + Mypvp
+M,,,;u]ul + Myw? + Myyrv + Myr - (Woz¢ >) W;, 24,) in @
= (W,Ge +) Wy eos ®@ cos $ - ZshroudXshrd + Mprop + Mthr

- *
+ Mais +1 Cigalal * Cogrdyw! + Cogpdy < + Cagl a

**
#[Cgglwla + Cggwiw] +6, AL o]

x
Terms are cancelled when w/qlL, >K, or w/ql,<K,- 1

*
Terms are cancelled when K,= w/ql, 2 K, - 1

Cig =F ply /41KS + (1 - Ke)" Zbrwt
Coq =5eL3 P20 +(1 - Ks) ] Z wiwl
C3q= - = pLs/6 Ziwi

Cag =5p2/3L, [-K, F(t = Ky) ZG
Ce, = p2/3L, [K> + (4 - Rep Z re
Cog= 5 ple /2[-Ky + (1 - Ke] Zhiq
Ch, = 5 ple /Al - Ky + (1 - Kyl") Ziaw

1130

Vehicle Dynamics Assoctated with Submarine Rescue

TABLE 2

DSRV (ML-493-03) Parameters and Coefficients

we, (436s ates I, 3678X 10° slug-#t
26 0.1335 feet Iy 4.52% 10° slug-ft*
i 49.33 feet Te 4550 40 alupor
L, 23.4 feet
L. 46.95 feet

FORWARD MODE

-3 2 |
-6.87X 10 p72. = -1.67X 10

|

i yd 2a ee tons 2.433 X 10° -1.24x 10°
| am -4.6 X10° ai i2 x 10"
L Zhu x10" 2.43
pa x 10° -1.95 x 10°
| LAT 208, AO -5,59X 10:
| Ye W422 Xa" ao = 13.36 X 107
| -9.41 X10 2.198 X 10° -~2.07 X 10°

NOTE * Indicates coefficient value is zero when w is negative,
** Indicates coefficient value is zero when |= | > 0.173.
(1) Under following conditions:

200# in each variable ballast tank,

93# in each mercury list tank,
remainder in reservoir.

1131

Sehretber, Bentkowsky and Kerr

TABLE 2 (Cont'd.)

Coefficient Non- Dimensional Dimensional

-1.2 x10° PY = -4. 44X10"

x 10° 1.20X10° 3.72X10°

-2 3

x 10 -3.60 X10

x 107 “4,32 x 10"

x 10> 1.04% 10°

x 10° 2.64 X 10°

-2 3

x 10 3.60 X 10

-2 3

x 10 -3.60 X 10

-3 2

x 10 -4.44 X 10

-2 3

-3.4 X40 -3.72 X10

-3 3 2

5.32X10 af2Li= 6.32 10
-4

5.4 5G10 1.20 X 10° -6.6 R19

2.3 X107 2.76 X10°

-2 3

-2.8 X10 =3,36 10

- 3

oO KAO" -3.6 X10

=> 2

4.98 X 10 5.98 X 10

é 2

Heo tO. pel -8.3 X10

-4 6 2

-1.3 x10 5.92 xX 10 = fel x 10

4. 2

-1.34X 10 -7.93 X 10

1.4 x10 8.3 40

2 2
1.34xX10° 7.93 X 10
2

i.3 X10° 7.7 X10

=1.3 7207" =7.7 M40"

1232

b

Vehtele Dynamics Assoctated with Submarine Rescue

TABLE 2 (Cont'd.)

Coefficient Non- Dimensional Dimensional
5.92 x 10°
ok Seto) 8.3 x 10°
-4 2
1 34.<40 7.93 X10
-4 2
=1e,10.. e110) -5.92 x 10
OL Ore 5.92 x 10.
0) 0)

-3 2
2210 2.64 X 10
-3 2
1.419. 40 4,42 <* 10
-4 4 3
-3.16 X 10 py Zine = -1.87 X 10
-4 6 2
1.5510 5.92 x 10 9.17 X10
-4 3
=2.354 ~ 10 -1.39xX10
-4.5 x10° 2266 40"
-4 2
1.3410 7.93 X10
-4 2
= 23) 3x 10 (ey eae 8)
x 2
ia tO" -8.3 X10
-4 2
1.4 X10 8.3 X10
-4 2
={.58 X10 -9.35 X10
n5le X40 322 10°
“i G4 io -7.93 X 10°
ee oe -7.7 X10"

-4 2
-1.34X10 -7.93 X10

-5 5 3
-1),55 X 10 p/2L = -4,53 X 10

0 2.92 X 10° 0

L133

Sehretber, Bentkowsky and Kerr

TABLE 2 (Cont'd. )

-1.13 X10
-3
-1.4 x10

-3
1.39 X10

0

-3
-1.4 x10

-3
-1.39 X10

-3

-8.05 X40
-2

-9.9 x10
5

-1.0 X10
-615 “TO

7.7 X107

3
5535 X 10

-2

-8.7 x 10
-3

5.35 X10
-4

x 10

-2

x 10

2.4 X10
-4

2652 Xf0
aa

-7.25 X40
-3

5.4 X40
-4

-7.25 X10
-3

8.4 X10

2
p/2L =

2.433 X10°

p/2L° =

5
1.20 x10

p/2L =

6
592 x 10

1134

Coefficient Non-Dimensional Dimensional
5 :

2.92 X10°

4
-3.3 X10
5

-4.1 X10
5
4.06 X 10
0

5

-4.1 X10

5
-4.06 X 10

|
-1.96X 10
-2.41 X 10°
-2.43
-1.58 xX 10°
9.24 10°

2
6.42 X 10

4
-1.04X 10
6.42 10°
566 SUG:
13.0 ° to"
2.88X10°
-3.67X 10°
-4,29x 10°

4
=e x 10

3
-4,29 X 10

4
-4.97X 10

Vehicle Dynamics Assoctated with Submarine Rescue

SHROUD RING FORCES

The control shroud is a circular movable wing located at the
aft end of the vehicle, supported by four struts space 90° apart,
Fig. 4. The force and moments on the shroud ring are obtained from
curves of lift and drag on the shroud as a function of total angle of
attack of the shroud, Figs. 4 and 5. Resolving the resultant force,
F, into the directions of.the vehicle's axis system provides the
components used in the equations of motion, Xgbroug» Yshrouqs 2nd
Z shroud’ The relative velocity of the shroud with respect to the water
in the directions of body axis system is

se u
Vop = vg f=|v- Fe
Ww, w + X,q

The ordered deflection of the shroud, 6p, a deflection in the vehicle
pitch plane followed by 6y a deflection about the pitched shroud
yaw axis results in the relationship

SAE SHROUD ANGLE OF ATTACK, ag, DEGREES

1.4
Uer4

on ae
0.6 IS sles
a Nia
a

oe 0.2 Pax

BS
REN SES
Pa Nay
roe al a
ee eS
ee ee
FSF
ae a
poi aa ea

Pm Viniaes
ee ae

0 120 140 160 180

iets Gok i atti. a, DEGREES

Fig. 4. Shroud Lift Coefficient, C, vs. Shroud Angle-of-Attack, aS

1135

Schretber, Bentkowsky and Kerr

DRAG = C, 1/2pVv?
Fibgc&s

Ue Vg Ws

8 FT
1,83 FT

0 20 40 60 80 100 120 140 160 180
SHROUD ANGLE OF ATTACK, Ac IN DEGS,

Fig. 5. Shroud Drag Coefficient vs. Shroud Angle of Attack

Us Us
ss = | Yes | = Teaae| Vs | = TsoeeYse
Wes Wg

between the velocity of the shroud relative to the water expressed in
the direction of the body axis system Vcsg and the same velocity

expressed in the direction of the deflected shroud axis system V,,.
Us We cos 65 O- - sin 6p]f ug,
are EE 0 4 0 Vs }
Wy sin 6) 0 cos 6, |L_w.

Wis

1136

Vehtele Dynamites Associated with Submarine Rescue

Sy
U2s
“Is —_
es) 7 Chie
Y2s
‘2s 28
Is
cos 6, sin by 0 cos 6, QO -sin 5p Ug
= |-sin 6, cos 6, 0 0 4 0 Vs
= 20 0 1Jtisin6, 0 cos 6pJLw,
Us cos dycos 6p sin by -cos éysin d5][ us
= Ts22s| Vs |= |-Sin 6ycos 6) cos dy sin dysin 6p |] vg | = Ts22s VsB
Ws sin 5p 0 cos 6p Ws

The total angle of attack at the shroud can then be expressed as

1/2

-lf (vag tw -I

= 2s =

shroud = tan [! a 2s) ] = tan | vns/‘tps|
2s

or performing the indicated operation

1/2
7 ( (-sinSycos dpus tcos Syvs tsindysin Spws )’ Hs inSpus tos SpweF| \
shroud =

(cos § cos dpu, tsind,v, -cos 6ysind,wg)

are then looked up on Figs. 4 and 5 and values of lift, L,, and drag,
D,, on the shroud are calculated from

2 z
L, = p/2S,6,.V and D, = p/25;CpV
with
S, = b,C,
2

Veutvtwe

b, = 8 feet

C,= 1.83 feet

The lift and drag on the shroud are then resolved into forces on the

shroud F,, and transformed into the body axis system, Fae °

1

Scehretber, Bentkowsky and Kerr

Np = Cc. cos @, + Cy sin a,

on sin C.e- Cy cos @,

—

Fog se V25/Vns Ne
-Was/VagNe

and

F Shroud = T2208 Fg

The moments on the shroud are the product of the shroud force and
the distance of the shroud center from the center of the axis system
as shown in the moment equations.

THRUSTER FORCES

The propulsion system of the DSRV consists of a single con-
ventional screw propeller for axial thrust and four ducted screw
propellers arranged in forward and aft pairs for lateral and normal
thrust. This system provides the vehicle with five degrees of
maneuvering freedom (heave, sway, surge, pitch and yaw) and pro-
vides forces and moments sufficient to meet hovering requirements
in currents of the order of one knot. The control of the sixth degree
of freedom, roll, is provided by the trim and list system. A com-
plete treatment of the development of the maneuvering system is
contained in Ref. 7. The following treatment will present the data
used in the simulation with a little explanation of its development.

Main Propeller

The main propeller is a 6-foot diameter, wake adapted,
three-bladed propeller with a blade area ratio of 0.24 and a maxi-
mum speed of 1.64 revolutions per second.

For estimates of the vehicle maneuvering performance the
propeller thrust and torque characteristics are required for ahead
and astern motion of the vehicle and for positive as well as negative
propeller rpm. "Behind-the-ship" tests of the DSRV propeller were
performed for all four operating modes at the Naval Ship Research
and Development Center (Ref. 8).

The curves of the thrust and torque coefficients, Fig. 6, are

typical for ahead and astern operation of a propeller and can be
expressed in the form

eee her

1138

Vehtele Dynamics Assoctated wtth Submarine Rescue

NOTE: DATA FROM MODEL TESTS
CONDUCTED AT NSRDC

THRUST COEFFICIENT, Kr, AND TORQUE COEFFICIENT, 10Kq

APPARENT SPEED COEFFICIENT, Ja = 745

Fig. 6. Characteristics Curves for DSRV Propeller
(Note: Data from model tests conducted at
NSRDC (Ref. 8) )

The coefficients a, b, c have been evaluated separately for each
quadrant of the propeller curves, and the resulting thrust and torque
coefficients are used as an expression of the steady and transient
characteristics of the propeller forces and moments. During
maneuvering the propeller may also experience velocities normal to
its axis. The resulting effect on the propeller thrust in the axial
direction and torque about the roll axis have been estimated and the
form of the coefficients can be rewritten including this effect as
follows:

2 2 2 1/2
KeatbJ, te] +a(se +3.)

re)

where

1139

Schretber, Bentkowsky and Kerr

vie ge
Jy= nd

ed
nd

u,v,w = X;y,Z components of vehicle velocity

= components of vehicle angular velocity relative to
yaw and pitch axis

RK
.Q
J

£ = distance of propeller from coordinate system origin

The coefficients of the y and z components of the propeller
force and the corresponding moment coefficients can be similarly
estimated and are proportional to J, and Jy respectively. For the
computations of the vehicle responses, all six force and moment
components have been considered. The resulting propeller force
and moment equations are:

Xpop = 755 n(n| - 58 un - 3.8 ae +26 mn (ye twee” 7 U=) 05 <= -0.21

= - 365 n?-172un- 45u>+26nlvatwe)’ ;u2z0,>8-0.21

= 155 7 + 60 un + 22 n” + 26n(ve + wa)!” 3u< 0, n=0
= - 365 n@ - 13 un + 22 n® + 26 n(ve + wo) 5 u<.0% mes
Yprop = - 30 nvp 5. ee 0
= - 12nvp ;n<0O
Zprop = - 30 wp nO
= - 12 nwp nex 0
2 2 /2 e
Pars a0 mln|- 6) Sai AOS ure eentve + we F231 A

n
= > -0.21

2 2 /2 °
Wag Fee SS wal 38a + 22(v, + wp)” +130 a 4%

>)

We =,0% = -0.21

e|p

1140

Vehicle Dynamics Assoctated wtth Submarine Rescue

2 @e
Ba0n #80 un + 35.200 + 22(vp + we)? Tn eee

it. <= 0, oy = 20

i

- 468 n - 8unt15,2u° + 22(ve two)! +131 a 5

u<0,n<0

Mprop= - 765 nwp - n= 0
= - 306 nWp ; n<0O
Norop = 705 NVp > n=O
= 306 NVp Di 0
where

Vp=v- 25.54

RS
"

Ww ot 25% 5 '¢q

Ducted Thrusters

There are two pairs of ducted thrusters used for maneuvering
in the pitch and yaw planes, as shown in Fig. 2. The four-bladed
propellers are 18 inches in diameter and have a maximum speed of
9.8 revolutions per second and produce side force through a combi-
nation of impeller thrust and a change in the pressure distribution on
the hull (Fig. 7 and Ref. 7). The thrust due to variation in pressure
distribution is very dependent on forward speed and the total thrust
coefficient for the steady state n= 0 condition was measured at
NSRDC as a function of forward speed (Ref. 9). Force coefficients
derived from this test are shown in Figs. 5 and 6 as a function of
u/|n| where n is the propeller RPS. The steady state force is
obtained from the relationships

2 *
Xthr mf = Nmfl 5
X thr ma = mat
Lehr yf = an [nv T

Yonr ya So

Scehretber, Bentkowsky and Kerr

Fig. 7... Ducted Thruster Schematic

*
Z the zt = ny¢|M2¢/T|
k
Zthr za = Hy, @se\s

*

Mebr 2¢ = 1-06 nz¢|nz¢ | M,

*
Méhr za = 1-13 Nzq|Nz9| M2

- 0.686 Ayatiyn

Mehr yn

*
N the yf = 0.98 ny¢ | ny] My

*

N = 1.04n,,|n,,|Mz

thr ya

N thr zn - 0. 686 De, len
where

Kone mana X force due to thruster mn

Yenr mn= * force due to thruster mn

1142

Vehicle Dynamics Assoctated wtth Submarine Rescue

Zehr mn= Z force due to thruster mn

M¢bhr mn = Pitch moment due to thruster mn

Neher mn= Y2W Moment due to thruster mn
M = Direction in which force acts

f for forward-thruster, a for aft-thruster

Dp
It

with the force eye ; Te : Ts Pay > and moment coefficients
(M,*, Mo*, My, M,*) shown in Figs. 8 and 9.

FORCE COEFFICIENT (LBS/RPS“)

FORWARD SPEED/IMPELLER SPEED, TnT (FT/SEC/RPS)

Fig. 8. Effect of Forward Speed on Duct Forces

1143

Sechretber, Bentkowsky and Kerr

MOMENT COEFFICIENT (FT LBS/RPS2)

FORWARD SPEED/IMPELLER SPEED inl (FT./SEC ./KPS)

Fig. 9. Effect of Forward Speed on Duct Movements

When the propeller in the duct is accelerating, the thrust
component due to the propeller is a function of the ratio of jet
velocity, Vj, to propeller speed, n. The thrust coefficient for a
propeller of this type was estimated from data taken from Ref. 10
and fit with a second order curve resulting in

V;
Tpmn= 10.63 nmn|Mmn|- 5-04 Vijmon| Vjmnn| ; ina a

= 10.63 nap|nmnal + 6-0 Mmal Vjmnl - 5+04 Vjmnl Vimal 3

Vimn < 9
mn

1144

Vehicle Dynamtes Assoctated wtth Submarine Rescue

where the jet velocity is obtained by integration of the expression

aVimn _ 9035 (T

at ~ 1.91 Vimnl Vjmnl)

pmn

The forces on the body due to pressure distribution changes are a
function of the jet velocity alone and are expressed as

Top sOeG55 Viet | Vime| (Ty - 2.93)
Tyma= 9+ 655 Vine | V. ett 2003)
The resulting forces and moments on the body are then expressed as
2 *
X thr mf — 0; 655 V jmt T3
2 *
Xehemareeeze Vim 14
Y the yn Tpyn i T hyn

Zthe zn = Tpz nt ie

? * ok

= Mig 2M; M,
Tio Ty Tio
M _ 2M r M ‘
22 2 2 29
ae T2 Teo

Mig s0%e88 Uiya-t 0573 nye

*
= Mio M20
Nene yt = 9-98 Tr +1.31 “T.* Tox bt
Riise * *
N = 1,04 M207 44,34 2Me2 oie zo
thr ya — T 5 pya T* a * bya
20 2 20

Nishi > 06238 Lin. 73 nee

* *
with Tio» Too » Mio» Mgo being the values of T,*, T,", M,” and
M, for 0 forward way (u= 0).

1145

Sehretber, Bentkowsky and Kerr

III SIMULATION OF DSRV/SUBMARINE INTERACTION FORCES

There are two forms of DSRV/submarine interaction forces.
The first is a mechanical type of force produced when the shock miti-
gation system touches on the deck and transmits a force to the DSRV.
The second type of force is caused by changes in the flow field cuased
by the bottom and the downed submarine and will be called flow inter-
action forces.

SHOCK MITIGATION SYSTEM

The shock mitigation system is primarily designed to protect
the transfer skirt and absorb shocks in the event of obstacle collision.
A secondary purpose is to act as a retractable base from which the
transfer skirt may be slowly lowered to contact the hatch mating
surface. ;

The shock system consists of a bumper ring concentric about
the transfer skirt (see Fig. 10). The ring is attached to eight struts
extending from four points on the outer hull. Each strut has a hy-
draulic piston/cylinder arrangement designed to attenuate impacts,
as well as extend and retract upon command.

A simplified model of the shock mitigation system is pre-
sented for purposes of simulating near normal impacts during the
controlled docking event. Figure 11 illustrates the DSRV with four
vertical legs extending down from the four hardpoints on the outer

FULLY

eae

FULLY a
EXTENDED

Fig. 10. Shock Mitigation System

1146

Vehtele Dynamics Assoctated wtth Submarine Rescue

Fig. 11. Simplified Shock Mitigation System for Simulation

hull. Each leg acts independently of the other three and is limited
to axial deflections only. The force elements in each leg consist
of a spring in series with either a damper or a constant force
element depending on both deflection magnitude and rate. Other
forces applied to the leg ends are due to lateral friction at the con-
tact surface. The equations that follow, approximate the force
effects on the vehicle due to near normal impact during docking.
The approximation is good if the deviation of the transfer skirt
mating flange plane from the plane of impact at instant of contact
is less than 10 degrees. Also, the vehicle velocity parallel to the
impact plane should be less than 0.5 fps at time of contact.

The resulting equations will give a disturbing force and
moment expression, X pist> Y pist> Z ois? Kost? Moist » and Noist
for application to the vehicle model equations of motion.

Using the direction cosine matrix

coswcos@ coswsinOsind-sinbcosd sinpsindtcossinOcos¢
[D] = | sinbcos®@ cosycos$tsinbsinOsind sinwsinOcos ¢-coswsind

sin 0 cosOsind cosOcos

1147

Schretber, Bentkowsky and Kerr

Lp Por

The forces on the i'” leg can be calculated:

a:

F pFerces
i

-1
Yy =[D] Yo

Zi Zi

Subscript V refers to vehicle axis systems.

Xi BoXy. +
Yvi * Yy sue
Zyi = Zy + Zi

Also,

1148

Vehtele Dynamics Associated wtth Submarine Rescue

In all cases,

{ Xvi (eo) X;
Miye LO wie=-< Yor Tl Dix Y
Zj Zvi fe) Zi,

Xyj

F,; = Fy = Fy = 0
if Z - Aj =O

Lg
iT
So

Zi = A> 6
Ai < 0.58 ft

= {K/c(z}- A,) “*}
Aj< 0.5 ft/sec

22
Fj = -CAj

F5j = - K(zj - 4;)

Zi AO
A,= 0.58
K(Z, - A,;) >11,000 lbs

Aji = Zi - 14,000/K if

A, = dA; /dt

Z; - A, >0
Aj < 0.58 ft
A,> 0.5 ft/sec

F,j = - 11,000 lbs
Ai iseZ ioe 000/K

j; - Aj> 0
if A;= 0.58 ft
K(Z; - A;)< 11,000 Ibs

F,j = - 11,000 lbs |

t
Aj -\ Aj dt
0
°2 1/2

ERAT Pe (X,/(X, +¥, Wie

1149

Schretber, Bentkowsky and Kerr

Fyi = bE /{¥,/(X, i ¥ 7}

Once Fy, Fyj and F, have been calculated, they are transferred
back into the vehicle frame of reference and summed:

Pyyj Fxi }
aM
Py = [D] Fy;
Poyi Fy
4
Xpist = > Pxyj
i= |
4
Ypist -) By
i=l
4
Zpist -) Fai
i=l
4
Koist -) ( By ln eh i)
i=l
= ( ly fs Ste)
Moist = (Fy, Zi - Fy, Xi)

Npist = » (- Ry Vit Fy Xi)

The physical and geometric properties of the system are shown below.

1150

Vehicle Dynamics Associated with Submarine Rescue

K = 310,000 lbs /ft

C = 44,000 Ibs-sec’/ft”

y= 0.4 lbs/lb for wet rubber on steel

TESTS TO MEASURE FLOW INTERACTION FORCES

Because of the complex flow phenomena, tests were required
to obtain measurements of the forces applied to the DSRV during the
mating sequence with probable currents of 0 to 1.5 knots, wherein
the flow forces are dependent on the approach attitude of the rescue
vehicle, the orientation of the bottomed submarine, and the proximity
of the two bodies.

In order to obtain meaningful experimental data, the following
test requirements had to be satisfied:

(1) The tests had to be performed at full scale Reynolds
number, and

(2) The environmental conditions had to be known.

Test Facility

The operating characteristics of existing hydromechanic test
facilities were investigated to determine the facility most suited
to conduct the test program. Because of the stringent combination
of test conditions , i,e., (1) test operating Reynolds number range of
2 to 6X 10° per foot, (2) rescue vehicle angles-of-attack up to 45°,
and (3) varying proximity of two bodies in the test channel, it was
determined that the test requirements extended beyond the operating
characteristics of all hydromechanics laboratories. However, the
National Aeronautics and Space Administration (Ames) 12-foot vari-
able pressure low turbulence wind tunnel was capable of meeting
all the test requirements. The Ames Research Center is located at
the Moffett Field Naval Air Station at Mountain View, California.
The DSRV test program was conducted at this facility, which operates
at subsonic speeds up to approximately Mach 1.0. The facility was

£151

Schretber, Bentkowsky and Kerr

operated by the Arnold Research Organization under contract with
NASA. The operating Reynolds number per foot versus Mach-num-
ber range of the tunnel is presented in Fig. 12. With a 1/30 scale
model of the bottomed submarine, Reynolds numbers up to 6 X 10&
based on the model diameter could be achieved at a Mach number of
0.2. This corresponded to the full-scale Reynolds number ina

4.5 knot current. Compressibility effects at a Mach number of 0.2
are known to be insignificant. Because of the large size of the test
channel and the available equipment and instrumentation this facility
afforded a unique capability for the DSRV program.

MACH NUMBER

STAGNATION PRESSURE,
LB/SQ IN abs

a 70i 75
ec ie

DYNAMIC PRESSURE
— LB/SQ FT

F 50]

0 l 2 3 4 5 6 7 8 9 LO} casei

REYNOLDS NUMBER PER FOOT (x10~°)

Fig. 12. Operating Characteristics of the Ames 12-Foot Pressure
Wing Tunnel

Test Section and Model Support System

The test section is circular in cross section except for flat
fairings. Figure 13 presents a schematic sketch of the general
arrangement of the test section and the DSRV model support system.
As illustrated, the sting-type model support consists of a fixed strut
mounted vertically in the wind tunnel to which is attached a movable
body of revolution carrying the sting and, in turn, the DSRV model.
The strut functions as a support and guide for the body of revolution
which can be pitched in the vertical plane by means of motor-driven
lead screws. The range of pitch angles is 10 to 20 degrees; however,

1152

Vehicle Dynamics Associated with Submarine Rescue

12-FT DIAMETER

SUNKEN 2 a
SUB 10 DEG
MODEL

~ as

SEA BOTTOM PLANE
| | FIXED SUPPORT STRUT
DOWNSTREAM ELEVATION SIDE ELEVATION

Fig. 13. Arrangement ofthe Test Section and Support System

pitch angles of 45 degrees were obtained using a bent sting.

The 1/30th scale submarine model was situated on a ground
plane installed to simulate the ocean floor, and was oriented both
into and normal to the current at 0, + 22.5, and + 45 degree roll
positions. For these mating positions, the DSRV model was set at
various attitudes when approaching the submarine model along its
longitudinal axis and from the side (athwartships). Photographs
of the actual model installation for various mating conditions on the
forward and aft hatches are shown in Figs. 14 and 15.

The displacement between the two bodies was regulated by
vertical movement of the sting mount which supports the DSRV.

Models

The 1/30-scale model of the bottomed submarine, shown in
Fig. 16, was constructed of poplar wood. The diameter of the
model was 1.07 feet and its overall length was 8.2 feet. The model
was constructed with a removable aft section in order to position the
DSRV sting-support, which is located vertically along the centerline
of the tunnel, over the area of the forward hatch. The model was
also provided with a small and a large sail in order to simulate
Permit (594) and Skipjack (588) class submarines. The forward
hatch locations with the small and large sails was as shown in Fig. 5.
Only one location of the aft hatch was considered.

£153

Bentkowsky and Kerr

Sehretber,

sooid0q 0
JO Suypeoay ouyzeurqns e YIM YEH WV oY} UO SuyJep[-uopyeTLe su] Tepoyy

‘Dy °Sta

1154

Vehtele Dynamites Associated with Submarine Rescue

sooid0q 06 FO
Zupeopy euyTAeUIGNS e& YIM YOIe_] PIeMAOT oY} uO Suyzep -uoqeTpeysu] Tepoywy

3

it55

Sehretber, Bentkowsky and Kerr

AFT HATCH
LOCATION

FWD. HATCH
LOCATION WITH
SKIPJACK SAIL

FED, HATCH

LOCATION WITH
REMOVABLE SECTION PERMIT SAIL

PERMIT SAIL

NOTE: THE MODEL SAIL ASSEMBLY
COULD BE ROTATED +22.5
AND 45 DEGREES ALL DIMENSIONS IN INCHES

Fig. 16. 1/30 Scale Submarine Model

The submarine model was mounted on a flat rectangular plate,
1/2" X 40" x 45", to distribute the load on the ground plane. The
roll positions of the planes and sail were adjustable to simulate sub-
marine roll angles of 0, + 22.5, and 45 degrees.

The DSRV model (1/30 scale), shown in Fig. 17, was con-
structed principally of aluminum. The diameter of the model was
3.3 inches and its overall length, 19.73 inches corresponding to a
full-scale length of 49.33 feet. The transfer bell extended 1.3
inches below the baseline of the DSRV hull.

Instrumentation

The overall steady-state forces and moments acting on the
DSRV model were measured by a strain-gauge balance mounted on
the end of the supporting sting within the model. A type T-0.75
(i,e., 3.4 inch diameter) six-component internal strain gauge was
used for the test program.

A seven-track FM tape recorder was used to record the

balance outputs and to provide a time code in order to locate specific
data for later analysis. Additionally, a switching network was pro-

1156

Vehicle Dynamies Assoctated wtth Submarine Rescue

DSRV MATING
BELL

ALL DIMENSIONS IN INCHES

Fig. 17. 1/30-Scale Model of the DSRV

vided for each data input to provide direct data entry to an oscillo-
graph as well as the conventional "Record onto tape/Play-back from
tape to graph" and for quick-look at model oscillation frequencies
where they occurred.

Test Conditions

The test program (Ref. 11) consisted of 258 runs, at Reynolds
numbers up to 6.0 X 10® per foot, and the DSRV positioned at 6 or
more locations (distances from the submarine model). The data
were recorded for about five minutes for each set of test conditions.

The vertical distance from the mating surface of the DSRV
rescue bell to the mating surface (hatch) of the submarine, Zg, was
varied from 0 to at least 12 inches, which corresponds to 30 feet, or
about one submarine diameter, in full scale.

As shown in Fig. 13, an angle adapter was used to obtain
different pitch angles, Orv, of the DSRV. For these test conditions,
the angle of attach, @ry, of the DSRV was the same as Ory. Angle
adapters of 0, 15, 30 and 45 degrees were used. When the sub-
marine model was removed from the tunnel to obtain DSRV free
stream conditions, the vehicle's angles of attack (pitch angles)
were obtained by a combination of adapter arrangements and pitch
of the strut mechanism.

1157

Scehretber, Bentkowsky and Kerr

REDUCTION AND PRESENTATION OF DATA

The proximity effects are described by a time independent
term and a time varying term in each of the six equations of motion.
These components are functionally dependent on the proximity to the
distressed submarine, Zq, the DSRV attitude angle, Orv; and the
orientation of the distressed submarine relative to the current. By
means of the tests conducted in the Ames facility, these effects
were determined primarily for two orientations of the submarine to
the current: head-on and athwartships. The tests were conducted
with the DSRV mating at both the forward and aft hatches with
various attitude angles of the DSRV and roll angles of the distressed
submarine. The DSRV yaw angle was zero for all test conditions.

Time Independent Interaction Forces

At the Ames facility the balance data were recorded by
printing devices, punched onto paper tape by a Beckman 210 com-
puter, and carried to the laboratory's computing center. The
resultant steady-state force and moment coefficients were computed
at the Ames center in the body-axis system. The force coefficients
were non-dimensionalized by the DSRV's maximum cross sectional
area; the moment reference arm for moment coefficients was the
maximum diameter of the DSRV.

In order to determine the DSRV characteristics in free
stream, the submarine model was removed from the tunnel, and
the forces on the DSRV were determined through an angle of attack
(pitch angles) range from - 12.5 to + 35.0 degrees. The resulting
normal force, pitching moment, and axial force coefficients versus
angle of attack are shown in Fig. 18. These results are shown to
correlate well with previous free stream tests conducted at Hydro-
nautic'’s ane o(Reft 5).

Interaction coefficients were determined by plotting the
measured data and extrapolating the curves to the free stream con-
ditions.

The interaction coefficients are:
Cy (-G)s Cy.» Cz (- Cy,) Force Coefficients

he Cy, A Cy Moment Coefficients
i i

where

wl Ory - Om
Cy; = CX Om TK {5 weLGe
Cz, = C26 + K Seve 2 Pp CLGs

Vehicle Dynamics Assoctated wtth Submarine Rescue

O DENOTES AMES DATA

© DENOTES DATA OF
REF, 2

AXIAL FORCE COEFFICIENT, Ca

Sat!
DSRV ANGLE-OF-ATTACK IN DEGS,

PITCHING MOMENT COEFFICIENT, C -f

NORMAL FORCE COEFFICIENT, Cy,

Fig. 18. DSRV Free Stream Characteristics

where C and K are functions of the vertical distance between
the bottom of the transfer bell and the hatch, Zg.

Ory is the pitch angle of the DSRV (in degrees). 6,, denotes
the mean DSRV attitude angle (the angle to which the data is refer-
enced) in degrees and can be converted to disturbing forces and
moments as follows:

wd max 1
4

2
X gist (lbs) = Cx; 2 pVc 57 CLC.

wd, 1

2
List (ft-lbs) = CL =e dinax > pV, JCC.

It was determined early in the experimental program that the flow
forces encountered during mating on the forward hatch with the
Skipjack sail configuration were more severe than those with the
Permit sail geometry, and flow forces encountered during mating
on the aft hatch are less severe than those encountered on the for-
ward hatch. Hence, almost the entire test program was conducted
with the Skipjack sail, and the data presented herein are repre-
sentative of the most severe flow forces that the DSRV will experi-
ence during mating with a downed submarine.

11F59

Sehretber, Bentkowsky and Kerr

When the DSRV is approaching the distressed submarine
along its centerline and headed into the current, a suction force is
applied to the DSRV when it is within a one submarine diameter of
the hatch, Fig. 19, because of the accelerated flow and the associ-
ated reduced pressures between the DSRV andthe hull. This suction
force increases as the displacement between the bodies, Zg, is
decreased. The maximum value of the suction force, Fyyction » for
ai knot current is 45.5 1bs. This result correlates well with the
theory presented in Ref. 12.

TUNNEL '€ = FWD. HATCH
LOCATION
B
GROUND PLANE gou8 ee
SUB
@ METHOD OF REF. 3 Gace
Zz,’ ni! vy,
-0.40
-0.30
= E
UO -0.20 -
-0.10

= 0
oo 5 10 15 20 25 30 0 5 10 15 20 25 30
Zp IN FEET Zp IN FEET 2 IN FEET

Fig. 19. Time Independent Flow Interaction Forces -- DSRV Mating
Parallel to Submarine Centerline

In contrast, when the DSRV is approaching the distressed
submarine athwartships and headed into the current, interaction
forces are applied to the DSRV when it is within 2-1/2 submarine
diameters (75 feet) of the hatch. Referring to the solid Cyn .5 curve
of Fig. 20, it is apparent that the maximum suction force is also
45.5 1bs, for this orientation in a 1 knot current.

Note: The subscript 6, denotes that the attitude angle of the DSRV
is zero.

1160

Vehicle Dynamics Assoctated with Submarine Rescue

40 a %o
=—
Kj

-—
Zp IN FEET

30 40
2p IN FEET

0)

Zp IN FEET

Zp IN FEET

Fig. 20. Time Independent Flow Interaction Forces -- DSRV Mating
Normal to Submarine Centerline

Although the DSRV was heading athwartships directly into the
current for this series, it is shown in Fig. 20 that lateral forces
were applied to the vehicle, i.e., Cys Cy,» and Cj; were not equal
to zero. This result is due to the fact that a cross flow results when
the current is deflected off the sail and the DSRV is therefore not
heading directly into the resultant flow. The magnitude of the side
force, Fejgg, in ai knot current is 300 lbs.

Time Dependent Forces

During the test program, the outputs of the six-component
balance (i.e., normal and lateral forward and aft gauges and the roll
and axial-force channels) were recorded over a five minute interval.
The long recording time was established to provide a high confidence
level during analysis of the unsteady effects (Ref. 13).

The information was digitized by data conversion and input to
an IBM 7094 force and moment conversion program. The output of
the 7094 program was then used as input for an existing LMSC Power-
Spectral Density Computer Program.

1161

Sehretber, Bentkowsky and Kerr

Plots of typical power-spectral energy versus frequency in
model scale are shown in Figs. 21 and 22 for the unsteady normal
force and pitching moments acting onthe DSRV. These conditions
were for the DSRV mated (Zy = 0) athwartships on the forward hatch
of a downed submarine with no roll.

The full-scale natural pitch period of the DSRV can range
from 41.5 to 72 seconds, corresponding to a BG value of 1 to 3
inches and a weight of 75,000 lbs. This information shown in terms
of model data for a 1 knot current in Fig. 22 indicates that there is
no significant concentration of energy near the DSRV natural fre-
quency; therefore, motion excitation at resonant conditions will not
be significant.

The standard deviation of forces and moments in model scale
were determined as the square root of the spectral energy, which

POWER SPECTRAL DENSITY LBS/CPS

FREQUENCY (CPS)

Fig. 21. Normal Force Power Spectral Density vs. Frequency
(Model Scale Values Shown)

£162

Vehtele Dynamtes Assoctated wtth Submarine Rescue

SUBMARINE

POWER SPECTRAL DENSITY LBS/CPS

FREQUENCY (CPS)

Fig. 22. Pitching Moment Power Spectral Density vs. Frequency
(Model Scale Values Shown)

was obtained from the integrated power density spectrum. Then
using Strouhal number and Reynolds number scaling, the full-scale
unsteady (a.c.) component standard deviation of forces and moments
were computed. For the corresponding test conditions, the inter-
action data were used to compute the full-scale magnitude of the
steady components. In the simulation the unsteady forces were
approximated by a white noise perturbation on the current used to
generate the steady forces. Based on the standard deviation of the
test data a ratio of unsteady (a.c.) to steady (d.c.) forces of 0.15
was selected in the frequency range from 0 to 0.082 V, Hertz.

The ducted thrusters of the DSRV 4re presently designed to
provide 830 lbs of normal and lateral force and are required to
maintain vehicle heading in a 1 knot current. The worst side force
condition, obtained during tests with the sail rolled 22.5 degrees

1163

Sehretber, Bentkowsky and Kerr

into a 1 knot current and mating on the forward hatch, was 690 lbs
well within the thruster forces available, Furthermore, during
mating, the operational procedure will be to head the DSRV, within
practical limits, into the actual flow (the resultant of the current and
the cross-flow due to deflected flow off the sail). This operation
will result in a reduction to the side force to a much lower force
level.

INCLUSION IN THE MATHEMATICAL MODEL

Attempts to mathematically simulate the interaction forces
by using the potential solution of flow around a cylinder on a plane to
generate the flow field were unsuccessful in the time available. In
addition, not enough submarine-current configurations were tested
to verify superposition techniques.

The parameters Cxg and K of the previously mentioned
interaction terms were apporixmated in the simulation by two-slope
nonlinearities. Figure 23 shows the two slope approximations of the

><

10

ee
15

20'~°25' "30° “35 ons TO" 1S “20 * 25, “Sema

<0.5 Spee ee) sy Sy d= 1B)
320 FORWARD HATCH 0.8

DISSUB -C
2.5 ROLL ANGLE: O 0.6 z8,

DISSUB

HEADING: 90°

Or 5 oO 15: 20 25° 40,035
ae 2 ee

Fig. 23. Interaction Force Parameters

1164

Vehicle Dynamites Associated wtth Submarine Rescue

experimental data equivalent to Fig. 20 with the initial conditions
starting with the DSRV 35 feet above the submarine 1.atch.

The method of simulating the steady and unsteady interaction
effects is shown in Fig. 24.

GENERATE C AND K PARAMETERS AS
XO

APPROXIMATED BY TWO-SEGMENT FUNCTIONS
OF DSRV ALTITUDE ABOVE HATCH Za

GENERATE UNSTEADY OCEAN CURRENT
(ALTITUDE INPUT

FROM VEHICLE)
VARIABLE
BAND PASS
FILTER
PERTUBATION INPUT
(UNSTEADY COMPONENT OF CURRENT)
COMPUTE Cy,'s FORCE AND MOMENT COEFFICIENTS: GENERATE DISTURBING FORCES AND MOMENTS:
; < F | Xeaist), ¥ (cist), a, ee
WHERE: : LeMans Now :
(PITCH ANGLE Gre C. ce (dist), © (dist) ©“ (dist)
FROM VEHICLE)

Ko ea 2
Qa 0, “AL (@ 8) X (dist) i Cxe ( rigs 5 Mery)

Ziaist) | OUTPUTS

fo}
* (aist) VEHICLE
M dist)
N(aist)

Fig. 24. Simulation of Interaction

IV. MANNED SIMULATION

Early in the DSRV program it was decided to initiate a
manned simulation study, whose primary objective would be the
investigation of the operation of DSRV under manual control con-
ditions, using minimum backup displays. The control system aboard
DSRV is relatively sophisticated, providing substantial pilot assist-
ance in the form of augmented stabilization, decoupling of degrees
of freedom, and automated control loops. Although the primary
operating modes of the DSRV were not to be manual, it was believed
that a manual control capability was essential for backup in the event
of failure or damage of the primary control system.

The simulation program was confined to the most severe

segment of the rescue mission, the mating of the DSRV to the hatch
of the distressed submarine (DISSUB). This segment starts when the

1465

Sehretber, Bentkowsky and Kerr

DSRV is approximately 20 feet above the deck of the DISSUB, and
ends when the DSRV is sitting on the deck and has positioned itself
to the accuracy required to assure a satisfactory seal. This simu-
lation would be used to determine the limits of current magnitude
and direction and distressed submarine attitude for which manual
control is feasible. Mating aids would be used as required.

The manned simulation program was undertaken sufficiently
early in the design program to permit some design investigation of
the parameters of various control elements. Several significant
design changes were made as a result of these investigations.

This computerized simulation program complemented the
simulations undertaken with the LASS (Lighter than Air Submarine
Simulator) vehicle. LASS operations (Ref. 14) had established the
feasibility of manual control. However, they did not permit con-
trolled variation of the environment.

The facility used for the manned simulation program was
located at the Sperry Marine Systems Division in Charlottesville,
Virginia. This facility had previously been used for simulation
studies of the NR-1 research submersible, and many of the programs
developed for the NR-1i were available for the DSRV studies.

Ron Rau, one of Lockheed's DSRV test pilots, served as test pilot
for the simulation study.

FACILITY DESCRIPTION

The computer facility utilized in the simulation combined
an Ambilog 200 hybrid computer with an EAI-231R analog computer.
The Ambilog 200 has a basic 4,096-30 bit word memory witha
memory cycle of 2ysec. The analog portion, used for multiplication
and division, has a 50 psec cycle. The Ambilog 200 was used to
simulate the vehicle, coordinate transformations, actuators and
effectors, current interaction effects and the mating aids. The
EAI-231R was used for display generation and for simulating the
ballast and trim systems.

In addition to the computer, the simulation facility included
a cab driven in two degrees of freedom (roll and pitch). The cab
contained a control station consisting of control sticks and other
system inputs, various meter type displays, and a TV display.
Figures 25 and 26 show the cab and its interior display arrangement,
respectively.

1166

Vehtele Dynamtes Assoctated wtth Submarine Rescue

Fig. 25. Simulation Cab

1167

Sehretber, Bentkowsky and Kerr

VE VELOCITY=
| RMR ATS PELL
(CURRENT MAGNITUDE =

TROLL RATE

_—

SWAY VELOCITY

Fig. 26. Instrument Panel

ELEMENTS OF SIMULATION

Vehicle

Because the mating operation is confined to low current
velocities, vehicle control is obtained by means of the main propul-
sion and thrusters; the shroud remaining locked amidships. The
relatively limited capability of the Ambilog computer necessitated
some approximations of the equations of motion to assure fitting
the entire problem to the computer. The major approximation used
ignored the stalling effect of the shroud.

The effect of this approximation is shown in the comparison
of the heave versus angle of attack curves of the analysis and simu-
lation models of Fig. 27. At high angles of attack the simulation
model provides higher heave forces than the analysis model, an
effect which tends to make the results of the simulation conservative.

The iteration interval used in the computation was 125 milli-
seconds. Since some of the degrees of freedom, particularly the
mating aids, require a higher frequency response, selected portions
of the simulation utilized a 62.5 millisecond interval. A relatively
simple integration routine was employed, as shown in Fig. 28.

1168

Vehtele Dynamtes Assoctated with Submarine Rescue

HEAVE VS ANGLE OF ATTACK

F/V2

+/ oe)

0 10 20 30 40 50 60 70 80 90
a, DEGREES -———>>—

Fig. 27. Comparison of Simulation and Analysis Models

TIME Noes | ical n

EQUATIONS OF MOTION

b= fF (u,v, W, Py r)

SOLUTION

Da

Un-y2! Vinee"? ETG)

AND vs Via + ut

ITERATION INTERVAL 62.5 OR 125 MILLISECONOS

Fig. 28. Computation of Vehicle Motions

1169

Sehretber, Bentkowsky and Kerr

Throughout the course of the program, vehicle responses
using the simulation model were compared to responses computed
from the more complete analysis model, which had been programmed
with a more sophisticated integration routine.

Vehicle Control System

The control effectors available are main propulsion for
surge, a pair of horizontal thrusters for yaw and sway, a pair of
vertical thrusters for pitch and heave, and mercury ballast control
for list and trim. Experience with DEEP QUEST and other submer-
sibles had shown that independent control of the thrusters was not
effective, since each horizontal thruster, for example, affected both
yaw and sway. To achieve effective control it is desirable to sepa-
rate the commands to each degree of freedom. Thus, a sway com-
mand would be applied equally to both fore and aft thrusters, while
a yaw command would be applied differentially to the two thrusters.

Vehicle control (except for trim and list) is obtained from
two hand controllers. The block diagram of the system, including
the actuators and effectors, is shown in Fig. 29. The output of

DSRV STICK SUMMATION AND THRUSTER SIMULATION

PROP RPM

EFFECTORS
| ACTUATORS
: STICK SUMMATION
Y x |
x|x|-—*

MAIN PROP

eo (nes) are
LEFT STICK y

|
T
iene Fe Bel fa

RIGHT STICK

T
AV | |
A THRUSTER F
x a l Ms l NIN REFINEMENT[ 7 OY
| | |

THRUSTER DESIGNATION

ML — MAGNITUDE LIMIT

v= SQUARE ROOT FH = FWD HORIZ THRUSTER
D.Z. — DEAD ZONE FY — FWD VERT THRUSTER
R.L. — RATE LIMIT AH — AFT HORIZ THRUSTER
PALG — MAJOR LOOP GAIN AV — AFT VERT THRUSTER

Fig. 29. DSRV Stick Summation and Thruster Simulation

1170

Vehtele Dynamtes Associated wtth Submarine Rescue

each command axis is fed into a "signed square" circuit, so that the
commands represent forces rather than propeller RPM. After
mixing the signals appropriately, they are fed into square root cir-
cuits so that the commands to the actuators represent RPM. Each
actuator is represented by a delay network which includes a maximum
motor acceleration limit. The effectors are represented by an
(RPM)? network (to convert RPM to thrust) and a "thrust refinement"
circuit. The latter accounts for the variations of thrust with the
forward speed of the vehicle. The approximations used in the thrust
refinement are shown in Fig. 30.

The actuator/effector simulation shown in Fig. 29 ignores
the lead-lag thrust effect of the thrusters which results from the
delay in accelerating the water in the thruster duct. Exploratory
experiments showed that, in manual control, the thrusters are used
in a bang-bang fashion, and the effects of ignoring the lead-lag
response of the thrusters are minimal,

In the more sophisticated control modes used aboard the
DSRV, the mercury list system control is integrated into the right-
hand stick, and the pitch control motion of the right-hand stick
affects both thrusters and mercury trim control.

NORMAL
FORCE

FORWARD

FORCE COEFFICIENT (LB/RPS“)

AXIAL
FORCE

MOMENT COEFFICIENT (LB FT/RPS*)

0 0.2 0.4 0.6 0.8 1.0 0! 0.2" OFA 0760058. 11.0
FORWARD VELOCITY FT/SEC FORWARD VELOCITY Epes
IMPELLER VELOCITY ’ IMPELLER VELOCITY ”

Fig. 30. Effect of Forward Speed on Thruster Forces and Moments

se BY a |

Schretber, Bentkowsky and Kerr

Displays

Two sensors are available aboard the DSRV for assistance
in performing the final mating maneuvers. These are a TV camera
which looks through a viewport in the midsphere lower hatch, anda
high resolution short range sonar (SRS) mounted to a retractable
boom in the transfer skirt. Only the TV was simulated. Details
of the performance of the SRS were not available at the time the
simulation study was performed. Also, with good visibility, the
TV is a much more informative sensor than the SRS.

The midsphere TV display was simulated by photographing
a model of a submarine. The photograph was then scanned by a
CRT, using a flying spot scanner. The size of the area scanned is a
function of the distance from the camera to the hatch. The dis-
placement of the center of the hatch from the center of the screen
is proportional to the distance between the hatch center and the inter-
section of the camera axis with the hatch plane. Because of the
relatively small angles between the DSRV and DISSUB planes, no
attempt was made to provide foreshortening effects. Reproductions
of the TV display are shown in Fig. 31. The four radial line seg-
ments at the extremity of the picture represent the staples on the
submarine deck surface to which a McCann Rescue Chamber can be
attached. These staples provide precise centering information for
the pilot.

Fig. 31. TV Display

1172

Vehtele Dynamtes Assoctated with Submarine Rescue

The most significant meter displays are those of doppler
velocity, attitude rates. The doppler sonar, located 8.9 feet aft of
the C.G. and 3.3 feet below the centerline, provides 3 axis ground
velocity data. Since the doppler sonar is offset from the center of
gravity, angular motions couple into the doppler signals, and in
some situations were interpreted by the pilot as translation veloc-
ities.

Displays were also provided for roll, pitch and heading angles,
and for sonar altitude above the DISSUB. Current magnitude and
direction indicators were available, but were not used in the simu-
lation, since the corresponding sensors were not installed in the
vehicle. The additional displays shown in Fig. 26 are associated
with the anchors and haul down winch control systems.

Shock Mitigat ion System

The shock mitigation system serves a dual purpose in the
mating operation. The primary one is that of dissipating the kinetic
energy of the DSRV when it lands on the DISSUB. The second function
was realized only after the simulation study was started. Prior to
dewatering, the DSRV is connected to the DISSUB primarily by verti-
cal thrust forces from the DSRV and coulomb friction. Because of
the existence of the shock mitigation system it is not necessary for
the DSRV to land precisely on target. As long as the shock mitigation
ring encloses. all the staples, the DSRV can slide on the DISSUB deck
until precise alignment is reached.

As described previously the shock mitigation system has been
simulated as four independent damped springs. The natural frequency
of the DSRV-shock mitigation system is approximately 11 radians per
second, which is too high to simulate with a 62.5 millisecond iteration
interval. Accordingly, the spring constant was reduced, with a re-
duction in the natural frequency to 1.7 radians per second. The
damping constant was also reduced to maintain essentially the same
percentage damping.

Anchor and Hauldown

Exploratory runs were made using both the anchors and haul-
down as mating aids. No help was obtained with the anchors, and
very limited assistance was obtained from the hauldown. Schedule
and budget limitations did not permit an intensive evaluation of this
problem at the time. Some digital simulation was performed at a
later date with the hauldown system, which indicated that it should
provide substantial assistance, particularly when the DSRV is re-
quired to mate bow up to the current. These results were confirmed
on simulated tests made with LASS.

1173

Sechretber, Bentkowsky and Kerr

CURRENT

Fig. 32. Mating Geometry

A TYPICAL SIMULATION RUN

The mating situation to be described is depicted schemati-
cally in Fig. 32. The DISSUB is rolled 225 degrees in an athwart-
ship current, so that the DSRV is required to mate bow up. With
the DSRV heading into the free stream, the interaction forces and
moments are as depicted in Fig. 33. Deflection of current off the
sail of the DISSUB causes a starboard sway force on the DSRV.

The corresponding yaw moment is counterclockwise at large separa-
tions, but becomes clockwise as the DSRV approaches the DISSUB.
The normal force provides a suction effect at relatively large dis-
placements, but becomes destabilizing as the DSRV approaches the
DISSUB. Thus the normal force, due to interaction, adds to the
force due to the free stream and tends to push the DSRV away from
the hatch. Pitch moments remain rather constant over the distance
Included in the run.

In performing the mating operation the pilot attempts to head
into the local stream rather than into the free stream. He sees and
"feels" the DSRV sway and adjusts his heading to minimize the sway
motions. Thus, the relative heading is not into the free stream but

14.74

Vehicle Dynamics Assoctated wtth Submarine Rescue

1 KNOT ATHWARTSHIP CURRENT, FORWARD HATCH, DISSUB ROLLED 22.5 DEG.
DSRV BOW UP (6 = DSRV PITCH ANGLE)

6 = 30° 6 = 22.5°

500
= Be
ui} 400 a
4 a
O 300 &
> uw
S 200 Z
wn 0 ~

OM 5 10-. 45920. 95° 30 re)

DISTANCE DSRV TO DISSUB (FT) 4

© 5 7410 18 -20°- 25- 30
DISTANCE (FT)
2 s
= os
a Z
Lau Li
= =
re re)
= =
uf
= ~4000 = 0
a 0 5 10 5 20 25 30 x 0 5 10 15 20 25 20
DISTANCE (FT) DISTANCE (FT)

Fig. 33. Interaction Forces and Moments

rather is in a relatively arbitrary direction. The effects of heading
changes on the interaction problem are illustrated in Fig. 34.

First of all, as the DSRV heading is changed, Euler angle
variations occur (Fig. 34a). Where initially the DSRV was not re-
quired to list at all, it must now both list and trim, and can no
longer make precise list and trim adjustments prior to landing.
Concurrent with these changes, the free stream current components
change with heading (Fig. 34b) which must be compensated appropri-
ately.

Assuming that the ideal heading corresponds to zero sway
force, the variation of optimum heading with distance to hatch is
shown in Fig. 34c. The optimum relative heading is approximately
30 degrees, with a significant heading change required as the separa-
tion is decreased. It will be noted from Fig. 34d that the zero head-
ing yaw angle does not correspond to the zero yaw moment angle,
This is due to the horizontal gradients in fluid velocity along the
length of the DSRV. Thus, there is no yaw plane equilibrium con-
dition, and yaw plane control becomes a more severe problem than

1175

Schretber, Bentkowsky and Kerr

DISSUB ROLLED 22.5 DEG

A PITCH & ROLL ANGLES REQUIRED
TO PARALLEL HATCH PLANE

8 8

C YAW ANGLE REQUIRED TO
MAINTAIN ZERO SWAY FORCE

ANGLE (DEG)
i)
°
YAW ANGLE (DEG)

0 5 10 15 20) 25) 30
DISTANCE TO HATCH (FT)

D
YAW MOMENT UNDER CONDITIONS
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Fig. 34. Counteracting Interaction Forces by Heading Changes

pitch plane control.

A six channel recording of the simulation test run under
these conditions is shown in Fig. 35. The variables plotted are the
displacements x, y and z of the camera of the DSRV from the
center of the hatch in the coordinate frame of the DSRV as shown
in Fig. 32. Also plotted are the roll, pitch and heading angles of
the DSRV, zero heading being into the free stream. For a one
minute interval during the run the roll, pitch and yaw angle traces
are replaced with RPM traces from the forward vertical, forward
horizontal and main propeller, respectively.

The run begins with the DSRV aligned with the free stream,
in a level attitude, and at a camera elevation of 20 feet (correspond-
ing to 15 foot distance between the DSRV seal and the DISSUB hatch).
This initial condition results in a more severe transient than would
occur on a mission, so that the advantages of the relative proximity
in distance is overcome by the necessity to restrore dynamic equili-
brium. At the start the pilot turned to port about 20 degrees to try
to maintain equilibrium. Simultaneously, the roll and pitch angles
are adjusted to try to maintain the DSRV sealing surface parallel to
the hatch plane. After about two minutes, the pilot descends about

1176

Vehicle Dynamies Associated with Submarine Rescue

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Big, 3

Schretber, Bentkowsky and Kerr

half the distance to the hatch, then attempts to maintain altitude
while adjusting his x and y coordinates. In the process the yaw
angle has been increased to about 35 degrees. The final descent is
made, the DSRV touching the deck somewhat in excess of 3.5 minutes
after the start of the problem. At touchdown the DSRV was almost
perfectly aligned inthe x direction but was nearly 2.5 feet off
center inthe y direction. Onthe deck, vertical thrust was applied
(trace not shown) in an attempt to keep from being lifted off by the
current. Roll and pitch angles were adjusted to try to position all
four legs in contact with the deck. Note that during this time the
DSRV continues to roll and pitch. The transients at impact caused
the DSRV to slip aft about 2.5 feet. This was corrected, as was

the misalignment in the y axis. After nearly 5 minutes the pilot had
the DSRV under control and could commence the final retraction of
the shock mitigation ring and start up the dewatering pump. At this
time the computer was placed into HOLD. The final misalignments
were 0,10 feet inthe x direction and 0.16 feet inthe y direction,
within the required tolerances. The final yaw pitch and roll angles
were 36, 6 and 16 degrees, respectively.

Unfortunately, this would not have been a completely success-
ful landing. At the time the pilot terminated the run, the DSRV seal
plane and the deck hatch plane were misaligned about 7 degrees in
roll and 2 degrees in pitch so that all four shock mitigation legs
were not in contact with the deck and a satisfactory seal could not be
made. No instrumentation exists on the DSRV to provide this relative
attitude data which could result in a serious operational limitation.
The problem can be alleviated in part by use of the hauldown system
which provides a larger stabilizing moment, reducing the offset
angles.

A major assistance was obtained by installing, in the simu-
lation, a set of 4 indicators which measured the stroke of the shock
mitigation hydraulic cylinders. Roll and pitch alignment could be
achieved with this system, while maintaining the horizontal plane
alignment.

A review of the thruster activity leads to two interesting
observations. First, the thrusters are operated in a bang-bang
fashion, that is, either maximum thrust or zero thurst is commanded.
This, despite the fact that an accurate proportional control system is
available. Second, the activity of the horizontal thrusters is much
greater than either the vertical thruster or the main propeller. This
was somewhat predictable from the environmental curves of Fig. 9.
The frequency of the horizontal thruster activity has implications
in the thermal design of the thruster motors.

1178

Vehtele Dynamics Assoctated wtth Submarine Rescue

DSRV

AFT_HATCH-FORE-AFT CURRENT FORWARD _HATCH-ATHWARTSHIP CURRENT
DISSUB ROLL ANGLE = 0 DISSUB PITCH ANGLE = 0

E
— — EXPLORATORY RUNS
FINAL RUN

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45 22.5 (2) 22.5 45
BOW BOW BOW BOW
DOWN UP DOWN UP
DISSUB PITCH ANGLE DISSUB ROLL ANGLE

Fig. 36. Experimental Results

RESULTS OF MANNED SIMULATION PROGRAM

The experimental results of the simulation study are sum-
marized in Fig. 36. Two sets of curves are shown, those for the
final runs, of which Fig. 35 was an example, and those of earlier,
exploratory runs which were made prior to the inclusion of the shock
mitigation system in the simulation. Performance of the final runs
did not meet those of the exploratory runs. Schedule constraints
did not permit much training with the use of the shock mitigation
system. The pilot believes that with such training the results of the
two sets of runs would match more closely.

On the basis of these results, we have tentatively arrived at
the following performance predictions.

a) Mating on the aft hatch is feasible for currents in excess
of one knot at all attitudes of the DISSUB (up to 45 degrees)
and headings of the DISSUB with respect to the local cur-
rent.

b) On the forward hatch, we have to distinguish between
longitudinal and athwartship currents. In athwartships
currents the goal of mating in a one knot current can be
achieved except possibly for very high DISSUB roll angles
which require the DSRV to mate bow up to the current.
With respect to longitudinal currents, on those submarine

£179

Schretber, Bentkowsky and Kerr

classes which have sufficient clearance between the sail
and the forward hatch to permit the alignment of the

axes of the two vehicles, mating is possible in currents
well in excess of one knot. For those submarine classes
which do not have sufficient clearance, mating is limited
to about 3/4 of a knot.

The manned simulation program yielded some important by-
product results which impacted on the vehicle design. The most
significant ones are as follows:

a) The splitter plate behind the transfer skirt was originally
incorporated to reduce flow separation behind the skirt
and minimize axial drag. Model testing unfortunately did
not verify this drag reduction. However, the splitter
plate was found in the simulation to provide sufficient
roll damping to permit mating without automatic roll
stabilization (automatic roll stabilization is, however,
provided even in the manual mode).

b) The shock mitigation system had originally been designed
to dissipate energy only for impact velocities in excess
of 0.25 feet per second. Below 0.25 ft/sec, the system
acted as a spring. However, because the DSRV is
neutrally buoyant, it will bounce off any spring unless the
impact energy is absorbed. As a result of observing this
phenomenon in the manned simulation study, the shock
mitigation system was redesigned to provide damping for
all impact velocities.

c) The relative attitude indicators required to assure angular
alignment have not yet been incorporated in the design.

The results of the Ames tunnel tests have been invaluable in
gaining an understanding of the problems involved in submarine
mating. In the early phases of the simulation program, before the
Ames results were available, mating runs were made under free
stream conditions. Although there had been apprehension about the
ability of the pilot to perform the 6 degree of freedom control function
manually, we found that experienced aircraft pilots, with nominal
DSRV simulator training, could control the DSRV with ease. Success-
ful mating to currents up to two knots were anticipated for virtually
all orientations and in excess of two knots for the most favorable
conditions.

The inclusion of the interaction effects, particularly on the
forward hatch, dampened our optimism. The performance goals
could be met, but at considerably reduced current magnitudes, and
requiring considerably more pilot training.

1180

Vehitele Dynamics Assoctated wtth Submarine Rescue

Prior to the start of the Ames test program, it was believed
that the most serious interaction effects would be in the pitch plane,
due to Bernoulli or "suction" effects. The experimental program
was organized primarily to determine those forces. As we have
seen, yaw plane interactions are more critical than those in the
pitch plane. It would be desirable to have additional data, particu-
larly with respect to interactions as a function of yaw angle and (for
the longitudinal current) as a function of lateral separation of the
longitudinal axes of the two vehicles.

Recognizing the above limitations in the test conditions, no
attempt has been made to use the Ames data quantitatively in the
control system design. The data has been useful in the following
areas:

a) It has provided an appreciation of the yaw plane problems
associated with mating.

b) The force and moment gradients observed have been used
to select and verify the static gain requirements of the
automatic control system.

c) The non-steady interactions have provided an input which
could be used to establish the dynamic requirement of the
actuators and effectors, in particular the pumping rate
requirements of the list system.

The DSRV is completing preliminary sea trials and will soon
be conducting mating trials. Before too long we will have some full
scale verification of the usefulness of the Ames test results.

IV. CONCLUDING REMARKS

This paper has presented an approach to the problems of
simulation of the dynamics of highly maneuverable submersibles.
All elements of the simulation are covered in considerable detail
to provide an adequate base to build on for others with similar
problems. No such comprehensive reference was available for our
use.

Although the model test data are not presented in their
entirety, a reasonably complete description of the test procedure
and results should allow determination of the usefulness of the data.
Several references are given for more complete test results.

It is hoped that this paper illustrates where Naval Hydro-
dynamics is a continually expanding field and must take into con-
sideration aspects of control system design, man-in-the-loop
analysis, and numerous other fields not normally considered as
relevant to the theoretician.

1181

Sehretber, Bentkowsky and Kerr

Finally a note about the methods used during the development
of the simulation. The work was not done in a theoretically rigorous
manner but rather the simulation was built upon the background of
the personnel involved in its development and their ability to apply
existing theories to the problem. A number of the elements in the
complete simulation of the vehicle dynamics reflect engineering
judgment and experience. The major output of the study was re-
quired on a tight schedule and relatively little new theoretical
analysis were initiated. Since the DSRV is presently preparing
for sea trials the validity of the simulation should soon be checked
and correlation between the sea trials data and the results of the
simulation should provide valuable insight for futher simulations.

REFERENCES

1. Johnstone, R. S., "A Mathematical Model for a Three Degree-
of-Freedom Simulation of the Underwater Launch of a Rigid
Missile," Lockheed Missiles and Space Company, TM5774-
69-21, May 1969.

2. Kerr, K. P., "Determining Hydrodynamic Coefficients by Means
of Slender Body Theory," Lockheed Missiles and Space
Company, IAD/790, 21 July 1959.

3. Hess, John L., "Calculation of Potential Flow about Bodies of
Revolution Having Axes Perpendicular to the Free Stream
Direction," Douglas Aircraft Company, Report No. ES29812,
October i, 1960.

4, Feldman, Jerome P., "Model Investigation of Stability and
Control Characteristics of a Preliminary Design for the
DSRV," David Taylor Model Basin Report 2249, June 1966.

5. Goodman, H. and Ettis, P., "Experimental Determination of the
Stability and Control Characteristics of a Proposed Rescue
Submarine (DSRV) Using the Hydronautics High Speed
Channel," Hydronautics Inc. Technical Report 511-4,
November 1966.

6. Bentkowsky, J., et ale, "DSRV Model for Analysis ML 493-03
Vehicle," Lockheed Missiles and Space Company Report
No. RV-R-0037A, May 1968.

7. Reichart, G., "A Propulsion and Maneuvering System for Deep

Submergence Vehicles," presentation at AIAA meeting in
Seattle, Washington, June 1969.

1182

Vehicle Dynamics Associated with Submarine Rescue

8. Beveridge, J. L. and Paryear, F. W., "Performance of a
DSRV Propeller on Four Modes of Vehicle Operation
(NSRDC Model 5128)," Naval Ship Research and Development
Center T & E Report 099-H-05, August 1967.

9. Beveridge, J. L., "Static Performance of a DSRV Ducted Pro-
peller Thruster at Discrete Pitch Ratios," David Taylor
Model Basin Hydromechanics Test Lab Report 099-H-03,
July 1966.

140. Chislett, M. S. and Bjorheden, O., "Influence of Ship Speed
on the Effectiveness of a Lateral Thrust Unit," April 1966.

11. Kerr, K. P., "Experimental Determination of DSRV/Submarine
Mating Forces Using the Ames Variable Pressure Wind
Tunnel," Lockheed Missiles and Space Company, TMOS-H-
67-62, June 1967.

12. Ogilvie, A., "Force on an Ellipsoid Moving Near a Wall,"
David Taylor Model Basin Report, 1967.

13, Moody, R. C., "Statistical Considerations in Power Spectral
Density Analysis," Technical Products Company, 1966.

14, Turpen, F. J. and Goodman, A., "Experimental Determination
of the Performance Characteristics of the DSRV Based ona
1,25-Scale Lighter-Than-Air Submarine Simulator (LASSI),"
Hydronautics, Inc., Technical Report No. 705-3, August
1968,

1183

es

AUTHORS INDEX

Bentkowsky, J., 1111 Newman, J. N., 519
Bessho, M., 547 Norrbin, N. H., 807
Brennen, C., 117 Ogilvie, T. F.5 663
Carrier, G. F., 3 Paidoussis, M. P., 981
Chan, R. K. C., 149 Paulling; J. RR. 1083
Coantic, M., 37 Savitsky, D., 389, 447
Dagan, G., 607, 625 Sawatzki, O., 275
actors, L.J:, 601 Schooley, A. H., 311
Havre, A., 37¢ Schieler, M., 3614

Fromm, J. E., 149 Schreiber, H. G. Jr., 1111
Hasselman, K., 361 Sharma, S. D., 601
Hogben, N., 446, 473, 540 Shwartz, J., 321
Holmquist, C. O., xi Street, R. L., 149

Hsu, B. Y., 11 Baylor, bs Wey OL 1g O59
James, E. C., 951 Tuck, E. O., 627, 659
Kaplan, P., 1017 Tulin, M. P., 321, 607, 626
Kerr, K. P., 1111 Van Mater, P. R. Jr., 239
Krishnamurti, R., 289 van Wijngaarden, L. ,235,287,622
Lackenby, H., 474 Verhagen, J. H. G., 955
Landweber, L., 449, 475 Wang, D. P., 189

Lee, C. M.,; 905, 951 Waters, O. D. Jr., xiv

Le Méhauté, B., 71 Weinblum, G. B., 599
Linden, T. L. J., 477 Whitney, A. K., 117
Maestrello, L., 477 Yih; €.7S.,° 219, 236
Maruo, H., 624, 658 Yim, B., 573, 604

Miles, J. W., 95 Yu> Hs Y., 22

Munk, W. H.; 217 Zierep, J., 275, 288

Neal, B.4 259

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