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STANLEY W. DOROFF
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Office of Naval Research
Department of the Navy
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FOREWORD

In March 1960, Captain B. F. Bennett, USN of the Office of Naval Research
London Branch Office contacted through the American Embassy in The Hague the
Central Organization for Applied Scientific Research in The Netherlands (T.N.O.).
As a result of this contact the Netherlands Ship Model Basin ( NSMB), one of the
institutes administrated by this organization, was invited by Mr. Ralph D. Cooper,
Head, Fluid Dynamics Branch, Office of Naval Research, to co-sponsor the third
Symposium.

Although time for preparation was rather short the NSMB enthusiastically
accepted this invitation.

The Symposium, dedicated by Dr. Th. Von Karman to Sir Thomas Havelock in
recognition of his valuable contributions to naval hydrodynamics, was held at The
Hague in September 1960. It was devoted to high-performance ships, such as
hydrofoil boats, ground effect machines, deep diving submarines, semisubmerged
ships and other special or unconventional configurations. This timely subject was
of great interest to both hydrodynamicists and shipbuilders. Though the opera-
tional value of these high-performance ships appeals primarily to naval interests,
it was in addition very refreshing for practical shipbuilders who deal with prob-
lems concerning commercial ships.

The size, speed and type of ships that will be used within the near future is
difficult to predict. To face the possibilities within our present day technological
abilities was the aim of this Symposium. In this respect the Symposium can be
qualified as very successful.

It is my privilege to express the sponsors’ thanks to those who have con-
tributed to the success of the Symposium:. to Professor L. Troost and Dr. Th. J.
Killian who presented the addresses of welcome to the meeting, the first on behalf
of the Central Organization for Applied Scientific Research in the Netherlands, the
second on behalf of the USN Office of Naval Research; to the Chairmen of the
Sections; to the authors and those who took part in the discussions; and to those
organizations and companies who participated in demonstrations with various types
of hydrofoilboats on the river ‘‘New Waterway’’ at Rotterdam and in front of
Scheveningen Harbor, namely:

Hydrofoilboat ‘‘Sea Wings,’’ designed and demonstrated by Dynamic Devel-
opments Inc., for the U.S. Navy;

Hydrofoilboat ‘‘Waterman” demonstrated by Aquavion Holland N. V.;

U.S. Navy Hydrofoilboat ‘‘High Pockets’? developed by Baker Manufacturing
Company for the U.S. Navy;

Hydrofoilboat ‘‘Shellfoil’’ designed by Supramar A. G.

Thanks are also due to Lips Ltd. at Drunen for the kind reception during the
excursion to their Propeller Works.

ill

The tattoo of the Marine Band of the Royal Netherlands Navy, the cocktail
party given by VADM. A. H. J. van der Schatte Olivier, Flag Officer Material,
Royal Netherlands Navy and the dinner speech by VADM. L. Brouwer, Chief of
Naval Staff, Royal Netherlands Navy were Dutch contributions which added to the
success of the social part of the meeting.

I should also like to express our gratitude towards the Board of Directors and
the Staff of the Netherlands Ship Model Basin for their active help in organizing
this Symposium in such a short time.

Finally I would like to add my personal thanks to those of the Board of
Directors of the NSMB to the Office of Naval Research for the confidence put
in our organization and the help and close cooperation we enjoyed during the prep-
aration of the Symposium.

Wageningen, May 1962
Prof. Dr. Ir. W.P.A. van Lammeren
Director NSMB

PREFACE

This Symposium is the third in a series, each of which has
been concerned with various aspects of Naval Hydrodynamics.
The first (held in September 1956) presented critical surveys of
those areas of Hydrodynamics that are of significance in naval
science. Subsequent meetings were to be devoted to one or more
topics selected on the basis of importance and need for research
stimulation, or of particular current interest. In keeping with this
objective, the second symposium (August 1958) had for its dual
theme the areas of hydrodynamic noise and cavity flow.

Still continuing with the original plan, the present symposium
selected for its theme the area of high-performance ships, thus
emphasizing among other things the interest in the current prob-
lems and latest accomplishments associated with hydrofoil craft
and ground effect machines.

The international flavor of these meetings has been preserved
and, in fact, enhanced in the present case by virtue of the setting,
the participation, and most particularly by the joint sponsorship
by the Netherlands Ship Model Basin and the U.S. Office of Naval

Research.

The background, objectives, and hopes for this meeting are
more than adequately described in the speeches of welcome by
Professor Troost and Dr. Killian, the dedication address by Dr.
von Karman, and the foreword to this volume by Professor van
Lammeren. Little remains to be said other than to echo Profes-
sor van Lammeren’s expressions of gratitude to all those who
contributed so much to the success of this symposium. However,
taking the liberty of speaking for both the Office of Naval Re-
search as well as the international scientific community of hydro-
dynamicists, I should again like to express our deepest apprecia-
tion to Professor van Lammeren and the Netherlands Ship Model
Basin for their efficiency and care in managing the many varied
aspects of this symposium.

RALPH D. COOPER, Head
Fluid Dynamics Branch
Office of Naval Research

Thomas Havelock

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DEDICATION TO SIR THOMAS HAVELOCK

Theodore von Karman
Advisory Group for Aeronautical Research and Development
Paris

_It is a great honor and a pleasure for me to give this address which dedicates this
Third Symposium on Naval Hydrodynamics to Sir Thomas H. Havelock. Unfortunately my
knowledge on wave mechanics, which was the main domain to which Sir Thomas made clas-
sical contributions, is restricted to the fundamentals. However, it is a remarkable coinci-
dence that in the first half of our century aerodynamics and naval hydrodynamics made
analogous progress, and this encourages me to talk to you on the contributions of T. H.
Havelock at least from the viewpoint of an interested amateur. Also I had the good luck to
obtain the support and assistance of excellent specialists through the good services of my
friend J. G. Wenzel, Vice-President, General Dynamics Corporation. I would like to express
my thanks to him, to Mr. Earl Uram of the General Dynamics Electric Boat Division, to Dr.
Paul Wyers of Convair, and to several members of the U.S. Navy Bureau of Weapons and of
the David Taylor Model Basin for helping me to obtain the necessary information. Dr. John
Vannucci, Technical Information Officer of AGARD with the assistance of Mr. Francis H.
Smith, Librarian, Royal Aeronautical Society, furnished me a complete list of Sir Thomas’
publications.

Sir Thomas Henry Havelock was born in 1877; he is four years my senior. Incidentally
we are both unmarried; according to Francis Bacon, to no wife the next best is a good wife.
Sir Thomas published his first paper in 1903; the first paper published in my Collected
Works is from the year 1902.

The list of the scientific papers published by Sir Thomas Havelock contains 83 items;
the last item appeared in 1958. The first eight publications refer mostly to general prob-
lems of wave propagation, as for example the “Mathematical analysis of wave propagation
in isotropic space at p dimensions” (Proc. London Math. Soc., 1904) or “Wave fronts con-
sidered as the characteristics of partial differential equations” (ibidem, 1904), but also to
investigations in quite different fields of application, like “The dispersion of double refrac-
tion in relation to crystal structure” (Proc. Roy. Soc., 1907).

The first investigation which has relation to the problem of ship waves is his paper
published in 1908: “The propagation of groups of waves in dispensive media, with applica-
tion to waves on water produced by a traveling disturbance” (Proc. Roy. Soc., 1908). Then
_ the next year Sir Thomas first called the child by its proper name: “The wave-making resis-
tance of ships: a theoretical and practical analysis” (Proc. Roy. Soc., 1909). However his
broad interest in applied mathematics and theoretical physics remained to occupy his atten-
tion at least until about 1930. We find in the list—in between papers dealing with hydro-
dynamics and wave resistance—dissertations like “The dispersion of electric double
refraction” (Phys. Rev., 1909) and “Optical dispersion and selective reflection with applica-
tion to infrared natural frequencies” (Proc. Roy. Soc., 1929).

Let us briefly review the fundamental principles of ship resistance as they were

established by Froude.

Vii

T. von Karman

Froude divided the total resistance into “frictional” and “residual” resistance. The
computation of frictional resistance was based on experiments with towed flat planks and
was assumed clearly dependent on viscosity and the roughness of the surface. Thus the
frictional coefficient for a surface of given roughness was assumed to be a given function
of the Reynolds number.

On the other hand the residual-resistance coefficient was assumed to be unaffected by
viscosity and to satisfy the similarity law for an incompressible, nonviscous fluid with a
free surface. From this similarity law it follows that the coefficient of the residual resist-
ance is a function of the Froude number F only.

We have to note that in fact the residual resistance includes not only the wave-making
and eddy-making resistances, but all interaction effects and especially the difference
between the true frictional resistance of the ship and the frictional resistance of a flat
plank of the same surface. Thus the residual resistance cannot be expected to be actually
free of viscosity effects as was assumed by Froude and likewise in the conventional
practice in naval architecture.

Sir Thomas Havelock is one of those rare combinations of an extremely astute math-
ematician with a feeling and understanding for the application of mathematics to practical
ends. He is the major contributor to the theoretical hydrodynamics involved in the calcula-
tion of the resistance due to the generation of waves by moving bodies, either on the
surface or submerged. Over the years, since 1908, he has published in excess of 40
papers in the field, and it is only conjecture how many publications in the field were insti-
gated by his findings and contributions.

One of his major contributions, and, interestingly, the first one that he made to the
field, was in 1908 in which he extended the work of Lord Kelvin concerning the waves pro-
duced by traveling point disturbances. Essentially, he considered the waves generated by
a disturbance as a simple group or aggregate of wave trains expressed by a Fourier integral.
He was able to obtain the solution for the wave group for any depth of the fluid rather than
the infinite depth of Kelvin’s theory. He also found that the significance of finite depth
was to introduce a critical velocity above which the wave crests emanating from the source
change their character from convex to concave. At the critical velocity, the transverse and
divergent waves coincide and the resultant wave is normal to the path of the disturbance.
This finding is completely analogous to a situation which exists in high-speed aerodynam-
ics. As early as then, although he didn’t realize it, he had developed an insight into the
analogy between the shallow depth water waves generated by bodies and their shock wave
patterns at supersonic speeds, which underlies modern day water table experiments.

Evidently motivated by his solution and its application to ship resistance calculations,
Sir Thomas Havelock embarked on concentrated efforts in this field. Many of his contribu-
tions over the next 10 years (1908 to 1918), oddly enough, were somewhat semiempirical in
nature. This is unusual for a theoretical mathematician and points up his interest in the
practical application of mathematics. In the series of papers during that period, published
in the Proceedings of the Royal Society, he attempted to obtain an analytical formulation
for calculating the family of curves which are indicative of the “residual resistance” of
ships as postulated by Froude. Initially, he considered a transverse linear pressure dis-
turbance traveling uniformly over a water surface of infinite depth and arrived at a relation
involving three universal constants (determined from experiment) and three parameters
which depend upon the ship form. A major problem involved with the semiempirical

vill

Dedication to Sir Thomas Havelock

relationship was the determination or prediction of the three ship-form parameters for some
arbitrary ship. Actually, the resistance relationship gives quite good agreement with
experiments when experimentally determined ship-form parameters are used.

Apparently, in view of the shortcomings of the semiempirical approach and Sir Thomas’
rediscovery of Michell’s work, he abandoned this track and resorted to obtaining a solution
to the basic formulation of wave resistance that was put forth by Michell in 1898 and which
still forms the basis of all modern theoretical analyses of ship wave resistance. Michell’s
formulation is based on the representation of the fluid velocity in terms of a potential func-
tion which is built up of the sum of simple harmonic functions in the coordinates of the
system and is coupled with several rather idealistic boundary conditions. These boundary
conditions specify a slender-body ship characterized by small slopes in both the water-line
and draft planes. The theory also requires that the wave slopes be small and does not
allow changes of ship attitude.

Sir Thomas, over the period from 1920 to 1930, investigated the representation of ship
bodies in terms of discrete and continuous source-sink or doublet distributions along the
centerline plane of the ship and the ramifications and results of such an approach to the
evaluation of Michell’s integral. He was able to investigate the effects of straight or hollow
bow lines, variations of entrance and beam for constant displacement, effect of parallel
middle bodies, effect of finite draft, effects of relatively blunt and fine sterns on wave
interference, and the variation of wave profile properties with systematic changes in ship
form among other items important to practical ship design. In this work, he was the first to
analytically determine and describe the influence of changes in water line shape on the
wave resistance.

The wave resistance curves Havelock obtained as a function of Froude number have all
of the characteristics of those obtained from ship model experiments. The location of the
characteristic humps and hollows are depicted extremely well but their amplitudes are exag-
gerated in the low Froude number range (below 0.3). The agreement of the calculated
resistance is excellent in cases of ship forms (especially “Michell ships”) which are
described by simple functions but relatively poor for actual ship forms.

In the 1930’s and 1940’s, Professor Havelock directed his efforts mainly toward the
calculation of the wave profiles generated by two-dimensional and three-dimensional bodies
as represented by source-sink or normal doublet distributions. He also derived relation-
ships for and computed the wave drag of such bodies in terms of the energy and work in
the waves.

Included in his work during this period, Havelock initiated the idea of accounting for
the boundary layer effects in a real fluid by modifying the source strength function by a
reduction factor which would vary with the form of the ship and with the Reynolds number.
He made calculations for both two-dimensional and three-dimensional bodies and found the
reduction factor had no significant effect on the bow wave but it did reduce the wave height
along the side of the ship, particularly near the stern. He also investigated the effect of a
modification of the lines of the ship in the stern region and found this device has the
greatest influence on the wave resistance at low Froude numbers where there was greatest
disagreement between theoretical and measured results. However, there still existed appre-
ciable discrepancies between measured and theoretical curves in the low Froude number
range. The effect of the modification was found to be insignificant at high Froude numbers.

ix

T. von Karman

Also, during the 1940’s and continuing until as recently as 1956, he worked on wave
drag relationships for submerged bodies of revolution (spheroids), as well as for a body
traveling in the wake of another body. He was also able to account for the effects of and
compute the wave motion and drag due to body acceleration or oscillations (rolling, pitching,
and heaving). In addition, he investigated the characteristics of a submerged body (sphe-
roid motion in normal and oblique wave systems) and demonstrated that pressure of the
waves does not necessitate a modification of the source-sink distribution representing the
body. This phase of his work gives valuable insight to problems associated with torpedoes
and submarines.

The work of Havelock has been recently extended by Inui in Japan to a point where the
accuracy of calculations relating to hulls of conventional width has been substantially
improved and Hershey of the U.S. Naval Weapons Laboratory, Dahlgren, is using a variation
of the theory to determine the wave pattern generated by a submerged body. Also, at the
present time, extensive use of the generalized theory is used by the U.S. Navy’s David
Taylor Model Basin in the performance of numerous classified projects associated with
modern submarine development.

This paper has not, and could not, do complete justice to all of Havelock’s works in
the limited amount of time available today. Nevertheless it is indeed an apparent and
accepted fact that Havelock is a giant in his field and one feels sure that his classical
contributions will be appreciated and utilized for a long time to come.

DISCUSSION

W.P.A. van Lammeren (Netherlands Ship Model Basin), after saying how much it was
regretted that Sir Thomas Havelock could not be present because of illness, read the follow-
ing letter, sent to the Office of Naval Research, Washington:

“I send greetings to the members of the Symposium.

I am extremely gratified by the proposal to dedicate the Symposium to myself;
it is indeed a high honour, and I do not know any more pleasing compliment than
to be remembered in this way by one’s friends and fellow workers in the same field.

I also wish especially to thank Dr. von Karman; I am very appreciative of the
great honour he has done me by giving the opening address.

I regret very much that it was not possible for me to attend the Conference
myself and express my appreciation more adequately in person; but I send warmest
thanks to all, and best wishes for a successful and pleasant meeting.

T. H. Havelock”

CONTENTS

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Theodore von Karman

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L. Troost
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T. J. Killian

HIGH PERFORMANCE SHIPS — PROMISES AND PROBLEMS .......+-+2eeeeees 1
Owen H. Oakley

SUA. YAP ANNI ASIST ID) OlD SISMUSTSy IN) AMEN OP MORU IS Soa Goo ono ooo Oe ae Zo
J. D. van Manen

ON THE EFFICIENCY OF A VERTICAL-AXIS PROPELLER...... sipsige) elle Wokese) loike fiayse 45
J. A. Sparenberg

A SOLUTION OF THE MINIMUM WAVE RESISTANCE PROBLEM ...........26..-. 67
R. Timman and G. Vossers

ON THE PROBLEM OF MINIMUM WAVE RESISTANCE FOR
STRUTS AND STRUT-LIKE DIPOLE DISTRIBUTIONS .........2-..-22+eeeees 75
Samuel Karp, Jack Kotik, and Jerome Lurye

THE HYDRODYNAMICS OF HIGH-SPEED HYDROFOIL CRAFT........... Nweliette 121
Marshall P. Tulin

ON HYDROFOILS RUNNING NEAR A FREE SURFACE ......-+-ceeeeeeseevsees 147
S. Schuster and H. Schwanecke

THE EFFECT OF SIZE ON THE SEAWORTHINESS OF
IBIVAD ISOM OME (ORY NDALES 15 ol d Goo ooo 3 Oo Goo. coool a 6 Oo So SHabemerteilcl aio teMet at oie 191
A. Hadjidakis

DESIGN AND INITIAL TEST OF ONR SUPERCAVITATING
HYDROFOIL BOAT XCH-6...... SO 5553 DOO UO OO Go oS exis feremalfonsnetiste = ord ofc 205
Glen J. Wennagel

DESIGN AND OPERATING PROBLEMS OF
COMMERCIAL HYDROFOILS .......... Hee aewehallseictcu nts catesl siistensMoniscisaene ueixkeas 233
H. Von Schertel

GROUND EFFECT MACHINE RESEARCH AND DEVELOPMENT
LIN ANSS; WINIMhIDD) SWWNMISS Sig og aeadoo0o uo od ano O 6 3 Suu Ghobes 3 Ouse Cro Casicen Ba
Harvey R. Chaplin

HYDRODYNAMIC ASPECTS OF A DEEP-DIVING

OCEANOGRAPHIC SUBMARINE ........ Ruleriaieioteh slic as iaifezia leietienis wioteleateLetcipeceite 307
P. Mandel

SUBMARINE CARGO SHIPS AND TANKERS ........2.2.2.2-. aboneMUiacticlicikerelteite Loumisear eis 341
F.H. Todd

xi

EXPERIMENTAL TECHNIQUES AND METHODS OF ANALYSIS
USED IN SUBMERGED ODORS HARIS Elim cic teilia) eiteioi =icell eis oreo) ia(itetcon cite)itelncMla nt incnisilteitenta 379
Alex Goodman _

A THEORY OF THE STABILITY OF LAMINAR FLOW
ALONG COMPLIANT PLATES ................ Goo GoM OUgOu Ga oG0000 451
F. W. Boggs and N. Tokita

RE RE NGH BA DES GAPE HPR'OG RAMs isc o7 i se) <itojiotuebiciieuleielsereciiet Glue uicicch couse) <iiet cue eie nae 475
Pierre H. Willm

A METHOD FOR A MORE PRECISE COMPUTATION
OF HEAVING AND PITCHING MOTIONS BOTH IN
SMOOTH WATER AND IN WAVES ‘a Sons ee a ss 8 a we ee ee el le ee te oe 483
O. Grim

SEMISUBMERGED SHIPS FOR HIGH-SPEED
OPERATION IN ROUGH SBAG YS ss Se we ates Se eee Sees eee 525
Edward V. Lewis and John P. Breslin
DESIGN DATA FOR HIGH SPEED DISPLACEMENT-TYPE
HULLS AND A COMPARISON WITH HYDROFOIL CRAFT ..............c.e--% 561
W. J. Marwood and A. Silverleaf

ANOIBSIOIR IUNIDITRS 505000005000 HOOP eOSooOOD OOO Ob AD OO OO DO OG Stols Glad o 6 6 . 613

xii

ADDRESS OF WELCOME

L. Troost
Central Organization for Applied Scientific Research in the Netherlands

The joint sponsors of the Third Symposium on Naval Hydrodynamics, the U.S. Office of
Naval Research and the Netherlands Ship Model Basin, have honored me by asking my
assistance in the opening of this four-day Symposium in the Netherlands by delivering a
brief welcome address on behalf of the host country. I am delighted to accept this invita-
tion, in the first place because I am going to meet so many old and personal friends from
the international scientific community concerned with the problems of naval hydrodynamics,
in the second place because the Central Organization for Applied Scientific Research
T.N.O. in the Netherlands which I represent, established by Law about 30 years ago,
through its Special Organization for Industrial Research, has a very warm and active inter-
est in applied and also more fundamental research in the field of naval architecture and
marine engineering. This T.N.O. Organization brings together the interests of science,
trade and industry with those of government under expert supervision, and channels research
contributions of private industry and government appropriately to the areas where they are
needed, encouraging, but not always attaining, a 50/50 ratio. We have the Netherlands
Study Center for Naval Architecture and Navigation which is administrated by the T.N.O.
Organization. This Symposium’s cosponsor, however, the Netherlands Ship Model Basin,
being somewhat older than the Organization itself, is an independent Foundation, self-
supporting for all practical purposes, but maintaining old and friendly relations with T.N.O.
in a joint and successful effort to produce more fundamental research in greater quantity
and of higher quality than would otherwise be possible. It should be stated here that the
research-mindedness and support of the progressive shipping and shipbuilding industries,
in a centuries-old tradition in this small Low Country-by-the-Sea, are as exemplary for all
T.N.O.’s industrial relations, as is the efficiency in the use of industrial and governmental
research funds by the Wageningen Institute. It is believed that the Dutch build more ship
tonnage per head of the population than anyone else and they are rightly proud of their mari-
time research facilities, including the new ones incorporated in the division of Naval
Architecture of the Delft Institute of Technology.

It may be about a year and a half ago that the group of naval and marine scientists of
the Massachusetts Institute of Technology, to which I then belonged, on the initiative of
Professor Harvey Evans discussed the possibility of a seminar or symposium on recent
developments in unusual ships like submersible and semisubmerged commercial vessels,
hydrofoil, and hovering craft, under joint sponsorship of the Society of Naval Architects
and Marine Engineers and M.I.T.’s corresponding Department. Through various circum-
stances the Society was unable to support the proposal in its then form. You may imagine
my personal satisfaction in seeing it realized after my departure from the U.S., here and
now, although under different sponsorship, because I consider the subjects under discussion
as entirely timely in a period characteristic of almost unbelievably rapid changes in tech-
nology, in which the wild-looking idea of today may be the usual thing within 10 or 15

xiii

L. Troost

years. If in a Western World of undersupply of young scientists and engineers this rapid
progress and the terrific amount of original thinking and imagination associated with it
would stimulate a larger number of undecided students to enter the maritime sciences, and
their teachers to a more and more imaginative approach of their subjects, an event as the
one that is being launched today would be more than justified, quite apart from the value of
its transactions of papers and discussions for the profession.

That our friends of the American Navy have considered the Netherlands as a place fit
for this grand-style demonstration of progressiveness in ocean transport research, and the
Wageningen Modelbasin as a worthy partner in this enterprise, is a source of deep satisfac-
tion to the Organization I have the honor to represent here, and to the Dutch people as a
whole. Traditionally devoted to welcoming foreign scholars and creative thinkers in a long
history on these crossroads of European trade and traffic, science and culture, there may
be few places where gratitude for assistance after World War II and for acceptance of
Western World leadership have matured to such an extent of warm understanding of the ways
of the American people and in general to such a degree of international orientation. These
feelings of genuine friendship and hospitality prevailing here have given rise to a remark-
able amount of international exchange of scientists, engineers, and students in our and
other professions, not only with regard to the United States but also to the great majority of
the countries whose delegates are assembled here.

Myself being an exponent of this international exchange, and having promoted it during
my entire career, I have found quite some opportunity to compare the ways of engineers and
scientists in our profession from this and the other side of the ocean. Let me say at once
that I found them basically and on the average to be the same. Internationally seen, their
characteristics, in a favorable sense, are defined by the fact that usually their road to
immortality is not paved with dollars or an equivalent currency. Many of them have lately
become a little more restless and inclined to change of occupation or surroundings, and not
only for the above reason. But all of them have one thing in common, and also with the
category of naval officers so happily represented in this audience: they have managed to
marry attractive and sensible girls, who in the atmosphere of art and science have matured
to the wonderful crop of gracious women here present, the unexcelled helpmates of those
whose scientific productivity and creativity are so greatly dependent on these fortunate
circumstances. It is to you, fair ladies, that my first word of welcome to the Netherlands
goes out as well as my wishes that you may feel happily at home, but delightfully different
from home, in this little country of vanishing windmills and wooden shoes, but vastly
increasing industrialization. As an inveterate conference-goer I have already seen at a
first glance that this one is going to be quite O.K. from the social point of view, which is
at least as important, if not more, than from that of the scientific program. It is greatly
dependent on your graciousness and understanding that the male attendants under your com-
mand come together in those friendly and personal shop-talking sessions which are so
important to the informal scientific give-and-take and essential to those stimulating inter-
national friendships and professional relations which, even more than the formal papers and
discussions, will define the lasting value of this Symposium.

To you, Officers of ONR and NSMB, sponsors of this important event, have already
gone my feelings of deep appreciation for your endeavors in locating this Third Symposium
in the Netherlands. Its dedication to the great Sir Thomas Havelock will be introduced by
a man far more famed and worthy to this task than anybody else here present. This befits
me only to the passing remark that someone has had a most wonderful idea in proposing the

X1V

Address of Welcome

incorporation of this mark of honor in the program of this Conference, at the same time
giving a special tribute to that great shipbuilding and seafaring nation to which the name of
our all-time scientific predecessor William Froude is forever associated. To the delegates
of this and the other countries and, last but not least, to those of the great nation over the
ocean that co-sponsors this Conference and that have made my and Mrs. Troost’s previous
nine years an unforgettable experience, goes the warmest welcome of the Dutch people, its
Royal Navy, its shipping and shipbuilding community, and its T.N.O. Organization. In
joining this welcome with cordial personal greetings to the many old friends and colleagues
here present, I wish you all the happiest of times in our country, and a most interesting,
stimulating and successful conference on the Scheveningen beach!

XV

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ADDRESS OF WELCOME

T. J. Killian
Office of Naval Research

It is a pleasure and a privilege to participate in opening a meeting that is a landmark
in the steady growth of a truly international scientific community. Although this symposium
is the third sponsored by the Office of Naval Research in the field of hydrodynamics, it is
the first of a whole series, I am sure, to take place in Europe.

Certainly Europeans have made outstanding reputations for themselves in advancing
hydrodynamics research. With enthusiastic support from private industry, especially in the
hydrofoil area, Western European laboratories are coming forth with a steady stream of
new ideas.

This week we will have the pleasure of hearing reports on recent progress and the
latest approaches to the basic problems in this field. I am sure you are anticipating with
great interest, as I am, the promising papers on the symposium agenda. You will hear from
the top people in hydrodynamics research on both sides of the ocean. Although there will
be some emphasis in the development of hydrofoils, other important areas will also be
discussed.

In the case of the U.S. Navy, you will hear from specialists in the Bureau of Ships
which, with the Office of Naval Research have joined together to solve the knotty problems
of achieving high speed hydrofoil craft. The Office of Naval Research or ONR, as we are
generally known, is also concerned with basic problems in hydrodynamics. Since ONR is a
unique organization and not well understood even in the United States and as Chief Scien-
tist of ONR, I think it would be appropriate to give you a brief review of our operation and
our raison d'etre.

The U.S. Navy is well aware that the revolutionary nuclear powered, guided missile
fleet of the future we now have under construction is more than anything else a product of
science. We recognize that the design and production of an advanced piece of equipment
does not begin until the scientist tucked away in his research laboratory has some time
before worked out the principle in theory.

This search for new knowledge, which we call basic research, is one of the primary
concerns of ONR. This organization was established at the end of World War II to make
certain that the Navy would have the advantage of the latest scientific knowledge in design-
ing and building its postwar fleet. At the same time, the Navy felt it was essential that
much of the valuable research undertaken by civilian scientists before the war should be
continued when the war ended.

XVil

646551 O—62 2

T. J. Killian

Thus ONR became the first agency of the U.S. Government to be established primarily
to support scientific research across the board in any discipline. It underlined the Navy’s
belief that all future attempts to maintain our national defense through seapower would
depend in large part on the cooperation of the country’s scientific community with the naval
authorities.

To achieve this goal, at its inception the office established the guiding principle that
it would foster fundamental research in universities and nonprofit institutions. Furthermore,
we decided that such support should be given whether or not the projects might hold promise
of immediately foreseeable applications for naval use. This was rather revolutionary
doctrine in those days.

In other words, the Navy is quite willing to risk its funds on research projects without
being certain we will get definite results for our money. The scientist himself, of course,
cannot predict his results in advance, even though he starts off fully confident that he will
throw some new light on his particular subject. On the other hand, itis quite possible that
after a year or two of research he will discover that he has been proceeding up a blind
alley. But ONR accepts the view that even negative results often have great value. The
point is that uncertainty is a fact of life in scientific research. As one American scientist
has stated, “the most important facts that wait to be discovered are those whose existence
we do not even suspect at the present time.”

There were other policies we inaugurated which were also innovations in support of
academic research by the military services. For example, we insisted that the freedom of
basic research from security restrictions was essential to the quality and rapid progress of
the work being carried on. Today only a slight percentage, less than 1 percent, of our
research contracts are classified, and these are not in the category of basic research. Our
early policy also stated that the Navy should cooperate with, rather than attempt to direct
and control the work of, the scientists engaged in research on its projects.

A brief summary of the administration of our contract research program today will
illustrate how we have continued to carry out this early policy. About 80 percent of the
funds in this program.are used to support contracts in the basic research category. These
contracts are awarded on the basis of proposals submitted to ONR, which are unsolicited.
Normally we receive many more proposals than we could find if we had to seek them.

Selection is made by evaluating the scientific merit of the idea, and the competence of
the principal investigator; finally we must have some idea as to how the results of the
research to be undertaken could conceivably influence the Navy of the future. The size of
the organization to which the investigator is attached or the type of organization, whether
academic, nonprofit or industrial, is not a factor in our consideration. ONR seeks only to
support the best research in any particular field.

We recognize also that few projects can be completed in one year. Therefore, research
contracts let by ONR are not limited to one year; they average three years’ duration or more.
Not infrequently they are extended. This has made possible long-term basic research pro-
grams where several years were required to achieve fruition. In referring to the policy and
operation of ONR, Dr. Nathan Pusey, President of Harvard, has said: “Backing of this kind
gives the scientist confidence and frees him from the burdensome and wasteful necessity of
making yearly special plans, special budgets, and special appeals for funds.”

XVili

Address of Welcome

Naval research is also guided by the Naval Research Advisory Committee, which is
composed of leaders in American industry as well as academic research. Utilizing the
thinking of such men as Nobel prize-winner I. Rabi, we are kept alert to areas on which a
greater concentration of research is needed.

ONR also recognized at the start that we would need to depend greatly on the European
scientific community, and for this purpose we established our branch office in London.
From there we maintain close liaison with all the important research centers in Europe and

the Middle East.

Other government and military research agencies later established, such as the National
Science Foundation, the Atomic Energy Commission, and the Office of Scientific Research of
the U.S. Air Force, I like to feel have followed the ONR pattern to a more or less degree.

In fact, the National Science Foundation even took our first chief scientist for its first and
present Director.

Naturally, the Navy is proud of pioneering in the widespread, large-scale support of
research by the U.S. Government: More important than that, however, is the fact that we
have helped the American scientific community in producing a steady flow of brilliant ideas
in recent years.

Finally, it is important to emphasize that while the products of naval research are
designed primarily for military use, it is our hope, even our burning dream, that their greatest
use will be their exploitation by industry to promote the peaceful progress of civilization.
This dream I am sure we share with all who have joined together this pleasant morning in
Scheveningen.

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HIGH PERFORMANCE SHIPS - PROMISES AND PROBLEMS

Owen H. Oakle
Bureau of Ships, U.S. Navy

The U.S. Navy’s interests in ships of maximum performance as regards
speed, motions, and quietness are discussed from the point of view of
design and application. A number of exploratory studies, and the de-
signs of developmental craft, are reviewed in this light. The promise of
each is considered in regard to the compromises required and the prob-
lems encountered in the course of development. It is concluded that,
while no dramatic “breakthroughs” are foreseen, the accelerated pace of
research in ship hydrodynamics will produce significant changes in the
shape and performance capabilities of naval craft in the next few years.

INTRODUCTION

The United States Navy is currently engaged in the study of a variety of design con-
cepts, some of which differ sharply from the conventional shape of ships. It is my purpose
to describe and discuss the pattern and direction of these efforts, less from the standpoint
of hydrodynamics than from the overall viewpoint of naval ship design.

If this viewpoint appears sometimes to emphasize the problems rather than the promises,
it is because it is the daily experience of design that the problems are often obscured by the
rosy glow of the promises.

Many of our present-day naval and merchant ships are a result of a slow process of evo-
lution. New designs have been based on previous successful ships; small modifications
were made to accomodate new needs. However, in recent years, it has become evident that
a new, fresh look was desirable. Modern technology presents greater capabilities in power
plants and electronic equipment. Basic research has given a better understanding of ship
environment and ship behavior in that environment. The question is being raised more and
more as to whether we need adhere to many of the compromises that have traditionally made
up standard design procedure. The need for high performance in ships is pressing and the
time is right for reappraisal of basic concepts.

Exploratory studies of the type under discussion may follow one of three paths. One
approach is to select a single facet of ship behavior and try to optimize it. Necessary com-
promises are then made with an eye toward retaining as much of the original gains as possi-
ble. Another method is to examine closely and carefully new and hitherto neglected scientific
achievements which may only lack development to bring them to the point of usefulness. The
third method is to take an overall look at various ship missions, pinpoint the major problems,

2 O. H. Oakley

outline broadly the possible solutions to these problems and then develop the most promis-
ing. The high performance ships discussed in this paper are products of all three methods.

At the risk of oversimplifying, we can state that there are three principal reasons for
exploring unusual hull forms as they apply to naval craft: to increase speed, to minimize or
control motions, and to improve stealthiness. The first two are neither new nor peculiar to
naval ships; men have been concerned with speed since the first log canoe and with motions
since the first case of mal-de-mer. The third feature, stealthiness, was concocted to cover
the functions of detection, localization, and classification of the enemy ships while remain-
ing immune to detection, localization, and classification by the enemy. One of the major
components of stealthiness is hydrodynamic quietness. This is a relatively new goal in
ship design and is, of course, important where sonar performance is concerned.

The three qualities of speed, motion, and quietness are obviously not independent, so
that changing one changes the other two. It appears, sometimes, to be a fundamental law of
nature that maximizing one quality usually results in degrading others. In a few cases this
did not occur; these are exceptions and like all good exceptions only go to prove the rule.

The phenomenon of surface waves is one of the more interesting properties of water. In
a sense it is the subject of this symposium, since it is by the avoidance or minimization of
wave effects that we hope to maximize speed and seakeeping performance. The energy
absorbed in the creation of surface waves to a large degree limits the speed of surface
ships. Likewise, it is the energy imparted to the ship by surface waves that limits other
aspects of ship performance. Quite naturally, therefore, exploratory studies have been
directed primarily at avoiding or minimizing these surface wave effects.

Fig. 1 illustrates a range of concepts and relates them to the free surface. In the top
group, two methods of avoiding the free surface are indicated, in the middle group the avoid-
ance is less complete, while in the bottom group, two surface ship hull forms are shown
which are designed to cancel wave effects rather than avoid them.

! /
ae
GROUND

EFFECT

SUBMARINE

NEAR SURFACE

HYDROFOIL

ee
SHARK FORM ESCORT RESEARCH

Fig. 1. Various high performance ships

High Performance Ships—Promises and Problems 3

This paper will discuss the potential of these concepts and some of the problems that
arose in expanding certain of them into practical ship designs.

SOME HIGH-PERFORMANCE SHIPS

Ground-Effect Craft

Ground-effect craft offer the contradictory capabilities of high speed and the ability to
hover above the surface. Speeds of well over 100 knots appear to be attainable and such
speeds are attractive for many applications. The ability to hover has obvious advantages,
for example, in amphibious operations. Recent theoretical and experimental evidence indi-
cates that in the high speed range of operation, these craft will not respond to wave action,
but can maintain level flight over ocean waves. These promises, when and if attained, indi-
cate applications for amphibious warfare, antisubmarine warfare, aircraft carriers, missile
carriers, cargo and personnel carriers, etc. The rapidly expanding literature on these ground-
effect craft is a measure of their apparent promise; it is also a measure of the many problems
to be resolved.

Ground-effect craft do not completely avoid the free surface, particularly the more
“pedestrian” versions of this concept. Fig. 2 shows four types. Various combinations of
these basic schemes are possible, and it would seem that every conceivable variation has
been proposed.

iL oe
ees

AIR CUSHION
RAM WING PERIPHERAL JET
SPEED > 100 KNOTS SPEED < IOOKNOTS
ZT WS
AIR CUSHION
SIDEWALL PLENUM CHAMBER
SPEED < 50KNOTS SPEED < 100 KNOTS

Fig. 2. Four forms of ground-effect craft

The “ram wing” shown in the top of Fig. 2, is actually a low-flying aircraft. The sig-
nificant reduction in induced drag achieved by flying close to surface should permit the
attainment of very high lift/drag ratios at high speeds. Values as high as 60 at a chord/
height ratio of 0.5 have been estimated.

4 O. H. Oakley

The two bottom illustrations of Fig. 2 have received a great deal of attention. The lit-
erature is too extensive for complete reference here; the International Symposium at Prince-
ton did present, however, a substantial cross section of the efforts going on in the world.

A large percentage of the research effort has been applied to the peripheral jet type. In
this case, air jets are located along the periphery of the craft. When starting, the jets hit
the water, and split, part of the jet goes inward and is trapped. The pressure in the air
cushion is thus built up. Equilibrium is reached when the pressure in the air cushion is suf-
ficient to bend the jets so that no more air goes inward. A simple analysis indicates that
the required lift power for a particular craft depends on the weight of the craft times the
velocity in the jet for a constant jet momentum. If a heavy fluid, such as water, is used in-
stead of air, constant jet momentum implies a higher mass flow and a lower jet velocity and
thus lower required lift power. Air, for the air cushion, must be supplied separately; the
water curtain simply provides the seal to contain the air.

The amount of power required to supply the air cushion is a function of many parameters;
among the most important is how high the craft flies and the peripheral length of air seal.
Flight over water permits the use of sidewalls extending into the water (lower left in Fig. 2).
The air loss can be greatly reduced by this device. The penalty for this is lower speed and
loss of ability to “fly” onto a beach. The drag of the sidewalls apparently limits this type
to about 50 knots, beyond which the peripheral jet offers a more efficient use of installed
power. Fig. 3 shows in a highly qualitative manner the relative speed/horsepower charac-
teristics for the four basic types.

L SIDEWALL

/ PLENUM
CHAMBER

ae

PERIPHERAL JET

HORSEPOWER

RAM WING

SPEED

Fig. 3. Speed-power curves for the four forms of ground-effect craft

The problems to be solved are numerous. As implied by the previous discussion, most
of the past efforts have gone into the determination of the mechanism and parameters affect-
ing hovering. An examination of dynamic effects associated with forward motion has just
recently begun.

High Performance Ships—Promises and Problems 5

The ram wing will encounter all the problems of takeoff, landing, and in-flight control
of ordinary aircraft, plus the precise control required in flying close to the surface. Take-
off or low-speed power requirements may well prove to be governing.

Stability and control behavior over waves is not well understood. Tests to date indicate
problems associated with running trim even in calm water. Theoretical insight is being
gained into the mechanisms of damping and resonant frequencies of motions over waves, but
this is not complete, nor do we have as yet very much experimental experience.

The scaling laws are being carefully examined, especially in light of the Sritish experi-
ments with “hovercraft.” Here, for example, large scale tests indicated severe spray prob-
lems, which did not arise in the small scale model tests.

The dynamics of water-wall craft have not been examined extensively. How can we
reduce air leakage through this water screen? We know that the screen is relatively good at
short distances down from the jet exit, but leakage is severe at substantial heights before
the jet hits the free surface.

In all of these craft, a major problem is how to design the light structure required and
still retain the ability to land and rest on the water in waves. Aircraft structure would be
too light to meet the landing in waves requirement, while standard ship structure would be
so heavy as to eliminate any payload. A blending is required.

Light-weight, high capacity air fans are required that will operate against the high
cushion pressure that is needed for ground-effect craft. It is estimated that for large, oper-
ational ships this back pressure may be in the order of 150 pounds per square foot. Such
fans are not shelf items.

From the designer’s viewpoint, the promises are meaningful and the problems do not
appear insurmountable. However, considerable expenditures of research effort and time are
required, U.S. Navy work in this field, is aimed at the construction of a large ground-effect
vehicle about 1963. The size and type await the results of research now underway.

Submarines

Just as the aircraft flies high and eliminates surface effects, the submarine departs
from the free surface by submerging. Undesired motions are reduced rapidly with departure
from the surface because wind-formed waves attenuate quickly with depth. Speed is also
benefitted because it is no longer limited by the creation of surface waves. The submarine
offers the ultimate in stealthiness. Thus, its principal promises are high speeds and a high
degree of undetectability.

Fig. 4 shows a comparison of the power to drive two hulls of about the same displace-
ment where one is a modern submarine and the other a destroyer hull form. Part of the sub-
marine’s advantage is its much superior propulsive efficiency. The remainder of the differ-
ence is largely attributable to the elimination of wave resistance. These two effects more
than balance the greater frictional resistance of the submarine which results from its greater
area of wetted surface.

6 O. H. Oakley

DESTROYER

HORSEPOWER

SPEED

Fig. 4. Speed-power curves for a submarine and a destroyer of about the
same displacement

Fig. 5 shows a comparison between the pitching motion of a submarine on the surface in
a moderate sea state compared to the submerged pitching. The data is presented in spectral
form and shows the large decreases in motions for fairly shallow submergence. (Note, how-
ever, that the ordinate is proportional to the pitch amplitude squared.)

f (PITCH AMPLITUDE)

FOUR DIAMETERS
BELOW SURFACE

FREQUENCY OF ENCOUNTER

Fig. 5. Energy spectrum of the pitch amplitude for
a submarine

High Performance Ships—Promises and Problems 7

Within the past 10 years technological advances have been exploited to make the sub-
marine the most fantastically mobile and powerful warship yet devised.

The foregoing are all promises realized, but progress in improving the performance of
submarines still encounters problems. The problems essentially consist of a search for
methods ofimproving this already powerful vehicle. The most significant improvement would
be to increase speed. The most direct way, of course, is simply to increase the power.
This would require a larger hull, so that some of the gain would be eaten up in added fric-
tional resistance. A simplified parametric study holding payload constant and using present
- specific weights and volumes for machinery, indicates that a submarine would require well
over 100,000 shaft horsepower to attain 50 knots. This is right at the upper bound of pres-
ent technology as regards the power that can be absorbed by a single propeller. Improve-
ments in specific weights and volumes of submarine power plants will naturally improve this
situation, but it is apparent that dramatic increases in speed will be hard earned. Other
methods of improvement, such as reduction of resistance by boundary layer control will be
discussed in a later portion of this paper.

Near Surface Craft

The near surface craft is intended to combine the stealthiness, speed, and motion char-
acteristics of a submarine with the air-breathing, air-communication aspects of the surface

ship.

At present, nuclear power plant weights are extremely high compared with the more
advanced air-breathing plant. If the submarine could be run sufficiently close to the surface
to breathe air through a slender strut, and still stay far enough down to avoid excessive
wave drag, a lightweight air-breathing plant could be used.

The near surface craft is intended to do this. Work on this type is currently being per-
formed by the Davidson Laboratory. Fig. 6 shows a series of speed-power curves, as a
function of depth of submergence, for an arbitrary submarine type hull form. In order to

4
‘ Aa
/
DESTROYER ie ee
/

ve

1.25 DIAS_/
1.58 eee

SUBMARINE SUBMERGED + 366

HORSEPOWER

SPEED

Fig. 6. Effect of the depth of submergence on the speed-power
curves for a submarine

8 O. H. Oakley

compete with a destroyer hull form of equal displacement, the near surface craft must travel
below a depth of about one and a half diameters to the axis. This implies a strut length
somewhat greater than one hull diameter plus additional length in order to project some dis-
tance above the free surface to allow for waves and depth control variations. Such a strut
must carry at least a periscope or closed circuit TV camera unit, communications and radar
antenna and a snorkel head valve. The size of the air passages appears to govern the size
of the strut. For a 20,000 shaft horsepower installation, some 80 square feet of intake,
exhaust and ventilation duct is needed. This results in a strut with a chord of some 26 feet
and a thickness of 6-1/2 feet, altogether not a small structure. The resistance of this strut
would degrade the speed performance and affect the stability and control.

Just as the ram wing needs controls like an airplane, the near surface craft needs con-
trols like a submarine. In fact, the controls have to be more effective to counter the effects
of surface waves. Also, the structure would have to be sufficiently strong to withstand
accidental deeper submergence.

Hydrofoil Craft

Hydrofoil craft promise primarily high speeds. It appears likely that speeds in excess
of 100 knots are possible. In addition, motions in a seaway should be considerably less
than buoyant craft of the same displacement. These craft can maintain high speed in sea
conditions considerably more severe than would compel a displacement craft to slow down.

The United States Navy has maintained a hydrofoil program since shortly after World
War IJ. Out of this has come considerable research and development, and a number of test
craft embodying some promising principles. Four of these are shown in Figs. 7, 8, 9 and 10.

Fig. 7. A test hydrofoil craft, the Baker “High-Pockets”

High Performance Ships—Promises and Problems 9

The Baker boat,* High Pockets,” exemplifies the use of surface-piercing foils (see Fig. 7).
This craft has the feature of essentially equal distribution weight on the forward and the
after foil systems. The Carl XCH-4 (see Fig. 8) embodies ladder foils, an “airplane” dis-
tribution of foil area fore and aft, and air screw propulsion. The Miami Shipbuilding Company’ s
“Halobates” (Fig. 9), utilizes submerged foils in an “airplane” distribution and a right angle
drive over the stern. Control in the version shown was achieved by the “Hook” system which

Fig. 8. A test hydrofoil craft, the Carl XCH-4

Fig. 9. A test hydrofoil craft, the Miami Shipbuilding Company’s “Halobates”

10 O. H. Oakley

used mechanical “feelers” to sense the waves ahead of the boat, and which, through link-
ages, adjusted the angles of attack of the forward foils. This craft was also successfully
operated using a resistance height sensing device on the forward struts and an electronic
“hydropilot ” The most successful of these craft was “Sea Legs,” by Gibbs and Cox,
shown in Fig. 10. “Sea Legs” has submerged foils in a canard arrangement and a sonic
height sensing system which actuates the control surfaces through an autopilot and hydrau-
lic servos.

Fig. 10. A test hydrofoil craft, the Gibbs and Cox “Sea Legs”

Based on the successful performance of “Sea Legs” the submerged foils-autopilot sys-
tem, was used for the antisubmarine hydrofoil craft “PC(H).” Fig. 11 shows an artist’ s
concept of PC(H), for the construction of which a contract has recently been awarded to
the Boeing Aircraft Co.

These craft are all of the subcavitating type. The maximum speed for subcavitating
craft for all practical purposes is limited by cavitation to about 55 knots in calm water. It
should be noted however that the XCH-4 attained a speed of about 65 knots. This craft,
however, was overpowered and lightly loaded, so that any added drag due to cavitation was
probably not limiting. In rough water, wave orbital velocities induce variations inthe angle

High Performance Ships—Promises and Problems ll

- Fig. 11. Artist’s concept of the antisubmarine hydrofoil craft “PC(H)”

of attack of the foils, which causes cavitation to occur at lower speeds. Fig. 12 shows the
variation of angle of attack with craft speed for a state 5 sea. Fig. 13 shows the degrada-
tion of cavitation-inception speed with increasing wave height for a thin hydrofoil. Consid-
erable work has been done on section shapes to delay cavitation, but only limited gains are
possible in this direction. If we are to attain the promise of high speeds, it must be by
utilizing supercavitating sections.

In the supercavitating regime we find that very high speeds or large angles of attack
are required for true cavity flow. In practical foil configurations, the cavity is likely to
vent to the atmosphere, either through tip vortices or down a strut. This leads to the con-
clusion that superventilation rather than supercavitation is the more practical mechanism
to consider.

It appears that in the subcavitating range we may expect overall lift/drag ratios of about
9 to 12, and in the superventilated range, lift/drag ratios of about 5 to 7. While the lower
lift/drag ratios require more power and demand more economy in structural and other weights,
this does not appear to be prohibitively restrictive.

In the course of designing PC(H) a number of problems were successfully resolved on
paper, while others required considerable test and reevaluation effort. For example, calcu-
lated predictions of lift and drag at top speed were in close agreement with model tests.

12

O. H. Oakley

ine)
oO

ip)
fe)

15
MEAN OF ONE-HUNDREDTH HIGHEST VALUES

MEAN OF ONE-THIRD HIGHEST VALUES
ROOT-MEAN-SQUARE VALUES

;

ANGLE OF ATTACK DUE TO WAVE MOTION (DEG)
)

fo)

20 30 40 50 60 70 80
SHIP SPEED (KNOTS)

Fig. 12. Variation of the wave induced angle of attack for a state 5 sea and
a hydrofoil 5 feet below the surface

(KNOTS)
S00
eo)

30

20

SPEED OF INCIPIENT CAVITATION

O | 2 3 4 5 6 7 8 9 10
WAVE HEIGHT (FEET)

Fig. 13. Degradation of cavitation-inception speed with increasing
wave height for a thin foil submerged 6 feet and a wavelength of
100 feet

High Performance Ships—Promises and Problems 13

This result confirms our confidence in the use of the extensive airfoil data that are avail-
able. Model experiments in takeoff drag and available thrust indicated satisfactory margins.
It was found, however, that very poor flow conditions existed at the intersection of the
strut, foil, and nacelle. This poor flow was found to be cavitation and not separated flow.
In analyzing the problem, aircraft technique was borrowed and simple consideration of add-
ing pressure distributions led to small changes in configuration; the resultant clean flow
was confirmed by experiment. The PC(H) represents the first hydrofoil for which the hydro-
dynamic coefficients were obtained experimentally at the David Taylor Model Basin. These
coefficients are needed for determination of stability and autopilot gains for vertical-plane
motion. Horizontal-plane and coupling effects are planned to be obtained in a similar
manner in the near future.

A few problems remain to be resolved in the subcavitating regime. Probably the major
one is the determination of the hydroelastic dangers, i.e., flutter and divergence. There is
no adequate theory for prediction of instability speeds, and practically no experimental work
in the marine range of interest. The flow problems at the intersections of strut, foil, and
nacelle need to be codified. Flap effectiveness at low submergences (as determined from
PC(H) tests) appears to be less than predicted by theory. These problems are all currently
under study.

Flying qualities in waves are being examined. In very high speed hydrofoil craft
(70 to 100 knots), the frequency of encounter with waves will be so high that it will be
impossible to “contour” the surface without inducing excessive accelerations. This implies
the necessity for “platforming;” that is, the craft’s trajectory must remain essentially hori-
zontal. This further implies submerged foils, because surface-piercing systems cannot
platform. In addition, long struts will be needed to keep the hull clear of the water and yet
insure continuous good submergences for the foils in rough water. In order to platform, the
autopilot system must move the control surfaces continuously and at rapid rates in order to
nullify the wave-induced disturbances. In some following seas it may not be desirable to
platform and here the control system will have to permit some vertical motion. This sug-
gests not only a wave-height sensing device, but also inputs to the autopilot from acceler-
ometers. Roll control will require additional displacement, velocity, and acceleration
inputs.

The major problems to be resolved are primarily in the supercavitating regime. Here we
have theory to guide us, but systematic experimentation on the effect of geometric charac-
teristics of foils is lacking. There are no corresponding airfoil data, of course, that are
useful.

As indicated earlier, the mechanisms of ventilation are not well understood and require
research starting with the basic physics of the phenomenon.

A simplified theory of hydroelastic instabilities has been developed, but there are no
experimental data. This theory, incidentally, indicates a greater likelihood of hydroelastic
instabilities than for a comparable all-wetted hydrofoil. Also, advances are sorely needed
in materials in order that the requirements in hydrofoils for very high strength, erosion, and
corrosion resistance, toughness, good fatigue life, etc., may be met.

In the supercavitating or superventilating range, takeoff may well be critical. Feasi-
bility studies have shown that for very high speed craft, takeoff thrust requirements are
incompatible with top-speed requirements. Means for achieving takeoff at lower speeds
need special attention.

646551 O—62——3

14 O. H. Oakley

Foil smoothness is, of course, vital to good performance. Present tests indicate that
paint (both anticorrosive and antifouling) is likely to peel off at high speeds. One promis-
ing approach is the Cox System of cathodic protection. In this method, high current densi-
ties are applied when the boat is motionless in the water. This plates out a combination of
magnesium hydroxide and calcium carbonate on the foils. The idea is that the fouling will
attach to this rather soft coating, and slough off at reasonably low forward speeds. A mini-
mum value of current density is applied at all times to eliminate corrosion.

The Navy’s present activity in the hydrofoil area includes an extensive research and
development program directed toward a large experimental subcavitating hydrofoil ship of
about 300 tons displacement and a small superventilating hydrofoil boat (about 15 tons) to
be in operation by 1963.

Shark Form

If we do not try quite so hard to avoid the free surface, and if we acknowledge that
certain armament and surveillance activities require topside space, then it is not a long
step from the concept of near surface craft to the shark form of Fig. 1. In effect, this is
simply a near surface craft with a relatively large strut piercing the surface. The shape and
location of this large strut should be such as to cancel as much of the main hull’s wave
resistance as possible and reduce the exciting forces and moments due to sea action. Thus,
the promises of the shark form are high speeds and small motions.

This concept was explored to some extent by the Germans during World War II, andthis
resulted in the Englemann craft. This form should have favorable motion characteristics,
especially in head seas, because of the long natural periods of pitch and heave, which
result from the short, fine waterline.

Fig. 14 shows a comparison of the shark form and a destroyer type hull form. At the
lower Froude numbers, the shark form is quite resistful; however, at high speeds, the shark
form is better than the destroyer form.

There are a number of problems to be resolved before the promises of this form can be
realized.

As shown in the case of the near surface craft, the submerged hull must be located
well below the free surface in order to minimize wave-making resistance and the exciting
forces of the sea. In common also with the near surface craft, the shark form requires
power plants of less specific volume than currently available in order to attain the speed-
power advantages shown in Fig. 14.

The amount of topside weight which can be carried is limited by the low transverse
stability. With very little waterplane, stability must be attained by keeping the center of
gravity below the center of buoyancy. This is not easy in a form such as this, in the face
of the demand for topside locations for equipment.

The shark form has, in common with the ground-effect craft, a problem of running trim
in calm water. Fig. 15 compares observed trim tendencies at zero angle of attack with those
calculated from Pond’s theory for Rankine ovoids. This comparison is, of course, not
strictly valid because of the difference in submerged hull shape and the fact that the theory
does not take into account the topside shape of the shark form. Even so, the trends do

High Performance Ships—Promises and Problems 15

/
DESTROYER /

HORSEPOWER

SHARK FORM

SPEED

Fig. 14. Speed-power curves of a destroyer and the
shark form

(+) BOW UP

SHARK FORM (ISLAND AFT)

fe)

MOMENT COEFF

FROUDE NUMBER

Fig. 15. Comparison of the pitching moment for the shark form with Rankine
ovoids, with the hull 1.5 diameters below the surface

appear comparable. The pitching moments are not small and will probably require large
control surfaces to counteract them. In view of the fact that the moments vary in direction

and magnitude with speed, the control surfaces may have to be controlled by an autopilot
system.

16 O. H. Oakley

Escort Research Ship

The escort research ship (see Fig. 1) is an actual design which was prepared to provide
a research vehicle emphasizing quietness and minimum motion. Hydrodynamic quietness
was sought by using a lightly loaded propeller and locating it deep and at the end of a
nacelle well removed from the influence of the hull. Fig. 1 also shows the large nacelle or
dome located at the bow to house sonar transducers and to provide damping and added mass
which tend to produce favorable pitch characteristics.

The aim of this unusual hull form was to avoid resonance in pitch by raising the natural
pitch period. It was hoped by this to permit operation in the supercritical region with
attendant low pitch response. This is illustrated in Fig. 16, which shows a single degree
of freedom type of response for a damped system and indicates the supercritical range in
which it is desired to operate.

| RESONANCE —

( AMPLITUDE )

f

SUPERCRITICAL
MOTION

FORCING FREQUENCY
NATURAL FREQUENCY

Fig. 16. The supercritical region of motion, showing the region of
operation under conditions of favorable pitch characteristics

The longitudinal moment of inertia of the water plane was made small; this, combined
with maximizing the longitudinal mass moment of inertia of the hull by moving heavy
weights toward the ends and the added mass due to the water entrained by the nacelles,
tended to produce a long pitching period.

The early models tested of this concept were largely exploratory and investigated the
locations of nacelles, etc. Model tests showed excellent pitch characteristics for the con-
figuration then under study. The short-dashed line in Fig. 17 shows the very favorable
behavior in pitch. This figure shows the pitch response in regular waves and is similar to
the previous figure except that the forcing frequency or period of encounter is represented
as a function of speed. In this instance the wave corresponds to the maximum energy wave
in a Neuman spectrum for a state 5 sea.

High Performance Ships—Promises and Problems 17

\ a ay INTERMEDIATE

abc
Wy a
TS

FINAL DESIGN Be

PITCH AMPLITUDE

ORIGINAL DESIGN min cee ce eee

FOLLOWING SEA O HEAD SEA
SPEED

Fig. 17. Pitch behavior of various designs of the escort research ship

When firm requirements were established for the design and the design adjusted to
accommodate them it was found that by designing in the new features we had also designed
out the favorable pitch characteristics of the original design. The long-dashed line in Fig.
17 shows the disappointing results. The pitch period had been shortened so that resonant
pitching occurred near operating speeds.

It was found that we could calculate the period of pitch, using simple single degree
of freedom relationships, to agree quite well with model tests by using F’. M. Lewis’ data
for added mass plus allowances based on simplified shapes for the end nacelles. This
proved to be a useful design tool and we set about to make changes to lengthen the natural
period to approach that of the original design. A period length 7/VL was chosen in
accordance with Mandel’s analysis which would place the design in the supercritical range
for all but very low speeds.

This was accomplished by shortening and deepening the hull, shaving away the ends
of the water plane to reduce longitudinal moment of inertia, and increasing the nacelle
sizes. For a single degree of freedom system the natural frequency is given by

a Kies
Ld | Cerra)

where f is the natural frequency in pitch, Twp is the moment of inertia of water plane,
I, is the mass moment of inertia of ship, and I, is the mass moment of inertia of
entrained water. The changes enumerated above obviously tend to lower the natural
frequency.

It is not easy to alter the natural frequency of a ship while still keeping other charac-
teristics unimpaired. For one thing the longitudinal radius of gyration in ordinary ships is
about 0.22L or 0.25, not counting entrained water. In this design a value of close to

18 O. H. Oakley

0.30L was needed. It is most difficult to relocate weights in a practical ship design to
accomplish this.

By enlarging the end nacelles well beyond the size required for items of equipment,
the entrained water was increased and this had the most pronounced effect in increasing the
length of the pitching period.

The changes did result in moving the resonant speed back to about zero and the final

design exhibited nearly as good pitch characteristics as the original, as the solid line in
Fig. 17 shows.

Fig. 18 compares the escort research ship with a destroyer and a destroyer escort.
This is a spectral presentation and compares the ships in the same seaway, and at the same
speed. ‘he results appear dramatic and in a sense they are, although it should be remem-
bered that the ordinate is a function of pitch amplitude squared, which in effect exaggerates
the difference in performance as regards pitch amplitude.

DESTROYER

DESTROYER ESCORT
| ee

—
—

f (PITCH AMPLITUDE) 2

RESEARCH

FREQUENCY OF ENCOUNTER

Fig. 18. Predicted pitch spectra in a state 5 sea at a ship speed of 20 knots

The problems and compromises which practical considerations impose on an ideal con-
cept are well exemplified in our experience in this design. One facet, however, did not
develop unfavorably. The speed-power relationships provided a pleasant surprise. Fig. 19
shows a qualitative comparison with a destroyer escort hull of about the same displace-
ment. As in the case of the submarine vs destroyer comparison given previously (see Fig.
4) a significant part of the difference in power is the result of the superior propulsive coef-
ficient of the escort research ship. Nevertheless, considering the fact that roughly a third
of the displacement of the research ship resides in the nacelles it is remarkable that they
cost so little in power.

High Performance Ships—Promises and Problems 19

DESTROYER
ESCORT

COMPLETE WITH NACELLE &

SONAR , DOME
ESCORT RESEARCH. SHIP WITH SONAR ME

BARE HULL

HORSEPOWER

Fig. 19. Speed-power curves for an escort research ship and a destroyer
escort of about the same displacement

Some Other High-Performance Ships

Spar Ship—This craft was designed to be an effective and inexpensive sonar ship. The
principal idea was to submerge the sonar dome as deep as possible on a surface ship. It
was also desired to have minimum motions. Speed was not an important parameter. These
ideas led to a ship having a vertical distribution of displacement rather than horizontal
(see Fig. 20).

Fig. 20. A spar ship

20 O. H. Oakley

In the course of exploring the feasibility of this concept a number of interesting prob-
lems arose. The speed was low, as expected, but model tests showed that the vertical
location of the center of resistance varied considerably with speed. This caused substan-
tial changes in running trim. The form was also highly directionally unstable as one might
expect.

Natural heave periods were calculated which indicated supercritical heave motions in
practically all seas. No tests were made, however. The practicability of the concept is
admittedly questionable. For instance, consider the problem of taking the ship in and out
of harbors. The only apparent way to do this would be to provide ballasting arrangements
to permit the ship to be brought to a horizontal position. The problems of reorienting through
a 90-degree angle are all too obvious and painful, but not impossible.

Catamarans (Planing and Displacement)—The yachting and small boat magazines have

had many articles praising the sea-keeping characteristics of planing catamarans. This was
all qualitative information. A testing program has been initiated at the David Taylor Model
Basin to develop quantitative effects. There are no results at this time. This configuration
has promise of allowing planing in relatively high waves, a characteristic that has not been
attained with conventional planing craft. Structural problems will require careful attention.

Hulls of Minimum Wave-Making Resistance—Theory by Weinblum, Martin, Kotik and
others have developed waterline shapes that should produce minimum wave-making for a
given Froude Number. In the speed range of interest, a Coke-bottle form appears to be
optimum. As this involves the danger of flow separation, tests are being formulated for
comparison of model and theory. Motion tests are planned for the future. If the model tests
check the theory, feasibility studies of practical ships will be made to determine applica-
tion possibilities.

DEVICES TO INCREASE THE PERFORMANCE OF CONVENTIONAL SHIPS
To Increase Speed

In addition to unusual hull forms, there are various devices by which high speeds may
be obtained. One such means might be boundary layer control to reduce frictional resist-
ance. This could be accomplished by sucking off the boundary layer at various places
along the length of the body. The objective would be to maintain laminar flow; this, ideally,
would reduce frictional resistance to about 15 percent of its normal amount. The problem
areas are: How much power is needed to suck in the boundary layer? If the boundary layer
is drawn off through a porous outer shell, how can clogging of the pores by marine life be
prevented? If slots are used, what should their size, shape and location be?

Another possibility is to enclose the body by a gas. Ideally, this would reduce the
frictional resistance to about 15 percent of its normal amount. The major problem here is
the stability of the gas film. If the gas goes into bubble form, experience has shown that
resistance is only slightly decreased.

A third method would be to use a coating that absorbs the energy in perturbations in
the water. A theory, as yet unpublished, by Boggs of the United States Rubber Company,
indicates that a coating could be devised to maintain laminar flow up to very high Reynolds
numbers.

High Performance Ships—Promises and Problems 21

These three methods of reducing frictional resistance may have application to midget
submarines, and possibly to larger craft, but theoretical and experimental evidence to sub-
stantiate claims and predictions is yet to be acquired. These methods appear to have more
promise in a relatively low range of Reynolds numbers than at the high Reynolds numbers at
which large ships and submarines operate.

Another approach to the goal of high speed is to increase the efficiency of propulsion
devices. No dramatic improvements in this area appear to be in the immediate offing. How-
ever, a number of promising new types of propulsive devices are under study and some older
ideas are being refined—to name a few: supercavitating or superventilated propellers, pump
jets, Kort nozzles, cycloidal propellers, oscillating fins, and two-phase flow jets. Details
of these and others, would require too much space for this paper to encompass.

To Control or Minimize Motions

From the control point of view, there are a few devices that hold promise for competing
with the highly efficient, standard ship rudder. One of these, the cycloidal propeller, is
well known. Another is the jet flap, currently under study for aircraft. Design data, includ-
ing mass flow vs rudder effectiveness, are being obtained. Yet, another competitor for the

rudder is the ring airfoil developed originally for aircraft. Preliminary results indicate
excellent control characteristics.

The problem of minimizing motions by using devices has been part of the research
effort of the Navy and Merchant Marine for many years. For roll control, gyrostabilizers,
active fins, and Frahm tanks are available. Reexamination of the last named, in recent
years, has shown that by good design, passive tanks can be quite effective in reducing roll.
Considerable effort has recently gone into pitch stabilizers. Fixed fins at the bow can
significantly reduce pitch. Under study today are moving fins at bow and stern and flapped
ducted propellers.

CONCLUSIONS

High performance ships are basic and very vital objectives of Naval Research and
Development. Within the last few years, these objectives have been pursued by special
attention to unusual hull forms. A number of these forms are nearing readiness to be incor-
porated in useful craft; notable among these is the hydrofoil craft.

Modern technical advances, especially in aerodynamics, hydrodynamics, and power
plant development are exerting strong influences on ship design. For example, a hydrofoil
craft was flown by Alexander Graham Bell at the end of the 19th century, but we would not
be investigating, so thoroughly, the promises of hydrofoils if we did not today have high-
power, lightweight machinery plants (marine gas turbines), an understanding of supercavitat-
ing foils, and strong, lightweight structure.

This is the time to look at our old compromises, and reevaluate them in the light of the
new technology. We are not depending on “breakthroughs,” but rather recognize that the
pace of advance has quickened and the breadth of the front has widened.

The broad spectrum of craft shown in Fig. 21 illustrates the many types of marine
craft, and their relative quasi-efficiencies compared to the well-known Gabrielli-Von Karman

22 O. H. Oakley

VELOCITY (KNOTS)
10 100 8-47 /
HELICOPTER
——
COMMERCIAL AIRCRAFT
100

SPECIFIC POWER (HP/ton)
°

10 100

Fig. 21. Specific power curves

limit line. Of course, power per ton is not the only parameter; in the Spar Ship, for instance,
this was considered of minor importance. Nevertheless, it does illustrate one facet of our
high performance ships.

This paper by no means covers all the possible high performance ships. Only some of

high promise have been discussed and these with the caution that for every promise there
arises a multitude of problems.

DISCUSSION

(Discussion of this paper is included with that of the following paper, by J. D. “
van Manen.)

SIZE, TYPE, AND SPEED OF SHIPS IN THE FUTURE

J. D. van Manen

Netherlands Ship Model Basin

INTRODUCTION

Owners and builders of merchant ships generally view radical new ship designs with a
certain amount of cautious reservation. These people frequently realize from past experi-
ence the expenses which accompany the development and application of new ideas in the
field of naval architecture.

In serving naval interests, however, the situation is different. Military considerations
and economic and psychological factors all play important roles. The seemingly uneconomi-
cal increases in the speed and power of tankers, and the development of nuclear propelled
submarines, hydrofoil boats, and hovercraft are all projects which have been strongly stimu-
lated by naval interests.

A speed increase of 18 to 20 knots is no longer being considered by the designers of
high speed boats such as hovercraft. Indeed, speeds of 40, 60 and even 100 knots are now
being discussed. Designers of these new and progressive ship types are being confronted
with questions such as: How can we develop a 100-knot ship for operation on, beneath, or
immediately above the surface of the water? or What is the maximum shaft power which we
can install on one shaft? One might perhaps just as easily ask: What is the maximum
length of a ship? The answer to this last question could, of course, be given as all the way
from New York to Amsterdam, since optimum performance at sea has here been assured and
guaranteed.

In all seriousness, however, the author hopes to convey that it can be nothing short of
refreshing for the shipbuilder who is firmly established in mercantile construction to take
an active part in and even take the very necessary initiative in the field of naval
development.

If we but look carefully at the developments of the past 20 years, the following facts
become unmistakably clear:

* The design of ships, which has heretofore been primarily empirical in character, is
clearly becoming more and more scientific and fundamental in nature. Increases in ship
speeds and ship dimensions and the development of new and different types of propulsion
systems have required ever more theoretical treatment.

Prior to 1940, the maximum shaft power which could be installed per shaft in merchant
ships was in the order of 10,000 metric horsepower. At the present moment 16,000 shaft

23

24 J. D. van Manen

horsepower is the normal case, and frequently 20,000-shp installations per shaft are encoun-
tered. Ship speed for tankers increased at the same time from 10 to approximately 17 knots.

The development of push boats has led to a large application of this type of transpor-
tation on inland waterways.

The rising importance of the hydrofoil boat is undeniable. The application of the
hovercraft principle to marine vessels at the present time offers unknown prospects.

These imposing developments have induced several investigators to make speculative
examinations of future possibilities. Gabrielli, Von Karman, Davidson, and others have
kept themselves busy with more or less philosophical considerations in the field of
transportation.

Representative of their ideas is the relationship between the dimensionless constant
P/V,A and the speed coefficient V;/A!/® for various means of transportation. The various
ranges of this relationship are clearly presented in Fig. 1 for displacement ships, sub-
marines, hydrofoil boats, hovercraft, and so forth.

The philosophy behind this presentation in Fig. 1 is that there will always be a require-
ment for new developments in a vehicle whose characteristics permit it to be placed in a
blank area of the plot. In accordance with this thought process both hydrofoil boats and
hovercraft are always meaningful. They may be expected to fulfill already existing and yet
to be established requirements.

Supertankers

Supertankers form a type of ship which is indeed one of the most spectacular examples
of increases in displacement and speed of ships. In Figs. 2a, 2b and 2c, several charac-
teristic properties of supertankers are given on the basis of cargo carrying capacity or
deadweight tonnage. The various points on these plots are derived from model tests con-
ducted at NSMB; the curves presented are faired lines drawn through these points as
accurately as possible, without any correlation analysis. Noteworthy in Figs. 2 are the
following:

The increase in the beam-draft ratio, B/T, with increase in deadweight tonnage. The
limitation in draft to a maximum of about 10 meters in connection with the expected depth
of water in harbors thus plays an important role.

The increase in the block coefficient 51; with larger deadweight tonnages is alarming
by comparison with standards which still were valid 15 years ago.

Finally, the power required for ships’ speeds between 15 and 17 knots and deadweight
tonnages above 50,000 metric tons deserves consideration. Considered from this point of
view, the development of diesel engines of 20,000 to 22,000 hp in one 10- or 12-cylinder
installation is not amazing.

The large power per shaft in these extremely full ships has caused serious vibration
phenomena and has led to damage. Experimental research has shown that extreme after-
bodies, for the obtaining of a uniform circumferential velocity field in way of the propeller,
and propellers other than the conventional screw must be taken into consideration.

25

Size, Type, and Speed of Ships in the Future

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3,0

B/t

2.5

300

250

Lyin m.

200

150

0 50000 100.000

Deadweight in tons (metric)

150.000

Fig. 2(a) Characteristic properties of supertankers

Size, Type, and Speed of Ships in the Future

0.85
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0.80
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a
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100,000
0
0 50,000
Deadweight

L.C.B. in percent of Ly

forward of '/2 LU

Displ.

100000

in tons (metric)

Fig. 2(b) Characteristic properties of supertankers

3.0

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1.0

150,000

J. D. van Manen

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40,000

30,000
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Deadweight in tons (metric)

Fig. 2(c) Characteristic properties of supertankers

0.75

Ntot

Size, Type, and Speed of Ships in the Future 29

The 2-percent saving in shaft horsepower obtainable with the cigar-shaped Hogner
afterbody provided with a complete ring nozzle propeller system becomes highly significant
when considered together with the 25-percent reduction in propeller diameter obtainable
over that of a conventional screw, and with the notably reduced vibration and cavitation
characteristics (Fig. 3 and Table 1).

Table 1
Effect of Afterbody Shape on Propulsion
Results of Resistance and Self-Propulsion Tests on a Model of a 39,000-ton Tanker
(loaded condition; speed, 16 knots; 16,900 shp)

Total
Number resistance
Afterbody of (percentages—
Blades afterbody
I = 100)

DHP Propul-
(percentages— sive
afterbody Coe ffi-
I = 100) cient

I (moderately U-shaped)
II (extremely V-shaped)
Ill (extremely U-shaped)
IV (Hogner form)

V (III + nozzle)

VII (twin screw)

VIII (Hogner + nozzle)

Fig. 3. Tanker model with cigar-shaped afterbody and a propeller
operating in a nozzle ring

Another question which must be asked for supertankers is: At which speeds and dead-
weight tonnages does the submarine tanker, through application of nuclear propulsion, become
competitive with its surface counterpart.

646551 O—62——4

30 J. D. van Manen

In general, it is to be expected that, for a cargo carrying capacity of 20,000 dwt and a
speed larger than 20 to 25 knots, the submarine tanker will have a distinct advantage over
the conventional tanker. The Electric Boat Division of General Dynamics Corporation has
made, under contract to the Maritime Administration, a study of nuclear propelled submarine
tankers for mercantile application. Neither military nor economic factors were considered
in this study, which embraces 27 different designs.

The 27 designs were carried out for ship speeds of 20, 30, and 40 knots, cargo carrying
capacities of 20,000, 30,000, and 40,000 dwt, and for three types of cross section, namely,
rectangular, limited in draft; rectangular, unlimited in draft; and circular. The designs
restricted in draft were made suitable for harbor depths of water of about 11 meters.

Of these 27 designs, two types of submarine tankers appear worthy of further develop-
ment: the most conservative, and the largest and fastest.

The Most Conservative

The most conservative design would be one of rectangular cross section, limited in
draft, for which the following numerical data would apply:

Cargo deadweight 21,189 tons
Speed 20 knots
Surface displacement 38,791 tons
Submerged displacement 42,671 tons
Loaded draft 35 feet
Length 583 feet
Beam 80 feet
Depth 40 feet
Total power required 35,000 shp
Number of screws 1

The Largest and Fastest

The following design is currently considered as an upper limit in ship speed and cargo
carrying capacity because of the present state of power plant technology:

Cargo deadweight 41,565 tons
Speed 37 knots
Circular cross section

Maximum diameter 80 feet
Surface displacement 91,903 tons
Submerged displacement 101,000 tons
Loaded draft* 67 feet
Length 936 feet
Total power required 240,000 shp
Number of screws 4

*Made possible by the addition of a parallel middlebody.

Size, Type, and Speed of Ships in the Future 31

Noteworthy here is that Electric Boat Division currently considers 60,000 shp the
maximum which can be installed per shaft.

TYPES OF PROPELLERS

After the numerical data mentioned in the preceeding section, a discussion of propul-
sion seems desirable.

A frequently used method of expressing the characteristics of a propeller type is the
relationship between Bp- 5 and 7p for optimum propeller diameter. These are defined as:

the design coefficient:

i i) 2 Sane
Bp yanisi 2 e J
the diameter coefficient:
NDE ie
Bayt yct.

This optimum relationship is given in Fig. 4 for various types of propellers, namely,
(1) supercavitating propellers of the type TMB 3-50, (2) wide-blade propellers of the type
Gawn 3-110, (3) propellers of the type Wageningen B 4-40, (4) propeller with nozzle rings of the
type K 4-55, with a length-diameter ratio of 1/2. The range of profitable application ofeach —
type of propeller is given on the basis of B,. The considerable difference between the
optimum diameters of supercavitating and conventional screws and screws in nozzle rings
is noteworthy.

Since an evaluation of types of propellers on the basis of Bp- 6 can be troublesome for
those who do not work with this type of data every day, a more convenient and instructive
method of presentation is given in Figs. 5(a), 5(b), and 5(c) illustrating the general tenden-
cies of the propellers discussed.

In Fig. 5(a), the power-rpm relationship is given for various speeds and diameters, as
derived from the optimum relationship presented in Fig. 4, for supercavitating propellers.
The cavitation thresholds, according to Tachmindji and Morgan, and the limit of efficiency
Bp = 3, beyond which this type of propeller would not be utilized, are given for 60, 80, and
100 knot speeds. Only one line of constant propeller diameter, i.e., D = 1 meter, is given.
Other lines of constant diameter greater than 1 meter would appear to the right in the dia-
gram. For reasonable application of a supercavitating screw with a diameter of 1 meter, a
minimum speed of 54 knots is desirable.

In Fig. 5(b), the optimum Bp- 6 relationship is presented in the same manner for conven-
tional screws of the Wageningen B 4-40 type. If the limit of 60,000 shp given by Electric
Boat is adhered to for installed shaft power, then this appears to lead to inadmissably large
propeller diameters for speeds of 20 knots. For 40-knot speeds with 250 rpm there still
appear to be possibilities.

The same data are presented in Fig. 5(c) for the K 4-55 series operating in ducted
nozzles with a length-to-diameter ratio of 1/2. The superior properties of this type of
propeller with respect to maximum installed power for relatively low speedsis clearly evident.

32 J. D. van Manen

0,70

250 0,60.
a
ic)
—
200 0.50
0.40

@ S.C. propellers TMB 3-50
@Wide blade prop. Gawn 3-110
@ Wageningen B 4-40

i @® “Screw + nozzle’’- prop.

1 Bp 10 100
—— S.C. prop. Conventional screw propellers “Screw + nozzle"’- prop.

Fig. 4. Optimum relationship of B, and 6 for various types of propellers

Further investigations will have to indicate whether increasing the rpm of the ducted
screw or of the supercavitating screw, or whether going to an entirely unknown field, ducted
supercavitating propellers, is preferable.

GEMS AND HYDROFOIL BOATS

With the aid of Figs. 1 and 5(a) it is possible now to make estimates of size and shaft
power for the design of high-speed craft such as GEMs (hovercraft) and hydrofoil boats.

Let us for the moment assume a reasonable and realistic power coefficient P/V,A =
0.20. Then it follows from Fig. 5(a) that for an 80-knot hydrofoil boat with a l-meter-
diameter screw the maximum installed shaft power would be in the order of 4,000 shp,
whence it follows that for a single screw craft a displacement of about 40 tons could be
propelled. Therefore, for 80-knot speeds, hydrofoil boats will thus be limited, even with
the installation of more screws, to cargo carrying capacities of from 40 to 120 tons.

33

Size, Type, and Speed of Ships in the Future

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If we continue this estimate of dimensions of prototypes in the same manner for a hover-
craft, such as, for example, the 400-ton Crewe and Eggington project, then a power coeffi-
cient of P/V,A = 0.15 appears to have been chosen by the designers (P = 40,000 hp, V, =
100 knots, A = 400 tons).

A rough weight distribution might appear as follows:

800 passengers plus 80 motor cars 160 tons
40,000-hp turbine, at 1.5 kg/hp* 60 tons
Fuel for 24 hours 130 tons
Hull 50 tons

With a diameter of 100 meters (air cushion pressure 50 kg/m?) the weight of the struc-
ture becomes 5 kg/m2. A further study of the lifting mechanisms and possible structural
forms will be necessary to indicate how much the hull weight has to increase at the expense
of the pay load.

If the supposition is made that 50 to 60 hp per ton of displacement are required for
hovering, then from 20,000 to 24,000 hp will be required to maintain the air cushion. How
much special provisions such as Weiland’s labyrinth seal can improve this situation remains
yet to be seen. We will therefore use 20,000 to 16,000 hp to give the 400-ton GEM a 100-
knot speed. The important question which then pre~>nts itself for the case of a seaborne
GEM is whether to use air or water propulsion. In Fig. 7 the optimum diameter and effi-
ciency of supercavitating propellers in water and ducted propellers in air are compared
against a basis of B,. From this it appears that the diameter of the ducted propeller in air
is approximately three times as great as that of a comparable supercavitating screw in
water. The efficiency in air is approximately 40 percent lower than the efficiency in water.

Proceeding from an assumed propulsive efficiency of 0.75 in water and 0.45 in air, then
for the available power of 20,000 hp and a speed of 100 knots, the following resistances per
ton of displacement can be overcome:

55 kg/ton for water propulsion
33 kg/ton for air propulsion.

Whether it will be possible to choose the hull form and the operating height of the GEM
so that these values can be reached, the future will have to show us. If we consider the
water resistance negligible at these high speeds and assume the transverse sectional area
of hovercraft and air cushion to be 600 m2, then a resistance of 55 kg/ton in the case of

water propulsion implies a drag coefficient of 0.23, which is comparable to that of an
automobile.

In view of the large difference in efficiencies for water and air propulsion for a 100-
knot GEM, water propulsion for a seaborne GEM should be considered. However, from these
calculations for high speed GEMs one might be inclined to consider displacements, as pre-
dicted for hydrofoil boats, in the range from 40 to 120 tons to be more attractive.

Summarizing this introductory paper to this Symposium on High Performance Ships the
following might be concluded.

*See Fig. 6.

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Size, Type, and Speed of Ships in the Future 39

For displacement ships (cargo), the increase in ship speed will stop between 20 and 25
knots. Above these speeds, the nuclear powered submarine will get its chance (future). The
maximum draft will be limited to 10 to 11 m by harbors and docking facilities.

For submarine vessels (merchant ships), the limitations in horsepower per shaft and in
available space for engines, especially athwartships, results in a maximum deadweight for
submarine of 40,000 tons at a ship speed of 37 knots.

For hydrofoil boats, extremely high speeds such as 60-80 and 100 knots seem to be
obtainable. The displacement of such high speed hydrofoil boats will be restricted to about
100 tons or less.

For GEMs, displacements greater than 100 tons combined with extremely high speeds
(100 knots) lead to serious construction and propulsion difficulties. Propulsion in water
seems to be attractive for a seaborne GEM.

DISCUSSION

R. N. Newton (Admiralty Experiment Works)

My remarks bear upon the papers presented by both Mr. Oakley and Dr. van Manen and
concern those patts which deal with hydrofoil craft and hovercraft.

The considerable experience already acquired with hydrofoil craft leaves no doubt that
one clear advantage of this type of craft over the orthodox displacement or planing types is
the ability to maintain higher sustained speeds in a seaway. The same advantage, associ-
ated with higher speeds still, is being claimed for the more recent hovercraft or GEM.

One question which has yet to be resolved, however, is the extent to which this partic-
ular advantage applies; that is to say, it is necessary to qualify the words “in a seaway” to
cast a true comparison with, say, the planing craft. The moral can be pointed, in humorous
and yet serious vein in the present context, by this quotation from the writings of Hilaire

Belloc:

“The water-beetle here shall teach
A sermon far beyond our reach;

He flabbergasts the human race
By gliding on the water’s face
With ease, celerity and grace;

But if he ever stopped to think

Of how he did it, he would sink.”

Whatever form the humanly conceived “water-beetle” may take, I submit the time has arrived
to think seriously as to what state of sea would force it to stop and whether it would, in
fact, then sink.

Assuming no breakdown of the engines to occur the craft can be forced to land on the
sea from two principal causes:

40 J. D. van Manen

1. Encountering waves of length and height in which it is not designed to operate.
Statistical or spectrum analysis of the sea conditions in the operating area can provide
much data to guide the design of the craft as regards lift, stability, control, and response.
It can also provide an assessment of the probability of meeting isolated severe waves, but
these cannot be predicted in the true meaning of the word.

2. Impact with such isolated severe waves, or even waves over which the craft is
designed to operate if, for instance, a defect develops in the control system. In this con-
nection, it should be noted that while models of the size normally associated with ship tank
work can be used to measure the external forces, an assessment of the stresses induced by
these forces is a matter for the structural designer. To relate the stresses measured in the
full scale craft directly to those measured in a model, the latter would need to be of the
order of 1/2 to 3/4 full size. So far as ] am aware, no attempt to carry out investigations
of this nature has yet been made, but presumably data available from flying-boat investiga-
tions would provide valuable guidance. It is significant also to add that the structure of
hydrofoils and GEMs can hardly be regarded as “robust” in comparison with displacement
or planing craft.

Having been forced to land in severe conditions from any such cause as | or 2, or due
to engine failure, the question arises: Can the craft withstand the normal bending and shear
forces imposed in a seaway, or the impactive forces due to slamming? The answer to this
question also has yet to be resolved but in the absence of data on the subject, it is fairly
sound to assume that present designs would need to be strengthened.

It is realized these observations may be regarded by some as being very pessimistic.
At the same time, it is necessary to place the possibilities of high performance craft in their
proper perspective in relation to what can already be achieved by displacement craft in
unexpected severe weather conditions. The simple fact is that the planing or displacement
craft can successfully sustain any sea condition, and proceed, even though the speed may
be considerably reduced. Actually the reduction of speed is not so serious as is sometimes
implied. Comparisons can be odious and figures can be exaggerated in the absence of fac-
tual data. Nevertheless it is my opinion that, for instance, size for size, the planing craft
will still be capable of proceeding at nominally high speeds in conditions when the hydro-
foil or the GEM is brought virtually to a standstill. In such conditions it is the planing
craft which has an overwhelming advantage. Incidentally it must not be forgotten that a
planing craft to carry 50 to 60 passengers at speeds exceeding 50 knots in calm water now
offers no design or constructional difficulties.

The conclusion to be drawn from these few comments on this one aspect of the possi-
bilities of high performance surface craft is that there exists a real need for investigations
to determine more precisely their limitations in irregular sea conditions. This entails not
only the advancement of theory and model experiment but also correlation with full scale
results in similar conditions at sea—a long and arduous task following the lines of what is
already being attempted for displacement craft. Costly and time consuming though the task
may be it is quite essential to carry it out if any real comparison is to be made between the
orthodox and unorthodox type of craft. If it is not carried out then the risk must always
remain that a craft designed for some specified state of sea will meet more severe condi-
tions and suffer serious damage or even loss, and as so often happens with new ventures,
this could cause a serious setback, if not cessation, in development.

Size, Type, and Speed of Ships in the Future 4]

E. V, Lewis (Davidson Laboratory, Stevens Institute of Technology)

I wish first to compliment Mr. Oakley and the Bureau of Ships on their far-sighted view-
point in considering such a wide range of possible vehicles for military use. The escort
research ship in particular I am certain will be a significant milestone in naval ship design.

In its own way the modern supertanker considered by Dr. van Manen has been a truly
spectacular development. In this connection it is rather surprising to find in Fig. 2(c) that
the author expects to maintain a constant overall propulsive coefficient of about 71 percent
in ships up to 150,000 tons deadweight. Presumably he is assuming single screw vessels,
and I would expect a reduction in efficiency as size continues to increase. My reason for
expecting this trend is that for geometrical similarity the propeller diameter should go up
and rpm down in order to maintain good propulsive efficiency. However, in actual ships
practical considerations seem to have limited propeller diameter and prevented reduction in
rpm. Dr. van Manen has apparently counted on the application of improved sterns and
nozzles to maintain the efficiency of the heavily loaded propellers of larger and larger
ships. Perhaps he will care to comment on this point.

J. D. van Manen

In reply to Prof. Lewis I will mention a total efficiency of about 71 percent in Fig.
2(c) is only based on statistical data.

H. Lackenby (The 3ritish Shipbuilding Research Association)

I should like to raise one or two points on Dr. van Manen’s paper concerning the speeds
and powers of displacement ships.

Firstly, it is stated in his Introduction that up to 1940 the maximum shaft horsepower
which could be installed per shaft in a merchant ship was about 10,000. I am sure the
author is referring here to single screw ships and I think perhaps that this ought to be
stated. For multiple screw ships much higher powers per shaft were, of course, installed.

In this connection it is also stated that 16,000 shp is now normal and that 20,000 is
frequently encountered. This is followed by the statement that at the same time tanker
speeds have increased from 10 to 17 knots. This is quite true of course, but I think it
might be mentioned that this speed increase was largely a natural consequence of increasing
size and that in many instances it was a question of keeping the speed-length ratio about
the same and increasing the speed according to the square root of the length. In other
words there was a significant speed advantage in having a longer ship.

The author’s references to the larger block coefficients now being associated with the
high deadweight supertanker are of particular interest, and it is interesting to recall how
views on this have changed, bearing in mind that between the wars one authority advised
that no seagoing ship should have a block coefficient greater than 0.75. According to Fig.
2(b) we find that the largest block coefficient for the Wageningen models plotted there is
about 0.825. It might be of interest to mention here that the British Shipbuilding Research
Association in conjunction with Ship Division, NPL, are now endeavoring to develop ocean-
going forms with block coefficients up to 0.85.

42 J. D. van Manen

Next, a point about the economic comparison between a conventional and a submarine
tanker. It is stated in the paper that for 20,000 tons deadweight and a speed of 20 to 25
knots the submarine tanker would have a distinct advantage. One has to be careful here of
the basis of comparison—I think this is intended to be a surface tanker of 20,000 tons and
20 to 25 knots speed. This of course is not conventional as regards speed and would cer-
tainly be less economical than a commercial tanker of this deadweight which would have a
speed round about 15 to 16 knots. I am speaking here, by the way, of economy on the basis
of cost per ton-mile of cargo carried. On the other hand, from the military point of view,
there may of course be some advantage in carrying the cargo at the faster speed.

J. D. van Manen

In reply to Mr. Lackenby, the 10,000 hp before 1940 refers to a single screw ship.

D. Savitsky (Davidson Laboratory, Stevens Institute of Technology)

The presentations by Mr. Oakley and Dr. van Manen on the subject of high performance
ships of the future were indeed stimulating and exciting. The variety of hydrodynamic forms
being considered by the authors is expressive of an open mind toward the consideration of
solutions to the problem of high speed transport across the ocean.

Common to each of the vehicle forms considered by the authors is the serious problem
of negotiating the rough surface of the ocean at high speed—a problem where severity
appears to increase exponentially with speed. Partial solutions to the rough water problem
are proposed by the use of submerged hull form, hull forms lifted clear of the water by hydro-
foils and by ground effect machines flying just above the waves. In each case the roughness
of the sea is still a serious disturbance, although to a lesser extent than for a displacement
vessel. Clearly, the solutions proposed by each of the authors consist in a vertical dis-
placement from the water surface.

I would like to carry this suggested solution to its logical conclusion and suggest that,
for high speed transport over waves, let’s leave the water surface entirely. If our mission
is to transport material or people from one port to another at high speed, there is obviously
no necessity for exposing the high speed vehicle to the serious disturbances of the sea for
the entire length of the voyage. Obviously this suggestion leads to the rediscovery of the
water based aircraft—especially designed for short range, low altitude, and relatively low
speed operation (for an aircraft)—perhaps 200 to 300 miles/hr. Ordinarily I would not pro-
pose such a concept at a meeting on Naval Hydrodynamics. However, since the authors
have discussed ground effect machines operating just over the waves at speeds of 100
mils/hr or more, I feel they have strayed sufficiently from the area of conventional hydro-
dynamic forms to allow me the freedom to do likewise. I feel that as the GEM concept is
developed further and further, its appearance will be more like that of an aircraft. If this
is to be the case, let us immediately consider the role of the water based aircraft for the
short range transport vehicle. A well designed seaplane can easily develop a lift-drag ratio
of 20 at altitude. Perhaps this can be increased by flying at low altitude (in moderate
ground effect) yet out of the range of the severe wave disturbances of a sea state. To
obtain equivalent lift-drag ratios, the GEM would probably need a static thrust augmentation
factor of nearly 40, which would necessitate its operation at very low height-diameter ratios.
For operation in a sea state of moderate severity, then, the length of the GEM would prob-
ably be upwards of 500 or 600 feet in order to achieve the required height diameter ratios

Size, Type, and Speed of Ships in the Future 43

and still just clear the waves. Clearly such a large GEM craft would offer many practical
design difficulties—and much research will be required to perfect its development.

On the other hand, research in the hydrodynamics of water based aircraft has been
supported by the U. S. Navy for over 25 years and the success of this concept has been
demonstrated time and again in the form of numerous prototypes which have been success-
fully designed. If the large GEM is to be considered for oceanic transport, I submit that a
very hard and serious reexamination of the water based aircraft be considered for the same
mission. Obviously the hovering ability of the GEM is not contained within the capabili-
ties of a specially designed seaplane—but I have not heard the authors contend that this
was an important consideration in their studies. Certainly, the GEM will find usefulness
as a special purpose vehicle, but as a general working member of the family of everyday
transport vehicles, perhaps other concepts will excel.

H. P. Rader (Vosper Limited)

The authors did not mention a type of high speed craft in which we are particularly
interested, i.e., the hard chine planing craft. I think this may be of some interest because
we have achieved speeds in excess of 50 knots with hard chine planing craft which have
been in service for some time. I cannot say offhand how the performance curve of our craft
compares with the performance curve of the other craft shown by the authors, because we
did not have the papers in time for carrying out such a comparison. May I, therefore, sub-
mit the following table for your information and consideration. The figures quoted apply to
a hard chine craft weighing about 100 tons.

Speed (knots) Horsepower/ton
40 71
45 87
50 105
55 126

One more point concerning propellers. In the section on Types of Propellers Dr. van
Manen states, “For reasonable application of a supercavitating screw with a diameter of 1
meter, a minimum speed of 54 knots is desirable.” In our opinion this minimum speed is
much lower, I would say between 40 and 45 knots, depending on the rate of advance or, to
be more precise, on the local cavitation number at which the blade sections have to work.

J. D. van Manen

In reply to Mr. Rader, although I failed to mention the hard chine planing aircraft in my
paper, I am sure they will be included in subsequent papers.

Owen H. Oakley

I agree with Mr. Savitsky that the water based seaplane operating in ground effect is a
logical configuration to consider for the ram wing GEM concept. The very high L/D values
which are theoretically obtainable are most intriguing, and I feel confident that this area
will receive increased attention as the GEM concept develops.

44 J. D. van Manen

For ignoring the hard chine planing craft in my discussion | apologize; it would have
been of interest to make a comparison of this type with the hydrofoil and the GEM. How-
ever, I have made a rough comparison of lift over drag based on Mr. Rader’s figures. For
this I took the liberty of interpreting horsepower per ton as shp over displacement and
applied an assumed propulsive coefficient of 0.5 to get at ehp and drag. This gave L/D
values as follows: 7.8, 7.2, 6.6, 6.1. These are about in the low hydrofoil range but prob-
ably exceed what hovering GEMs may be expected to attain.

With regard to Mr. Newton’s comments concerning control, motions, and performance in
a seaway for all these craft, I must agree these matters are critical. I support his plea for
more research in this area. The strength of ground effect craft resting on the water is of
great importance. Since the weight available for structure in these craft is not great, sophis-
ticated structures similar to those of aircraft will have to be used. Even with the advanced
technology of the aircraft industry this problem promises to be formidable.

Mr. Newton quoted an appropriate piece of poetry regarding the ability of the water
beetle to stay afloat as long as he did not stop to think about it. I wish to respond in kind
with a verse which has a similar theme but which bears on a situation we find ourselves
faced with in this delightful country, namely the quantities of delicious food we encounter:

“Kat all kind nature doth bestow
It will amalgamate below;

But if you once begin to doubt
The gastric juice will find it out.”

ON THE EFFICIENCY OF A VERTICAL-AXIS PROPELLER

arenberg

J. A.
Netherlands Ship Model Basin

In order to minimize the kinetic energy left behind in the wake of a
vertical-axis propeller the oscillating motion of the propeller blades is
discussed. The blades are assumed to be infinitely thin and infinitely
long. The chord of the profiles is small with respect to the curvature of
the cycloidal orbit of the blades. Also the case of a propeller with many
blades or with a high rotational velocity is considered.

1. INTRODUCTION

The oscillatory motion of the blades of a vertical-axis propeller is governed by a system
of rods and hinges [1]. This system prescribes the angle of incidence of the blades when
they perform their rotational motion. The forces which act on the blades have been calcu-
lated by Isay [2], making approximations which give rise to some doubt and which we dis-
cuss at the end of section 6 of this paper.

One of the difficulties in a theoretical investigation is the complicated wake which is
crowded with vortex layers. Another difficulty, which is discussed in section 2 is that for
rigid blades with a finite chord no rigorous linearized theory for the perturbation velocities

can be given. This is caused by the varying radius of curvature of the cycloidal orbits de-
scribed by the blades.

In this paper we do not assume a prescribed oscillatory motion of the blades. We de-
scribe a method to construct the angle of incidence of the profile, as a function of position,
in such a way that the kinetic energy which remains in the wake becomes a minimum. The
following simplifying assumptions are made. The blades are infinitely thin, tip effects are
ignored, and the chord of the blades is assumed to be small with respect to the smallest
radius of curvature of the orbits of the blades.

The kinetic energy of the wake is expressed in terms of the bound vortices. Then by
the Ritz-method this energy is minimized. The following property (section 4) of the bound
vorticity of the blades, under the conditions just stated, will be proved: In order to obtain
the highest efficiency, it is necessary that the sum of the bound vorticities of a blade in its
two positions on a straight line parallel to the direction of translation of the propeller, be

Note: The results presented in this paper belong to research sponsored by the Office of Naval Re-
search under Contract N 62558-2630 with Ned. Scheepsbouwkundig Proefstation.

45

646551 O—62 i)

46 J. A. Sparenberg

constant. For instance, this constant may be taken equal to zero; then the blades pull as
much as they push. It is also possible to give the constant such a value that the profiles
are more active when they are in the front position.

Also the case of a vertical-axis propeller with many blades or a high-rotational velocity
is considered. Here the bound vorticity and the vortex layers in the wake are approximated
by continuous vortex densities. Then we find the following condition on the bound vorticity:
In order to obtain the highest efficiency it is necessary and sufficient that the difference of
the bound vorticity density for two points lying on a straight line parallel to the direction of
translation of the propeller is a constant (section 6). In this case it is possible to write
down explicitly the angle of the incidence of a blade as a function of its position.

It is intended to compare in a future paper numerical results of this theory with experi-
mental results.

2. STATEMENT OF THE PROBLEM

We consider an unbounded fluid which is at rest relative to a Cartesian coordinate sys-
tem x, y, z. In the direction of the positive x axis moves a circle with radius R (see Fig. 1).
Its centre is on the x axis and has a velocity
U. On the circle are M equally spaced identi-
cal blades perpendicular to the x, y plane.
The blades rotate with a constant angular
velocity @ around C. Besides this they ex-
ecute an oscillatory motion around the point
T in order to preserve the desired angle of
incidence. We simplify our considerations,
as stated in the introduction, by assuming
each blade to be two-sided and infinitely
long.

The orbits of the pivotal points T of the
M blades are the cycloids C_, with

x
II

R (uy 4 siny - rms \

us M
Fig. 1. Scheme of a vertical-axis -
i b
propeller with M blades y, = Ros y, “= . (2.1)

Several types of cycloids, as a function of p, are drawn in Fig. 2.

As in lifting surface theory for airplane wings or ship screws we try to refer the profiles
to a surface which moves through the fluid without disturbing it. For an airplane wing this
surface is the projection of the wing on a certain adjacent flat plane with zero angle of at-
tack. For a ship screw it is the projection of the blades on an adjacent helicoidal surface
which does not disturb the incoming fluid when it rotates with the rotational velocity of the
screw. In the case of a vertical-axis propeller a rigid reference surface does not exist be-
cause the curvature of the cycloids is not constant. For this reason we start with a deform-
able line segment A — B covered with bound vorticity (Fig. 3), which moves exactly along
the cycloidal orbit of the turning point T.

Vertical-Axis Propeller Efficiency 47

| o (Mf (0,242 |

Fig. 3. The deformable profile A — B
along the cycloid

On this cycloid the turning point T corresponds to the parameter value f = m = wt. Al-
though the segment A — B changes its radius of curvature we assume the distance s of some
point of the profile to the point T, measured along the cycloid, to be constant. Trailing and
leading edge are given by s = a and s=a+

The magnitude of the velocity V(Q) of the points of the profile is assumed to be equal

to the velocity of the pivotal point 7. Using the relation @ = wt for the parameter of position
of this point we find

Veo)i= oR /f + 22 Sen cos © (2.2)

The density of the bound vortices I'(~,s) is a function of the two variables 9 and s
with

48 J. A. Sparenberg

l'(9,s) + 0, azs<¢a+t+4. (2.3)

In the following we will use the notation

‘ a9, 2
SF LE: {SL =i’ (ons). (2.4)

When the elementary bound vortex I'(~,s) ds changes its strength when 9 increases, it
leaves behind at the place 9,s free vorticity of the density

" F(9,s) ds
2 1 (2.5)
RV1 + u* + 2n cos @

The total density y(P,s) of the free vorticity which is left behind at the point 9,s by
the bound vorticity of the profile which has passed this point is then given by

att =
¥(9,S) = - | : l'(@,0) do

EE OD Gee o (2.6)

After the partial differentiation in the integrand of (2.6) we have to consider } as a
function of 9, s, and o by

c= s+R[{ Jit p+ Qu cos x ox. (2.7)

6

A necessary condition on the bound vorticity follows from the mean value K of the x
component of the forces which act on the bound vortices. This mean value has to possess
some prescribed value which depends on the velocity of the ship. Calculating the angle X
(Fig. 3) and using the fact that the profile has the velocity V(p) we find for the M blades by
the law of Kutta-Joukowski =

27 att
/ 2
K = —a | | l'(9,'s) sin ge Vit w+ 2 cos Oo 4, do 6 (2.8)

0 2 V1 + uw? + 2u cos

where 8 = 0(9,s) by (2.7) with o = 0.

Another formula which will be used later on gives the difference of the value of the
velocity potential ¢(x,y) across the cycloid (points D,, D_, Fig. 3) when the profile has
passed. This difference at the place w is

ath
ew) - 0) = [ T(8,0) do. (2.9)

where @ = &(/,c), defined by (2.7) when we puts = 0 and ~ = w in that equation. In order to
use a potential for the whole fluid it is necessary that the lines covered with free vorticity

Vertical-Axis Propeller Efficiency 49

y(~,a) do not intersect each other. This can be obtained by a simple artifice; viz., in the
neighbourhood of points of intersection of the cycloids we assume that the bound vorticity

of the blades becomes zero. Then the bound vorticity of the profile is left behind as free
vorticity; however, a little farther, free vorticity of opposite strength is created which com-
pensates the effect of the first one. Then (2.9) can be used with arbitrary accuracy when the
distance between the two free vorticities just mentioned becomes small enough.

We now introduce two independent order quantities, viz., the length 4 of the chord of
the profile and I the total strength of the bound circulation of the profile at a reasonable
point. We consider the case

2
l1 - pl

an , as &4) (2.10)

hs te are 4 << R

where p # 1 means that the cycloid has no cusp, and the second condition means that the
chord of the profile is very small with respect to the smallest radius of curvature of the
cycloid.

The function I'(9,s) exhibits, as a function of s, large variations over the chord. In
fact, it becomes infinite at the leading edge s = a + % and zero at the trailing edge s = a.
As a function of 9 it is clear that the variations of I'(Q,s) are almost everywhere small when
4 + 0 and 9 changes with an amount of the order 4. In formulas:
(9, ste) + I(9,s) + eF'(9,'s),
‘ (2.11)
T(o+e,'s) = [(9,'s) + el(9,s), e= &4).

From this we can develop, with respect to 4 and I’, Eqs. (2.6), (2.8), and (2.9), which
are important for our theory. Only lowest order terms are taken into consideration.

We find from (2.6) for the density y(y) of the free vorticity at the end of the profile,
s = ain (2.6), within higher order terms

at+Z e
def DP. do
Mey = ylwed) == J vied (AMA) i EO serdenb
A RV1+pu?+2u cos 6
att

= ce Eas Sa | P(p,o) do
Ya

RV 1+ py? + 2u cos

Hegiry oe te PG) Fs OP):

212
RV1+ pp? + 2y cos w oe

The mean value of the component K in (2.8) becomes

277

RM
K = a | (9) sin 9 dg + QI), (2.13)

50 J. A. Sparenberg

while (2.9) changes into

P(Y) - d.(Y) = TW) + OCP). (2.14)

Hence with respect to the wake and to the mean value of K we can treat the profile with
a sufficiently small chord as a concentrated vortex (9). This is no longer true when we
calculate the angle of incidence of the profile; then we have to take into account the varia-
tions of the induced velocities over the chord (section 5).

3. THE KINETIC ENERGY LEFT BEHIND IN THE WAKE

We want to derive the potential of the cycloidal free vortex layers with density y(w)
(2.12). To this effect we start from the potential ¢*(x,y) of a row of equally spaced vortices
of equal strength. The potential of such a configuration is known [3, p. 186]. Because in
the remaining part of this paper we do not need the third space coordinate z, the symbol z
will be used to denote the complex variable x + iy. We find

$*(x,y) = Re xT ln sin a (z- (Gets) (3.1)

where y is the strength of the vortices, h their mutual separation distance, z the point in
which the potential is considered and €,, the location of some vortex (see Fig. 4). It can be
easily seen that this potential is not periodic, we find

-by, y>In£,
¢ (xth,y) - ¢*(x,y) = (3.2)
Vey ert

bNo|=

When we take h = 2myR and let the points ¢, =(€, + in,,) with m=1, ..., M, describe the
parts of the M cycloids lying in the interval 0 < x < 27pR, the other points of the rows de-
scribe the parts of the cycloids outside this interval. In this way we cover the whole wake
of the propeller with free vortices. The potential }(x,y) of the wake flow can be written as

y

Ym
()

Fig. 4. Row of vortices with streamlines

Vertical-Axis Propeller Efficiency 51

= Re aX Bia A ste: oo ila Ae ae che tings
(X,Y) = eg), fm n (sin Rp p+ 2p (3.3)
m=0 0
where
Cm = (Sm + ing) = R (ua + sin 6 - art 1 cos ®). (3.4)

Because the intensity (yw) of the bound vortex is a periodic function of w, the same holds
for the free vortex density y(~), and besides this the mean of y(w) over a ail O0<W<2z,
or over an interval of length 27pR in the x direction, is zero.

Then it follows from (3.2)

f(x+ 2mpR,y) - &x,y) = 0, ly] >R. (3.5)

When, however, |y| < R we have for the mean vorticity between y = +R and y over the period
27uR in the x direction

+ M{ (2m Y) = r(yy} if 5 eee (3.6)
Here by (3.2) we have

H(x+ 2muR,y) - oxy) = W{T(2n-y) - Te}, lvl < R. @.7)

Our aim is to determine the function I'(W) in such a way that the energy left behind in
the wake becomes as small as possible. To this end we consider the kinetic energy E of
the fluid in the strip 0 < x < 27pR, —0 < y <+00, which by Green’s theorem can be writtenas

HAGE) (Bla be festa. oe

The surface integral extends over a vertical strip of width 27yR, the line integral along
the vertical boundaries of the strip and along both sides of the cycloids (Fig. 2, dashed
lines). The normal derivative is directed into the enclosed regions.

We consider first the integrals along both sides of the cycloids. These integrals for M -
different cycloids have the same magnitude; hence we find by (2.14) for their combined con-
tribution E, to the kinetic energy E,

p, = - eft ij rey) 2S (yy J + u? + Qu cose ay (3.9)

where the line of integration and the direction of the normal to this line are denoted in Fig.
5. As in (3.9) we will not mention in this and the next section the order of accuracy of the
formulas. This will be considered again in section 5.

From (3.9) it follows that we have to determine the normal derivative 0¢6/dn. The unit
normal of the cycloid is

52 J. A. Sparenberg

1
1+ up? + 2u cos

n=(siny, 1+ cosy) * (3.10)

The velocity (x,y) which belongs to the poten-
tial } (x,y) (3.3) is

M-1 27
ee, ake
rr Amu j ¥(#)
m=0 0
* (Oe as) (=>)
$1 Tapp ese el) | ee
pR pR
sh s
Fig. 5. The path of integration HR LR

along the cycloid for m = 0

- J/1 + u2 + 2 cos 8 dd (3.11)

where €,, and 7,,, are defined in (3.4). By (3.10) and (3.11) we find for the normal velocity in
a point x, y, of the cycloid with m = 0 (2.1)

M-1 27
od 1
a ee
m=0 0
sin W sinh [2] - (u% + cos W) sin =
‘ OZ sau) (%0- Sm)
cos Pu Reds - cos a eat

abies tt py? + 2u cos # dp. (3-12)
v1 + py? + 2u cos W

By (2.12), (3.9), and (3.12) we obtain

M-1 27 27

e.=- oD | f rote

m=0 0 0

Sin w sinh vee) - (% + cos W) sin Sony
e -—|d@dy (3.13)

(Vos ia) (X,7 Sn)
ap ae ein os

cosh c
ns ‘aaqR brie
where x,, y, and €,,, 7, are defined by (2.1) and (3.4).

Next we consider the contribution of the vertical boundaries of the strip (Fig. 2) to the
line integrals (3.8) of the kinetic energy E. Because 0 (x,y)/dx is periodic for all values

Vertical-Axis Propeller Efficiency 53

of y and (x,y) is periodic for |y| >R the contributions of the integrals cancel each other for
ly| > R. This is no longer true for |y| < R. We obtain for the part E,, of E which belongs to
the vertical boundaries

+R
E,=- 5p i {$(0,y) - $(2muR, yy} FO) dy. ay

Using (3.7) and (3.11) we find after some reduction

. y-
et FP 0) U8) jain eae! sin py
Eo = - £— a Ee dt
Vv 87 (y'=7,,) “ ie ap ’ (3.15)
m=0 0 0 cos ar ae cos uR
where y = R cos y and €,, and 7, are from (3.4).
Hence we find for the total kinetic energy
27 © ,277 :
F=E,+E,= | I ay) T(&) Liy, 8) dp dé (3.16)
0 0
where
Cis a) eee
ol M-1) sin W sinh a a - (“+ cosw) sin (to usa!
ee ee eae
(¥,6) Sr = bei (Yo~- 7m) ii (x,-&n)
n= eo Ss —_—__—
pR HR
|
ae A me a
RSE ae Te a aS a a) ee
C4) be Saas
cosh ——_~ - cos —
LR

in which y = R cos w and x,, y, and €, 7, are from (2.1) and (3.4).

It can be seen that for p = $ this function possesses a singularity of the form

: aed 1
eae L(y,8) ~ 4 a (3.18)

Besides this singularity there are other ones which are the points of intersection of the
cycloids with themselves (yz < 1), with each other (M > 1), or with the vertical lines
x= 2knpR. The first term yields besides the singularity mentioned in (3.18) also singulari-
ties for

Veo Nes See Gos PATER, I Sy,

(3.19)
®< 27, yt, k=0, 1,

54 J. A. Sparenberg

and the second term yields singularities for
Y= 7, €—,= 2kapk, O <b, 8 < Qn, * k =hOscthink aoe

where as many integers k = 0, 1, ... have to be used as are compatible with the condition
0<w, 8< 27. The order of these latter singularities is

o{[(v. - p)* + (8, - 9)?] aa (3.21)

where WS, , denotes the singular point in the w,# plane.

4. THE VARIATIONAL PROBLEM
We want to minimize the energy E left behind in the wake over a length of a period
27pR. It seems most simple to start from (3.16) by using Ritz’s method. First, however, a

general property of the function I‘(y) will be derived.

We can split I'(W) into an even and an odd function with respect to = 7,

ry) = Te) + TQ), Porte) = Pry) .

(4.1)
The condition (2.13) then becomes
27

fee —— | Po(p) sin y dp= K (4.2)

0
which does not yield any restriction on I',(w). Using (4.1) the kinetic energy (3.16) becomes

27 277

Ds | {[r.w P(e) + Py) Pe) # [row r (8)

0 0
+ Top) ro] L(y,8) dp dd. — (4.3)
From (3.17) it can be derived that
L(,8) = -L(2n-, 27-9). (4.4)
By this (4.3) reduces to

27 27

E = | | rw) P(e) + Pop) r)| L(y,®) did. — (4.5)

0 0

Both terms between brackets in (4.5) yield a positive contribution to E, because each of
them represents the kinetic energy which belongs to some bound vortex function I",(W) or

ry).

Vertical-Axis Propeller Efficiency 55

From this result and from (4.2) we find that we have to take

ry) =0. (4.6)

Hence I(w) has to be an odd function with respect to w. We suppose

N
Mw) = Teg) 2 ah a, sin mp. (4.7)
n=1
Condition (2.13) yields
2K
a : P
Laas RM (4.8)
Using (4.7) the energy E becomes
N 27 27
5 Si De qa, 4, | | sin pw cos qét L(y,#8) dy dé. (4.9)
p,q=l1 0 0

Differentiation of this expression with respect to a, (n = 2, ..., N) yields

N
Poa
0a, q

277
q=1 0

27
J (q sin ny cos qé
0

+ n sin qy cos n@) L(y,#) dy dé = 0, Gnesi Dye eda do N) a w(4 0)
These are N — 1 linear equations for the N — 1 unknowns a,, ..., ay.

Without altering the kinetic energy in the wake we may add a constant value to the cir-

culation I'(w).

From the fact that the optimum bound vorticity (yw) has to be an odd function of W plus
an arbitrary constant we find the following property, mentioned already in the introduction,

(y) + T(-W) = const. (4.11)

5. THE DETERMINATION OF THE ANGLE OF INCIDENCE

Our aim is to derive formulas which determine the angles of incidence of the profiles
as functions of their position.

Because all blades are equivalent we consider the profile whose turning point T moves
along the cycloid C,. On this profile we introduce a parameter s* (Fig. 6) which measures
the distance of a point of the profile to the turning point 7; s* is positive in the direction
of V(~). We can calculate the normal velocities on the profile caused by the translation
V() and a rotation around 7, which have to be compensated by a suitable vortex distribu-
tion on the profile. Next we assume a kind of linearisation procedure which consists of
compensating the normal velocities not on the real profile, but on the cycloid. The points

56 ie ¥ J. A. Sparenberg

Fig. 6. The profile in the neighbourhood of the cycloid

~

of the real profile are represented by points on the cycloid in such a way that s* = s. The
leading and trailing edge of the profile are denoted on the cycloid by p and 9,.

First we consider the velocities induced by the bound vorticity of the other blades and
by the free vorticity along the cycloids C,,, m=1,...,M-—1. It is clear that for these ve-
locities and hence for their resultant component v, ,(9,,s) normal to the profile, we have
the relation

v,1(9,s) = QI), a<s<atf, (5.1)
At the end of this section it will turn out that this information about these velocities is suf-

ficient with respect to the accuracy with which we have approximated the basic formulas
(2.6), (2.8), and (2.9).

Next we consider the velocity 3,(9,z) with components v,, and v,,, induced by the
free vorticity along C,,

x,z y,z 1 J es C(@)]

<
J
hay
<
il

CCH.) :
T'(@) dl

Dinas i. [z- €(8)] R(uw + cos 0 - i sin #) © oe

The second integral formulation has the form of a Cauchy integral. We have to consider
the behaviour of this integral when z is in the neighbourhood of €(9,). This is a well-known
problem [4, ch. 4]; we find

tit dt vd Lennar oltre nb ook (Oeil I Zieral- Bedi
. i) pieleansio Are R(u + cos 9, - i sin 9,) the) ne)

Vertical-Axis Propeller Efficiency 57

where ¥/(z) is a bounded function for z + €(9,), which is of Q(T). Making use of (3.10) we
find for the normal component v, ,(9,s) of this velocity,

1 FC9,) In (s- a)

vv (9,8) = — - —_
Lin 21
RV1 + uw? + 2 cos on

‘S- a2

+ O(r) “S! wg) In (“Z*) + Ory . (5.4)

It will be assumed that the profile is infinitely thin and cambered. On the profile we
assume a vortex layer with density I, ,(9,s) which represents the sum of bound and free
vorticity. Hence we have the following equation:

att Feo (0:2) do

1 a 2 os
°° ri J Tata aay TEN Gag YOO) law) + POO ls ce "(oye

- wo) In (757) + 6) (5.5)

where Q@(Q) is the angle of incidence of the profile, f(s) is the camber, and 6(9) represents
the angle of the profile with respect to a line of fixed direction, for instance the x axis (Fig.
6). The angle of incidence (9) can be defined as the angle between the tangent to the pro-
file at the trailing edge and the velocity V(~). The angles a, 0, and 0; to be defined in
(5.6) are positive in the counterclockwise direction. The angle @(¢) can be split into two
parts:

e. =p. “4 sin Q
ADS Ore Os CN EA Sera gt (5.6)
For small values of ¢ Eq. (5.5) can be written in the form
1 att Poel O97) do , , ,
on | ERS SoS V( 9) [a( @) + f (s)] + [wO (9) + wa (9)] Ss

- wo) In (“F#)+ OF). 65.7

The inversion of this integral equation is well known [4, section 88]:

a+

oe A(9) 2 { : :
Pe(@s) = —— S++ v f
oo 0S) Stee Wan = SW eee ere ae | (9) [a(9) + #()]

= 949) + wa"(o)}o + wo) In (“Fe )

+ orp Yea eee vatt-o de | (5.8)

(so)

58 J. A. Sparenberg

where

att
pte) = | Tyo¢Co,) ds. (5.9)

We consider the asymptotical representation of I"
After some analysis we find from (5.8)

tot/?25) in the neighbourhood of s = a.

att

£9) 2 {
1 It a
SMG tie se pemeth reeroeee | Vo) [a(o) + F'(o)]

~ [63(9) + wa"(o)]o + wo) In (2)

" ory }VAt2=* do + 7? wo) VE ys- a |. (5.10)

We have to satisfy the condition that the velocity of the fluid remains finite at the trail-
ing edge. This is equivalent to the demand that the total vorticity I, ,,(?,s) for s = a equals
the free vorticity in the wake. Hence we find from (5.10)

att

B(9) = 2 | { vo) [a(p) + f'(c)] - @O;(9) + oa'(9)]

ee fei upie
+ w(9) In ae + ocr} vette? ae, (5.11)
and
- 27 Wo) = ¥(9,)- (5.12)

Equation (5.11) determines 8() in terms of the still unknown angle of incidence 9), while
(5.12) is satisfied automatically (Eqs. (2.12) and (5.4)).

The last condition we have to satisfy states that the integral over the bound vorticity
of the profile has to have the known value I'(9?). Hence first we have to calculate the bound
vorticity I'(9,s). We start from the definition of ’,,,(9,s)

Peot( S) = ¥(9,'8) + PQ, s) (5.13)

where y(Q,s) is the free vorticity passing along the profile. Using (2.6) we obtain

att
Poel S) Ss | (9,0) do

2 RV1 + pw? + 2ucos 6

+ T(o,s) + Q(ry 6.14)

Vertical-Axis Propeller Efficiency 59

where # follows from (2.7). However, the first term on the right-hand side of (5.14) is Q(I).
Hence we find, making use of (5.9),

att a+
(9) = i l'(9,'s) ds = | Peo t( OS) ds
+ OCT) = B(o) + OCT). (5.15)

From (5.11) we obtain

' me I'(9) AN te : ' =
) aca) ey = iii Ser ct

CS Wee
on V2(9) In R + 6(T),. (5.16)

Because 0= Q(['4"') the second term on the left-hand side can be disregarded. Besides
this by the choice a = —(1/4)¢ the coefficient of this term as well as the coefficient of
6,(9) vanishes. Then equation (5.16) changes into

i 2 [ . [2 ( s 3) | Se + Q(T). (5.17)

Hence, for a more easy analysis of the propeller, it can be recommended to place the
turning point T at one quarter of the chord length from the trailing edge.

From this we see that, within the accuracy of the theory, we do not have to take into
account the induced velocities (5.1), which give rise to a change of the angle of attack by
an amount of Q(I).

6. VERTICAL-AXIS PROPELLERS WITH AN INFINITE NUMBER
OF BLADES OR QUICKLY ROTATING PROPELLERS

When the number of blades or the rotational velocity of the propeller increases, the wake
of the propeller becomes crowded with free vortex layers. Hence it seems natural to replace
this complicated system by a continuous distribution of free vorticity. The bound vorticity
can be replaced by a continuous distribution over the circumference of the circle. We as-
sume for its strength per unit length the function I".(), where the index c indicates a con-
tinuous distribution.

For the free vorticity per unit length in the x direction, which is left in the strips be-
tween y and +R (see Fig. 7) we find

60 J. A. Sparenberg

Fig. 7. The propeller with a continuous
vortex distribution

+ = (9) = (27= 9} , y=Rcos 9. (6.1)

The velocity induced by two infinite strips with vorticity of strength given in (6.1) at
y=R cos 9 is

-= [r.co) - 1 (21-9)] . (6.2)

By this the energy in an arbitrary rectangle of unit length in the x direction between y = +R
and far behind the propeller becomes

R 2
= — [r. (9) - nie )| sin 9 dg. (6.3)
2" J,

The condition for the total component in the x direction of the force acting on the bound
vorticity is

27

pw R? T.(9) sin 9 dp = K. (6.4)
0

We have now to minimize (6.3) under the condition (6.4). Introducing the perturbed bound
vorticity

D.(9) + ©, 8,() + &, 6,(9), (6.5)
where g,(@) and g,(9) are arbitrary functions of Q, into (6.3) we have to differentiate E with

respect to €,, while condition (6.4) must be taken into account. Applying the multiplication
method of Lagrange we obtain

Vertical-Axis Propeller Efficiency 61

R aT
a | [r.(@) r r,(2n- 9) [ 4409) = é,(27- 9) sin 9 do
0
+ Apw R? \ 6&,(9) singdg=0 (6.6)
0

where A is a still unknown constant.
This equation can be written in the form
1 17 27
Fe za | [r. (o) 3 r(27- @)| 6,(9) sin 9 do
fe 0 7

27

+ A\wR 6,(9) sin gp dg=0. (6.7)

o

Because g,(9) is arbitrary we have

0<qQo<7

. 6.8
pe (6.8)

s [F. (@) = r,(27- 9)| == hau

From this it follows that the only condition which we have to satisfy is: the difference of
the bound vorticity at the front and the back of the circle for the same values of y is a con-
stant. Hence we may for instance assume I",(9) = -I",(27—@) = const. From (6.4) it follows

0 < <7
y (6.9)

K,
EE CO) eae Ha > :
This agrees with the result of section 4 where it was found that the optimum bound vorticity
was an odd function of 9 plus some arbitrary constant. In this case of continuous vorticities
it is allowed to add an arbitrary even function of 9 to the distribution in (6.9).

From the above it follows that the vorticity in the wake is concentrated wholly on the
lines y= +R. The strength y per unit length in the x direction becomes

ws iK
ae aa y= FR.
i 20 RU _

This means that the induced velocity u in the wake is, parallel to the x axis, independent of
y and amounts to

iain aS
2p RU : (6.11)

646551 O—62——_6

62 J. A. Sparenberg

This result can be checked easily by calculating the external force necessary for the in-
crease of momentum which follows from (6.11).

Isay [2] uses a formula, Eq. (1) in his work, which is not in agreement with our formula
(6.1). He gives also another formula, Eq. (12), which he thinks to be faulty, but which, in
the opinion of the author, is the correct one. That something is wrong with Eq. (1) of Isay
follows from the fact that in it the velocity of the incoming fluid does not occur. It. is clear
that when this velocity increases, the density of the free vorticity in the wake decreases,
when the bound vorticity distribution remains the same.

7. THE ANGLE OF INCIDENCE IN THE CASE OF
CONTINUOUS DISTRIBUTIONS

We first consider the induced velocity by the two half infinite equidistant rows of vor-
tices which form the wake. ee elementary calculations we find that the component of the
velocity v, 1, in the direction 7, normal to the cycloid, at some point 9 of the circle ESS:
7) amounts to

Mea pik AO ea UM NE ER le" sin 9
8oRU 1 + uw? + 2u cos op

1 1+ cos 9
Enea ae
— (4 + cos 9) In (js 2),

Pe: Oe OM ONES
Boas bl et ae ae (7.1)

The component of the velocity normal to the cycloid, induced by the bound vorticity (6.9) on
the circle, becomes

v = cos ta 1 + cos 9
n,2 ~ ~ ga ey 1 - cos 9 (7.2)
V1 + up? + 2 cos o

Finally we write down the normal component of induced velocities by a constant density Fs
of bound vorticity on the circle,

Py e
<b sin 9
Eo (7.3)

2/1+ uw? + 2 cos o

The velocities (7.2) and (7.3) are the mean values of the velocity components normal to the
cycloid in points just outside and just inside the circle. Hence we find for the total induced
normal velocity

Vina Vian estan (a at MB ene (7.4)

When the propeller consists of M blades, each blade has to possess a bound vorticity

Vertical-Axis Propeller Efficiency 63

an t+ 12) RS aE

D9) = = (t# 4 RU LA on Me ly (7.5)

The constant bound vorticity [’,* is still arbitrary and can be used also in this case to
determine the activity of the blades in the front or in the back position.

With respect to the tangent at the cycloid we obtain for the angle of incidence 4(9)

i Bla

Vip) =7 V9) 4

ap) = a, - (7.6)

where ©, is the angle of zero lift, V(@) follows from (2.2), and 4 is the length of the chord
of the profile. The angles 0(() and @, are positive when they open to the left with respect
to the direction of V(9).

Because we have not taken into account the velocities induced by the oscillating motion
of the blades we have to place the turning point T of the profiles at one quarter of the chord
length from the trailing edge, as follows from the end of section 5.

REFERENCES

[1] Mueller, H.F., “Recent Developments in the Design and Application of the Vertical Axis
Propeller,” Trans. S.N.A.M.E., 1955

[2] Isay, W.H., “Zur Berechnung der Stromung durch Voith-Schneider-Propeller,” Ing. Arch.
XXIV Band, 1956

[3] Kotschin, N. J., Kibel, I. A., and Rose, N.W., “Theoretische Hydromechanik,” Band I,
Akademie Verlag, Berlin, 1954

[4] Muskhelishvily, N.J., “Singular Integral Equations,” 2nd ed., transl. P. Noordhoff, N.V.,
Groningen, 1954

DISCUSSION

G. Weinblum (Institut fiir Schiffbau, University of Hamburg)

I wish to raise the question, who should decide when mathematicians disagree? I have
not understood what you have done. Have you calculated some numerical values so that we
can compare what comes out?

J. A. Sparenburg

This is a case in which mathematicians do not agree upon the interpretation of the
basic formulae needed for the theory. In my opinion the taxation of the various ways of ap-
proach has to be made by executing experiments, not only of the over-all results but also of
the basic formulae. We have made some calculations based on the simple theory of sections
6 and 7. Before publishing results, however, we shall compare them with results based on
the more refined theory of sections 2-5 and on experiments.

64 J. A. Sparenberg

W. H. Isay (Institute of Applied Mathematics of the German Academy of Science)

In reply to Dr. Sparenberg’s discourse on my papers* dealing with the theory of the
Voith-Schneider propeller I should like to remark the following:

Equation (12) of my paper II mentioned by Dr. Sparenberg is obtained by the usual meth-
ods of potential flow analysis if it is assumed that the free vortices move to infinity down-
stream of the propeller without losing their intensity. In reality, however, these vortices
will decay in the real turbulent flow downstream of the propeller and both their influence
and their induced velocity will decrease accordingly. The use of Eq. (12) will therefore
yield grossly erroneous results, particularly for the blades on the downstream half of the
propeller which are hit by the free vortices produced by the blades on the upstream half of
the propeller. This is because the influence of the free vortices is greatly exaggerated by
Eq. (12) of potential theory.

On the other hand, any attempt to arrive at an actually comprehensive theoretical de-
scription of the turbulent mixing and decay of the free vortices would have little chance to
succeed as the process is not only different at different points but also subject to random
effects. It was therefore deemed important to evolve a physically reasonable equivalent rep-
resentation of the velocity field of free vortices which may be used also for numerical calcu-
lations without excessive difficulty. Equation (1) of my paper II, or Eq. (3) of my paper I, is
satisfactory under this aspect though naturally different from the conventional formulae of
potential theory. This formula yields useful results which are in satisfactory agreement with
force measurements. A full discussion of the difference between Eqs. (1) and (12) along with
numerical examples can be found in sections 2, 5, and 6 of my paper II.

The problem thus encountered will be of interest with all types of propellers and turbo
equipment in which part of the blades operate within the wake vortices produced by other
blades. This, however, shall not imply that the method I have shown for Voith-Schneider
propellers can be applied without change to other types of hydrodynamic machinery.

Contrary to the opinion held by Dr. Sparenberg the induced velocity uy of the free vor-
tices decreases with increasing incoming velocity u, also in the case of Eq. (1) of my paper
II. This is caused by the resulting change in the distribution of circulation as can be read-
ily seen from the numerical examples given in my papers. Furthermore, this reduction in uy,
for increases in u,, is not more pronounced with Eq. (12) than it is with Eq. (1) as might
erroneously be concluded when considering the initial factor w#R/u, without at the same time
observing the behaviour of the circulation. The latter becomes apparent only from the bound-
ary condition of flow past the profile, i.e., from the integral equation.

Naturally it is possible also with my theory to determine — similarly as has been shown
for the problem treated by Dr. Sparenberg in his paper — the distribution of the angle of inci-
dence for a prescribed relationship for the change of blade circulation as the blade proceeds
on the propeller circle. Also in this instance the difference between Eqs. (1) and (12) of my
paper II becomes apparent. If, for example, a simple cosine law is prescribed for blade cir-
culation (in which case the propulsive force produced by the upstream and downstream halves
of the propeller would be about equal), then the incidence angle distribution obtained from

*(1) “Zur Behandlung der Strémung durch einen Voith-Schneider-Propeller mit kleinem Fortschritts-
grad,” Ing. Arch. 23:379 (1955); (II) “Zur Berechnung der Stromung durch Voith-Schneider-Propeller,”
Ing. Arch. 24:148 (1956); (III) “Der Voith-Schneider-Propeller im Nachstrom eines Schiffsrumpfes,”
Ing. a ge (1957); (IV) “Erganzungen zur Theorie des Voith-Schneider-Propellers,” Ing. Arch.
26:220 (1958).

Vertical-Axis Propeller Efficiency 65

Eq. (1) is quite reasonable. In fact it is found that, apart from the obvious change in sign,
the angle of incidence for the blades on the downstream half of the propeller must be about
1.3 to 1.8 the value for the upstream half, depending on the advance ratio and the total lift.
On the other hand, if Eq. (12) is employed, the resulting incidence angle pattern for the

downstream side is completely senseless as it includes abrupt changes and even infinities.

It must further be noted that one might consider describing the mixing and decay of the
free vortices not by Eq. (1), but by introducing the known Lamb’s law* describing the decay
of vortices in a real liquid into Eq. (12). I have abstained from this for two reasons: First,
Lamb’s law holds only for isolated vortices in purely laminar flow whereas propeller flow is
turbulent, and second, the mathematical realization of this method would have been rather
complicated without enabling a definitely better physical description of the problem.

While this problem is discussed here to such an extent, it is scarcely considered in Dr.
Sparenberg’s paper. Equations (5.1), (5.2), and (5.3) used therein for angle of incidence de-
termination will render approximate solutions for the upstream half of the propeller blade cir-
cle only. One must keep in mind that the blades on the downstream side are exposed to the
free wake vortices induced by the upstream blades, and that on the downstream side the
velocity field of these free vortices cannot justly be omitted from the boundary condition for
the flow past the profile since it exerts a decisive influence.

In closing I should like to remark that in my opinion the theory of quickly rotating pro-
pellers evolved by Dr. Sparenberg in sections 6 and 7 with circulation (6.9) and free wake
vortices (6.10) has the disadvantage of introducing excessive simplification. It is therefore
probable that this theory will not render physically reasonable results. This can be seen
already from the fact that Eq. (7.6) for incidence angle @ yields & = exactly for 9 = 0 and
Q = 7, whereas in reality the value of @ will be relatively small or even zero for these values
of 9.

J. A. Sparenburg

Equation (1) in paper II by Prof. Isay is the basis of his theoretical work. This formula
describes the velocities induced by free vortices which are shed by a system of rotating and
periodically varying bound vortices, placed in a homogeneous stream. The density of the
free vorticity decreases when the velocity of the incoming stream increases. Hence it may
be expected that the induced velocities of the free vorticity decrease in this case. However,
in the above mentioned Eq. (1) the incoming velocity does not occur, which is surprising.
The remark of Prof. Isay, that the increase of the incoming velocity causes a decrease in
the free vorticity by changing the bound vorticity, describes a secondary effect. Here the
basis formula itself and not its relation to a special boundary value problem is discussed.

Prof. Isay further questions the validity of author’s formulae (5.1), (5.2), and (5.3) for
the downstream side of the circle. When we look at (5.2), for instance, we see that a pole
of the integrand crosses the line of integration when the point z, where we consider the in-
duced velocities, crosses the vortex layer. This means a discontinuity of the value of the
integral, which corresponds to the discontinuity of the velocities at the vortex layer. Hence
with respect to potential theory, this formula is valid also in the downstream region.

*H. Lamb, “Lehrbuch der Hydrodynamik,” 2nd ed., Leipzig, 1931, p. 669

66 J. A. Sparenberg

The remark of Prof. Isay that the theory for the quickly rotating propeller (sections 6
and 7 of my paper) is too crude may be true. It does not follow, however, from the fact that
the angle of incidence becomes infinite at the places p = 0 and 9=7. The regions where
@ is large are so small that they can be neglected by the realisation of a model. As is well
known in many linearised theories, there occur singularities at points where the theory does
not hold, for instance, the endpoints of the range of contact by indentation problems and the
leading edge by lifting surface theory. This does not affect the applicability of these the-
ories, when the influence of the singularities is restricted within narrow limits. Moreover
the results of the simple theory are not devoid of sense and show the behaviour mentioned
by Prof. Isay in his contribution.

It is, as Prof. Isay stresses, highly recommendable to investigate the interaction of
turbulence and free vorticity. However, in my opinion this must be done on the basis of
experiments or theory and not on the basis of a law stated a priori.

A SOLUTION OF THE MINIMUM WAVE RESISTANCE PROBLEM

R. Timman and G. Vossers

Netherlands Ship Model Basin

INTRODUCTION

The problem of the determination of ship hulls with minimum wave resistance has been
the subject of numerous investigations. The usual way of approach, based on a classical
paper by Weinblum (1930) is to consider a ship hull in the form of an infinitely deep cylinder
and to solve the variational problem of minimizing the resistance integral with prescribed
horizontal cross section of the ship. In Weinblum’s original paper a Ritz method is used;
the waterline is represented by a polynomial with unknown coefficients which are determined
from a minimum problem for a function of a finite number of variables. Computation of the
Weinblum-functions (1955) facilitates the procedure.

The method, however, is open to some criticism, since it is not a priori clear, that it
yields a good approximation to the variational problem.

It is known that Pavlenko (1934) reduced the variational problem to the solution of an
integral equation of the first kind. It is known that its solution has certain singularities,
which cannot properly be represented by a polynomial.

In this paper the problem is reduced to an integral equation of the second kind, obtained
by the following consideration. The resistance integral in the thin ship approximation is
known to be a quadratic functional of the hull function. If the additional condition is also
expressed by a quadratic functional, the problem is simply equivalent to a principal axis
problem and the solution exists, as is known from the general theory of quadratic functionals.
Essentially, it is assumed that the cross sections of the ship are similar in shape, which
(with a slight modification) produces the required result. This means, that the ship consid-
ered has a finite draft.

In order to obtain a simple integral equation, the influence of the bottom in the evalua-

tion of Michell’s integral is neglected, although a more correct treatment lies within the
power of numerical methods.

MICHELL’S FORMULA FOR THE RESISTANCE INTEGRAL

We introduce a coordinate system with the x axis along the axis of the ship, the y axis
athwartships, and the z axis downward. The ship hull is represented by a function

y= 2x, Zz) ; IES BIS & BID A QO Koes T

67

68 R. Timman and G. Vossers

where L is the length of the ship and T the draft. The ship is moving in the x direction with
constant velocity c. The disturbance potential 9, from which the disturbance velocities
u=—d9/dx, v=—09/dy, w=-09/dz are derived, is determined by the condition that at
the hull of the ship |

2 _. Of

Ape ee

The linearization, introduced by Michell consists in applying this condition at the plane of
symmetry y = 0, in which case the potential, satisfying the Laplace equation V7 = 0 and
the free surface condition at the plane z= 0:

Px ~ KO, = O

where
&
K = — e
'c2

The solution of this boundary value problem is then

9(X,¥,2Z)

2
a
en OsezD i a.
Ic ante z Of(x,, 21) fe e * *” sin [<x Xt y oa? 13]

Seek dx, [ 2 ax, ; ; a da.
-L/2 0 K USK
2 Oy
Fe To aay abuse
Le pes Pe (a posers Cag ipa: d
= xX, Zy ae cos a(x-x,) da
-L/2 0 4 0 Kk? - a?
L
2c ae f Of(x,,2,)
er | dx, | dz, ax J cos @(x-x,) da
DS 2 0 : 0
0
[2,2
x | eo" *” cos (nz-€) cos (nz,-€) dn
0 a? + n?
where
ne
tan ——
An

The pressure is determined by Bernoulli’s equation

1 i 2 Dp
-9, + 5|(-'c+ 0) Paget o2| rE eR oho as

Minimum Wave Resistance Problem Solution

or, in linearized approximation for steady flow,
Pi- P= SPC WL + pez
The force in the x direction, exerted on a surface element of the hull is

dK = - p 2 dxdz

where dxdz is the projection of the element on the x,z plane. The total resistance is

+L/2 C4) +L/2 @
Bay of( x, z) bes a of
ne ae | { p dxdz = 2pc { f So 2 axdz.
“*L/2 ‘0 -L/2 0

Substitution of the expression for @ gives for the resistance integral

+L/2 +L/2 @ ©

c2 OF(x, DECK. nz
R = ae dx dx. dz dz oe 2) pas 4
7 1 i Ox Ox,
-L/2 LL /2 0 0
2
a - 2 (2424) a? cos a (x- x,)
x e cP a ee Vee aie
K (ea ca

We introduce dimensionless coordinates

and the resistance integral takes the form

p'c2B2T2x2 +1 +1 +1 +1
ee Gri | bien | eaeas. ce ced) Bes, C,)
Beate Lv Lp ae eo ge

0 2
-A°KT(C+E,) NL
x e ‘ cos —5~ (€-€,)

d? dr

69

1 VA? - 1

70 R. Timman and G. Vossers

The volume of the ship is given by

+L/2 T

V= | dx | £(¢x,2)) dz.
“L/2 0

The problem then consists in the determination of the function f(x,z), which minimizes the
resistance at a prescribed volume.

SIMPLIFIED TREATMENT OF THE VARIATIONAL PROBLEM

We put the condition that f(x,z)=0 for x= +L/2 and transform the resistance integral
by integration by parts:

© @

‘c2B2T2,2,4 a
Coe te ae (ae ee a
-1 -1 0 0

NK T(CH,) =
= er Seeds as (é>4,)

1 py |

In order to simplify the problem, a special class of ship hulls is considered which is so de-
termined that both R and V appear as quadratic functionals of a function f(£). We assume

/y _ £2
f(E, 0) = rey a [OAS g |.

f(¢)

where f(&), the waterline function, is to be determined and g(y) is a fixed function of p
which is zero for » greater than a certain value. If the factor /1—€2 did not exist, the as-
sumption would mean that all cross sections of the ship were similar. If f(€) vanishes at
the front and the stern of the ship, also the draft would vanish there. This is prevented by
the present assumption, and moreover it will appear later that the addition of this factor ac-

tually yields functions f(€) which vanish at these places. The volume of the ship is given
by

! +1 (4) +1 2 oe
v=LT( ae. at (2 ‘[ohee | = LT) FO) Ge'( teqaylaae
2 | j f(é) 2 iy 1- €? j

The resistance is

Minimum Wave Resistance Problem Solution fil

A ee a ee +1 +1 oo 4) 1- 2
peur Ye | hs ye el ey nen «(=F
Li 1

f(é,)
0 0
i MK T( CHL) LK 4 dr
x e s

co - —___—.,
1 2 - 1

In order to obtain a simple integral expression, we neglect the dependence of g on € and
assume infinite depth. An estimate of the validity of this procedure is obtained by consid-
ering the case, where the cross sections are rectangular:

B(e) = 1, OPS a
&u)=0, w>il.
Then
ECS)
ViLes
bvi-€? \ rere titers caamsiciens
at 8 dt = e dC
RCE). S A
-\2KT£(E)
= i 1 => 3 V1 = 2
2 KT 2

which integral is simply replaced by 1/A°xT.

If necessary, the method can be refined, yielding more complicated integrals which can
be treated by numerical methods. The simplified expression for R is

2p2y7 2,2 . ™ cos ¥(6=€,)r
——— im «fo Be oe
1 2-1

or, using the formula

V[y(é-6))] = - 2( —— dn,

pic?B2L 2,2 +1 +1
ps eee ee { dé | a, M2) 4.) Y, [v.(E- £1] |

LD R. Timman and G. Vossers

where

depends on the Froude number. We introduce variables ® and ®, by cos #= €, cos #, = &
and have to minimize the integral

7 77
j= | ao { de, f(#) f(8,) sin ® sin @, ¥,(y,|cos 8- cos 0,1) = J(f,f)
0 fr)

with the condition

f(97) do=C.
0

REDUCTION TO AN INTEGRAL EQUATION OF THE SECOND KIND

Suppose that f(®) is the minimal function and consider a neighbouring function
f(®) + eh(*)
where € is a small parameter. Then
JCf+ ch) = J(f,f) + 2eJ(f,h) + ©7J(h,h)
and
V(ft+eh) = Wf,f) + 2eV(f,h) + €?V(h,h).
Since J(f,f) must be a minimum for f, with constant V, we consider
JC f+ eh) - VC f+ ch)
where \ is a Lagrangian multiplier. The expression
J(£+ eh) - W( f+ ch) = JC f, f) - WCF, F) + 2e(JCf,A) - AV(E,A))
+ e7[J(h,h) - \V(h,h)]
can only be an extremum if

JCf,h) - VC f,h)

Minimum Wave Resistance Problem Solution 73

vanishes for any choice of h. This means that f must be such that

77 77

| de | dé, f(®) h(#,) sin ® sin @, ¥.[7o(cos ® - cos 9,)|
0 0

- rf f(%) h(®) d® = 0
0

for all functions h(#), or

Tr T

\ h(#,) d#,<sin @, | f(%) sin $ Y,|7.(cos $- cos $,)] dé-)f(#,) >= 0.
0 0

This can only be true, if f(#) satisfies the integral equation of the second kind:

TT

df(®,) = |

f(®) sin ® sin ®, ¥,|7.(cos ® - cos *1)) dé.
0

Since the kernel is integrable, this is an ordinary Fredholm equation and the functions f(#)
are the eigenfunctions of the corresponding equation, while » is the corresponding eigen-
function. Apparently the solutions vanish for = 0 and $= 7, as was required. Moreover,
the eigenvalue 2 is the value of the quotient J(f,f)/V(f,f) for the eigenfunction. For the
ship hull only the first eigenfunction, which hag no zeros, will come into consideration.

SOLUTION OF THE INTEGRAL EQUATION

The integral equation

7
Af(#,) = ‘ f(®) sin 6 sin #, ¥,(y.leos $ - cos 9,1) dé
0

resembles an integral equation satisfied by the Mathieu functions. Therefore, at first a solu-
tion in terms of a Fourier series in #, with coefficients depending on y, was looked for.
This method, however, is only feasible for very small values of y,. For this reason, a di-
rect numerical approach was used.

{¢ 2 ROG

ai
e CaP! i pal a
a Seyret b Phun ny siigiinys ie Me denis or i Miss ny ” - #
o Miri whi ey Sit a talrd BAR ae Lean Ph Aye tht i) We

oO os OV ts ‘ys, (H yh) sg

% ‘

+ ' ea b en Fy Pitan ar ae ey ee ot ere ae 4 ‘ ‘ i
’ UY i ie { MB HARB: Be ih Mig F i a ned Hoe
‘i MT i

it ye) Worn ic a bicep ¢ #09), o ¥ a Al cay 1 - ale i)
fe ; Lh

f ; pee hs gt ae ae | Rie
By thntd bkooom od? to anitwnye leyeiel odd metedine (OY Hi niet iy
ey thant hi dis RC ACG ih HEMET M (Chk a R ce sey cp mee
L, J ee
Dagon Nisin CW) rEg, yoclvekur ake orn tl « aba adamant Fe

(OP) sigan odd hast omitenps web ays: seine thee oe el eeatd eohchinteevnial wi { sat
sericea gniiouc qanrioo al? ei A oblided ol geben Neel aves vinon pil lo etch ana

_ same cbantwpae ww ee.) a Gi Mine Oe & sol gaicaw aincilow add 1 sng
iat 94, yon coatia areal mene wet Lk AY nyt tamitomp mht a ember amp atl A ow
woltadoblemos otnt enon fiw vmowais seth dt rive poieroi’l ‘wait ds ody

A 2: 7

¥ { i eb wa 74 ri if 7 ae, : ey Yel? ie “i ra e

Sur

: es: sala Tt OLA oe ee get toe see Me Vid dpi .
hy (1,® eo2 #09 spe 1 & Sy i re e Ce) ;
| 4 Bin ie Pie angi hae iW
nh M ; ¢ seal it if , "|

asi sd ide cna ae wll Bet id, boitnitan panes ce
ak Bekool wan yy mo gat bewatols ednnk Tania 1 ni denial wba of

Dee, Sighs it aia 0, Oh bi i a in ier 3 i ives Hit,

pe

‘Seal a

j
ee}

thot! Coe Tn cake ee ans 4
‘ i ‘ he

Leth, i Rn

Sa
CEA SRS 00

ON THE PROBLEM OF MINIMUM WAVE RESISTANCE
FOR STRUTS AND STRUT-LIKE DIPOLE DISTRIBUTIONS

Samuel Karp,* Jack Kotik, and Jerome Lurye
Technical Research Group, Inc.
Syosset, New York

In this paper we consider the problem of minimizing the wave resistance
of a strut of fixed length and volume-per-unit depth at a given Froude
number. The work of others has shown that a satisfactory solution of
the problem within the framework of the classical linearized theory is
not possible. In this paper we show that by regarding the dipole dis-
tribution rather than the form as the unknown function, and by correcting
the classical thin-ship relation between the dipole distribution and the
form, a satisfactory solution can be found. The universal minimum curve
of Cw vs f is shown, as well as Cy vs f for a number ofoptimum and non-
optimum forms. Among the problems left unanswered in this paper are
the influence of three-dimensional effects and other corrections to the
linearized theory.

1, INTRODUCTION

The linearized theory of ship wave resistance has been developed by a series of stu-
dents beginning with Michell [1]. Accounts of this theory together with extensive bibli-
ographies will be found in Refs. 2-4. Based on the assumption that the fluid is inviscid and
that the ship is thin enough to generate only waves of small amplitude, the theory imposes
the linearized free surface condition on the velocity potential that characterizes the flow.
By means of these assumptions, the problem is made mathematically tractable and expres-
sions for the wave resistance are derived. The expressions have the form of integrals involv-
ing either the functions that define the shape of the hull or the functions that define the dis-
tribution of sources and sinks by which the hull is generated. Consequently, if we wish to
find the hull of minimum wave resistance within the linear approximation, we have to solve
a problem in the calculus of variations. Specifically, a quadratic functional (the integral
representing wave resistance) is to be minimized subject to a suitable side condition. The
need for imposing some constraint on the minimization process becomes apparent upon ob-
serving, for example, that a hull of zero beam has zero wave resistance.

Previous studies have been of two types. One type depends on the fact that Michell’s
integral for the wave resistance can be evaluated in terms of tabulated functions for ship

Note: This work was supported by the Office of Naval Research under Contract Nonr-2427(00).
*Institute of Mathematical Sciences, N.Y.U., New York, New York. Prof. Karp’s contribution was
made in his capacity as consultant to TRG, Inc.

75

76 Samuel Karp, Jack Kotik, and Jerome Lurye

forms defined by a certain class of polynomials. The wave resistance is then a function of
the polynomial coefficients, and minima can be found by simple computations (see Ref. 5).
The other type of investigation has been concerned with minimization within the class of
vertical struts of infinite depth having fixed length, fixed volume-per-unit depth, and an
otherwise arbitrary form. The latter problem, with which this paper is concerned, has been
considered by a number of authors. Their results have been unsatisfactory for one or more
of the following reasons:

1. The problem was found to have no solution at all.
2. The shapes obtained were infinitely wide at bow and stern.

3. Shapes (which in some cases have negative width) were obtained by solving an in-
tegral equation numerically without regard for the fact that solutions of the equation are
known to be (in general) singular.

The cause ofthese difficulties is a nonuniformity in the accuracy of the perturbation
procedure used to linearize the problem. As in the case of thin air foils (see Ref. 6) for
fixed position along the air foil the results are arbitrarily accurate for sufficiently small
thickness. However, the convergence is not always uniform with respect to position, the
difficulty occurring at the ends of the foil.

In this paper the problem is formulated with dipole density instead of strut shape as the
unknown function. The dipole density satisfies one of the integral equations studied by
previous workers, and is generally singular at the ends of the interval in which it is defined.
The associated shape, defined by the closed streamlines in the flow generated by the di-
poles, is approximately proportional to the dipole density (as in the usual theory) except at
the ends. The shape has the required length and volume-per-unit depth, to first order in the
perturbation parameter, and finite positive width. Hence, although refinements can still be
made in the method of calculating the shape the basic problem is regarded as solved.

In Section 2 we formulate the problem and derive the integral equation for the dipole
density. In Section 3 we discuss the optimum shape for large Froude number. In Section 4
we describe the numerical solution of the integral equation. In Section 5 we determine the
shape of the optimum form. In Section 6 we compare the shape and wave drag of various
forms.

2. FORMULATION AND THE INTEGRAL EQUATIONS

We reproduce here the usual theory of the thin strut of minimum wave resistance in order
to motivate the formulation given later in this section. Throughout the discussion, the fluid
is assumed to be incompressible and inviscid and the flow to be irrotational. Referring to
Fig. 1, we introduce a right-handed rectangular coordinate system (x, y, z). The x-z plane
(y = 0) represents the upper surface of the fluid, the acceleration of gravity is downwards
and the flow velocity is in the +x-direction.

It is convenient also to introduce the dimensionless coordinates x,y,z defined by

x = Lx, y = Ly, z= Lz (1)

where L is the length of the strut’s horizontal section in the x-direction.

Minimum Wave Resistance for Dipole Distributions 77

ee
FREE STREAM
VELOCITY

Fig. 1. Coordinate system fixedin athin strut.
The X - Z plane (Y = 0) represents the upper

BE AVA
surface of the fluid, the acceleration of grav- LZ
ity is downward and the flow velocity is in wy

:

the +x - direction.

Nin
Nis -

“<

Now let the strut be represented by the equation

2S +0(%).*

(2)

The extremities of the strut are given by x= +L/2, so that c(t L/2) =0. Since we wish
to perform a perturbation about the strut of zero thickness, we introduce V/2L as a pertur-
bation parameter, where V is the volume-per-unit depth, and write

ees Ve owc® (3)

Since V is the cross-sectional area, V/2L is the average half-beam. The wave re-
sistance of the strut is given by

2
R, = 5 00? (xr C (4)

where C,, is the wave resistance coefficient. Michell’s theory gives [3, p. 115]

mee y ES)
Cae \ j SE SE Kl(e- 8] aa (5)
-L/2 %-L/2 OX dx
where v = g/c2 and the function K(t) is defined by
(+)
Kt) = 2 | COS NE (6)
1 \2Vd? - 1

*The strut is supposed symmetrical about the x - y plane.

646551: O—62——7

78 Samuel Karp, Jack Kotik, and Jerome Lurge

If we introduce the dimensionless variables x and x’ into (5), we get

1/2

1/2

d d

C= 4 | | — aise, ) K [F(x- x ')] dx' dx (7)
1/2 1/2

where C(x) is defined by
tx) = C,(Lx) (s)
and

uh a |
fark FE he pn" (9)

The quantity f is called the Froude number.

The problem is to find the function ¢€(x) that minimizes Eq. (7) subject to a suitable
constraint. In the case of a strut, a natural constraint is the requirement that a strut of
given length L have a given volume V per unit draft. The side condition is thus

L/2
20(x) dx = V. (10)
-L/2
In terms of €(x), this becomes
1/2
G(x) Mdxa= 1). (11)

-1/2

The problem therefore is to minimize Eq. (7) among all functions ¢(x) that satisfy the
following conditions:

1. d¢/dx must be integrable in —1/2 < x < 1/2 (otherwise Eq. (7) would be meaning-

less).
2. €(x) must satisfy Eq. (11).
3. ¢(+1/2) =0.

It is at this point that the central difficulty of the problem appears, in that the minimiz-
ing ¢(x) will now be shown to obey an integral equation that has no solution. Using stand-
ard procedures in the calculus of variations, we have at once that if ¢(x) satisfies the three
above conditions, it must also satisfy the equation

1/2
| BED Erle. | ees (12)
1/2

dx'

Minimum Wave Resistance for Dipole Distributions 79

where the constant & is determined by Eq. (11). Differentiation of Eq. (12) twice with re-
spect to x yields

Y,(F|x-x'|) dx’ = 0 (13)

| dx'
-1/2

where Y(¢) is the Bessel function defined by

@
2 cos At dv
Y(t) =- 2i Se
1 h? -1
From Eq. (6)
d?K(t) _
ape = Fol ts

Dorr [7] has solved Eq. (13) with an arbitrary given right-hand side; i.e., he has solved

1/2
| h(x') ¥)(F|x-x'|) dx’ = p(x). (14)
-1/2

The change of variables x =—1/2 cos B, x'=—1/2 cos B’ converts Eq. (14) into

| HCE.) Ys (F lcos B- cos p'|) dB’ = P(B) (15)

0

where H(B8') = (1/2 sin B') h(—1/2 cos B') and P(8) = p(—1/2 cos 8).

Dorr then proves that the eigenfunctions of Eq. (15) are the even Mathieu functions of
integral order, ce,(8, F2/4). Since over the integral (0,7) these are closed [8], and the
kernel Y,(F/2|cos B — cos B'|) is quadratically integrable over the square 0< B' <7,
0<B <z, it follows [9] that there is at most one solution to Eq. (15) in Ly. Moreover,
there are no solutions to Eq. (15) not in Lo, as may be deduced from the work of MacCamy
[10], who has shown that the only singularities of h(—1/2 cos 6’) in the interval 0< B'<7
occur at 8’ = 0 or 7, these singularities being of the form 1/sin 8’. Thus

H(B') = (2 san 6") n(-4 eae 6’)
is bounded in the interval 0 < B' <7 and is therefore in L,.

Summarizing, we have that Eqs. (15), (14), and (13) all have at most one solution. It
follows that the unique solution of Eq. (13) is zero. Since any solution of Eq. (12) is a

80 Samuel Karp, Jack Kotik, and Jerome Lurye

solution of Eq. (13), we conclude finally that Eq. (12) has no solution at all.* Thus, the
seeming|ly natural way in which the problem of the strut of minimum wave resistance has
been formulated turns out to be inadequate. We will show that the functions ¢(x) satisfying
the conditions 1, 2, and 3 comprise a class which is too severely restricted for the minimiz-
ing C(x) to be found within it.

The problem can also be approached as follows. Since we require C(+1/2) = 0, two in-
tegrations by parts in Eq. (7) lead to

/2 1/2

1
CS 4F? | | U(x) U(x!) Yo(Flx-x"|) dx'dx, Qe)
= / 22

and the corresponding integral equation is
1/2

| C(x") Y,(F|x-x'|) dx' = a on -

-1/2

Nl =

1
| a7)

where @ is determined so that Eq. (11) holds. As mentioned above it is shown in Ref. 10
that the solutions of Eq. (17) are generally infinite at x = +1/2, so that the derivation of
Eq. (16) is suspect and the solution physically unacceptable. Nevertheless Pavlenko [12]
has solved Eq. (17) numerically by replacing it by a set of algebraic equations and imposing
C(+1/2) = 0. According to Wehausen, for Froude numbers f < 0.325 he finds negative ordi-
nates near the ends. For higher Froude numbers his solutions are fairly good except at the
ends, as we shall see. Shen and Kern [13] solved Eq. (17) numerically by assuming that
¢(x) was a polynomial with three undetermined coefficients. Their results are discussed in
Section 6.

We now show that Eqs. (16) and (17), suitably interpreted, are still the correct equations.
Let (27,2; V/2L) be the potential for the flow generated by a strut advancing into still
water. Assuming the strut to be fixed and the free stream directed along the positive x axis
with velocity c, we have cx as the potential of the free stream.t Then

Alva aa V V Og a as a
U (8.9.8: =| = OE WX,Y,Z) + cx (18)

where Vf/2L is the “perturbation potential” expressing the amount by which the strut dis-
turbs the free stream. The well-known [3, p. 113] linearized boundary conditions on ¢ for
a thin strut are

(19)

at the mean free surface y = 0 and

*The exceptional case k = 0 in Eq. (12) leads to the nontrivial solution ¢ = constant, but this fails to
satisfy the condition C(t 1/2) = 0 unless @ =0. Sretenskii [11] has concluded that there are no square-
integrable solutions, but his reasoning has been criticized by Wehausen [3].

tThe flow velocity, v, is given by 0 = grad W.

Minimum Wave Resistance for Dipole Distributions 81

dt ,(%)
Sr fe Co a a

oz dx
along the longitudinal center section of the strut, defined by the half-strip -L/2 < % < L/2,
720,02 = 0.

In terms of the dimensionless variables x,y,z, and the function $(x,y,z) = d(Lx,Ly,Lz),
Eqs. (19) and (19a) become

Ox? oy
0d( x,y, 0+) dC( x) 1 1 i.
Se eee ay cere My Sy eae pat Oe S205)

¢ is thus determined by the potential Eqs. (20) and (20a), and the usual conditions at
infinity.

We introduce next a Green’s function G(x,y,z;x‘y',z "* representing the potential at
(x,y,z) of a point sink at (x',y‘,z’) in a uniform flow under a free surface. Then G satisfies
Eq. (20) in the variables x,y,z and x,y,z‘, and the potential 4(x,y,z) can be expressed in
terms of G as follows:

Cs) 1/2
pve Benes ' ~20'(x! -x! y! /
Yoxyne iw | dy Py 20'(x") G(x,y,z; x',y",0) dx’. (21)
=b/2

¢ as given by Eq. (21) satisfies Eq. (20) because G does. It can also be directly veri-
fied to satisfy Eq. (20a). Thus the flow around a thin strut in Michell’s theory is charac-
terized by the potential

V
b(xy.eg) = cLx + 57 $(X,Y,Z) (22)

where ¢ is defined by Eq. (21).

Note that the ship form associated with the potential ¢ is not really VC(x)/2L, which is
only the approximate form to first order in V/2L; the actual ship form is defined by the sur-
face of closed streamlines of the full flow whose potential is the w of Eq. (22). Thus if we
denote the actual form by

*The coefficient of the singularity in G is unity, i.e., G =[(x—x42 + (y—y')2 + (2—2')2]71/? plus a
regular function. This coefficient is positive for a sink and negative for a source in contrast to the
usual situation, because we take for the flow velocity the gradient of the potential, rather than the
negative of the gradient.

82 Samuel Karp, Jack Kotik, and Jerome Lurye

a _ : k V
ZS zz a) ’ (23)

its expansion in powers of V/2L is assumed to be (see, for instance, Ref. 14)

Byped Ziy Wags
s(x.¥: a mya C(x) + smaller terms. (24)

Equation (24) reflects the fact that the so-called strut is a strut only to the first order
in V/2L. The actual shape as determined by the closed streamlines of Eq. (22) has a depth
dependence; however, we expect that the shape approaches a strut as a limiting form for
great depths.*

At this point it is important to observe that the leading term in the approximation to C,,

for the shape s(x,y;V/2L), as V/2L + 0, is given by Eqs. (7) and (16).
The basic idea, on which all our results depend, is to consider a class of potentials
more extensive than that defined by Eq. (21) but including the latter as a subclass. Specif-

ically, let

(0 1/2

V c V OG(x,y,z;x',y' ,0
aL M(X.¥+2) = ar ar | dy' | &(x') —— dx' (25)
0 -1/2
where g(x) is integrable and satisfies Eq. (11), i.e.,
1/2
ACD) he = ie (26)
-1/2

Otherwise g(x) is to be arbitrary. Equation (25) resembles Eq. (21) integrated by parts.
Evidently, p satisfies Eqs. (20) and (20a) with d¢/dx replaced by dg/dx. The derivative
dg/dx need not be integrable, and g(+1/2) need not vanish; hence the potentials p form a
larger class than the potentials ¢, since the functions ¢(x) appearing in Eq. (21) were as-
sumed to have integrable first derivatives and to vanish at x = 1/2. (Obviously, the class
of function €(x) is included in the class g(x).) Our expectation is that among the ship forms
defined by this larger class of potentials there will be found one with a minimum wave re-
sistance. That this is so will be established shortly.

As in the case of the forms s(x,y; V/2L) associated with qd, the extended class of forms,
a(x,y; V/2L) say, associated with p, are made up of the closed streamlines! of the flow

*For instance a depth equal to a wavelength of a surface wave having velocity c.

tAlthough we have not proved it rigorously, we believe that there will always be a closed body (i.e.,
a surface of closed streamlines) formed in the flow defined by Eqs. (25) and (27) whenever the total
moment of the dipole distribution on the x axis is negative. Since, as discussed above, the density
of the distribution is —(c/2m7)(V/2L) g(x), the assertion is that there will be a closed body if

1/2
{ g(x) dx > 0.
-1/2

We now observe that all the distributions considered satisfy this latter condition because they all
satisfy Eq. (26), the right side of which is always positive. Thus, all the flows defined by Eqs. (25)
and (27), subject to Eq. (26), lead to closed bodies.

Minimum Wave Resistance for Dipole Distributions 83

whose potential is 0(x,y,z; V/2L) where
V V
0 (xv 2: z = cLx + By Sande (27)
If we could expand o(x,y; V/2L) in powers of V/2L, we would expect to find
Z=+o0(x Aides saps é(x) + smaller terms = o, |x Vie itis Bigs (28)
= ep a) 2L 2L 1 »V> 2L

analogous to Eq. (24).

Just as V¢(x)/2L in Eq. (21) is the first order approximation to s(x,y; V/2L), as shown
in Kq. (24), so Vg(x)/2L = 0, in Eq. (25) is the first-order approximation to o(x,y;V/2L), as
shown in Eq. (28). However, in contrast to C(x), g(x) need merely be integrable and can
therefore have integrable singularities, so that the first-order approximation o = Vg/2L is
not always uniform in x. On the other hand, in spite of the nonuniformity, we expect that the
integral in Eq. (26) when multiplied by V/2L still gives, to first order, the volume-per-unit
draft.

Before we can find the function g(x) in Eq. (25) that minimizes the first-order drag coef-
ficient of the shape o we need an expression for this coefficient. (Equations (7) and (16)
apply to the restricted class of shapes, s(x,y; V/2L) but not necessarily to the extended
class, o(x,y;V/2L).) To obtain the drag coefficient for o, we note that -dG(x,y,z;x',y '0)/dx'
in Eq. (25) gives the potential of an x-directed dipole located at (x‘,y',0).* Therefore, we
may interpret p in Eq. (25) as arising from a distribution of such dipoles over the half-strip
-1/2<x«'< 1/2, y'>0, z'=0, the density of the distribution being —(c/27)(V/2L) g(x’).

It is possible to calculate the force exerted on any one of the above dipoles by the com-
bination consisting of the remaining dipoles and the uniform flow. The force on the entire
distribution, i.e., the wave resistance, is then the product of the density function, —(c/27)
(V/2L) g(x’), and the force on a single dipole, integrated over the distribution. When the in-
tegration is carried out and the integral divided by (p9c?/2)(V/2L)2, the drag coefficient of
the distribution and therefore of the shape o(x,y; V/2L) is

1/2 1/2
cy = ~4F? [ J 800 ex") Yo(Flx-x'|) dx'dx. 9)
1/2 1/2

This result is proved in Appendix A.

Comparison of Eq. (29) with Eq. (16) shows that the expression for the wave drag of the
shape o(x,y; V/2L) has the same form as that for the drag of the shape s(x,y; V/2L) with the
restricted functions ¢(x) replaced by g(x). Note, however, that Eqs. (7) and (16), which were
equivalent for the restricted class €(x), are no longer so when € is replaced by g. Indeed,
if dg/dx is not integrable, such replacement makes Eq. (7) meaningless. Thus, only Eqs.

*Recall that G is the potential of a sink.

84 Samuel Karp, Jack Kotik, and Jerome Lurye

(16) or (29) represent valid formulas for the drag of the extended class of shapes
o(x,y; V/2L).*

To minimize this drag among all shapes o(x,y; V/2L), we must determine a function
g(x) that minimizes C,, in Eq. (29), subject to the side condition Eq. (26). This is a
straighforward problem in the calculus of variations and leads to the following integral equa-
tion for the minimizing function g(x):

1/2

| é(x') Y,(F|x-x"|]) dx’ = A (30)
-1/2

where A is a (constant) Lagrange multiplier whose value is determined by the use of Eq. (26).
It is easily shown that Eq. (30) is the necessary and sufficient condition for g(x) to minimize
Eq. (29).

Once Eq. (30) has been solved, subject to Eq. (26), the resultant g(x) is substituted
into Eq. (25) and the potential (x,y,z) obtained. The minimizing shape, o(x,y; V/2L), is
then given by the set of closed streamlines associated with the potential cLx + (V/2L)
u(x,y,z). As already noted, such a shape will not be a true strut; it is a strut only to the

first order in V/2L.

It has been proved [10] that any solution to the integral Eq. (30) must become singular
at the endpoints x = +1/2, the singularity being of the form

Thus the solution cannot have a first derivative which is integrable over the closed interval
-1/2 < =< 1/2, and therefore cannot belong to the set of functions C(x). It is for this rea-
son that we introduced the extended class of functions g(x).

Since the minimizing g(x), i.e., the solution to Eq. (30), becomes infinite at x = +1/2,
it may seem that such a function can have little to do with the actual minimizing shape,
o(x,y;V/2L). However, Eq. (28) indicates that (V/2L) g(x) = 0, is the first-order (in V/2L)
approximation to a. The explanation is that the approximation of o by o, is not uniform in
x, and gets worse and worse toward the endpoints x = +1/2. Thus, even for very small
V/2L, the shape will be well approximated by o, only if x is not too near the ends. The
shape is determined by a set of streamlines, as stated above, and its complete determination
is discussed in Sections 3 and 5.

We have interpreted g(x) as a measure of the density, —(c/2m)(V/2L) g(x), of the dipole
distribution that gives rise to the shape o(x,y; V/2L). In this view, the minimization process
consists in the following: among all distributions of x-directed dipoles in the half-strip al-
ready defined, whose density is a function of x only and satisfies Eq. (26), we choose that

*Note that if g(x) = const. the body in an infinite fluid would be a Rankine oval, and delta function
singularities would have been necessary in Eq. (7).

Minimum Wave Resistance for Dipole Distributions 85

distribution which has (and hence produces a shape o(x,y;V/2L) which has) the least
wave resistance.*

Since the minimizing density is infinite at the endpoints x = +1/2, the stagnation points
of the flow defined by Eq. (27) lie outside the interval —-1/2 < x<1/2. Hence the minimiz-
ing shape o(x,y; V/2L) actually extends beyond the limits of this interval. However, as
V/2L ~ 0 the stagnation points approach +1/2, as shown in Sections 3 and 5.

Finally, we remark that any solution of Eq. (30) is an even function of x, so that if free
surface effects are disregarded the minimizing struts may be said to be symmetrical fore and

aft. The above discussion summarizes our formulation of the problem of the strut of minimum
wave resistance.

3. THE MINIMIZING SHAPE FOR LARGE FROUDE NUMBER
When the Froude number is large, it is possible to derive an algebraic expression for
the limiting form assumed by the minimizing shape at great depths. In this section we indi-

cate briefly how this can be done.

For large Froude number the parameter F is small, so that the kernel in the integral
Eq. (30) can be approximated in the usual way by the formula

~ 2 1 7 t
Y,(F|x-x'|) = — log (57 F| x-x |) (31)

where log y’ = y = Euler’s constant = 0.577... .

Thus, the integral equation becomes

1/2
2 | &(x') log ce y'Flx-x'|) dx' =). (32)
-1/2
The solution to this equation is an elementary function. Specifically, we have [15]

(x) = oh
2 (33)

where A is a constant that depends on A and F, i.e., it depends on the side condition Eq.
(26) and on the Froude number, f. Naturally, when A = A(F) is chosen so that g(x) satisfies
Eq. (26) then A = constant independent of F. Wehausen [3] regards the singularity at

|x| = 1/2 as depriving the solution of physical reality. This criticism is valid in the context

*We recall, that, to first order in V/2L, Eq. (26) fixes the volume-per-unit draft. Thus within the limi-
tations of first-order theory, the minimization is still being performed among all struts having a given
carrying capacity per unit draft.

86 Samuel Karp, Jack Kotik, and Jerome Lurye

of Pavlenko’s theory in which g(x) represents the form of the strut. In the present theory
g(x) represents a dipole density and the criticism does not apply.

Now, according to the procedure developed in the previous sections, we can obtain the
minimizing shape o by substituting g(x) from Eq. (33) into Eq. (25) and determining the
closed streamlines of the potential defined by Eq. (27). However, this procedure can be by-
passed if we desire only the limiting form assumed by o at large depths. At such depths,

the flow is very nearly two-dimensional (horizontal) and can therefore be characterized by a
complex velocity potential

Q(T) = cLr + + wT) (34)

where 7 = x + iz and w is the complex perturbation potential arising from the presence of the
strut.

In terms of its real and imaginary parts,
wT) = w(x,z) + i€(x,z) (35)

where pz is the real perturbation potential of Eqs. (25) and (27) and € is the associated real
stream function.

We now verify that w has the specific form

ik

wT) = ——..
1 36
Vi- ma (36)

In Eq. (36), K is a real constant to be determined. The definition of w is completed by
stipulating that the 7 plane be slit along the x axis from —1/2 to +1/2, with the square root
given the positive determination on the side of the slit for which z = 0+.

Differentiating Eq. (36) with respect to z, we get

iKT oT
( 1 ) 3/2 02 (37)
Fight

Letting z + 0+ in Eq. (37) and introducing Eq. (35), we have

ou( x,0+) pis 0&(x,0+) Bh cook wn ck
oo Te (5 zh x2) ats (38)

It follows that

Minimum Wave Resistance for Dipole Distributions 87

Ou x,0+) Kx sail t,
oz ia a/a 0 2 3 A
(1 i <?) (39)

dx (2- se ; (40)

Now we set K=—Ac. Then, upon comparing Eqs. (39) and (40), we get

Ou x,0+) _ mA dg

0z dx ay

which is what Eq. (20) becomes when ¢ is replaced by p and € by g. (The process z + 0-
reproduces Eq. (41) with a negative sign on the right, in agreement with Eq. (20).)

We conclude that for large Froude numbers and at large depths, the perturbation poten-
tial, p, of Eqs. (25) and (27) is given by

=) |
U(x,Zz) = Re sc ele Im ae

42
Set ee
The associated perturbation stream function, &, is therefore
-Ac

yi- of (43)

We recall that the constant A appearing in Eqs. (42) and (43) is determined by the
Froude number and the side condition Eq. (26). But

E(x,z) = Re

1/2 A
| ————. dx = liimplies A= 7}.
~-1/2 1/4 - x?

From Eqs. (34) and (43) it follows that the stream function, I"(x,z), for the total flow at
large depths and large Froude number has the form

T(x,z) = cLz + M. Cx tay = "Lr" = Vovpe =

2L 2L ; 44
= 72 (44)

88 ‘Samuel Karp, Jack Kotik, and Jerome Lurye

The closed streamline defining the minimizing shape, a, is given by
Cx.zy = 0. (45)

To see this, note that the flow is symmetrical about the x-y plane, so that the part of
the x axis outside the shape o belongs to the streamline in question. Now as x + © (which
implies 7 + 0), ['(x,z) + cLz, as is evident from Eq. (44). But cLz=0 on the x axis, prov-
ing that the desired streamline is defined by Eq. (45).

Thus the limiting form assumed by the minimizing shape, o, at large depths and for large
Froude number, is represented by

cl inde igs any é
cLz aL Re ; = 0 (7 = xX + 2 (46)
REE 7)
4

If we define a thickness parameter € = V/2L? = B/2L,* where B is the average (full)
- width, we can rewrite Eq. (46) as

(47)

In bipolar coordinates! € and 6 we have

1
; sin 0 2 sinh €

“cosh £ + cose nae % > eoshwettteosnG

letting the line 0 = 0 have length 1. Also

= e&(cos 6 + i sin 6)

and Kq. (47) becomes

5 sin 6 ‘ aa g:
cosh € + cos Bg 2) CEL PED Gy 9

*€ is also equal to the horizontal-plane area coefficient multiplied by B/2L, where B is the usual max-

imum beam.
tSee, for example, Morse and Feshbach, “Methods of Theoretical Physics,” New York: McGraw-Hill,

1953, p. 1210.

Minimum Wave Resistance for Dipole Distributions 89

Finally,
0 7 646? 4
sin > = Pee | 1 + 1 + cosh & (48)
Z 8e cosh 2 (as

which is a convenient form for computation. It is easily seen that € = 0 corresponds to

x= 0. In order to get an idea of the range of & we observe that at the stagnation points we
have y = 0, |x| > 1/2, and hence 0= 7. Solving Eq. (48) with 0 = 7 approximately for €,,,,>
the value of & at the stagnation point, we find

1/3
Geter ae cosh’! Pane (49)

and

The midship beam is given to first order by

2e€
z(0) rc} a 5

Since 2€/z is the beam given by the Michell theory it is interesting to compare z(0) with
2€/m. Table 1 shows that the relative error is less than 0.008 for € < 0.10.*

Table 1
Comparison of z(0) and 2€/7

0.00318

0.00637
0.1273

0.03183
0.06366

The values of x,,,, indicate the extent to which the form fails to have the preassigned
length. Figure 2 shows the optimum forms for five values of € ranging from 0.005 to 0.100.
They resemble dogbones.

*Typical values of € are 0.04 for a destroyer and 0.048 for a tanker, based on the waterplane section
in each case.

900 Samuel Karp, Jack Kotik, and Jerome Lurye

O ——.

w 2 3 4 a 6
X= xX €= Var? = B/AL X

Z-Z2/L

Fig. 2. The optimum shape (at large depth) for large Froude number for various values of the thick-
ness parameter € This form is obtained by considering the streamfunction of Eqs. (47) and (48). We
have Eq. (48), where z = (1/2) sin 6 /(cosh € + cos @) and x = (1/2) sinh €/(cosh € + cos 9),

Equation (28) implies that

1/2 1/2 1/2

| 2(x) dx = | o,(x) dx = | = E(x) dx = _ (50)
-1/2 -*-1/2 -1/2
or
1/2
| z(x) dx = €
-1/2

To see how well the approximate side condition specifies the volume we have compared

max
| ZC axe
“x

max

computed by numerical integration, with €, for various values of €. The results are as
follows:

2(x) dx

Expansion of the exact volume for small € yields

1/2
z(x) dx = « [1 - 0.44 <1/3| (50a)
-1/2

Minimum Wave Resistance for Dipole Distributions 91

in which the coefficient 0.44 is approximate. The general form (specifically the minus sign)
of this relation is in agreement with the theorem of G. I. Taylor relating the dipole moment,
the volume, and the added mass in the x-direction, for a line dipole distribution in a flow
without free surface.

4. NUMERICAL SOLUTION OF THE INTEGRAL EQUATION

Equation (30), with A = —1, was solved by numerical methods as described in Appendix
B. The solution depends on Froude number and is denoted by g,(x) or g,(x;f). Solutions
for other values of A are given simply by Ag_(x)/—1. The desired solution g(x) which satis-
fies Eq. (26) is given by .

6,(%;5 f)

a(xsf) = CE (51)
where
1/2
I ,(f) = 6,(x;f) dx. (52)
-1/2
I,(f) and g(x;f) have been computed from g,(x;f). As indicated in Appendix B
h
6(x) = we ae
dois 2 (53)
4

where h(x) is a bounded function. The behavior of h(+1/2) as a function of f (see Fig. 5) is
of interest since when h(+1/2) is large (small) the bulge in the shape at bow and stern is
also large (small), as shown in Section 5. Figure 4 shows g(x;f) for various values of f.
Figure 5 shows A(x;f) for various values of f. Figures 3, 4, and 5 show that bluntness is a
predominant feature at high speed, but less so at low speeds. However, the forms appear to
have bulbs at all values of f yet examined (f> 0.3). Note that (in Fig. 3) A(+1/2) + 7-1 =
0.3183, as f + 00, in agreement with Eq. (43) et seq.

The function h(x) (Fig. 5) is of some interest. We have h(x) + 77! as f + 0, as explained
in Section 3. Hence one can qualitatively define a high-speed range as that in which
h(x) ~7-1, Figures 4 and 5 suggest that f > 0.6 is a reasonable definition of the high-speed
range. Below f= 0.6 there is a rapid transition from the dogbone shape characteristic of
high speed to shapes whose maximum beam is amidship.

‘

Let Chip be the wave resistance coefficient at Froude number f of the form which is
optimum at Froude number f’ . cf (f), shown in Fig. 6, is then a universal optimum for
struts. Note that Ch if) is a maximum at f = 0.65. In Fig. 6 we show also C}-%f) and
Cr: Clif) and C1-°%f) are very close for f > 0.6, which substantiates our identifica-
tion of f > 0.6 as the high-speed range. The divergence of Ci (f) and C1-°(f) below f = 0.6
is substantial. chip) and C9-5(f) are close in the range 0.46 < f < 0.6.

92 Samuel Karp, Jack Kotik, and Jerome Lurye

Fig. 3. Graph of Al +1/2) vs f. Theoptimizing dipole distribution
has the form -(c/27) h(x)/(1/4 = x2)1/2 = <cg x)/27; see Eq
(53). Al +1/2) is the coefficient of the edge singularity, andisim-
portant in determining the shape at the ends.

As indicated in Appendix B the interval of integration (half a boat-length) was divided
into six subintervals (seven points). One of the factors affecting the accuracy of the nu-
merical scheme with seven points is the value of F, since as F increases the number of
oscillations of Y,(F|x|) increases. We believe that all our numerical results for f < 0.4
should be regarded as questionable. For some lower values of f the numerical solution of
the integral equation and the wave resistance coefficient had the wrong sign. Figure 3
shows h(+1/2) decreasing rapidly for 0.3 < f < 0.4, with intentions of becoming negative.
Also, a separate computer program was written for calculating C,, for any strutlike dipole
distribution. The mesh in this program was variable, and it was found that seven points was
not sufficient to calculate C,, for f= 0.4 with an accuracy of the order of 5 percent whereas
seven points gave an accuracy of better than 1 percent for f= 0.5. A more powerful com-
puter program is in preparation for work on the three-dimensional problem, and we expect
that it will allow us to extend the range of accurate calculation to lower values of f.

5. DETERMINATION OF THE SHAPE NEAR THE ENDS

Having found the optimum dipole distribution we are left with the problem of finding the
form. While digital methods were available and will play a role in future work we preferred
to determine the shape approximately by analytic means. We expect that z = Vg(x)/2L
should be a good approximation to the true shape away from the ends, at least for sufficiently
small V/2L* = € and sufficiently far from the free surface. Figure 7 shows the exact shape
and the linearized shape g(x) in the case of f=, for €= 0.05. The agreement even away
from the ends is not as good as one would like” but no attempt will be made to improve this
aspect of the theory.

*See also Fig. 10. For additional discussion see Section 6.

Minimum Wave Resistance for Dipole Distributions 93

3 f=3162 F=.100 3 £=100 F=/.00

‘
Ce ey Se SY a 0 eZ iy 5

3 £269 F=2367 2 #255 F=3,306

(a) (7)

Oe MS a! YO A CT Ae Ri PaO RP Re HYPO Fee UG as
Soe x—

2 £296 F=4726 3 #242 F= 3.669

BE SoA eb ae ay? 2) mle > Sere eC
x—

3 £=40 Fe6.25 £=38 F=6.925

a9 -@ 23 -2 -/ O fF 2 FF FF “5 -$ 3 32 OO th 2 TOG OS

x— Xe
Fig. 4. Graphs of (x) vs x for eight values of f.-cg(x)/27 is the optimum dipole distribution.

646551 O—62——8

04 Samuel Karp, Jack Kotik, and Jerome Lurye

#*=Z/62 F=./00

6
h(x) .5 Aix) |

4

2

2 2

| .

=

damm deuy isha cua Swen ah “5 -#$--3 32 7 10 / 23 4 «6

#=/00 F=/00

iO NH WA W ON
x:
ie

7 SAE AGT- 7 £2.55 F-33506
6 6
hix) 5 Ai) 5,

up A

oo 2

2 2

iy “s

He) me

-5 -$ -3 -2 </ Oo O/ <9 ap =2) = 20) Pape Bans) 2

£46 F=4726 #2 F-26669

olf

ic ier a
h(x) 5

4 4

2 2D

2 2

all Af

4) @

hp ee ee =

= FO -=6.27 #:.38 F=6.922
Ah) ¢ 4 a b

3

4 4

3 ZI

i 2

| /

Oo =

«5 <4 =5 =2 =/ oO a% 4B a2) =) 10. Uf 2) S50 See
x

Fig. 5. Graphs of A(x) vs x for eight values of f. A(x) = 6(x)(1/4 - x2)!1/2, Note that A(x) for
f>0.65 is almost independent of f.

Minimum Wave Resistance for Dipole Distributions 95

eo gp = E90)
&ND APPROXIMATION
~~"-- ~~ EXACT SHAPE

f=0o €=,.05

ws. ws Tem 7g. /i2 38 x

Fig. 6. Cyf( f), 6C,1-9( f) and 6C,9-5( £) Fig. 7. Graphs of the exact optimum shape, €4( x),
vs f, Notice the changes of f-scale at for f = © and an approximation to the shape. The
f = 0.7 and 1.0.C,£( f) is the universal min- figure shows the extent to which the approximate

shape at the ends matched the linearized shape, and

i . The fi i
imum curve e figure shows that a dipole die carayisa watch Weg etied Bien

distribution which is optimal for f = 1 per-
forms well for 0.6< f<, but ceases to do
so as f decreases. Likewise, a strut which
is optimal for f = 0.5 performs well in the
range 0.46< f <0.6. We now prove that the shape at each end is

determined by the singularity of g(x) at that end

in the following sense. Suppose we have a dis-

tribution of x-directed dipoles whose strength
is 0 for x’ >O and —9(x')/n(x')}/? for x'>0, where x' = x + L/2 and

q(x') = a Ay Ae 1 ON an rey Be (54a)

If we let x’ =7 cos 0 and Z' =7 sin O, the stream function of the two-dimensional flow is
given by

A
SN ei aL 0 a a1/2 g a3/2 30
é.= er sind (2, cos 5 + A,r cos 5 ~ A,r cos ~ si (55)

Let us introduce dimensionless variables r=r/L, &= EL, so that the dipole density is

q(x’) Lec '
- —— = - —— _ (B + Bx iy, ata 54b
aie ner: ° la one
where
A
Ba =
cL
1
: AL /2

96 Samuel Karp, Jack Kotik, and Jerome Lurye

and the dimensionless stream-furction is given by

B, cos 6/2
Guz ce nin oo (MES ayr'2 con Se .. - (56)
r

If we solve Eq. (54b) when &= 0 (the streamline passing through the stagnation point) for r
as a function of 6 we obtain the following expansion of r in terms of 0 and €:

2/3 € as 2 1/3 € - <3
r=) (xaiwo7a) +38 Si (zeae) to OM

where the successive terms involve ascending powers of €.* We conclude that for small €
the shape at one end is determined mainly by the singularity of the dipole distribution at
that end. The series Ay + A,x' +... will have a radius of convergence equal to L, the dis-
tance to the other singularity. In order to use Eq. (57) to find the approximate shape at the
ends we relate the coefficients B, and B, to the solution g(x) of the integral equation. Since
€g(x) generates the desired unnormalized shape shrunk by a factor of L in the x-direction,
and since the dipole density corresponding to €g(x) is - Lcég(x)/m, we have from Eq. (54b)

Le _ Le h(x) Le hx" )
- — g(x) = -— € Sr =e ee
7 ae ee
Lec !
=- CB. + Bex ot yee)
nxt 0 1 (58)
from which
h(x') = (By + Byx' +...) V1 - x’ (59)
and therefore
h(x'=0) = h(x=-i)=B
(x'=20) = A(x=- 5)= 8B, (60a)
and
dh dh 4 1
ane ax Gr Pad Bie (60b)
x'2Q x=-1/2

These equations determine By and B, in terms of A(x). Since we have only a digital approx-
imation to h(x) we were only able to determine B, and B, approximately. Figure 7 shows the
extent to which the approximate shape at the ends matched the linearized shape, and the
way in which we joined the two. This example is for f = « but it is fairly typical. Figure 8
shows the resulting shapes for a number of values of f.

*The next term involves all the B i:

97

Minimum Wave Resistance for Dipole Distributions

Fig. 8. Approximate optimum shapes at sufficiently large depth for eight values of f.

98 Samuel Karp, Jack Kotik, and Jerome Lurye

6. COMPARISON OF VARIOUS FORMS

In Section 4 we examined the resistance of some optimum dipole distributions. We now
compare these with some others. Figure 9 shows the universal minimum Cf (f), C08 f) and
C,,(f) for a parabolic dipole distribution vanishing at the ends (labeled C(f)). It is known
that a dipole distribution —cg(x)/27, where

- 2 sin 22 (1-4x2)”?
&( x) = Se eS (61)
(1+ 2x)4/? + (1- 2x)*/? - 2(1- 4x?) cos =

generates (in two-dimensional flow) a lens given by

ZC) = + (pore i - 4x? + cot ) ; (62)

For p = 1.812 the full interior lens angle is (2 — p)# = 0.1887, while

1/2 1/2
| z(x) dx = 0.05 = and | &(x) dx = 0.057.
-1/2 -1/2

Figure 9 shows C,,(f) for a dipole distribution — cg(x)/2m (labeled chp) and also C,,(f)
for a dipole distribution —cz(x)/27 (labeled CL5/(f)). This last quantity is Michell’s wave
resistance for the shape 2(x), since —cz(x)/27 is the linearized dipole distribution defined
by the shape 2(x). Figure 10 shows g(x) and z(x), and we see that even for these thin forms
the approximation g(x) ~ z(x) is not very accurate. Comparing CcLD(f) with CLS/f) in Fig. 9
we find a discrepancy of up to 25 percent. We notice that C2-5/f) is about half of CF (f),
Chap): or CL5/f) over a considerable range of f.

Some information on Pavlenko’s work was obtained from Refs. 4 (p. 115) and 12. The
forms shown by Pavlenko are reminiscent of ours, and show the same trend as a function of
f (for f < 0.43, the maximum f considered in these references). Shen and Kern [13] solved
the integral equation approximately, by determining three coefficients in a polynomial ap-
proximation to the solution. They carried out the computations for F = 0.1, 0.2, 0.4, 0.6

Fig. 9. Graphs of
: P L 127-LS
efi, *cl'ey, chip, [C52 (p, andl 2h sig

2 vs f. The upper curve and the points nearby show C.(f) for a
parabolic dipole distribution

fo)
: (s[--I)
27 L6 \4 ’

4
2 for the dipole distribution generating a lens (in an infinite

fluid), and for the linearized dipole distribution defined by the
O lens. The lower curve and nearby points show the universal
ee ‘ Mg Mite Ee ne “n “7 minimum C.(f) and also C,,(f) for the dipole distribution which
SOC i atals Bini is optimum at f = 0.5. Areduction in C,,(f) by 50 percent has

6.8 y “i
xxx C,(#) C2) 446 Co) been achieved.

Minimum Wave Resistance for Dipole Distributions 99.

900 =2717% [DIPOLE DENSITY]
ZO = EXACT SHAPE GENERATED BY 30%)

Fig. 10. Graphs of the shape function for a lens and
-2n/c times the exact dipole distribution which gen-
erates it in an infinite fluid.

(f = 3.162, 2.236, 1.581, 1.292). The forms shown by them are somewhat reminiscent of
ours. The forms of Pavlenko and of Shen and Kern differ from ours in having finite entrance
angles.

In a series of pioneering papers Weinblum has considered the problem of minimizing the
wave resistance of more realistic, three-dimensional forms. He has defined various families
of polynomials, imposed constraints on the coefficients by specifying some geometrical
properties of the form, and used the remaining degrees of freedom to minimize the resistance.
Again, in spite of these differences in approach, our forms show the same general variation
with Froude number as his waterplane sections.

(It should be noted that all wave resistance coefficients of dipole distributions are nor-
malized with respect to dipole moment (multiplied by —27/c) rather than volume. Hence to
the extent that volume differs from dipole moment (multiplied by —27/c) a comparison of wave
resistance coefficients does not provide an exact comparison of resistance values based on
equal volume. This discrepancy, like the length discrepancy for optimum forms, does not
affect our theoretical results, which are asymptotically valid for € + 0.)

7. SUMMARY

A universal curve of minimum wave resistance for infinite strutlike dipole distributions
has been found (see Fig. 6). The optimizing distributions have been found, and turn out to
be infinite at the ends. Despite this last mentioned feature the corresponding shapes, at
large depths, are determined. The shapes are blunt at the ends, which extend beyond the
interval on which the dipoles are distributed (see Fig. 8). Attention is called to the dis-
tinction between shape and dipole distribution, a distinction which is of fundamental impor-
tance in the present context. Attention is also called to the discrepancy between the wave
resistances calculated using Michell’s approximate dipole distribution proportional to shape
and the exact dipole distribution yielding that shape, especially for ships with nonzero en-
trance angle (see Figs. 9 and 10). Some of the objectionable features of the analysis given
by previous investigations have been removed.

100 Samuel Karp, Jack Kotik, and Jerome Lurye

ACKNOWLEDGMENT

We wish to acknowledge the valuable contributions of Mr. V. Mangulis, who helped us
in some of the analytical calculations, Mr. D. Cope, who heads the computer group at TRG,
and Mr. G. Weinstein, who wrote the main digital computer programs.

Appendix A
PROOF OF THE WAVE RESISTANCE FORMULA
We wigh to show that Eq. (29) gives the correct formula for the drag coefficient of a dis-

tribution of horizontal dipoles in the half-strip —1/2 < x< 1/2, 0< y< oo, z=0, the density
of the distribution being —(c/2m)(V/2L) f(x) with f(x) integrable but not necessarily bounded.

In what follows, we revert temporarily to the unnormalized variables x and y. Consider
first a finite number, n, of sources located in the above half-strip at the points (x., ¥ i, 0)
(s=1, 2,..., m) and having strengths q,(s =1, 2,..., ). Then the wave resistance, R, of

such an assemblage in a steady stream flowing in the x direction with velocity c, is shown

by Lunde [2] to be
oo
2 2°. 2 2
R= 167pv | (1; + J,)) cosh?u du (Al)
0

where p is the fluid density, vy = g/c?, and

: 2 UR cosh’ u
I= q, cos (vx! cosh u) e (A2)
rs
= : my -vyt cosh7u
Jaz q, sin (vx, cosh u) e 3 (A3)

a=l

We now evaluate /, and J,, when the source celleceions is arranged to form a finite set
of x-directed dipoles. For the | purpose, we must assume that n is even, and we locate the
Hee at the points (x47 p> 0) (p =1,2,..., 2/2). The coordinates Bo9¥p are related to

%.,Y, by the pauaions ae

op = 1) x

x, the x €s-= 2p) p= 1,2,..455- (A4)
y, = yi (s = 2p - 1 or 2p)

‘The quantity hisa emall distance which will eventually approach 0.

Minimum Wave Resistance for Dipole Distributions 101

The dipole moments, m,, are related to the source strengths, 7,, by

a (s = 2p- 1)
Gari |G s = =
h % e
(A5)

-—=4q, (s = 2p).

Introducing Eqs. (A4) and (A5) into (A2) and (A3), we obtain

n/2 me _2
1 a ~ a “vy ,cos u
I= z Mm, {eos (vx, cosh u) - cos [v< x, + h) cosh ul} e (A6)
p=1
we a o -vy cosh*u
Jn 1 m,{sin (vx, coshu) - sin ck, + h) cosh u]} an * , (A7)
h p=1

For small h,

cosh u)

cos|v(X, + h) cosh u] = cos (VX,

- hv cosh u sin (vx, cosh u)_ (A8)

sin[v(%, + h) cosh u| = Sin (vx, cosh u)

+ hv cosh u cos (vx, cosh u). (A9)

Therefore in the limit, as h + 0,

n/2 K, -v¥_cosh?u (A10)
I, = v cosh u m, sin (vx, cosh u) e th
p=1
me A -vy cosh-a
J, =7v cosh u a m, Cos (vx, cosh u) e E ‘ (All)
p=1

These equations represent the forms assumed by /, and J, when the original sources
(and sinks) are combined so as to form a collection of n/2 x-directed dipoles in the x-y
plane, the dipole moments being m, (p=1,2,..., /2). The wave resistance of such a
collection is obtained by substituting Eqs. (A10) and (A11) into (Al). Before substituting,
we go to the limit of an infinite number of dipoles of infinitesimal strength, continuously
distributed with a density m(x) in the half strip —L/2 <%<L/2, 0<y<~, Z=0. To this
end, it is convenient first to imagine the n/2 dipoles arranged in a rectangular array of r
rows each containing g dipoles. Then (A10) and (Al1) can be replaced by the double sums

102 Samuel Karp, Jack Kotik, and Jerome Lurye

r
a -v} ,cosh7u

I,=vecoshu )| )) m, sin (vx, cosh u) e (A12)
isl jel
Z = wy -vy.cosh7u
J, = 7v cosh u Ms Ds m, cos (vx, cosh u) e / (A13)
i=l jz=1

Passing to the continuous case with the y integration extended to infinity, we get, in-

stead of (Al2) and (A13),

L/2 @ a 2
I = v cosh u { m(X) sin (vx cosh u) B per ead dydx (Al4)
-L/2 “0

L/2 © a 5
J = -v cosh u | | m(X) cos (vx cosh u) eu tte Sard es dydx. (A15)
-L/2 “0

L/

Note that in Equations (A14) and (A15), we have assumed only the integrability of m(x)
and not its boundedness.

The y integration can be performed separately, so that (Al4) and (A15) become

L/2
ve 1 a x a a
[= are J m(X) sin (vx cosh u) dx (A16)
-L/2
E72
1 a a
dh Fas: Sean | m(X) cos (vx cosh u) dx. (A17)
-L/2

Finally, we reintroduce the normalized variable x = x/L. Then

1/2
ces ale :
I= Pantin i M(x) sin (Fx cosh u) dx (A18)
-1/2
L 1/2
Oe me Rania: fi M(x) cos (Fx cosh u) dx. (A19)
-1/2

In these equations, M(x) is defined to be m(Lx), while F = L = gL/c? as before.

Minimum Wave Resistance for Dipole Distributions 103

According to (Al) and the above discussion, the wave resistance of a continuous dis-
tribution of x-directed dipoles in the aforementioned half-strip is

te)

R = 167pv? | (I? + J?) cosh2u du (A20)
0

where / and J are given by (A18) and (A19).
Squaring (A18) and (A19), we get
1/2 1/2

2 bE? ' :
: | M(x) M(x") sin (Fx cosh u)
sol faz “= 1/2

* sin (Fx' cosh u) dxdx' (A21)

1/2 1/2

L2
de S Beet J M(x) M(x') cos (Fx cosh u)
cosh“u Las

* cos (Fx' cosh u) dxdx’. (A22)

Thus

L? 1/2 Wrz
=) ft) HO)
cosh“u wp aac age

* cos [ Foxx") cosh u| dxdx'. (A23)

ee y=

Substituting from (A23) into (A20), we have

Comey /22 1/2
= 167pF? | | | M(x) M(x")
0 “-1/2 “1/2
“cos [ Fox-x") cosh | dxdx'du. (A24)

We perform the u integration first, using the identity

oe)

| cos (t cosh u) du = a eet rel): (A25)
0

104 Samuel Karp, Jack Kotik, and Jerome Lurye

Then (A24) becomes

1/2 1/2

R = -877F? | | M(x) MCx') Y((F|x-x'|) dxdx'. (A26)
-1/2

-1/2

Now let the density M(x) =—{c/27)(V/2L) f(x), as in Eq. (25) of the text. Then

1/2
R = -20c? iy se
Gorge VOE

-1/2 -

The drag coefficient, C,,, is obtained by dividing R by

2
sg
2 2L

1/2
f(x) f(x') Y((F|x-x'|) dxdx'. (A27)
1/2

Thus, the final result is

j f(x) f(x') Yj(F|x-x'|) dxdx' (A28)
-1/2

in agreement with Eq. (29).

Appendix B
NUMERICAL SOLUTION OF THE INTEGRAL EQUATION

The equation to be solved is Eq. (30) of the text, viz.,

1/2
[69 ¥CFlxx! |) dt = 1. (B1)
-1/2

It is easily shown that g,(x) = g,(—x), so that (B1) can be written

1/2 :
| 6,(x') {y,(Flx-x'|) oY SIR G@at a) dx" ==) (B2)
0

As already noted, MacCamy has proved [10] that g (x) has the form

Minimum Wave Resistance for Dipole Distributions 105

&,(x) = uncrseit p7 x) (B3)

J1/4 - x?

where h(x) is a regular function in -1/2<x< 1/2. The singularity in g(x) at x = 1/2 in-
troduces a loss of accuracy into the numerical solution of (B2); we therefore convert (B2)
into an equation for the regular function, h, by means of the following change of variables:

x= 172 sin £, x = 21/2) sin 6. (B4)
Equation (B2) then becomes

a/ 2

H(B') rae |sin 8 - sin f' )

i)

fale [F(sin B+ sin ey] dB' = -1 (B5)

where H(8) = h(1/2 sin 8). Thus the unknown in (B5) is a regular function.

To perform the integration in Eq. (B5) numerically, we first divide the interval of in-
tegration, 0 < B' < 7/2, into an even number, n, of equal subintervals of length 7/2n. We
denote the values assumed by f' at the endpoints of these subintervals by Bj =1, 2,...,
n+1). Then 8, =0and B,,,= 7/2. We further denote by H; the values of the unknown
function H(8') at B,, i.e., H; = H(8;). Finally, we allow the variable 8 to assume the
values B(i=1,2,...,n+ 1). We thereby obtain n + 1 linear algebraic equations for the
n+ 1 unknowns H,, each equation corresponding to a different value of the index, i. These
equations are constructed as follows:

The part of the integration in (B5) that involves Y,[F/2(sin B + sin B’)] can be per-
formed at once by Simpson’s rule:

1 /

|

0

2
H(B') ¥, [Fcsin 6, + sin B")| dg' = 7 {i [Ecsin B,+ sin 6,)|
+ 4H,Y, Hes B; + sin p.)| + 2H,Y, | (sin B, + sin p3)|

Fa.» gH ¥,|£(sin 8, + sin Pau] (1 = 2,3, .<. mt 1)s Fe)

(We have excluded the case i = | in (B6) because it is discussed separately below.)

On the other hand, the logarithmic singularity in Y, prevents the remaining integral in
(B5) from being represented by a formula like that of (B6); when i = j, the expression
Y,(F/2|sin 8; — sin B,|) is meaningless. We therefore break up the integral and write
(when i = 1 or n + 1)

106 Samuel Karp, Jack Kotik, and Jerome Lurye

1/2 F
| H(B') ¥,(Flsin 8, - sin 6'|) dB'

0
fe F
H(B') v,(Zlsin By 7 sin 6'|) dg'

Bist F
+f H(B') v,($ sin Bo San 6 |) dp’
Biiy

Bnei
ml H(B') ¥,($ |sin 8, - sin 6'|) dp' (i = 2,3,...,m). (B7)

i+1

Since the total number of subintervals, n, is even, the first and third integrals on the
right of (B7) (these contain no singularity) either both extend over an even or both extend
over an odd number of subintervals. In the even case, the integrals are evaluated by
Simpson’s rule. In the odd case, they are evaluated by the trapezoidal rule over the first
subinterval and by Simpson’s rule over the rest. (Note that when i = 2, the first integral on
the right of (B7) vanishes, while when i = n, the third integral vanishes.)

The second integral on the right of (B7) is approximated by first writing

Biay F
H(B') vy, (F |sin 8, - sin e't) dB'
Bi.

Bi+y F
© H, I Y ( |sin 8, - sin 6 |) dB’. (Ba)
i-1

This approximation is sufficiently accurate for our purpose when n> 6. The integral on
the right of (B8) is evaluated by expanding the Bessel function about the point B'=B8. The
result is

Biks 2F2 2
ie ae 2 my! F il
(elfen ane =n tana

2p2 2
2: 2 1 - , )- 1 cos B, , B
oe Wan pied) CLE BY Awe s : 9
= ie + s(t + tan“6, ein (1 at ,n) (B9)

where log y’ = y = Euler’s constant = 0.577... .

Minimum Wave Resistance for Dipole Distributions 107

By these procedures, we can perform numerically the integration on the left side of (B7)
for i= 2,3,..., 2. Adding the result to the right side of (B6) and (in accordance with the
integral equation (B5)) equating the entire sum to —1, we obtain n — 1 linear algebraic equa-
tions for the n+ 1 unknowns H;(j=1,2,..., 2+ 1), each equation corresponding to one of
the i values, i = 2,3,..., 7.

The remaining two equations, associated with i = 1 and i =n + 1, are arrived at sepa-
rately. When i = 1,

y,(E |sin B= Sin 6'|) = AG |sin By + Sin 6'|)
since 8, = 0; thus (B5) becomes

mn / 2

2H(B') (4 sin 2’) dp’ = -1 (i = 1). (B10)

This equation can be rewritten

Bo - Bn+1 :
2H(") ¥, ( sin 6’) dB’ + | H(A") Y, (sin 6" dp’ = -1. (B11)
By B

2

The first integral on the left is approximated by setting H(8‘) = H, and expanding Y,
about 8'=0. The result is

5 Rid é ea 2H 1y'F 72F2
i 7% ȴ6(5 fe s') Te Pian oe ( on ) ( i rtd

2H, er Oy F2
sien haw iong Bot BS B12
n [ arr (4 =) roe

The second integral in (B11), which is free of singularities, extends over an odd num-
ber of subintervals and is therefore evaluated by the combination of the trapezoidal and
Simpson’s rules already described. Upon adding the result to the right side of (B12) and
substituting into (B11), we obtain the algebraic equation corresponding toi = 1. Finally,
when i = n+ 1, (B5) becomes

Bn+1 F
j we'yy, (4 lsin Bue isin a"|) dB'
1

Ba+t
+ \ H(B') ¥, E (sin 6,4, + sin 6’) | dp'=-1 (i= n+1). (B13)

1

108 Samuel Karp, Jack Kotik, and Jerome Lurye

The second integral on the left of (B13) is given by (B6) with i=n+1. The first in-
tegral in (B13) can be written (since sin 8 ,, = 1)

Bnet Bn F
|, H(B')Y, 5 |1 - sin e'\) dBi | H(B') (Fl - sin 6'|) dp'
1

Bn+1 F
¥ |, me'y¥,($ - sin 6'|) dp’. (B14)

n

The first integral on the right of (B14) contains no singularities and extends over an
odd number of subintervals; we therefore evaluate it by the combination of the trapezoidal
and Simpson’s rules. The second integral must be specially dealt with because the first
derivative of 1- sin B' with respect to 8B’ vanishes at B' = 6,,, = 7/2. If we are to main-
tain consistency in order of accuracy to which the integral is evaluated, we cannot merely
replace H(B') by H, 4, in the interval B, <B'< B41- Rather, a linear approximation to
H(B') must be employed, viz.,

: : Hae a H,
H(B') %H_ + Bisa aS GS ere)

n

= H+ yy H(A + -F) (BS BY < Byyy)- BD)

Upon substituting from (B15) into the integral and expanding Y, about the point
B' = 7/2, we obtain the following approximation:

Bnei

: ’ F ‘ , ; alte y' Fr? a2
j, H(B DY. (Fla - sin B ) dB © On toe 22) 1 = a:

n
H " Fr2 2
AS | jigs (—)- 3-— (B16)
2n 32n2 288n2

Addition of this result to the first integral on the right of (B14), followed by substitu-
tion from (B14) into (B13), gives the final algebraic equation for the quantities H;. This
equation corresponds toi =n+1. All our calculations were made with n = 6.

Having obtained g,(x) the computer finds

I cf) = | 6,(x;f) dx

Minimum Wave Resistance for Dipole Distributions 109

and g(x) = ie (f) go(%; f). The wave resistance is found easily from
f 2_-1
Cit) = 4k lf). (B17)

which follows from (29) and (B1). Since one can prove that Eq. (29) is a positive definite
expression it follows that /,(f) > 0, which is not obvious from the integral equation (B1).

REFERENCES
[1] Michell, J.H., “The Wave Resistance of a Ship,” Phil. Mag. (5) 45:106-123 (1898)

[2] Lunde, J.K., “The Linearized Theory of Wave Resistance and Its Application to Ship-
Shaped Bodies in Motion on the Surface of a Deep, Previously Undisturbed Fluid,”
Technical and Research Bulletin 1-18, The Society of Naval Architects and Marine En-
gineers, New York, July 1957

[3] Wehausen, J.V., “Wave Resistance of Thin Ships,” in Proceedings of the Symposium on
Naval Hydrodynamics, Publication 515, National Academy of Sciences-National Re-
search Council, pp. 109-137, 1957

[4] Birkhoff, G., Korvin-Kroukovsky, B.V., and Kotik, J., “Theory of the Wave Resistance
of Ships,” Trans. Soc. Naval Arch. Marine Engrs. 62:359-396 (1954)

[5] G. Weinblum, “Schiffe Geringsten Widerstandes,” Proc. Third Internat. Congr. Appl.
Mech., Stockholm, pp. 449-458, 1930; also “Schiffe Geringsten Widerstandes,” Jahrbuch
der Schiffbautechnischen Gesellschaft, Vol. 51, 1957

[6] Lighthill, M.J., “A New Approach to Thin Aerofoil Theory,” Aero Quart., vol. 3, p. 3,
Nov. 1951

[7] Dorr, J., “Zwei Integralgleichungen erster Art, die sich mit Hilfe Mathieuscher Funk-
tionen losen lassen,” Zeits. angew., Math. Physik 3:427-439 (1952)

[8] Erdelyi, A., et al., “Higher Transcendental Functions,” vol. 3, NewYork:McGraw-Hill,
p. 132, 1955

[9] Tricomi, F.G., “Integral Equations,” New York:Interscience, pp. 143-144, 1957

[10] MacCamy, R.C., “On Singular Integral Equations with Logarithmic or Cauchy Kernels,”
J. Math. and Mech. 7:355-375 (May 1958)

[11] Sretenskii, L.N., “Sur un Probleme de Minimum Dans la Theorie du Navire,” C.R. (Dokl.)
Acad. Sci. USSR(N.S.) 3:247-248 (1935)

[12] Pavlenko, G.E., “The Resistance of Water to the Motion of a Ship,” State Publishing
House for Water Transport, Moscow, pp. 188-192, 1953 (in Russian)

[13] Shen, Y.C. and Kern, G.E., “Development oy of a Subsurface Ship,” Aerojet-General
Corp. Report 1535, 1958

646551 O—62——9

110 Samuel Karp, Jack Kotik, and Jerome Lurye

[14] Stoker, J.J., and Peters, A.S., “The Motion of a Ship, as a Floating Rigid Body, in a
Seaway,” Institute of Mathematical Sciences, New York University, Report IMM-NYU
203, 1954

[15] Magnus, W., and Oberhettinger, F., “Formulas and Theorems for the Functions of Math-
ematical Physics,” Chelsea, 1949

[16] Yourkevitch, V., “The Form of Least Resistance,” A.T.M.A., 31:687 (1932); see also
the Shipbuilder and Marine Engine Builder, p. 235, Apr. 1933

[17] Karp, S., Kotik, J., and Lurye, J., “On Ship Forms Having Minimum Wave Resistance,”
TRG, Inc., Report TRG-119-SR-1, 1959

DISCUSSION
C. Wigley (London)

Unfortunately advance copies of these papers were only available yesterday. This pre-
vents any criticism of the mathematical work, but there are some points of importance which
should be mentioned regarding the practical use of these calculations of forms of minimum
resistance.

Firstly the wave resistance as calculated by the Michell or equivalent formulas is that
wave resistance which would exist in a perfect fluid.

Secondly it is assumed that for a ship the wave resistance is to be simply added to an-
other resistance due to viscosity which does not change greatly with a change of form.

Regarding the first point raised, in fact, except at very high speeds where the Froude
number is above 0.4, the effects of viscosity on the wave formation are serious, causing a
decrease in the efficiency of the afterbody as a wavemaker.

Regarding the second point, the nonwave resistance may be considered as consisting
of the sum of two terms, the first term depending only on the wetted surface and speed and
being some 90 to 95 percent of the total, and the second term, sometimes called the form re-
sistance, depending as well on the shape of the form. Very little is known as to the varia-
tion of this form resistance with speed, although its value at very low speed is shown by
the difference between the calculated frictional resistance and that actually measured, since
the wave resistance can be neglected at such speeds. It is suggested by M. Guilloton (see
Trans. I.N.A., London, vol. 1952, pp. 352 and 353) that the form resistance is increased at
the higher speeds owing to the effect of the wave motion on the flow round the form, a symp-
tom of this influence being the change of attitude of the form during motion.

The effects of these considerations on the actual minima of resistance are well shown
by the simple question of the optimum position for the center of buoyancy. From the math-
ematical theory, as given by these papers today, the optimum position of the center of buoy-
ancy would be amidships at all speeds. In practice, it is found that, at the lower speeds up
to a Froude number of about 0.25 where the wave resistance is small the optimum position of
the center of buoyancy lies forward of amidships, since the form resistance is thus dimin-
ished. As the speed increases the optimum position moves aft and lies aft of amidships for

Minimum Wave Resistance for Dipole Distributions lll

a range of Froude number up to about 0.4, owing to the lower efficiency in wavemaking of
the afterbody, which evidently causes an advantage when displacement is moved from the
forebody to the afterbody. At still higher speeds the best position tends to agree with the
theoretical position at amidships.

It may be of some interest to compare the results for struts of infinite draft found by
Messrs. Kotik, Karp, and Lurye with some experimental and calculated results for bulbous
bows I published in the Proceedings of the North-East Coast Institution of Engineers &
Shipbuilders in 1936. These calculations were made for a spheroid added to a calculable
ship form, and the experiments were made with a fairing between the spheroid and the form
which would tend to diminish any additional form resistance. Also the change of form was
hoped to be insufficient to change appreciably the frictional correction to the wave resist-
ance. Under these precautions the calculated and experimental curves showed, for the best

position and size of the bulb, a definite decrease in resistance over a range of Froude num-
ber from 0.25 to 0.5.

The conclusion of these comments is that, owing to the uncertainty of the application
of such results as those in the papers under discussion, it is advisable that they should be
checked by actual measurements before any practical use be made of them.

G. Weinblum (Institut fur Schiffbau, Hamburg)

May I express my sincerest thanks to the authors of this paper and of the preceding
paper for the aesthetic pleasure presented to me. Some general remarks may be to the point:

1. It is a well-known proposition that ship theory can become too difficult for naval
architects.

2. It is therefore advisable for naval architects to love mathematicians.
3. This love, however, should not be unreciprocated.

4. The present speaker feels much obliged that his mathematic colleagues have given
such a kind credit to his earlier work. Obviously these attempts were formally rather poor;
nonetheless, they embrace some physical and technical ideas which have proved to be fruit-
ful in the further development. Besides the authors mentioned, Dr. Guilloton has contributed
to the discussion of the problem which is thrilling both from the point of view of ship hydro-
dynamics as well as mathematics.

Twenty-five years ago a contribution was made by Prof. von Karman at the 4th Interna-
tional Congress in Cambridge, England, in which he pointed out difficulties encountered
when dealing with the exact solution of the minimum problem. Among other things, he found
curves with infinite horns shown in the paper by Prof. Karp and others. This finding caused
a lot of confusion in the professional world. Unfortunately, Prof. von Karman has forgotten
to publish these interesting investigations.

A decisive progress has been made in the meanwhile by distinguishing between the
shape of the singularity distribution and the actual body form. Although this difference is
well known and has been clearly illustrated by Havelock, e.g., in the case of the general
ellipsoid, no use was made of this fact in many earlier publications on our subject. I sug-
gested in vain a thesis on this topic some 12 years ago but it was not completed. So we are

112 Samuel Karp, Jack Kotik, and Jerome Lurye

essentially indebted to Prof. Inui, because he has dealt with the problem at stake in a con-
sequent and efficient if approximate manner. His severe remark that Wigley and I have ham-
pered the progress in wave resistance research by neglecting the distinction between hull
form and distribution is perhaps too hard since our conclusions are primarily based on the
sectional area curve and this, fortunately, is less affected by the difficulties mentioned
than that of the actual ship form.

I am dwelling at some length on this subject as a warning example: one should not
proceed too long in a well-established groove of thought and therefore lose the connection
with facts.

Prof. Timman and Mr. Vossers have, as far as I understand, rigorously remained within
the concept of the Michell ship. We are looking forward for explicit results. Prof. Karp and
colleagues have followed rather the way indicated by Havelock, distinguishing between
body form and distribution. Although the results apply to a rather restricted case of a
deeply submerged strut, they are in principle extremely interesting. The optimum cross
sections plotted correspond to some extent to those which have been derived by approximate
methods, except, however, that they all are more exaggerated and show a rounded nose which
could not be obtained by the low degree polynomials used before. It would be interesting to
have similar cross sections for Froude numbers below 0.38.

This lower range appears to have important practical consequences. Using my earlier
work my collaborator Kracht has investigated the influence of the bulbous bow (and stern).
Contrary to experimental results and earlier findings by Mr. Wigley and myself he has estab-
lished that moderate bulbs may have a beneficial influence on wave resistance even at low
Froude numbers. We are continuing with these investigations. To get an independent check,
wholly submerged bodies of revolution moving in the vicinity of the free surface have been
considered. I have already treated this problem several times earlier. However, while in
the case of the Michell surface ship it is plausible from physical reasoning to assume zero
end ordinates for the distributions, there is no need to introduce this restriction for bodies
of revolution. Therefore, together with Dr. Eggers and Mr. Sharma, I have investigated sin-
gularities systems which include continuous line doublet distributions, concentrated sources,
and, following Wigley, doublets located on the axis of the body.

The plots in Fig. D1 show the optimum longitudinal distribution calculated for a con-

stant area coefficient 9 = 0.60 (except for plot (g)), three depths of immersion ratios 2f/L,
and two Froude numbers.

The symbol <2, 4,6,8 D> indicates the powers of the polynomial terms; D stands for
concentrated dipoles at the ends of the axis with a doublet moment ap.

Computations have been made under three assumptions:
1. The end ordinates of the dipole distribution, (+1) = 0.

2. End ordinates —7(+1) are not prescribed but ap = 0; a result 7(+1) > 0 means a con-
centrated source at the ends.

3. The dipole moment a, + 0. In this case the area of the circle at the ends of the
axis represents the dipole moment to the scale of the distribution curve.

Figures D1(a)-(c) show the optimum doublet distributions for y, = 1/2F* = 6, F = 0.289.
These distributions coincide sometimes nicely for different 2f/L (Fig. D1(a)), sometimes

Minimum Wave Resistance for Dipole Distributions 113

P (2.4.6.8)

ee eee
y = 06

70.2, 4,68) a ee
Ro? 0 iD a0

Fig. D1(b). Optimum longitudinal distribution

114 Samuel Karp, Jack Kotik, and Jerome Lurye

7 (26680)

Fig. D1(c). Optimum longitudinal distribution

they differ. Occasional erratic behavior has been disregarded for present purposes. We are
primarily interested in the variation of the shape with the three assumptions listed. Figure

D1(a) (n(+1) = 0, ap = 0) shows the well-known swan-neck form; the curve for 2f/L = 0.25 in
Fig. D1(b) may be doubtful.

In the range of low Froude numbers y, = 15, F = 0.183 we find in Fig. D1(d) the orthodox
hollow distribution (y(+1) = 0, a, = 0); in Figs. D1(e) and D1(f) we see the relatively small
end ordinates and doublets respectively.

Figure D1(g) finally shows the optimum forms for ahigh prismatic 9=0.80 and y, =1/2F7=15.
In general the results do not present great surprises except for the appearance of a concen-
trated doublet at small F and the large reduction in resistance due to the former as compared
with the assumption @p =.0. An explanation can be found by comparing the generated body
shapes which display marked differences. The resulting forms and data for other Froude
numbers will be the subject of a more elaborate report.

The numbers R* stand for dimensionless wave resistance values. Obviously, they indi-
cate that the latter is negligible for 2f{/L = 1 and in some cases for 2f/L = 0.5 at the Froude
numbers considered. Thus the investigations presented by the authors have contributed to
clarify a basic problem of ship theory. For purpose of practice we shall be forced to inves-
tigate more general forms than the Michell ship. E.g., in the range of higher Froude numbers
conclusive experiments have shown that by using a flat stern a definite improvement can be
reached. A further step in this direction may be the investigation of the so-called Hogner
interpolation formula although some essential difficulties must be overcome.

Minimum Wave Resistance for Dipole Distributions 115

2X2, 6, 6,8)

Fig. D1(d). Optimum longitudinal distribution

7 0,2,6,6,8))

are sy
a
S

Fig. Dl(e). Optimum longitudinal distribution

116 Samuel Karp, Jack Kotik, and Jerome Lurye

+ Q005 003 0,002

10°°
09 16 24

7 (2 4, 6, 8 2»

%
f
at > 10 05 025
29
Rt

Fig. D1(f). Optimum longitudinal distribution

vo 1S
o
y¥ = 08
af. 40 as Q25

= Q072 09006 Qo0é

Fig. D1(g). Optimum longitudinal distribution

Minimum Wave Resistance for Dipole Distributions 117

J. Kotik

Dr. Weinblum has asked a question as regards an arithmetical error in Fig. 10. The pa-
rameter € is defined as the average half-beam divided by the length, rather than the average
beam. Perhaps this factor of 2 will accommodate Prof. Weinblum’s doubts. In reference to
Dr. Weinblum’s descriptions of his work on the wave resistance of submerged boats, or of
dipoles distributed on a submerged horizontal line segment, I have the following comment.

It turns out that if one fixes the length and submergence of this horizontal line segment,
distributes dipoles on it, and tries to minimize the wave resistance within the family of di-
pole distributions having a fixed dipole moment (or approximately a fixed volume), one is led
to an integral equation which has no solution. We cannot say at present my this apparently
well-posed problem has no solution.

S. Karp

I would like to ask Prof. Timman whether the solution of the integral equations were
Mathieu functions.

R. Timman

There is still a big gap between the mathematical approach to the problem of minimum
resistance and the engineering point of view. With regard to the influence of viscosity, I am
quite sure that the purely mathematical approach is not feasible.

T. Inui (University of Tokyo)

I should like to make some general remarks regarding future possible developments in
our “practical” study of wavemaking resistance.

Firstly, I wish to point out the necessity of improving our measuring techniques such as
so-called stereophotography for recording the actual wavemaking phenomena which we can
observe in our daily model basin work. Figure D2 is one example of such stereo slides. It
is more than five years since I first tried to take a picture of model waves at the Tokyo Uni-
versity Tank.

Secondly, I should like to point out the special importance of coordination between
theory and experiment in our study of wavemaking resistance. If we had no theoretical
basis, we could not go any further in our study, even if we have succeeded in taking beauti-
ful pictures of model waves. In this connection, Sir Thomas Havelock’s contribution in the
field of wavemaking theory is most valuable.

Thirdly and finally, I wish to mention our desire of having an international panel to
make use of high speed computers for preparing many kinds of mathematical tables which
may be valuable not only to our basic studies but also to our practical design work.

118 Samuel Karp, Jack Kotik, and Jerome Lurye

No 8
C-20/
V=1317™%
F=0:267
Kel. =(F

Fig. D2. A black-and-white reproduction of a colored stereo slide

John M. Ferguson (John Brown and Co., Glasgow)

The authors of these two preceding papers must be admired for the ease and sureness
of handling the complicated mathematics of this problem of minimum wave resistance. Yet
the feeling of admiration is tempered with a little disappointment. In what way do these
papers, as they stand, assist the practical designer in his daily work of producing the best
form for a given set of conditions.

A typical example is as follows: A form is designed. To this form a model is made and
tested. The form may not come up to expectation. In what way can this model be altered to
improve its performance? Somewhere in R. E. Froude’s published work he states that the
quality of performance of a model depends mainly on the shape of the sectional area curve
and the load water plane. If the mathematicians could devise some method of analyzing the
sectional area curve so that the analysis could indicate the features of performance and
thereby suggest the manner in which the area curve could be modified for the better, then
their mathematics would be really worthwhile.

The mathematician has all the time he needs. The practical designer often has to pro-
duce the answer today, if not sooner.

Some years ago, a young colleague of mine was interested in this problem. He devised
a method in which a sectional area curve was analyzed by Runge’s scheme into a sine and
cosine series. It was found that the coefficients of the sine series could be related to cer-
tain values of the speed-length base and that the humps and hollows of the coefficients bore
some close relationship to the humps and hollows of the typical resistance curve. If a hump
on the resistance curve occurred at, say a trial or contract speed and there was a hump in
the Runge coefficient at that point, it was suggested that a reduction of the hump in the re-
sistance curve might result from a reduction in the value of the nearest Runge coefficient.
The series would then be rearranged to give the original value of the prismatic coefficient

Minimum Wave Resistance for Dipole Distributions 119

of the sectional area curve. This rearrangement of the series then resulted in an alteration
in the sectional area curve from a rebuilding of the series. In two or three cases where this
was tried there was some improvement in performance, but there were some failures as well.

In one particular example where a drastic change was made in the sines with the inten-
tion of completely removing a large hump in the resistance curve, a reversal of the analysis
gave a sectional area curve with a large hollow in it at or about midships. From the prac-
tical point of view this was laughable. Yet some work by Prof. Weinblum and others have
produced such sectional area curves.

An intriguing thought pertaining to this hollow is provided by the fact that the sectional
area curve for the model in motion, taking the wave profile into account, can show such
peculiar hollows.

I throw these thoughts to the mathematicians in the hope that they may be able to adapt
their powerful mathematical methods to the daily needs of the less expert practitioners. It
might be better if these experts could recognize the needs of their humbler brethren — said
in all sincerity. On the other hand it would be as helpful if we more practical people paid
greater attention to the more general but more fundamental work of the mathematicians.

dt he live

THE HYDRODYNAMICS OF HIGH-SPEED HYDROFOIL CRAFT

Marshall P. Tulin
Hydronautics, Incorporated

Rockville, Maryland

A general discussion of some important hydrodynamic problems associ-
ated with the operation of high-speed hydrofoil craft is presented. The
difficulties that arise from the necessity to either avoid cavitation or
design for supercavitating operation, and at the same time to deal with
those high-gust loadings that accompany flight in a seaway are empha-
sized. Theoretical results pertaining to the effect of a seaway on foil
loadings and cavitation inception are given, as well as results describ-
ing the influence of the free surface on inception speed. The importance
of flaps is stressed and new theoretical results concerning flap effec
tiveness in both sub- and supercavitating flows are presented. Finally,
the effect of near-surface operation on the lift-drag ratio of supercavitat-
ing foils is discussed and some new theoretical results are given.

INTRODUCTION

I should like at the very beginning to state two warnings. The first concerns the title
of this paper which was poorly and immodestly selected. Rather than “The Uydrodynamics
of High-Speed Hydrofoil Craft” it should read “Some Hydrodynamics of High-Speed Hydrofoil
Craft.” In excluding from discussion many important problems, I have sought to emphasize
others which are, in my opinion, most needy of discussion at this time. The second warning
is addressed to the very small number of you who not only expect from me but even look for-
ward to highly mathematical content. I am sorry to announce that not a single integral sign
is to be displayed here. For this reason, I have already been accused of a kind of scientific
degeneracy. Such a charge must, of course, be vigorously denied and | hasten to point out
that between the lines and behind the figures lies, I think, enough mathematics to satisfy all
but the most jaded theoretician.

The hydrofoil boat together with the airplane has been evolving now for over 50 years.
It has appeared in many and varied configurations and performed with mixed success; until
recently its future as a marine vehicle remained uncertain. The inevitably growing need for
high-speed transport over water would seem to be well enough recognized at this time, how-
ever, so that the serious and continued development and use of the hydrofoil boat may be
considered safely assured.

The advantages of the hydrofoil craft over displacement or planing craft for high Froude
number operation is well-known. Only the edge jet vehicle (or hydroskimmer or ground effect
machine as it is sometimes called) offers competitive performance and then, we believe,
only in large sizes, say several thousand tons displacement. In Fig. 1 are outlined on a

121

M. P. Tulin

122

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Sud} ul $yB1aM
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Hydrodynamics of High-Speed Hydrofoils 123

speed versus displacement diagram the regions in which various marine vehicles, including
semisubmerged ships in addition to those already mentioned above, would seem to offer
optimum powering performance. This diagram is based on available data [1, 2] and powering
calculations we have carried out at Hydronautics, Incorporated. Also shown are design
points for various hydrofoil boats that have been constructed or are under construction.
Study reveals that both with regard to needs and capability, the hydrofoil boat seems partic-
ularly fated for high-speed operation, that is for speed in excess of 40 and perhaps as high
as 120 knots.

The engineering problems involved in the design and construction of high-speed hydro-
foil boats are exceedingly severe with respect to power plant, transmission, structure, and
hydrodynamic aspects. It is the purpose of the present paper to discuss some of the impor-
tant hydrodynamic considerations involved, to underline certain problems, and generally to
give perspective to the situation that the hydrodynamicist faces.

3oth sub- and supercavitating craft intended for sustained operation at sea are con-
sidered. It is emphasized that high-speed craft involve very high static foil loadings and
additional severe seaway loadings. The hostility of the sea environment is strikingly illus-
trated through comparison with the atmospheric environment through which the airplane
flies. The problem of preventing cavitation on subcavitating craft is discussed and the
effect of seaway motions on inception is predicted. The importance of flaps for loads and
motion control is underlined and some new theoretical results relating to the effect of the
free surface on two-dimensional flap effectiveness for both sub- and supercavitating foils
are presented. Other new theoretical results pertaining to the effect of the free surface are
also given and, in particular, its beneficial influence on lift-drag ratios of two-dimensional
supercavitating foils is revealed.

Many staff members of Hydronautics, Incorporated, have participated in the preparation
of the results indicated above.

WING LOADINGS

Considering the essential similarity between the hydrofoil boat and the airplane, it is
natural to compare the problems that have faced their development and, of course, to take
maximum advantage of mutually useful and pertinent information and experience. This last
statement will almost universally be translated as meaning that the naval architect should
take advantage of the knowledge of the aeronautical engineer—as, most naturally, he often
does. Of course, seacraft and aircraft operate in vastly different environments. Whereas
air is so light that it is still a matter of amazement to many that anything substantial can
support itself therein, every successful hydrodynamicist early acquires a great deal of
respect for the weightiness of his own particular subject. A consequence of this difference
in densities is illustrated in Fig. 2. In interpretation of this figure it is useful to recall
that the wing or foil loading (displacement over He area) equals the wing lift coefficient
(Cy) multiplied by the dynamic pressure 7, (QUp 2/2), and that the optimum lift coeffi-
cient for a given design depends primarily upon hydrodynamic considerations relating to
minimum powering requirements; practical design lift coefficients for both seacraft and air-
craft lie in the range 0.1 to 0.35. The very much larger wing loadings experienced by high-
speed seacraft in comparison to those for supersonic aircraft are apparent from the figure.
It is striking that the wing loadings of 60-knot-plus seacraft are at least one and almost two
orders of magnitude larger than for aircraft of contemplated design. At the same time, the
hydrodynamic demands for thinness of underwater structures are just as severe as in the

124

Dynamic Pressure, q

M. P. Tulin

Gy 20 40 60 80 100 120

Seacraft Speed in knots Aircraft Speed in knots x 10"
Propulsion Barrier (Hurdled About 1954)

upercavitating (Achieved About 1960) ~/\—=

Subcavitating

Propulsion Barrier (Hurdled About 1944)
Supersonic (Achieved About 1950) —‘\/\\>

Subsonic

Fig. 2. A comparison of some parameters for seacraft and aircraft

Hydrodynamics of High-Speed Hydrofoils 125

case of aircraft; for either a supercavitating or supersonic foil of given length, drags are
incurred proportional to the square of the foil thickness. It is rather difficult to put this
comparison on a quantitative basis, but suffice it to say that a 5 percent diamond-shaped
strut has the same drag coefficient (about 0.1 based on thickness) operating at mach number
2.3 in air, as when operating at 60 knots and 5 feet submergence in water while ventilated
to the atmosphere.

Similarly, for subsonic or subcavitating foils a great premium is placed on structural
slenderness, since the speeds which may be attained in either case without impairment in
operation increase with thinness of structure. An attempt is made to illustrate this fact in
Fig. 2 (bottom) where the critical pressure coefficient for aircraft and inception cavitation
number for seacraft are plotted, together with the minimum pressure coefficient for a wing of
about 7 percent thickness ratio and 0.1 lift coefficient. Such a wing would encounter,
according to Fig. 2, drag rise and buffeting difficulties at 500 knots plus in air and 65 knots
plus in calm water. However, a wing of 4 percent thickness and 0.1 lift coefficient could
probably achieve about 530 knots in air and 80 knots in calm water before impairment of
operation. The crucial observation to be made is that the seacraft structural designer must
thin his wings at design dynamic pressures and wing loadings some 20 to 50 times greater
than would exist in the case of an aircraft.

It is interesting, having put the static loading problem in some perspective, to note
some analogies between propulsion and hydrodynamic problems for the airplane and hydro-
foil boat. In these analogies, which it will be immediately recognized cannot be taken too
literally, high-speed problems incurred in air due to the finite speed of sound become
equivalent to difficulties caused by cavitation in water. In both media, screw propellers
are commonly used at low speeds, with their use being finally impaired unless radical
change in form is made at speeds of the order of 400 knots in air and 40 knots in water.
The propulsion barriers that existed as a consequence, have both been successfully hur-
dled at this time, by the introduction of jet propulsion in one case, and of the supercavitat-
ing propeller in the other [3, 4]. It is taken for granted that high-speed hydrofoil craft
will utilize supercavitating propellers with efficiencies at design in the range 0.65 to 0.72.

At speeds somewhat higher than those at which the low-speed screw propeller runs
afoul of efficiency loss and severe vibration, the supporting foils of vehicles in both media
begin to suffer those same difficulties, and at speeds in excess of 750 knots in air and 75
knots in water, it is almost essential that the design of wings and foils be based on entirely
different principles than for low speeds. As is indicated in Fig. 2, the first hydrofoil craft
utilizing supercavitating foils is being test flown this year (1960); it has been designed and
constructed by Dynamic Developments, Inc., of Babylon, New York, under contract to the
Office of Naval Research of the U.S. Navy. The cavities attached to these foils are not,
of course, filled with water vapor, but are ventilated through the free surface to the
atmosphere.

THE HOSTILE SEA

In a very important respect and in addition to considerations of static wing loading, the
sea is a much more difficult, even hostile, environment in which to fly than the atmosphere.
Most naval designers and ship’s passengers have acquired an appreciation for the severity
of the wind-driven motions which must statistically be expected to exist near and on the
ocean’s surface; and all of us who allow ourselves to be transported from place to place in
aircraft have awareness of the turbulence in the atmosphere—we may even have observed

646551 O—62——10

126 M. P. Tulin

that such turbulence is generally much more intense near the ground than at high altitudes.
I do not think, however, that ever before has a quantitative comparison of gust intensities
just beneath the surface of the sea, and in the atmosphere been made. In Fig. 3, the hori-
zontal axis represents the root-mean-square of the vertical velocities (in feet per
second) either in air or water, and the vertical axis the probability of exceeding a given
value of root mean square velocity. Shown plotted are curves for the North Atlantic at a 5-
foot depth averaged the year around and curves for averages for the atmosphere at low alti-
tudes (0 to 10,000 feet) and moderate altitudes (30 to 50,000 feet). The values for air have
been taken from Ref. 5. The values for the North Atlantic at 5-foot depth were theoretically
derived from experimental all-year observations of significant wave heights in the North
Atlantic made by the U.S. Weather Bureau, and have been taken from Ref. 6. It will be
observed that a root-mean-square value of 1 foot per second is exceeded only 17 percent of
the time at moderate altitudes in the atmosphere, 53 percent of the time at low altitudes, and
89 percent of the time at a depth of 5 feet in the North Atlantic. For an rms value of 3 feet
per second, the respective percentages are 3, 16, and 28 percent. Only for high-intensity
gusts (which occur infrequently) is the atmosphere at low altitudes statistically gustier
than the ocean. The intensity of motions in the sea of course varies with depth of submer-
gence, and in Fig. 4 this dependence is shown for a sea generated by a 20-foot-per-second
wind, which about corresponds to a state 3 sea. It is to be noted that the rms vertical
velocity is about twice as great at the surface as it is at 5-foot submergence and is further
reduced about 50 percent as the depth increases to 10 feet.

The motion of the environment induces loads on the vehicle structure and subsequent
rigid body motions and structural flexing. The seacraft, operating as it does in such close
proximity to the sea surface must maintain its altitude relative to the instantaneous sea
surface within very strict limits lest the hull impact or the foils and propeller broach. For
that reason it is generally desirable for the craft to “follow” or respond to waves which are
as long or longer than the craft itself. Both surface-piercing and variable-incidence foils
are designed with such response in mind. Of course, the actual response of the vehicle to
a certain sea will particularly depend on the frequency with which it encounters the waves
that compose that sea. It so happens that the spectrum of a real, fully-risen, wind-gen-
erated sea seems to be sharply peaked, most of the energy residing in waves traveling with
a celerity close to the wind speed and with corresponding lengths [7]. The consequence of
this fact is that a boat traversing such a sea will sense a dominant frequency of encounter,
which will depend on the wind speed that generated the sea, the boat’s speed, and its direc-
tion relative to the wind. In Fig. 5 is plotted this frequency of encounter versus boat speed
from motion into (to the left) and with (to the right) the wind, for wind speeds from 7 to 23.5
knots, corresponding to sea states from 2 through 5. These curves are based on studies
made by F’. Turpin and M. Martin [8]. It is to be noted that a response to frequencies, w,
in excess of 2.5 radians per second or about0.4 cps (averaging the “with” wind and “into”
wind encounters) is required for a 60-knot boat operating in state 5 sea, and even more
rapid responses for faster craft. The rms accelerations experienced by a boat responding
to such a sea are 0.707 w24%p, where ‘Ip is the semiamplitude of the vehicle’s heaving
motion relative to fixed axes.

Such accelerations must themselves be limited especially on account of human comfort
limits. It turns out, in fact, that our tolerance to vertical accelerations is quite small.
Shown in Fig. 6 are curves based on recent studies of factors influencing hydrofoil boat
handling qualities by P. Eisenberg [9] which delimit acceleration and frequency ranges for
which vertical motions are perceptible, uncomfortable, and intolerable. If an rms acceleration
of 0.15 g is taken as an upper design limit, then in a state 5 sea, Ho could not exceed
about 6 inches. However, the rms wave semiamplitude in the same sea is about 3 feet and
the 10 percent highest waves have an rms semiamplitude of 5.4 feet. It would seem that

Probability of Exceeding Vamsi” percent

Hydrodynamics of High-Speed Hydrofoils

Vrms, Root Mean Square Vertical Velocity in feet/ second

Fig. 3. Statistical vertical velocities at sea and in the atmosphere

127

128 M. P. Tulin

RMS Vertical Velocity in ft/sec

Depth Below Free Surface in feet

Fig. 4. Effect of depth on vertical velocities
in a seaway

Frequency of Encounter, w, in rad/sec

6
Sea State
2
4
2 Ya
fe) a a
80 60 40 20 (0) 20 40 60 80

—— Into Wind Boat Speed in knots With Wind —=—

Fig. 5. Frequencies with which a boat encounters waves of
various sea states corresponding to various wind speeds

Hydrodynamics of High-Speed Hydrofoils 129

INTOLERABLE

gs

RMS Heaving Accelerations in

PERCEPTIBLE

Ol 1.0 10
Frequency of Heaving in cycles/sec

Fig. 6. Human sensitivity to heaving oscillations

such a sea cannot, for reasons of comfort alone, be followed to any extent, but must be
plowed. Such an effort, of course, requires a boat with sufficient hull clearance and foil
submergence for the hull to pass over crests and the foil system to extend under wave
troughs. Variable incidence foil systems particularly recommend themselves for plowing,
since a typical surface-piercing foil would during such motions experience fluctuations in
load due not only to the orbital subsurface velocities associated with waves, but also to
changes in wetted area.

It is of interest to calculate the fluctuating loads acting on a plowing fully submerged
foil system, without trim changes, flap or spoiler actuation (in other words, a passive foil
system), in order that the problem of controlling these loads by one or another of the means
just mentioned can be properly assessed. These loads will actually depend on the rigid body
motions of the vehicle and the structural flexing. Aircraft experience [10, 11] would lead
us to believe that such dynamic effects tend to increase the loads calculated on the basis
of a rigid structure and quasi-static analysis such as we make here. Thus, alarming as our
results may be, they are probably not to be relieved by more rigorous calculations, but only
through a relaxing of the sea conditions that we have referred to. The dynamic g loadings
(in addition to static loadings) induced by the seaway on a fully submerged wing are, accord-
ing to quasi-static and rigid-body theory, approximately equal to

(CL,/CLU0)7

130 M. P. Tulin

where Cy, is the foil lift curve slope, Cz is the equilibrium lift coefficient, Up is the
speed of advance, and v is the seaway-induced vertical velocity. If rms g loadings are
desired, then rms values of V are to be used in computation. The quantity

Cr /CLU0

at design speed is a characteristic of the craft, while the quantity V depends on the envi-
ronment. Clearly, if minimum dynamic loadings are desired, then the parameter

CL, /CLY

should be minimized in design.

In Fig. 7 are shown calculated values of this parameter as a function of speed, for a
family of subcavitating and supercavitating hydrofoil wings. The former was selected with
wing lift coefficients, thickness ratios, and aspect ratios varying from 0.35, 0.11, and 7
(respectively) at 45 knots to 0.15, 0.05, and 3.3 (respectively) at 70 knots and in such a
way as to maintain approximately constant wing stress and satisfy cavitation inception con-
siderations at each speed. The operating depth of submergence is for all wings taken as
one mean foil chord. The supercavitating wings were taken in two versions, utilizing in
one case thin low-drag foil sections (t/c =~ 0.08) with aspect ratios and lift coefficients
ranging from 5 and 0.18 respectively at 60 knots to 3.4 and 0.16 respectively at 90 knots,
and in the other case thicker foil sections (t/c =~ 0.16) with aspect ratios and lift coeffi-
cients ranging from 8.1 and 0.26 at 60 knots to 5.6 and 0.23 at the higher speed. All of the
supercavitating wings considered are designed for approximately equal maximum nominal
stress and are optimized with respect to lift-drag ratios.

The dynamic loads parameter is seen to be generally larger for subcavitating wings,
mainly as a result of the higher sectional lift curve slopes of subcavitating foils relative to
supercavitating sections. It will be recalled that at infinite depth, the former are about

Current Prop Transport (300 knots)

Current Jet Transport (530knots)
Projected Jet Transport (2000 knots)

Dynamic Loads Parameter, C, . /C. Up, in sec /ft

60

Hydrofoil Craft Speed, Uy , in knots

Fig. 7. Dynamic loads parameter Cz, /C,Uo for a family of hydrofoil craft

Hydrodynamics of High Speed Hydrofoils 131

four times the latter in value. The proximity of the free surface tends to reduce this ratio,
however, and for the wings considered, the Cz, were in an average ratio of about two to
one. The increase in the dynamic loads parameter for subcavitating craft between 60 and 70
knots is entirely due to the decrease in lift coefficients demanded in order to prevent cavi-
tation inception. For the optimized supercavitating wing families a general decrease in
dynamic load parameter accompanies and is due to increasing speed. The family with
thicker foil sections enjoys a smaller value of the parameter because of higher optimum

lift coefficients. The values of

Cre /Oni

for three transport aircraft [12] are also shown on the vertical ordinate and are seen to be
considerably smaller in value than for the best supercavitating wing at 85 knots; the speed
of these aircraft range from 300 to 2,000 knots.

The results of the calculation of the dynamic loads parameter shown in Fig. 7 and
which have just been discussed, are combined with earlier statistical results, Fig. 3, relat-
ing to the character of the North Atlantic and of the atmosphere, to determine the probability
that a particular craft with passive foil system (no load alleviation) will suffer dynamic rms
g loadings exceeding a certain level. These new results are shown in Fig. 8. The extremely
marked effect of decreasing values of dynamic loads parameter,

CL, /CLU

in reducing the probability of high g loadings is to be noted. As this parameter varies from
a value of 0.04, which is a little less than for the best supercavitating wing at 90 knots, to
a value of 0.12 which is a little less than for the subcavitating wing at 65 knots, the prob-
ability of exceeding an rms g loading of 0.2 increases from 4 to 70 percent.

The results for aircraft utilizing the low-altitude gust probability curve of Fig. 3 for
the propeller transport and the high-altitude curve for the jet are interesting in themselves,
but they have been especially included in order to afford through comparison a striking
illustration of the magnitude of the dynamic loads problem that must be faced by the
designer of seagoing high-speed hydrofoil craft.

LOAD ALLEVIATION AND FLAPS

The results shown in Fig. 8 make it abundantly clear that extremely effective load-
alleviating devices must be provided if the design limit of 0.15 g rms vertical acceleration
set according to comfort criteria is to be met. These devices must counter vertical orbital
velocities in both upward and downward directions and for 60-knot plus craft at encounter
frequencies of the order of 2.5 radians per second and higher. If load alleviation is sought
through tail foil trim control, then angular accelerations leading to vertical accelerations in
excess of the design limit would, according to our calculations, seem to be implied. As an
alternative, control of lift at the main foils without essential change of trim suggests itself
and has the additional advantage of affording more rapid response than in the case of trim
control.

The use of flaps has especially been considered, and in that connection we present
here the results of recent calculations on the behavior of two-dimensional flaps for both
subcavitating and supercavitating foils, both operating at very high forward speeds. It must

132 M. P. Tulin

Hydrofoil Craft (North Atlantic)
Current Prop Transport

Probability of Exceeding RMS "g" Loading

fo) 0.2 0.4 06 08 1.0 1.2

Root Mean Square “g" Loading

Fig. 8. Statistical dynamic loading for seacraft and aircraft

be understood that the values of flap effectiveness and hinge moments shown must be suit-
ably modified in practice to take into account real fluid and finite aspect ratio effects;
these results do, however, show the magnitude of the effect of near surface proximity.

In Fig. 9 (left) are given curves of subcavitating flap effectiveness versus flap-chord
ratio for depth-chord ratios from zero to infinity. A composite curve for a range of flap-
chord ratios, graphically illustrating the near surface effect is also given in Fig. 9 (right).
The important conclusion is that theory predicts about a 4-percent loss in ftap effectiveness
relative to infinite depth at one chord submergence, and a 13-percent loss at 1/2 chord
depth. The loss at zero submergence in the planing condition is 50 percent. These calcu-
lations assume that no cavitation occurs on the foil in the vicinity of the hinge; such occur-
rence would of course lead to loss in flap effectiveness. The results for surface effect on
subcavitating flap effectiveness were obtained by my colleague C. F. Chen in a very inter-
esting way. He determined that a particular flow reversal theorem which had previously
been derived for flows without free surfaces [13] was applicable in the present case, and he
was thus able to calculate the force on a flapped foil in terms of the flow about a flat plate
at incidence under the free surface; the latter flow is exactly the flow about the lower foil

|

Hydrodynamics of High-Speed Hydrofoils 133

Flap Effectiveness, C, Je

050 1.00
Flap-Chord Ratio, f/¢ Depth-Chord Ratio, hy

Fig. 9. Free surface effect on two-dimensional subcavitating flap effectiveness

of a two-dimensional biplane and has been known for decades [14]. This calculation con-
stituted the first application to my knowledge of flow reversal theorems for flows with a
free surface.

The flap effectiveness of supercavitating flaps at a given depth and for a particular
flap-chord ratio is less than that of subcavitating flaps, but the difference in effectiveness
decreases with depth and finally disappears at zero submergence. This situation corresponds
to the fact that subcavitating flap effectiveness decreases as the free surface is approached
whereas supercavitating flaps become more effective. J. Auslaender [15] has recently devel-
oped pertinent theory and calculated the performance of supercavitating flaps including the
effect of the free surface. In Fig. 10 are presented curves of flap effectiveness versus flap-
chord ratio for depth-chord ratios of 0, 1, and infinity. The results for infinite submergence
had been obtained previously [16], at which time it was observed that supercavitating flaps
at infinite depth reached a maximum value of flap effectiveness for flap-chord ratios less
than one. It will be observed in Fig. 10 that for a flap-chord ratio of 0.25, the supercavitat-
ing flap gains 17 percent in effectiveness at 1 chord submergence (relative to infinite depth)
and 47 percent at 0 submergence in the planing condition. Also, presented as Fig. 11 are
hinge moment coefficients for supercavitating foils composed of a forward flat section oper-
ating at various angles of incidence 4, followed by a 25-percent flap, all operating at a
depth of 1 chord. These results and many more of the same nature have been obtained by
Auslaender using linearized theory for zero cavitation number [16-19]. Itis presumed that
practical foils will be ventilated to the atmosphere and thus operate at a cavitation number
close to zero, so that the present results are meaningful practically.

The action of flaps or other load alleviation devices must, of course, be supervised by
a suitable sensing and control system. Such systems, involving acoustic wave sensors and
similarly sophisticated devices—plus the ubiquitous black box or two—are apparently under
active development. I will not dare to say anything about these things, but it is perhaps
worth speculating in the present context that high-speed boats must mainly plow through
rather than respond to the sea, and that the alleviation of dynamic loads, rather than the

134 M. P. Tulin

2.0

a

2/, x Flap Effectiveness, 2/a Ci /e
°

(0) 0.2 0.4 0.6 0.8 1.0
Flap-Chord Ratio, f/,

Fig. 10. Free surface effect on two-dimensional
supercavitating flap effectiveness

-10

)

M
Yo? v2 c2

Hinge Moment Coefficient, Cy' x 10°
( Cyr =

fo) 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Foil Lift Coefficient, C, (6,€)

Fig. 11. Supercavitating hinge moment coefficient

Hydrodynamics of High-Speed Hydrofoils 135

control of motions, becomes the task that must be successfully accomplished by the flaps,
black boxes, etc.

CAVITATION SUPPRESSION

As boat speeds increase beyond 40 knots the danger of cavitation occurrence on foils,
vertical struts, and propulsion nacelles rapidly increases. Because of the erosive effects
of cavitation it does not seem attractive to design for operation with small regions of local
cavitation occurring on the underwater structure, nor does it seem really feasible to allow
extensive cavitation short of supercavitation because of the buffeting that very often accom-
panies such “transcavitating” flows [20]. The designer must thus strive to suppress cavi-
tation, a process which as earlier mentioned leads to underwater structures which become
increasingly thin.

A great deal has been written about the estimation of minimum pressure coefficients on
hydrodynamic bodies, and I will here only add several small straws to the camel’s back.
The first concerns the point that it would seem to us generally dangerous in design, even in
the earliest stages, to consider the inception properties of components by themselvesrather
than in combination. This might seem to be a rather obvious remark, but nevertheless there
has been considerable discussion of the inception properties of components, and very little
on combined configurations. In Fig. 12 are shown theoretical inception speeds for a modi-
fied spheroidal nacelle and a NASA 16 series strut both separately and in combination, as
a function of pod length-diameter ratio. The strut maximum thickness was assumed to be
1/6 of the nacelle diameter and its length 67% of the nacelle length; since such a nacelle
may at times operate in practice right up to the free surface, the calculations were made for
zero depth although the relieving effect of the free surface was not taken into account.
These calculations were made for illustrative purposes only, and although thought to be
realistic, do not necessarily represent optimum configurations; further they neglect the
influence of the propeller and horizontal foils should they be called for, and the additional
influence of pitch and yaw. Even so the relation between component and configuration
inception speeds is indicated.

In considering the inception speeds of components it would seem very important that
the effect of wave motions be taken into account, for the minimum pressure coefficients of
high-speed foil and strut sections are very sensitive to incidence and yaw. Shown as Fig.
13 is an estimation of the probability that local cavitation will occur on a hydrofoil wing of
aspect ratio 4, with Series 16 sections of 4-percent thickness and a design lift coefficient
of 0.15, while operating at 5-foot depth in the average North Atlantic environment described
in Fig. 3; it was assumed that cavitation occurs during motions involving fluctuating angles
of attack for values of rms angle of attack equal to steady inception angles, and this is
believed to be a conservative assumption. It is observed that this wing, which could accord-
ing to theory make 70 knots in calm water, will cavitate over 50 percent of the time during
year-round operation at speeds as low as 50 knots, and over 75 percent of the time at 60
knots. These predictions of average performance do not, of course, show how inception
speed varies with sea state, so in Fig. 14 is presented (solid line from the left) a curve of
inception speed versus sea states (and corresponding wind speed) for the wing described
above, assuming again that the rms angle of attack corresponds to the steady inception
angle. The dashed curves are based on the slightly different assumptions that the average
of one-third highest and maximum angles of attack in the seaway correspond to the steady
inception angle. It is of interest to know how much faster than inception speed a wing may
be driven before the cavity grows long enough to cause serious buffeting, and the right-hand

136

Cavitation Inception Speed in knots

M. P. Tulin

Pod Alone

140 ei j
(Modified Spheroid)

120

Pod Max. Diam. =6x Strut Max. Thick.
Strut Length = 0.67 Pod Length

100

Strut Alone
(16 Series)

Strut and Pod
Combination

4 5 6 iG 8 9
Pod Length- Diameter Ratio

Fig. 12. Cavitation inception speeds for streamlined
pods and struts

Probability of Cavitation Inception in percent

(0)
80 70 60 50 40 30
Speed inknots

Fig. 13. Probability of cavitation inception vs speed for
North Atlantic operation

Hydrodynamics of High-Speed Hydrofoils 137

— — — — For Max. Vertical Velocity in Waves
—- —— - ——_ For |/3 Greatest Velocity in Waves
For RMS Velocity in Waves

70

wn
2 60 Danger of Buffeting
=
Uv
®
a
a
7p)
= 50
ra

40 ‘

Sea State |
| 2 3 + 4 5 6
30
(0) oy 10 15 20 25 30

Wind Speed in knots

Fig. 14. Wing cavitation inception vs sea state

boundary in Fig. 14 outlines the region of forward speeds and sea state wherein severe foil
vibrations might be expected; this prediction is entirely theoretical as far as operation in a
seaway is concemed, being based on observations under steady angles of attack [20].

In order not to end this discussion of cavitation suppression on too despairing a note,
I would like to present as Fig. 15 the results of calculations that we have carried out on
the effect of the proximity of the free surface on the minimum pressures due to the thickness
portion of a thin plane wing, and on the minimum pressures acting on spheroidal bodies of
revolution. For the smaller depth-diameter ratios, the latter refer particularly to the pres-
sures on the upper portion of the body. These calculations indicate that very modest gains
in inception speed may result in practical cases due to the proximity of the free surface.

SUPERCAVITATING OPERATION

It has already been pointed out that the propulsion of high-speed hydrofoil boats
depends upon the utilization of supercavitating propellers, and it would seem clear, in addi-
tion, that the 70-knot-plus potential of these craft will depend upon the utilization of super-
cavitating foils and struts. A great deal has recently been written about practical bodies
operating in cavity flow, and particularly good references to this work are the collective
papers presented at the second Office of Naval Research Symposium on Naval Hydrodynamics
(1958), which was the immediate predecessor of the present Symposium.

The Langley Laboratory of the (U.S.) National Aeronautics and Space Administration
have in the past conducted especially valuable theoretical and experimental studies of
supercavitating wings and struts operating in ventilated condition [17-19, 21]. These
results provide at least a lower bound to the values of lift-drag ratios to be attained for
high-speed craft, and it is believed that further significant increases in foil efficiency will
be realized, if at all, only as the result of detailed and intensive research.

M. P. Tulin

138

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“sapawipig Apog wnwixow /Apog 40 doy 0, yydaq 9/y ‘01¥Dy P4oyd yn4JS - yydaq
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UOIJN]OAaY JO Salpog

Hydrodynamics of High-Speed Hydrofoils 139

For foils of optimum design, the sum of induced and cavity drags seems to be con-
siderably greater than the friction drag. The division of induced and cavity drag depends
very much on the wing aspect ratio, the cavity drag being generally larger and thus more
important for aspect ratios of 2 or 3 and higher.

Theoretical studies [16, 17], now confirmed by experiments [18, 22], have led to the
design of low-drag supercavitating foils such as are used in propeller design. We have
recently developed a theory and made calculations to reveal the effect of near-surface
proximity on the shape of optimized low-drag foils, and on the resulting lift-cavity drag
(L/D ,) ratios. It may be recalled that I had some time ago shown that, according to linear-
ized theory, there exist for supercavitating foils at infinite submergence, optimum values of
L/D, inversely proportional to the foil lift coefficient. These ideal optimum ratios are not
realized in practice partly because of the necessity to design foils of sufficient strength,
and thus develop lift from incidence as well as camber; nevertheless, these ratios do pro-
vide information as to the efficiency with which lift can be developed through camber. In
Fig. 16 are shown together with previous results for infinite submergence, theoretical two-
dimensional foil efficiencies at a submergence of one chord. The optimum ratios of L/D,
are seen to be almost four times higher at this moderately shallow submergence than for
deep submergences, but for various reasons these same ratios for practical foils and lift
coefficients are reduced to about 1.5. As a caution against any undue optimism that might
be generated by this finding, it should be recalled that active load-alleviation devices such
as the supercavitating flaps earlier discussed will, it seems safe to predict, inevitably
extract their own penalty in terms of increased cavity drag.

Theoretical Lift-Drag (Cavity) Ratios, Lp,

40 Practical" Two Term’
Foils
asian
zeae
10
0 0.1 0.2 03

Lift Coefficient, C,

Fig. 16. Theoretical two-dimensional lift-drag (cavity) ratios

140 M. P. Tulin

CONCLUSION

The subject of the hydrodynamics of high-speed hydrofoil craft has been far from exhaus-
tively discussed here, even with regard to a cataloging of important problems. An effort has
been made, however, to provide a little perspective as to the task that faces the designer
and, in particular, to emphasize the problem of seaway induced dynamic loads, the related
problem of load alleviation, certain problems connected with cavitation suppression—espe-
cially in a seaway, and, finally, the problem of the varied effects of near-surface proximity
on foil performance. An attempt has also been made to provide some new and useful infor-
mation relating to these problems; it will be understood that much of this information is
based on theoretical developments and calculations carried out by the staff of Hydronautics,
Inc., which are themselves the subjects of separate papers, and could not for reasons of
space be gone into in any greater detail here; it is a pleasure to acknowledge that a good
many of these results that I have just referred to were obtained under a research program

sponsored by the U.S. Navy’s Bureau of Ships.

Finally, lest some of the conclusions that have been arrived at are interpreted as
throwing some doubt on the feasibility of high-speed ocean-going hydrofoil craft, let us
affirm our personal conviction that such craft will inevitably be successfully developed and
will before too much time passes find their unique place in the transportation scheme of
things.

REFERENCES

[1] Saunders, H.E., “Hydrodynamics in Ship Design,” Vol. Il, New York: The Society of
Naval Architects and Marine Engineers, 1957

[2] Geyer, L.A., and Wennagel, G.J., “A Feasibility Study of Hydrofoil Seacraft,” Pre-
sented before The Chesapeake Bay Section of the Society of Naval Architects and
Marine Engineers, 1958

[3] Tachmindji, A.J., Morgan, W.B., Miller, M.L., and Hecker, R., “The Design and Per-
formance of Supercavitating Propellers,” David Taylor Model Basin Report C-807, 1957

[4] Tachmindji, A.J., and Morgan, W.B., “The Design and Estimated Performance of a
Series of Supercavitating Propellers,” p. 489 in “Second (1958) Symposium on Naval
Hydrodynamics,” Washington: Office of Naval Research, 1960

[5] Press, H., Meadows, May T., and Hadlock, I., “A Reevaluation of Data on Atmospheric
Turbulence and Airplane Gust Loads for Application in Spectral Calculations,” NASA
Technical Report 1272, Washington, 1956

[6] Brooks, R.L., and Jasper, N.H., “Statistics on Wave Heights and Periods for the North
Atlantic Ocean,” David Taylor Model Basin, Sept. 1957

[7] Neumann, G., and Pierson, W.J., “A Comparison of Various Theoretical Wave Spectra,”
Proceedings of the Symposium on the Behaviour of Ships in a Seaway (25th Anniversary
of the NSMB), H. Veenman and Zonen, Wageningen, The Netherlands, 1957

Hydrodynamics of High-Speed Hydrofoils 141

[8] Turpin, F.J.,and Martin, M., “The Effect of Surface Waves on Some Design Parameters of
a Hydrofoil Boat,” Hydronautics, Inc., Technical Report 001-3, Jan. 1961

[9] Eisenberg, P., “Considerations of Hydrofoil Boat Handling and Motions Qualities,”
Hydronautics, Inc., Technical Report 001-4, Oct. 1960

[10] Bisplinghoff, R.L., “Some Structural and Aeroelastic Considerations of High-Speed
Flight (Nineteenth Wright Brothers Lecture),” Journal of the Aeronautical Sciences 23
(No. 4), Apr. 1956

[11] Bennett, F.V., and Pratt, K.G., “Calculated Responses of a Large Sweptwing Airplane
to Continuous Turbulence with Flight-Test Comparisons,” NASA Technical Report
R-69, Washington, 1960

[12] The Staff of the Langley Research Center, “The Supersonic Transport—A Technical
Summary,” NASA Technical Note D-423, Washington, June 1960

[13] Heaslet, M.A., and Spreiter, J.R., “Reciprocity Relations in Aerodynamics,” NACA
Technical Note 2700, Washington, 1952

[14] Glauert, H., “The Elements of Aerofoil and Airscrew Theory,” New York: Macmillan,
Ist ed., 1944

[15] Auslaender, J., “The Behaviour of Supercavitating Foils with Flaps Operating at High
Speed Near a Free Surface,” Hydronautics, Inc., Technical Report 001-2, 1960

[16] Tulin, M.P., and Burkart, M., “Linearized Theory for Flows about Lifting Foils at
Zero Cavitation Number,” David Taylor Model Basin Report C-638, Feb. 1955

[17] Johnson, V.E., Jr., “Theoretical Determination of Low-Drag Supercavitating Hydrofoils
and Their Two-Dimensional Characteristics at Zero Cavitation Number,” NACA RM
L57Glla, Washington, 1957

[18] Johnson, V.E., Jr., “Theoretical and Experimental Investigation of Arbitrary Aspect
Ratio, Supercavitating Hydrofoils Operating Near the Free Water Surface,” NACA RM
L57116, Washington, 1957

[19] Johnson, V.E., Jr., “The Influence of Submersion, Aspect Ratio, and Thickness on
Supercavitating Hydrofoils Operating at Zero Cavitation Number,” p. 317 in Proceed-
ings “Second (1958) Symposium on Naval Hydrodynamics,” Washington: Office of
Naval Research, 1960

[20] Numachi, F., Tsudoda, K., and Chida, I., “Cavitation Tests on Six Profiles for Blade
Elements,” Inst. High Speed Mech., Tohoku Univ., Report 8, pp. 25-46, 1957

[21] Wadlin, K.L., “Ventilated Flows with Hydrofoils” (NASA), Presented at the Twelfth
General Meeting of the American Towing Tank Conference, Berkeley, California,

Sept. 1959

[22] Waid, R.L., and Lindberg, Z.M., “Water Tunnel Investigations of a Supercavitating
Hydrofoil,” California Institute of Technology, Hydrodynamics Laboratory Report 47-8,
Apr. 1957

646551 O—62 11

142. M. P. Tulin

DISCUSSION
S. F. Hoerner (Gibbs and Cox, Inc., New York)

The Froude number is generally used in the consideration of displacement-type ships.
As far as hydrofoil boats are concerned, it is known that such craft will be able to operate
efficiently at higher Froude numbers. The size of hydrofoil craft is limited, accordingly,
and to mention a number, the limit might be in the order of 500 tons. However, there is also
an upper limit to the Froude number as far as hydrofoil craft are concerned. To explain this
briefly, consider a craft of 200 tons operating at 50 knots, which is believed to be feasible.
We may now put into that craft heavier machinery, propelling it at 100 knots, which is twice
the original speed. As a consequence, to support the same weight, we will only need 1/4
the original foil area. The size of the struts supporting the weight of the hull above the
foils is, on the other hand, a function of the hull weight and of drag and other forces such
as impact which are more or less proportional to that weight. In other words, the struts
remain the same size as those of the original craft. Since struts are roughly accountable
for half the drag of a foil system, the faster boat will have higher drag. This result can be
said to be the consequence of high Froude number. The argument applies to plain configura-
tions of hull plus foil system, while cavitation is not yet taken into account.

Marshall P. Tulin

It is certainly true that all studies of the speed and power of hydrofoil craft have
revealed that there is a maximum practical Froude number beyond which hydrofoil craft may
not be economically or otherwise feasible. Such studies, of course, include those made by
Dr. Hoerner himself—which have proved very valuable to students of hydrofoil craft during
the last few years. The actual value of a maximum Froude number depends on structural
and power plant weights and its existence is not to be argued, but I would like to suggest
that this Froude number may be an increasing function of time and that its estimation is
thus not particularly easy. I might further comment on Dr. Hoemer’s remark with regard to
the drag of struts. In particular, I don’t think I can agree that the strut drag is simply about
half of the foil system drag; I believe that very much depends upon the configuration of the
foil system and that the strut drag may be considerably less than half of the total for certain
systems.

S. F. Hoerner

My statement that strut drag is roughly half the foil system drag applies to fully sub-
merged foil systems at maximum speed. Generally, struts in such systems represent approxi-
mately 1/2 the parasitic drag, which mean approximately 1/4 the total drag, at maximum-
range speed.

F. S. Burt (Admiralty Research Laboratory)

I endorse the point Mr. Tulin made that there is a considerable shortage of systematic
data of components in association with one another. It is more important that tests on these
associated systems should be carried out at the correct cavitation number than at the correct
Froude number. In fact you are in the usual difficulty; you want all your numbers right,

Hydrodynamics of High-Speed Hydrofoils 143

which can only be achieved at full-scale, and under these conditions it is very difficult to
get really systematic measurements under completely controlled conditions. There are very
few facilities available which can run at sufficient speed to get the correct cavitation num-
ber and a reasonable approximation to the Froude and Reynolds numbers. We have, in fact,
been doing some work in the rotating-beam channel at the Admiralty Research Laboratory
because we can run at large scale and very high speeds, up to 100 knots. There are avail-
able in the world quite a number of seaplane tanks which are going into disuse at the
moment and I would suggest that these are ideal facilities for this type of work, particu-
larly as a lot of them have the ability to make tests in simple wave systems, which is a
very vital part of the whole problem. I suggest that more use might be made of these facil-
ities by our various governments for hydrofoil craft experiments.

Marshall P. Tulin

I am very happy to hear Mr. Burt’s comment on the effects of interactions on inception
based upon his own experience and presumably on experimental measurements of cavitation
inception. With regard to facilities, he has an admirable point with regard to the utilization
of seaplane tanks where they exist. As I mentioned in my talk, the Hydrodynamics Division
of the former National Advisory Committee for Aeronautics, now the National Aeronautics
and Space Administration (NASA), carried out most valuable research on supercavitating
hydrofoils and other related problems very pertinent to the development of hydrofoil craft;
we all hope in the United States that those fine facilities which are no longer being used
for seaplane research will continue to be put to use for high-speed hydrofoil and other
marine testing.

Glen J. Wennagel (Dynamic Developments, Inc., Babylon, New York)

Mr. Burt talked on the further use of existing high-speed towing tanks and certainly
these should be used. In fact, with the decrease in interest in high-speed seaplanes in
recent years, perhaps hydrofoils have given these tanks a new lease on life. However, it
is not always easy to test in the existing towing tank facilities. Sometimes they are far
away and scheduling difficulties exist. Also, they are expensive. We went, about a year
ago, to facilities like the Whirling tank and the Pendulum in order to have something rela-
tively cheap in our own back yard, where we could use relatively inexpensive small models,
make changes quickly and put ourselves in the right ball park on a hydrofoil or strut design.
Then we would look forward to using some of the high-speed tanks, such as those at
Langley Field, Virginia, after a foil design has been optimized. Let me give you an idea
of the dollars and cents involved here. The whirling tank is worth about 100,000 dollars.
You can use that amount of money up rather quickly in testing with larger scale models. To
get the accuracies we wanted, a single model tested at Langley Field was worth about
12,000 dollars. You do not have to build many of those to use up the cost of a smaller
facility.

P. Kaplan (Technical Research Group, Inc., Syosset, New York)

I have some questions as to the validity of the simple gust-load formula for the g load-
ing of a hydrofoil system. The numbers appear to be too pessimistic, at least from my quick
judgement of them, and we must remember the fact that the real hydrofoil craft is a coupled

144 M. P. Tulin

system where there is important effect of pitch and you have really two foils at different
positions relative to the center, but quite different from a simple aircraft and what I have
done was to compute some values of this gust-load factor according to the quasi-steady
formula and then compare that with some values obtained from some experiments at the
Model Basin. This bore out the fact that the original formulation appears to be quite pessi-
mistic. However, the Model Basin tests were at a rather low Froude number, so perhaps
that may cover some of the difference. Also, Mr. Tulin mentioned that he did not cover all
of the various problems associated with the hydrodynamics of high-speed craft, and in par-
ticular I would like to put forward the fact that in this case of high-speed and supercavitat-
ing foils it appears that there may be a rather distinct possibility of the hydroelastic insta-
bility, that is, flutter. Some theoretical results carried out about a year ago at Stevens
indicated that there is a greater possibility of obtaining a hydroelastic instability, that is,
flutter, for supercavitating foils as compared to fully wetted foils. This sort of problem
should be looked into very carefully in the design of these craft.

Marshall P. Tulin

With regard to Dr. Kaplan’s question about the calculation of g loadings (which were
admittedly based on the simplest sort of analysis), there is no question that very much more
sophisticated calculations for hydrofoil craft should be carried out. In my paper I refer to
several sophisticated calculations carried out for aircraft and I would suggest that these
calculations should serve as a model with regard to hydrofoil investigations. To say that
coupled motions are involved in such calculations is a gross understatement. In the case
of a recent aircraft investigation, five degrees-of-freedom were considered, including torsion
and bending of the wing, which are very important for high-speed aircraft and should cer-
tainly be as important in the case of the hydrofoil craft with its high wing loadings. It was
particularly this aircraft investigation which prompted me to make the remark in my paper
that the simplest kind of analysis leads, if anything, to conservative results regarding load
estimations. Of course, the damping of hydrofoil craft motions is not necessarily the same
as in the case of an aircraft, and for that reason alone we cannot come to any rigorous con-
clusions from aircraft experience. I hope that the kind of calculations, as | have indicated
above, will be carried ‘out; I can say that we plan to make some calculations ourselves.

Problems of hydroelastic phenomena are well-known to be important for high-speed
supercavitating foils whether used in hydrofoil craft or in propellers. Such problems, par-
ticularly with regard to leading edge flutter, have already been experienced, and some stop-
gap solutions have been obtained. Certainly research on the subject of flutter and on strut
divergence, which may be very important for high-speed craft, should be the subject of
future investigations.

Edward V. Lewis (Davidson Laboratory, Stevens Institute of Technology)

Mr. Tulin has pointed out some of the serious problems of hydrofoil craft design for
rough sea conditions. He expresses optimism regarding the possibility of solving these
problems and suggests possible solutions.

I should like to return to a more basic problem of the hydrofoil boat which pertains to
fundamental limitations on this type of craft. For this purpose we may compare a hydrofoil
boat with an airplane, in which the boat hull and airplane fuselage are comparable, but the
hydrofoil wings are in water instead of air. At any particular speed the increased density

Hydrodynamics of High-Speed Hydrofoils 145

of water makes it possible for the wing area to be proportionately reduced for the hydrofoil
boat. The total drag and hence overall L/D would be about the same as an airplane for an
idealized hydrofoil craft in which resistance of appendages and supports is neglected. How-
ever, the increased density of water makes it possible with the proper “wing” area to obtain
sufficient dynamic lift for the hydrofoil boat to “fly” at lower speeds than the airplane in
air. When the hydrofoil craft is then designed to operate at a speed intermediate between
that of an airplane and a ship, there should theoretically be a reduction in hull or fuselage
resistance in air which is proportional to the square of the speed. Hence, the initial appeal
of the hydrofoil boat was the hope that it could fill the speed gap between ships and air-
planes, with a higher L/D or lower P/WV than the airplane. Unfortunately, the strut system
required to put the wings in the water adds more to the resistance than is saved by the
reduced speed. Hence, even the most enthusiastic hydrofoil man does not expect as good

an L/D for the hydrofoil craft as for the airplane.

An important difference remains between the hydrofoil boat and the airplane. At its
lower speed, the hydrofoil craft requires much less power, which should be advantageous.
But the lower speed is less efficient for propulsion, and the drive system becomes very
complex. Hence, no gain in payload from reduced power requirements can be expected in
practice. Insofar as fuel is concerned, for the same specific fuel rate the total fuel con-
sumed for equal distances is the same or greater.

Hence, it appears that the goal of the hydrofoil boat designer is simply to equal the
performance of aircraft, but at reduced speed. The possibility of doing this in terms of L/D
and gross weight (payload) appears remote. In short, a seaplane can outperform a hydrofoil
boat, and attain higher speed at the same time. We must recognize therefore that the hydro-
foil craft is inherently a special purpose vehicle.

I realize that it is dangerous in this company to appear antihydrofoil, hence, I hasten
to add that I am enthusiastic about this development. My reason is simply that it is already
stimulating technica! progress in naval architecture, bringing advanced hydrodynamics,
better structures, and more efficient power plants into the picture. No matter what happens
to the hydrofoil itself, naval architecture will never be the same again.

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ON HYDROFOILS RUNNING NEAR A FREE SURFACE

S. Schuster and H. Schwanecke
Versuchanstalt fiir Wasserbau und Shiffbau

Berlin Towing Tank

For the steady state of flow the results of pressure distribution measure-
ments dependent on speed, depth of submergence, angle of attack, and
roll angle—both for a flat and for a dihedral hydrofcil—are presented, as
well as the results of six-component measurements for two dihedral foils.
Below the Froude depth number 1, related to the depth of submergence,
the flow and hence the pressure distribution changes according to the
shallow water laws; above this figure no significant speed dependency
exists. Here the pressure distribution is influenced only by the distance
between the hydrofoil and the surface in relation to its profile length. For
this situation, which is normal for hydrofoil boats under service condi-
tions, simple theoretical rates for calculating the hydrodynamic forces are
given, as well as the results of these calculations in the case of a
parallel submerged hydrofoil of great span. By analyzing the separate in-
fluences the relation to results of wind tunnel tests can be set up. By
means of the results from theoretical and experimental examinations, rela-
tions are derived for the vertical and the lateral stability both of flat and
dihedral hydrofoils as well as for the influence of sideslip motion.

INTRODUCTION

One of the most important suppositions for the design of hydrofoil boats is the precise
knowledge of the flow forces occurring on the foils under given service conditions. There
is no doubt that many conceptions are transferable from aerodynamic research but some es-
sential properties like lift distribution and all values of driving performance and running be-
havior connected herewith will be more or less influenced by the proximity of the water sur-
faced. During the last twenty years this problem has been treated very often, but further
treatment, based on new experiments in the Versuchsanstalt fur Wasserbau und Schiffbau
(the Berlin Towing Tank), may be expected to clarify and complete the picture. During
these examinations the pressure distribution both for completely submerged hydrofoils and
for those piercing the water surface, the deformation of the surface, and the flow forces have
been measured as well as theoretically treated, all angles being varied.

PRESSURE DISTRIBUTION MEASUREMENT

Two brass hydrofoil models with special borings have been used for measuring the pres-
sure distribution, one of them being a flat hydrofoil and the other a dihedral hydrofoil (V-
form), both idealized with circle segment profiles of constant length without distortion (Fig.
1). On the starboard side the flat foil was fitted with 68 borings and the dihedral foil with

147

148 S. Schuster and H. Schwanecke

Fig. 1. Hydrofoil models simplified for pressure distribution measurement

49 borings of 1.5-mm diameter. The holes were arranged in sections parallel to the longitu-
dinal axis and connected to each other parallel to the lateral axis. For measurement all the
sets of holes but one have been covered with thin plastic strips. Thus, successively, the
pressure distribution around the profile for 6 cross sections for the flat foiland 7 cross sec-
tions for the dihedral foil has been measured and photographically recorded during each run
in the deepwater channel. For the flat foil the angle of attack has been varied from —1 degree
to +6 degrees, and the depth of submergence from 30 mm to 240 mm at a speed of ug = 3.7
m/sec; for a special series the speed was also varied, from 0.4 to 3.7 m/sec. For the di-
hedral foil the angle of attack has been kept corstant at +] degree while the rolling angle
has been varied from 0 to 33 degrees, one arm of the hydrofoil being parallel to the water
surface when the rolling angle was 33 degrees. The measured pressure in relation to’ the
ram pressure has been plotted versus the profile length for every section. Integration of the
pressure curves results in the local lift coefficients, which in turn by integration over the
span supply the total coefficient. For checking these values and for finding the drag three
component tests have been made.

Flat Foil

The variation with speed for the flat foil first of all confirmed the shallow water effect
as found by Laitone [1], Parkin, Perry, and Wu [2] and Plesset and Parkin [3], which occurs
especially at the critical speed u, = gh (Fig. 2), where h denotes the distance between
the trailing edge and the undisturbed water surface.* At a speed sufficiently above this
value the ratio between depth of submergence and length of profile only is authoritative for
the course of the pressure curve. Samples are shown in Figs. 3 and 4. Herein the charac-
teristics of the diagrams versus h/c and © are different. The local lift coefficients indeed
change in the same sense as depth of submergence and angle of attack, but in the first case
the pressure distribution is unchanged, while it is varying much in the second case. There-
fore, approaching the surface cannot be substituted by a reduction of the angle of attack.

*A nomenclature list is given at the end of the paper.

Hydrofoils Running Near a Free Surface 149

a
ee
2
=
=
eS
ers

P : ait ae ;

aeee
ae

oe
eee

cae
_

ae

aes
Be
eased ae

Fig. 2. Pressure data for the flat foil of circle segment profile
at subcritical and supercritical velocity

150 S. Schuster and H. Schwanecke

AEE
Ala

EEG AE Pa

= Ee
Se es S25
ee

Se
ee

Eva

Fig. 3. Pressure distribution for (= 2 degrees and
varying h/c ratios for the flat foil

Hydrofoils Running Near a Free Surface 151

HHL ge
HEE

Fig. 4. Pressure distribution for h/c = 2 and
varying angles of attack for the flat foil

152 S. Schuster and H. Schwanecke

The lift distribution over the span can be seen in Fig. 5, the lift coefficients for the
complete hydrofoil are shown in Fig. 6, and the components of the suction side as well as
those of the pressure side and the ratio between these components are shown in Figs. 7 to
9, where the C; values are related to conditions in great depth (infinite medium). In ap-
proaching the water surface the lift of the foil running at supercritical speed decreases
nearly along a parabola with the depth of submergence, especially for h/c < 2, the effect of
the suction side being diminished and the effect of the pressure side being augmented. This
change will decrease for increasing angles of attack. It is remarkable that the drag coef-
ficient is decreasing with the! decrease of depth in spite of the increasing waves (Fig. 10).

Summing up, the results of these tests with the flat foil were as follows:

1. Owing to the proximity of the water surface an additional velocity is induced which
is contrary to the direction of flow. This results in an increase of the lift component on the
pressure side and in a decrease on the suction side. Due to the suction side’s greater part
of the total lift in the deep submerged position, a diminution of lift thus results. Owing to
the minor flow velocities at both sides the profile resistance decreases.

2. Owing to the curvature of flow along the upper side the effective profile curvature is
diminished. Thus the angle of zero lift will be shifted toward positive values, and the effec-
tive angle of attack and thereby the lift coefficient will become less. Relatively this influ-
ence decreases with increasing angle of attack.

3. Owing to the continual lift diminution in the range 2 > h > 0 a submergence stabiliz-
ing seems to be possible even for hydrofoils not piercing the water surface.

Dihedral Foil

The measured lift distribution for the dihedral foil can be seen in Fig. 11. The pressure
distribution over the profile is similar to that for the flat, fully-submerged hydrofoil. The
joint of the two foil halves at the keel point, as well as that between foil and struts, causes
an increase of lift on the suction side, and in the first case simultaneously decreases the
pressure at the lower side. When piercing the surface a diminution of the local lift occurs
within the region of the surface approach in such a manner as though the results of flat
parallel submerged foils had been transferred stripwise.

Figure 12 shows the lift decrease of both foils plotted versus the relative submergence.
This diagram shows also the test results of Land [4], which were found at substantially

higher speeds.

The lift diminution for the complete dihedral hydrofoil caused by the surface effect can
be found by integration of the local values

Coun

within the range of h = 0 to h= yee} The mean of (C,,/C,_,) amounts to approximately 0.5
for hy.e1= 0.5, to 0.8 for hy...) = 1.5, and to 0.9 for hy.) = 5-

Summing up, it follows from the tests with the dihedral hydrofoil that:

Hydrofoils Running Near a Free Surface 153

Fig. 5. Lift distribution over the span of the flat foil for varying h/c ratios

154 S. Schuster and H. Schwanecke

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Fig. 7.

ef
a0) coer jdeuigeeetarere BIR Pee
ee pec eee =
fl ft 0 Ea ee Ea ct eed Brel
Lift coefficients of the upper sides
of the flat dihedral foils

Hydrofoils Running Near a Free Surface 155

Ec WT oe ee
ee

:

Ea ne are aL
as Seg eS
a
a Seate

ae

ae
a

275i

—— eens
So i
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iC

aerial mele
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=
iS

Es

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ig

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a

ee

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if
eee Eat t
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arnt area

seafieasjeactae
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cee |
go ee ee aie ee reese eee ee
alee sees Se ee eeieeie

RU TRT
ae a
ete Hea Bc

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age Gales Renee eae ee
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8
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ee a
(2 SORE SD SESS ss= nae af
= FS

Fig. 8. Lift coefficients of the lower side of the flat foil

S. Schuster and H. Schwanecke

156

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E eee)
eRe es Ss
apo fe
a PAR
=o agg8 .
a ay panes CUES = ae
= ae toe oe : 23
esas ee ss z aN eae a AEE] 3s
a “| El SALA Lg) Ss
SS a
eg
3) CT eal
Hs Se SUA ag
= a ce a5
= on . Ae
a: hi! 2-4
eae hia Se
a ete
PO 4
NG OV
eo
- fx

Fig. 10. Drag coefficients of the flat foil

Hydrofoils Running Near a Free Surface 157

Fig. 11. Lift distribution for the dihedral foil

646551 O—62——12

158 S. Schuster and H. Schwanecke

Fig. 12. Comparison of lift coefficients versus relative depth

1. The test results of fully submerged flat foils can be transferred to emerging foils by
means of the strip method. The lift decrease toward the outer regions becomes also measur-
able for h/c < 2 and essential for h/c <1. Enlarging the profile length in the outer regions
therefore seems to be less practical since the h/c value will be smaller and the influence of
the water surface will be extended to larger regions of the span. According to item 2 in the
summing up for the preceding subsection, it would be better to increase the angle of attack,
i.e., to distort the foil.

2. Since the strip method can be used for the rolling dihedral hydrofoil too, there is no
special need for investigating the influence of the roll angle. Even for hydrofoils in a side-
slip motion the measurement of the pressure distribution can be retained, for in this case
only the local angles of attack are changed, the influence of which has been determined for
the flat parallel submerged foil.

STATE OF FLOW EXAMINATION

If the existence of a shallow water effect could be concluded from the pressure measure-
ment, then the transition from the subcritical to the supercritical range of the Froude depth
number must be perceptible by a change of the surface deformation. Tests with a flat paral-
lel submerged hydrofoil, extending sideways to the walls of a circulating water channel,
proved that there is indeed a transition from a steady flow to a rapid flow with a hydraulic
jump between. With regard to the speed increment above the foil and to the slope of the
water surface connected herewith, a local Froude depth number

u 3/2
Bin, = U (3 + alll «)
h lsh! 2

where h’ denotes the minimum depth above the foil, can be defined as a barrier. The defor-
mation corresponds to pictures shown in the publication of Parkin, Perry, and Wu [2].

Hydrofoils Running Near a Free Surface 159

Below F,’ = 1, i.e., in the subcritical speed range, the depth of submergence of any
foil plays a similar part for the circulation as the water depth does for the orbital velocity
of waves in shallow water. Thus correspond the C, reduction on the upper side and the
wave speed reduction

Crh ad ee
Lo re)
Hence
2
F, =
follows.

To be sure, this can be seen exactly only at small depths, for example for h/c = 0.37
as in Fig. 13. For larger depths the transition is smoothed at F, = 1 and disappears gradu-
ally in the same manner as it does in deep water. Above F, = 2 no dependency of the Froude
depth number exists. Hydrofoil boats generally run by dynamic lift at a speed more than 25
knots and at a submergence less than 2 m. This corresponds to a smallest Froude depth
number of about 3.

Summing up, the results of examination of the state of flow are as follows:

1. The hydrofoil is susceptible to the shallow water effect. For operating conditions
of hydrofoil boats the flow pattern is a supercritical one. For theoretical considerations by
analogy from wave theory, rapid flow is presumed in the direct neighborhood of the hydrofoil
in place of orbital motion for waves.

2. For individual hydrofoils no critical speed barrier can exist as it does for displace-
ment ships in shallow water, given by speed and depth. But for the complete craft fitted
with several hydrofoils such a barrier, of course, can exist, since the wave formation behind
the hydraulic jump follows the Froude depth number given by the water depth and changes
the flow against the next hydrofoil in line.

3. The only significant quantity for the influence of the water surface on the circulation
around a hydrofoil in the supercritical range is the relative depth of submergence of the foil
element, h=h/c. It is independent of the configuration and speed of the hydrofoil.

FLOW FORCES MEASUREMENT

In 1941 six component tests on hydrofoils already had been made in the circulating water
tunnel of the firm Gebr. Sachsenberg AG., Berlin, under the direction of G. Weinblum. The
results were published at that time only fs a restricted number of persons.

In one case a model of a dihedral foil with a constant circle segment profile of a con-
stant length { =.96 mm and a thickness t = 6.4 mm was used [6]. The dihedral angle could

160 S. Schuster and H. Schwanecke

SoscesesreeeeREERE CE
i ee Spel Ae
SINE fiers Cane

mises Sree eae : i
ees ee peng node ee a
one ee

: i oF, : a0

Fig. 13. Local lift coefficients of the upper and lower sides of
the flat foil for different h/c ratios

be changed between 15 degrees and 50 degrees. Variations of span from 200 to 600 mm, of
roll angle from 0 to 10 degrees, and of angle of attack from —5 to +7 degrees were made. All
the values of C,, Cp, and Cy, were measured for a constant speed of 4 m/sec. For exam-
ple such a could be drawn as can be seen in Figs. 14 and 15 for lift and drag, or in
Figs. 16 and 17 for the roll moment. The results for the mean span of 400 mm were C,

0.36, dC; /da = 2.72 to 3.14 related to the effective angle of attack, 4(C, = 0) =—4.5 me...
Cp min = 0-015, and €,,;, = 0.07.

A comparison of the results for the different variations showed with regard to lift, drag,
drag-lift ratio, and lateral stability:

1. The aspect ratio A = 52/ F possibly should be chosen not less than 4. Raising it
beyond 5 will give but little additional profit.

2. As long as the proximity of the surface does not influence the complete foil, i.e., as
long as h,,,, > 1, the dC,, ,/@0 value will increase for decreasing dihedral angles, for
hax < 1 the ratio diminishes rapidly. For the aspect ratio 4.15 the maximum relative depth
is 1 for a dihedral angle of 27 degrees. The optimal dihedral angle for this aspect ratio is
around 30 degrees.

Hydrofoils Running Near a Free Surface 161

Fig. 14. Influence of the aspect ratio on the lift coefficients (upper curves)
and drag coefficients (lower curves) of a dihedral foil

OR=0°
Op-5°
@ p= 8°
DF = 15°, Apron 45

Fig. 15. Influence of the rolling angle on a dihedral foil

162 S. Schuster and H. Schwanecke

03

° @p- 5° 3
© p 10° | a-20
Ofp= 5° Laer
or} = 5° i e
@p-1" \s 2

01 02 ~ 03

Fig. 16. Influence of the dihedral and rolling angles
on the rolling moment

15

O Ham? Apa nb t5
© v-w * -:
@ v-w - .

A

Fig. 17. Lateral stability of dihedral foils

3. By rolling, the flow, being symmetrical before, becomes nonsymmetrical. When the
span is kept equal the aspect ratio and the angle of inclination between foil and water sur-
face increase for one part as much as they decrease for the other. Indeed rolling has the
same effect as superposing the influences of aspect ratio and dihedral angle. At small roll
angles C, and C, are only slightly changed. At greater angles lift and ascent of lift drop
more and more and the resistance increases.

Hydrofoils Running Near a Free Surface 163

The results of this first test series were complemented in 1958 by six component meas-
urements in the circulating water tunnel of the VWS[7]. The dihedral hydrofoil model al-
ready described for the measurements of the pressure distribution with a fixed dihedral angle
® = 33 degrees was used. The flow speed was about 2 m/sec. The influence of rolling and
sideslipping in particular were examined.

First of all Fig. 18 shows the geometrical variation of the effective angle of attack of
the pressure-side chord along with the roll angle and with the sideslip angle. For instance
if such a hydrofoil is running straight-on with 0 = 1 degree, then for the profiles, 0.8 degree
is the effective angle. If there does exist a sideslip angle of y = 10 degrees toward star-
board — for instance when turning to port — the effective angle of attack will increase on the
starboard side to +6 degrees while on the port side it will decrease to —4 degrees. If there
is an additional roll angle of about 8 = +20 degrees toward the center of the turning circle,
then this angle of attack difference will be increased to +8 degrees and —7 degrees respec-
tively. For a rolling angle of 8 = —20 degrees to the outside, a diminution of the effective
angles of attack to +2.5 degrees and —1.5 degrees respectively will result. An augmentation
of the sideslip angle causes a nearly linear augmentation of these angles of attack.

Since therefore for hydrofoil boats under service conditions the angle of attack can
spread over a very wide range, the course of the coefficients for lift, drag, and pitching
moment has been measured for a 45-degree variation of the angle of attack (Fig. 19). Here
the flow separated at about +8 degrees on the upper side and at about —10 degrees on the
lower side. The lift coefficient will increase, of course, after such a sudden breakdown but
the drag-lift ratio € = Cp/C, then will be three times as high as it was before (Fig. 20).
The fact thatrelatively high negative lift values can be obtained is of no importance for prac-
tice. In turning circles for the inner side, i.e., on the port side when turning to port, a de-
crease of lift may, of course, be desired, even down to zero. But it should not reach nega-
tive values, since an adequate raising of lift on the outer side beyond twice the value as
before seems to be unlikely considering the augmentation of drag, the diminution of speed,
and the smaller submerged area of the outer foil, which is reduced by rolling. For running
with a following sea — a situation when negative angles of attack will occur easily — such
a negative lift also will be dangerous even if it occurs only for a short period.

Figures 21 and 22 demonstrate the variation of forces and moments versus the sideslip
angle and the roll angle. The coefficients for lift, drag, and roll moment are practically con-
stant; the coefficient for the yawing moment is insignificant. The coefficients for lateral
force and roll moment increase linearly from zero upward with increasing sideslip angle and
roll angle. For instance a sideslip of +8 degrees toward starboard causes a lateral force
toward port of eight times the value brought about by -8 degrees rolling toward starboard;
the roll moment due to a sideslip is 70 percent higher. Both roll moments relative to the
keel point are stabilizing, i.e., any rolling will reduce itself and a sideslip will produce a
rolling toward the inside of the curve.

Simultaneous systematic variations of the angles of attack, of rolling, and of sideslip
produced only unessential differences relative to drag and lateral forces compared with the
separate variations. But the situation deteriorated relative to the lift. For a sideslip angle
around 10 degrees the flow separated throughout. Lift then broke down much more com-
pletely as the additional rolling increased (Fig. 23).

Summing up, it can be stated:

1. The range of the angle of attack of the profiles of a dihedral foil with a circle seg-
ment characteristic which can be utilized is about 4 degrees < @ <+6 degrees.

S. Schuster and H. Schwanecke

164

psoyo 0y3 JO YouIIe Jo a[Fue saNoosyy “ST “SIL

Ny /
[é] 94 290 ; /| )

Fa . YY

td, Zs
wa
LAS

Yi Z "0¢ 1

MO/{ JO vol{Ia/Ip

|S
i

[02 suis +}y-6)|¢/s02)Bus+s0urs 8 503] 509

0 / na E v

G7] Woy SY

5s sees

hd +t 06-8
7 |

of4

Hydrofoils Running Near a Free Surface 165

7

ace. fo

ae
aw
———— Ss
oe
as
s
Wot Gy 75

Fig. 19. Lift, drag, and pitching moment for a dihedral foil in a symmetrical flow

166 S. Schuster and H. Schwanecke

acc. to (7)

Fig. 20. Polar of a dihedral foil in a symmetrical flow

2. Lateral stabilizing can be brought about much more effectively by the moment due
to sideslip than by that due to rolling. While the rolling is less dangerous for the lift, a
range of only +8 degrees can be admitted for a sideslip angle.

3. Additional rolling of a dihedral foil in a sideslip motion acting in the same sense
as the moment due to sideslip will raise the risk of a lift breakdown.

EXAMINATION OF ROTARY INSTABILITY

With a model of a dihedral foil the relation between rolling and sideslip has been tested
thoroughly in a small VWS tank using a special device (Fig. 24). For a constant lift, the
model at u, = 4 m/sec could immerse freely. Relative to rotary motions both about the lon-
gitudinal and the vertical axis the model could either be fixed or set free.

The sideslip angle was measured each time when the lift collapsed. For free rolling,
sideslip angles up to 11 degrees could be brought about; rolling angles then amounted to
about 10 degrees acting in the same direction as the moment due to the sideslip. Enlarging
the sideslip angle to 11 degrees caused a lift breakdown at the foil which had been more
emerging and a rotary instability was the result. This procedure is not reversible, i.e., the
sideslip angle has to be totally reduced to regain the flow and to start with sideslip again.
For a fixed rolling of 20 degrees, sideslip angles of 16 degrees and 9 degrees could be ob-
tained; the smaller sideslip angle is conjugated to rolling in the same sense as the moment
due to sideslip, the other one is conjugated to rolling in the opposite direction.

167

Hydrofoils Running Near a Free Surface

‘so[3ue dijsepis [[BWs 10} SyUoWOU pue S9dI0q "1% “31 yq

[ é] 0290

NY

0
Any

/
501-560 = ay
Pat

s20+

o¢co+

S. Schuster and H. Schwanecke

168

[é] oj 990

S9[3ue [[OI [[eWs 10} syUoUIOU pue sedI0q °ZZ °3Iq

aad

—_ —_—>

——_.

——— b- G6 os ey

oS

Ol+
Mo|j JO uoljIa4Ip

on

Kh<
UK a

GLOt ~

Wo ldo

<=—_——

070 +

$20 +

0€0 +

169

Hydrofoils Running Near a Free Surface

x
dy yr] ssepuorsueutg gz “Bry

[4 Jo4 220 o

WB sounjd yog- . t
N w76Q. . . 7
YZ euojd GB ay fo buyswaosd 1
MO/J JO UONOJOdaS Asas UOl{O/IJUa/

2/9045 jou si kyjosauab moj jo
a4ojs ay4 sabuos Burzi] ay, wiygiyy

=
~
“S

bap ¢+» p:ajdwos

170 S. Schuster and H. Schwanecke

Fig. 24. Arrangement for rotary instability tests
of a dihedral foil

To sum up:

1. Rolling into the inside of the curve for hydrofoil boats as well as for planing craft
and airplanes such that the principal plane coincides with the apparent perpendicular is
more disadvantageous relative to security than is rolling into the outside of the curve as
occurs with displacement craft.

2. Hydrofoils already rolling to the outside at the beginning of the curve will run a
tighter circle (with higher sideslip angles) than those rolling to the inside.

3. The preceding result contradicts the conception that in a circle a foil rolling to the
outside will stall easier because of the higher load. Therefore the whole situation needs a
thorough investigation.

Hydrofoils Running Near a Free Surface wel

THEORETICAL CONSIDERATIONS

The velocity field of a hydrofoil of great span running at the depth h=h/c = h*/2 be-
neath the undisturbed water surface in deep water can be represented by the well-known re-
lations (for instance, Ref. 8)

u(x,z) _ u,(x,z)
+1
1 z + h*
= cat 7 ee ip aan A
ee o aya hal Zt) 2
+1
1 z — ht
os ey ye SE
"On J, es ee
+1
+ © exp i (ht | {| «oa, ber rest) dé (1a)
Uo %6 a4 Uo
wOX,Z) _ wy(x,Z)
Uy 4G
: +1 Z
aie Fe aaa ae ee
of E Ge ees
1 +1 é
a oe f ee ea ses Li, RR MOR
7 [ ‘a Ga) ee)

0 0 0

+1
age Po (- ee = j f(2)G, ao ae ab)
u u I u

where u, denotes the running speed and f(£) the pressure distribution over the profile length.
At the free surface the velocity field fulfils the boundary condition resulting from the line-
arized Bernoulli law

u,? du,(x,0)
0 1 2 =

The shape of the surface results as well from the Bernoulli law by linearization to

172 S. Schuster and H. Schwanecke

(3)

The pressure distribution at the profile, i.e., the function f(€) with f(x) being the meanline
of the profile, can be found from the relation

u(x,-h*) f'(x) = w(x,-h") (4)

with w and w taken from Eq. (1).

The boundary condition (2) and hence the velocity field (1) is valid for smooth water
above the foil as well as ahead and aft, i.e., for a condition which permits the existence of
deep water waves.

Within the technically interesting range of the Froude depth number F, = u, gh there
is as shown by experimental investigations rapid flow prevailing above the foil with the ex-
ception of the direct neighborhood of the leading edge. For this case the boundary condition
(2) and therewith the terms for the velocity field (1) are no longer valid. Behind the foil,
with the exception of the direct neighborhood of the trailing edge, the flow is tranquil again,
for here the physical suppositions for rapid flow are missing due to the water presumed as
being deep. For this range Eqs. (1) and (2) are valid again.

The consequence of the rapid flow is that the water surface above the foil remains
smooth, while for small distances between foil and water surface the water flows parallel to
the upper side of the profile, whereby the well-known apparent camber reduction results. That
means above the foil there is, dependent on the Froude depth number, at relative low speeds
already a state of flow which will occur behind the foil only at considerably higher speeds.
Therefore for the range of the foil (-1 > x > +1) in Eqs. (1) the terms containing g/u,?2 may
equal 0. The equations simplify to

+1 5
Sale e | £(E) (aa aes

e0 ie (x- €)2 + (zt+h*)y?

+1

1 (z-h") dé
a f soe ee a
= | Oper ReeeSTT (5a)

+1
w(X,Z) 1 Gesejeds
= = —— f Be ee NO
Uy ar . () (x-€)2 + (z+h*)?
ie (x= €) dé
tay ; OP So Se Slo BES b
arr fe 4 (x= Gy + (a- hy? a

At the surface (z = 0) the additional velocity u, in Eqs. (5) disappears, so that in the case
of rapid flow at the upper side of the profile the boundary condition

Hydrofoils Running Near a Free Surface 173

u(x,0) _
caer ae 1 (6)

will be satisfied.

By inserting the velocity components (5) into (4) the following integral equation for de-
termining the pressure distribution f(€) results:

+1

Mig ats i 1 Ce a SE 9)
Df" (x) = if Ke] = DEO dé. (7)

The integral

+1
{ FE) ae
be aS

is the known relation for the downwash of a thin profile in an infinite medium. The part of
the relation (7) dependent on the depth of submergence h* represents the influence of the
surface, which results in a diminution cf the effective angle of incidence for the profile.

For the solution of Eq. (7) the following function with unknown coefficients will be
chosen:

f(é) = 2a, +4 Se eae ee) ee (8)

The coefficients a, will be found by complying with Eq. (7) for three points of the chord.
For these points x = +c/4 and x = +0 may be chosen. By means of the a,, which are a func-
tion of h*, the lift of the profile follows from the known relation

C= €,th*)|= 2 |ag(h*) + a,(A*)] (9)

If the distance h* is considerable, i.e., if the profile is deeply submerged, it then follows
from (7) that

+1
-2nf'(x) = | ae
-1

ee

(7a)

that is, the Ackermann-Birnbaum integral equation for evaluating the pressure distribution of
thin profiles in infinite flow. If the distance h* disappears, i.e., if the profile runs at the
surface, it then follows from (7) that

+1
-nf'(x) = | f(E) = — sae (7b)
=
646551 O—62—_13

174 S. Schuster and H. Schwanecke

The lift then has only one-half the magnitude it has in great depth; that means in this case
only the lift of the lower side is effective.

Now a curved profile may be considered, for instance a circle segment profile with
camber m and angle of incidence @ eom Of the chord. For this the contour of the chord is
given by

HCO) = toe aca (10)
Then
Ge) — hee. 2f,x (11)

where f, and f, may be of small values:

fy = etn
f, = -2m.
For this case Eq. (1) can be written
i i) Gaerne
Pay 1 = Ie F(é) eee + cee dé (12a)

1 (226) =4h fox,
a (12b)

f SS
es F Ee O? + 4h?

= 4a, EX = \
ty

Equation (12a) gives the lift of a flat plate,
C,(A*) = nla (h*) + ay 9(h*)|
and Eq. (12b) gives the additional lift produced by the camber,
C, (h*) = 2rr|ag ,(h*) ta, (h")|
The total lift then amounts to
Cy(h*) = C, (AY) + C (h)

and is shown in Fig. 25 for a flat plate as well as for a circle segment profile.

For the profile approaching the surface an apparent camber reduction at the upper side
occurs whereby the lift of the profile, already reduced by the proximity of the surface, will
be reduced further. In the actual case of a circle segment profile the lower side is flat; it is
therefore equal to the lower side of a flat plate and delivers a lift taken as independent of

the depth h*:

Hydrofoils Running Near a Free Surface 175

8
re Fey 1. (x)= —0,0526 x (plate - profile) |
2.f (x)= —Q05%x-007%5 x? (circlesegment-profile)
ee | | :
x geom =+3
1 2 3 4

Fig. 25. Decrease of lift near the surface without regard
to the apparent camber reduction

rs Ag 9(™)
C,(h") ae: Sg) RIA ne re (13)
Cel aes
C.(o) De
The lift of the upper side can be written
C,(h") = C,H) + C, (A)
* Ago(™) ~
= DTT, aac ic “h Biga + aig(h )
h* * *
+ o (4) [20 1(h ) + a, Ch }} (14)
C,(h") 1 a, (0)
L Bee ce ele =e Od eee * wh 00 *
C,(@) re 0,5 anata) + alge) {oat ) amt Ay9(h")

+ (2) [ 20168") + 20")

where o(k*/2) stands the apparent camber reduction. The total lift of the profile then
amounts to

176 S. Schuster and H. Schwanecke

C,(h*) = C Ch)

tuGaG@he)

us

4s

2n | 00th" tae ig (2a) o (A fag (A) - acs yf (15)

*

* * h * *x
Cain ye Agg(h’) + a, (A) + o() [ay Ch ) + a,(h )]

C1 (©) Ayo(®) + 4,,(©)

where
= 01 ED pti sien
411 (@) = 2m = -f
In Eq. (15) only the factor o(h*/2) is unknown. It can be calculated theoretically, but then
separate mathematical operations will be necessary. By experiment this camber reduction

factor can be found easily from the reduction of the no-lift angle occurring in an approach to
the surface

pe MLD age nat
se Deere as (0) ~ m()

(16)

where independence of the shape of the profile can be supposed. Figure (26) shows the fac-
tor o(h) as found by experiments. With the aid of this factor the lift reduction of the profile
can be found as a function of the depth h from Eq. (15) as shown in Fig. 27.

Figure (28) shows the calculated lift of the upper side and of the lower side of the con-
sidered profile dependent on the depth h, as well as the corresponding values taken by meas-
urement. The actual increase of lift of the lower side and the decrease of lift of the upper

05 20

Fig. 26. Apparent camber reduction of a circle segment profile
as found experimentally

Hydrofoils Running Near a Free Surface 77

("
it (@) experimental results
, corrected fo infinile aspect ratio
fet
Q5

7 2 3 4 =

Fig. 27. Decrease of lift for a circle segment profile near the free surface

(@) experimental results corrected
to infinite aspect ratio
al

1 2 3

h——

Fig. 28. Lift coefficients of the upper and lower sides of a circle segment profile.

178 S. Schuster and H. Schwanecke

side, being higher than calculated, cannot be represented by these simple formulas. But
since these influences compensate each other for the most part, the calculated total lift

nearly equals the measured lift. For calculating the total lift, therefore, this method may be
used as an approximation.

INTRODUCTION OF THE MAIN PARAMETERS

Laying out foils for hydrofoil boats needs not only information about the decrease of lift
due to the free surface but also about the influence of the aspect ratio and eventually of the
dihedral angle. The above-mentioned measurements proved that for supercritical flow above
the foils the spanwise distribution of lift is only slightly dependent on the depth h. For su-
percritical flow, therefore, it is possible to consider the influences of the surface and of the
aspect ratio separately without being greatly mistaken. Moreover this gives the possibility
of calculating the lift decrease of dihedral foils.

The main parameters will be introduced now. The influence of the finite aspect ratio
will be covered by the relation found by Weinig [9] by means of the cascade theory:

dC
—L = K, OP 6

da (17)

The factor K, is shown in Fig. 29. The aspect ratio of a dihedral foil has to be taken as
the wetted aspect ratio, i.e.,

03

| | |
die K |
Ez =) 12K (ace. to Weinig [9])
k = #4 tanh (4)

+

Ka

8 10

A —

Fig. 29. Dependence of the lift gradient and the increase
of the lift gradient on the aspect ratio

Hydrofoils Running Near a Free Surface

A ;
A eee
cos

179

The parameter for the influence of the water surface Kz for parallel submerged foils equals

the term

C uw
ees
Lo

as shown in Fig. 30. For dihedral foils or slanting flat foils of constant profile length, the
tips of which run at the depths h, and hy (hy > hy), a mean value valid for the whole foil can

be found by integration of the term

Cc w
L
ee,
Che
over the depth of submergence:
1 hy
By Wash le
K; = i, = joer (h)dh.
SHR Snes Sam NN Rg

015

01

7 2 3

h —e

Fig. 30. Dependence of the decrease of lift and the
gradient of the lift decrease on the depth ratio

(18)

180 S. Schuster and H. Schwanecke

In case the foil pierces the surface, h, = 0 and the term shown in Fig. 31 is found to be
i ee kak 2
kK =) | on (h) dh. (18 ')
Lo

One more parameter takes care of the difference between the geometrical angle of inci-
dence Q -eom generally defined for a system moving with the foils and the geometrical angle
of incidence Qt eom actually existing in a system fixed relative to flow. It may be stated

(see Fig. 18)

geom cos $
a a ~ sin a

[cos y sin Qe eam * Sim y(cos Bltan 9|

eom
+ sin 8 cos tell « 1@'9)

For a nonrolling and nonsideslipping dihedral foil with the dihedral angle ®,

Ky = cos #.

Together with the known profile efficiency factor Kp the increase of lift for a hydrofoil
becomes

10
| a ee
08 | 2 ne

fol Mies Lt
06 > |
06 /
C
aa 7
0 40
10 20 40 ”
i 7
hb, -o
- af -2 [Sayan ;
Ky, i he Jat)
K geom = +3

Fig. 31. Decrease of lift for a dihedral foil

Hydrofoils Running Near a Free Surface 181

bi.
ie aK KOK, Kee (20)

By means of this relation the results of wind-tunnel tests or of theoretical calculations can
be compared with the results of hydrofoil measurements.

VERTICAL STABILITY OF HYDROFOILS

For the sake of good driving properties a suitable vertical stability of hydrofoils is very
important. However, it should again not be too great since the foils then will respond to
minor disturbances, for instance to small waves on a surface otherwise calm; that means
such foils will run stiff. Therefore not too steep a dL/dh curve is desirable.

For a nonrolling surface-piercing foil, the lowest point of which is at depth ho with
wetted aspect ratio A, the total change of lift

ab. db + 2h

dA + — =
= oh

(21)

0,h,

is dependent on the variations of aspect ratio, span, and depth of submergence. From the
geometry of the foil it follows that

ee eee
sin }
so that
dL oL 1 oL 1 oL
z ~ @A sin ® 35 Si es
LN sete. A g ad a
From
L = putbeC, KK, cos @ K, Ky (22)
it follows that for each panel of the foil
CS ais | ey OK ,
Se pwitakhos cLakakr £O5;% com eA. Ry (23a)
OL} Pare y raat : 1
5B _ = gy PY bcC, K Kp cos z K, Ky (23b)
b A ho
e 1 pu2beC, °K_K ee
Se oO c cos ¢ =
Silo. 2 °° ee he ORG) ie oe

182 S. Schuster and H. Schwanecke

The quantity 0K 4/0A may be taken from Fig. 29. Furthermore

Ne ae “a oe y
dL 2 A hy ho
dh Ay;byho i A SLOMt A b sin @
1
A 1, hy the
and
dc OK * h 7 h
"lh ™! h
L A 2 2
— = C, K_K, cos 8 = ; K
- Loa P aA A a
dh ae f sin } 4. \B sing
K- - K=
h h
+ — (24b)
h ho

with C, ., = 27 sin (@ + 2m).

In the case of a submerged dihedral foil or a flat parallel submerged foil the change of
lift due to changes of submergence will be remarkably less than for the surface-piercing
dihedral foil, because the first two terms of Eq. (24b) disappear. For a flat foil running at
depth hg the depth-gradient becomes

oK>
= no = pug bee, KK, “= if (K, = 1) (25a)
A;byhy A ho
and
dc, OK;
ain tallies Cra KpK, aia (25b)
A,byho A hy

It can be seen here that for the range of a small depth of submergence (A, < 1) a vertical
stabilization of a flat foil is quite possible as confirmed by experiments made in the Berlin
Towing Tank with such a foil in a seaway. The submerged dihedral foil proved to be much
more unfavorable since only those parts running near the surface are strongly influenced
by the surface effect. In this case with the lowest point of the foil at depth h, and the
highest point at depth 4,,

Hydrofoils Running Near a Free Surface 183

on one
ae =" fu beC, K,Kp cos 6 K, = (26a)
dh Aybyh ho A oh hy, ho
Thus
OK; hy
= = soe == : = | 1 Ges dh 7 b ; (Ki rs cs Ks;
Rata: hh. = base iy
oh ee oh 1 ire sin 2 1
This gives
Ke “7
i+ ee
ee = C. Kk. K; cos 0 K Edie DS al a (26b)
205 at it Lo a P A b sin ‘iY
A,b,h,,h, $

An improvement for the submerged dihedral foil will be possible by arranging horizontal
auxiliary foils at the upper ends as near as possible to the surface such that they are within

the range of substantial OK_/ dh (Fig. 30).

ROLLING STABILITY

A surface-piercing nonrolling dihedral foil with the dihedral angle #, constant profile
length, and wetted aspect ratio A may be examined now. For an inclination df a differen-
tial stabilizing moment relative to the lowest point is produced according to

om
gl sp =

a M
db + —* dh.
A ob h

(27)

The differentials dA, db, and dh can be obtained from the geometry of the foil:

: 1

+d -= dA
ee

a
o
T

Q
=a
iT}

+

+ dBb7 cos ¢

with the dimensionless distance from the lowest point 7 = 7/6. For evaluating 0M,/0A,
OM ,/0b, and 0M ~/0h the spanwise lift distribution will be given by the term f(y, h) = f@) fh),

where with sufficient accuracy an elliptical law may be chosen for the function f (7):

za as 4 =
(mq). = 6 (7)i 6,0 Ky, vl - 4 -

184 S. Schuster and H. Schwanecke

The function f (7) takes care of the depth of submergence, i.e.,

where
h = h,(1 - 7)

=h,-b7sin®.

Herewith the moment M, may be written as

= 4 = =2
My = pu, be Ci ka kpPK, =) TN Laat K; dj (28)
0
The derivatives of the moments relative to the single parameters are
OM 1 2 OK, 4 1 ee =
ame 2 = 5 Puy be*C, Ky Kp b ay a gS ool ae ae
A,5,h, A 6

OM io Girne SES e

= = 5 Pug? bo? Cy KRpK, oT | 7) 1-77 Kr _ dj (29b)
Ob | 4.8,h A ye J

151h9

om 1 b an 4 : = —2 OK; =

Ba! Brin uy bc?C, K,Kpb K, s/ Avi q> eae (29c)
At ane oh |;

Herewith the total change of moment relative to the lowest point of the dihedral becomes

om -2 4 OK , Fy
x = -opu*bc?C, K.K, cos 8b) —|=— :
dp ae (Day Loa” P m | 0A |, sin @
QF, i
+ it 30
a A \b sin 8 < yee
and
dc
M OK F
z ‘ Bey (anata ye LE, d +, (30b)
“dB AB, = Cri ie cos#b , oA|, sin 6 i= 4

Hydrofoils Running Near a Free Surface

with

F, = | AV1- 7? K;-| d7
x n
1
OK;
F, = | 7? V1- 7? Sen? «
0 oh 5

185

Together with the stabilizing moment a lateral force is effective, the magnitude of which is

oP | ns a es
dP, = 7 dA+ — db +| ae.
e ob oh

The general term for the lateral force is given by

Py _ = pu, bc Chak Kp sin 6K,

A,b,h

>
3)

With the differentials

dA

I
+!
Q

D

and
oP OK z
c 1 Bis: : A| 4 2
eA = 0u-- be C.K K sind a oe
aA ae 2 ft) Loa’ P 0A aoe , h 7
oP 7
= | ; 1 é. ea!
aE = 5 Pug be Ci aK Kp sin 5 MA =| 1-7* Ky
A,byh, 2 g 7
oP i ree
1 4 =
Pa = % Ug be Ce ee ke sin $ K, A 1 - 7? +
dh |, cc ee oh 5
A,;byho A 0 7

the resulting change of lateral force is found to be

dj.

dn,

(31)

(32)

(33a)

(33b)

(33c)

186 S. Schuster and H. Schwanecke

dP, ae = OKA , 1
ap Pelli) iu be C, Kk Kp cos #? b TA Rr + K, oes i.
A,b,h, 2 A i hy
+ as sin § F (34
7 3 a)
and
dc
a = Ce K ale cos ® b ae Kz + K, + Kz
p A,byh, 2 a i hy
OTB 3
with
1 OK
a = h at
Bake | Dee aie eee ea
b iy

Being the terms dCy /dB and dCp y/ 4p the position of the initial lateral metacenter above
lowest point can be “found [7]:

eu y (35)

Analogously, stabilizing moment and lateral force for any inclination B can be found. The
different values K,, and dK ,/dA have to be taken [7] considering the wetted breadth for
both panels relative to the profile length. In addition to that a dependence on B appears,
i.e., for the moment there has to be written:

Mt ees M
Lee he Ea ee

fi oA A oh 2B

For the case of the submerged dihedral foil as well as of a submerged flat foil all the terms
taking care of the change of the aspect ratio and of the breadth will disappear. The rolling
stability of submerged foils therefore is principally less than that of dihedral foils piercing
the surface.

As in the case of vertical stabilization, a lateral stabilization might be possible for
submerged foils as well if the depth of submergence is low or if horizontal stabilizing plates
are arranged near the free surface.

Hydrofoils Running Near a Free Surface

187

For a submerged dihedral foil the lowest point of which is running at the depth hi, with
the upper ends at the depth 4, the gradient of the rolling moment in the upright position be-

comes
dC 1
M ~2 dK;
aa Bg) toe KK. cos fb ca | Wolo ley
Seed 3 A 4 dh |%
and the gradient of the lateral force becomes
dC 1
Py : 4 = =2 —
—ag| .. _ = CrokoXpcos @ singb4 x, | qvi- 7? —*| a7
Ag Bi Rip itis Aycaly =
with
H < hy) < hh, .
For a parallel submerged foil running at the depth Tes
dC 1
x a Of Sg
Raph), 3) = -C,.Kpb ae =. | 7m? V1- 7° dy
1514/9 0
= dk-~
Bie wi ello
le ats tals
and
aC p
a =
= 0 (dA = @gljo= 0)
dp A,byho

(36)

(37)

(38)

(39)

The lateral stabilization due to sideslipping will not be developed here since this can be
done in a manner corresponding to the treatment of the vertical and of the rolling stability.

NOMENCLATURE

b semispan of the hydrofoil
br0j = 5 cos 8

c chord length

188

h

h!

h=h/c

h* = h/(c/2)
A = 2b/c

S = 2be

CH)

ey

Wy

n= uo/Vgh
O peom

9

m

C. = L/pu 2be

E= C,/C,

S. Schuster and H. Schwanecke

depth of submergence of the foil

minimum local water depth above the foil

aspect ratio

area of the hydrofoil

running speed

induced velocity in the direction of the undisturbed flow
induced velocity perpendicular to the direction of the undisturbed flow
Froude depth number

geometrical angle of attack of the chord

no-lift angle

camber

lift coefficient

drag-coefficient

transverse-force coefficient

rolling-moment coefficient

dihedral angle

rolling angle

sideslip angle

drag-lift ratio

REFERENCES

[1] Laitone, E.V., “Limiting Pressure on Hydrofoils at Small Submergence Depths,” J. Appl.
Phys. 25:623 (1954)

[2] Parkin, B.R., Perry, B., and Wu, Y.T., “Pressure Distribution on a Hydrofoil Running
Near the Water Surface,” J. Appl. Phys. 27:232 (1956)

Hydrofoils Running Near a Free Surface 189

[3] Plesset, M.S., and Parkin, B.R., “Hydrofoils in Noncavitating and Cavitating Flow,”
Cavitation in Hydrodynamics, Nat. Phys. Lab., 1955

[4] Land, N.S., “Characteristics of an NACA 66, S-209 Section Hydrofoil at Several Depths,”
NACA W.R.L.-757, 1943

[5] Schuster, S., and Schwanecke, H., “Uber den Einfluss der Wasseroberflache auf die
Auftriebsverteilung von Tragflugeln,” Schiffstechnik 4:117 (1957)

[6] Schuster, S., and Logothetopoulos, J., “Sechskomponentenmessungen an einer Wasser-
tragflache zur Untersuchung der Einflusse von Streckung, Kielung und Krangung,”
Versuchs-Abtlg. d. Gebr. Sachsenberg AG. No. 5a, 1942 (unpublished)

[7] Schwanecke, H., “Uber Sechskomponenten-Messungen an einem V-Wassertragflugel bei
symmetrischer und unsymmetrischer Anstromung,” Schiffstechnik 6:93 (1957)

[8] Isay, W.H., “Zur Theorie der nahe der Wasseroberflache fahrenden Tragflachen,”
Ingenieur-Archiv 27:295 (1959/60)

[9] Weinig, F., “Beitrag zur Theorie des Tragflugels endlicher, insbesondere kleiner
Spannweite,” Luftfahrt-Forschung 13:405 (1936)

DISCUSSION
P. Kaplan (Technical Research Group, Inc., Syosset, New York)

I noticed, or it appeared to me, that the influence of roll was only static. I wonder if
you have considered or done any work, not reported here, as to the effect of the roll damp-
ing, that is, what happens when the foil is performing not just a static roll, deflection, but
is actually performing a roll angular motion? This is important since it is the roll damping
that determines the major behavior of the roll motion. According to the results of the free
motion studies, there was quite different behavior of the system as compared to an aircraft,
so I wonder just what influence a dihedral will have on the roll damping and if any consid-
eration was given to this problem.

S. Schuster

Our results given here of the tests and formulas are valid only for the steady state.
Even the test for finding the point of breaking down of the lift shows only the beginning of
the motion. Damping forces and moments can only be found during the motion itself. We are
beginning the investigation for the unsteady problems, especially for hydrofoils in waves,
but at present we are not able to give you an answer to these special questions. I think that
damping is higher for rolling than for yawing, but for the whole system of two or more hydro-
foils, we have to consider sideslip motion too. Perhaps there will be something to report
after finishing our investigations on oscillations of a propeller in the axial direction and in
the rotational direction.

646551 O—62——14

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THE EFFECT OF SIZE ON THE SEAWORTHINESS OF HYDROFOIL CRAFT

A. Hadjidakis
Aquavion Holland N.V.,. The Hague

The pitching movements and the vertical accelerations of hydrofoil craft
with surface-piercing foils are studied in a very simplified way to show
the effect of size as a function of wave parameters, and speed or Froude
number. Considerations are given to variations of static pitch angle,
heave, natural frequency, and damping ratio—the latter based on a line-
arized equation of motion. The analysis indicates that at wave lengths
of 2/3 and 3/2 of the length of the hydrofoil craft, extreme values of
vertical accelerations are to be expected, which decrease rapidly with
increasing lengthofthe craft, or decreasing speed, which leads to the con-
ae that the seaworthiness increases with the length of the hydro-
oil craft.

INTRODUCTION

The seaworthiness of hydrofoil craft is sometimes doubted, while the comfort offered to
the passengers is often considered to be insufficient, when going on a seaway [1-7]. The
purpose of this paper is to give in the most simple way some insight into the behavior of
hydrofoil craft on sea, and to determine the most unfavorable conditions, which are compared
for craft of different sizes.

This, of course, needs some explanation. The high cost involved in constructing big
hydrofoil craft makes it necessary to obtain experience regarding seaworthiness, stability,
and comfort, as well as structural loadings and other technical aspects, by means of scale
models or small prototypes. It is therefore desirable to prove that the results obtained in
this way can be interpreted for craft of greater size.

Today a hydrofoil craft does not present problems on relatively smooth water. Its
behavior under unfavorable conditions is thus decisive for its suitability for passenger
transport at sea.

Therefore it is necessary to gather the existing knowledge which may contribute to a
prediction of the properties of a big hydrofoil craft, based on the known performance and
behavior of a small craft, so that a project for a seaworthy passenger ferry can be started
in full confidence.

The following considerations and calculations have been formulated in a general way,
making them valid for a large number of existing hydrofoil systems. However, they are not

valid for systems which cannot be compared to a spring-and-mass system. Thus they do not

191

192 A. Hadjidakis

apply to hydrofoil craft with fully submerged foils, lacking a definite position of equilibrium
relative to the water surface. Their stability depends only on a human or electric brain,
controlling the lift developed by the different foils. The behavior in a seaway of this latter
category depends merely on the degree of intelligence of its governor.

Furthermore, only the most unfavorable conditions have been considered. ‘Going against
the waves, the vertical accelerations were found to be critical; when going with the waves,
this was the case with the maximum negative pitch angle.

Both critical values depend on design characteristics of the different hydrofoil systems,
as there are: the lift reserve of the forward foils, the natural frequencies for pitch and
heave, the damping ratio, etc. These design characteristics, being independent of size,
will not be discussed in this paper.

The many purposely introduced simplifications, of course, create deviations. Thus it
is necessary to apply the results only to two or more craft of the same hydrofoil system,
differing essentially in a scale factor only, for it is only in that case that the deviations are
in the same sense for all units, so that they will largely compensate each other.

THE SEAWAY

The waves are assumed to be of sinusoidal shape, where the waveheight H is equal to
o times the wavelength A, and o indicates the rate of steepness of the waves. The craft’s
speed V forms an angle y with the vector of the speed of wave propagation c. Then the
forced frequency or excitation frequency of the forces, trying to disturb the craft from its

equilibrium position, is
ce V. : Xr
w = 27 [2- Fees where C = aa (1)

The amplitude of the static pitch angle on a given wave pattern can be determined as
follows: The maximum static pitch angle y, is attained in the position shown in Fig. 1.
Its value is

ee a Jen (ed (2)

which can never exceed zo.

Fig. 1. Maximum static pitch angle

Hydrofoil Size and Seaworthiness 193

Similarly, the amplitude of the static roll angle is found to be
Po = — nn (r 28122) : (3)

The vertical movements of the center-of-gravity of a hydrofoil craft can also be deter-
mined: The dimensionless static amplitude of the center of gravity is (see Fig. 2)

5p Fh cool D) .

Ze

Coosy
A

Fig. 2. Static amplitude of the center of gravity

Both values are shown in Fig. 3 as a function of wavelength divided by craft length.
The scales for wy and Z,/% are valid for o = 0.05. For other values of o the scales have
to be adjusted.

The roll angle @ will not be discussed in more detail, because experience has shown
that rolling is not a critical factor. Moreover it can be treated in the same way as the pitch
angle.

Figure 3 shows both the static pitch amplitude wy and the dimensionless static verti-
cal amplitude of the center of gravity Z)/as a function of the relative wavelength

\/tcos y. If

r Dee Dn joe
Wiss 2 De aE , etc., then Wo = 0
but Z,/ 4, shows an extreme value, while if

re :
Le eee erc., then Dafa = 0

and wy attains its extreme values. The effect of these parameters on the craft’s behavior
will be discussed later.

It should be noted that the values of the excitation frequency resulting from Eq. (1)
should be regarded with some reserve. Practice has shown results to be sufficiently accu-
rate when going against or with the waves, but when going along the waves higher values of
@ are actually found due to the irregularity of the seaway [8].

194 A. Hadjidakis

=
—<-—~ <GRITICAL CON-
DITIONS IF y20

“4 or Pra l9
Laer Ce w 0.45

Fig. 3. Wave characteristics as shown in Figs. 1 and 2 (a list of symbols is given at the end
of the paper)

THE HYDROFOIL CRAFT

The seaworthiness of a craft might be specified as a combination of characteristics
which guarantee safety and comfort for passengers and crew while traveling at sea. Some
of those characteristics are: static stability, dynamic stability, constructional properties,
proper means of navigation, and lifesaving equipment. Variation of size of a certain type

Hydrofoil Size and Seaworthiness 195

of hydrofoil craft will not affect its static stability. Constructional properties, proper means
of navigation, and lifesaving equipment are (although very important) not to be discussed.

Hence the following will be limited to the dynamic stability and its influence on the
comfort of hydrofoil craft. The main factors which define the movements are: static ampli-
tude, (already discussed), excitation frequency, (already discussed), natural frequency, and
damping ratio.

The motions of hydrofoil craft are defined by a number of basic equations. There is
equilibrium when the weight of the craft equals the combined lift forces of the foils:

moa= 2b.

The lift force of a foil is proportional to dynamic pressure, lift coefficient, and submerged
foil area:

L = 5pV2q F. 6)

A deviation in pitch from the equilibrium position creates an extra lift AL which tries to
restore the craft to its original position. The restoring moment is called R. It will be
clear that: AL :: mg since L :: mg (Eq. (5)), so R:: mgl. Further, for the moment of
inertia, / :: ml? if the mass distribution can be supposed to be similar. Thus the natural
frequency of the craft can be calculated:

Auf nn aha
ae ee
where the index p is used for pitch, or
Way = ken ve : (7)

In an analogous way the natural frequency for heaving motions can be found to be

Wap ken, VE (8)

The dimensionless damping ratio in case of pitch, d,, is the damping N divided by the
critical damping:
Neus is 1
d,= 9/tTR Where V: as : (9)

In this equation D, is the damping force:

1
D, = zpV2AC) Fa (10)

in which Ay = W/P and Wc. 00" soN = VOR, -

The foil area Fy, contributing to the damping, is of course proportional to the foil area F:
Ni: VA7F , (ny
It was already stated that / :: ml? and PR :: mgl, so

ae eae VE
a Vml2mél mV 6)” (12)

in es

Combining Eqs. (5) and (6) with (12) leads to

kg
Jee eR
Similarly it can be found that in case of heave
ka
d, = C| Fr. (14)

Consequently the damping ratio is inversely proportional to the lift coefficient and the
Froude number for both types of motion.

This result might seem unexpected, the damping being proportional to speed, while the
damping ratio is inversely proportional to the Froude number. The explanation is simple:
an increase of speed of 10 percent corresponds with a decrease of submerged foil area of
20 percent, if the lift coefficient remains constant (see Eq. (6)). For similar hydrofoil craft
of differing size, 4,,,, kay» kid a and kq, are constant values. They may vary, however,
for different hydrofoil systems.

All elements necessary to calculate the motions of a hydrofoil craft are now available

if the craft may be considered as a damped linearized spring and mass system. The dynamic
amplitudes are then determined by the equations

1
Wg Z (1- w/ wat)? + 4d, W/ Wap (15)
1

z (16)
Akay = V(1- w/w2,)7+ 4d,?w/ wz, §

These equations have been plotted in Fig. 4 for various damping ratios.

It is now possible to determine the vertical accelerations at a certain point of the craft.
This point has been chosen near the bow at a distance from the center of gravity of 1/3, no
passengers or crew being expected to be carried at a more forward point. The resulting
accelerations have been calculated by adding vectorially the vertical acceleration caused
by a variation of the pitch w and the vertical acceleration caused by a variation of heave
Z. This is based on the assumption that pitch and heave are entirely independent of each
other, which, although not correct, gives very acceptable results, as has been shown by

Abkowitz [9].

Hydrofoil Size and Seaworthiness 197

{
“he 5 ) - V Gal?) , daly?

-) 0s 40 13

wh, —

Fig. 4. Amplitudes of forced vibrations for various degrees of damping

BEHAVIOR OF THE CRAFT

Conceming the course of a hydrofoil craft relative to the waves, experience has shown
two conditions to be of decisive importance for its seaworthiness. The first occurs if
y = O, i.e., on a following sea. In that case, the extreme value of w plays a dominant role.

Figure 5 shows the influence of the orbital motion of water particles in a following
sea. This orbital motion creates a moment which tries to pull the nose of the craft down.

Fig. 5. Influence of the orbital motion of water particles in a following sea

198 A. Hadjidakis

As the pitch increases with increasing wave height, it will be clear that at a certain
wave height the combination of a high value of negative pitch, combined with the orbital
effect, makes the bow touch water. This phenomenon introduces a braking effect which
causes the craft to slow down, so this wave height represents the limit up to which the
craft can remain foilborne on a following sea. The lift reserve of the bow foils of a certain
hydrofoil system evidently determines the limiting wave height.

However, in Fig. 3 the dynamic pitch amplitudes of two similar hydrofoil craft differing
only in size have been indicated as functions of the relative wavelength. The critical con-
dition for the craft with the lower Froude number is seen to occur at a greater relative
wavelength. Consequently, of two similar craft the bigger one (having the lower Froude
number) will encounter its critical condition on a following sea on relatively higher waves.

When going along the waves, i.e., when y = 7/2, a hydrofoil craft will attain its
greatest roll angle 9, - However, experience has shown that this is not a critical condition
at all; therefore it will not be discussed in more detail.

When the craft is going against the waves, i.e., y = 7, the pitch amplitude is not criti-
cal, because the orbital moment will help to surmount the next wave, but the maximum value
of the excitation frequency is obtained. Therefore this can be called a critical condition as
far as vertical accelerations are concerned, these accelerations being proportional to the
square of the excitation frequency. While discussing vertical accelerations, y will hence-
forth be assumed to be equal to 7z.

Applying the foregoing theory to an example, a comparison is made between an existing
small hydrofoil craft (J = 6.73 m, V = 30 knots) and a designed bigger craft (J = 36 m,
V = 45 knots). Of the existing craft (Fig. 6), the natural frequencies for pitch and heave
and the damping ratio are known. With the aid of Figs. 3 and 4 a curve could be obtained
indicating the vertical acceleration at the bow as a function of the relative wavelength. The
natural frequencies and the damping ratio of the designed bigger craft have been calculated
according to Eqs. (7), (8), (13), and (14), which made it possible to establish an equivalent
curve for the bigger craft. Both curves are shown in Fig. 7.

It will be noted that there are two critical values of the relative wavelength A // where
the vertical acceleration at the bow Z;/g attains peak values. These critical wavelengths
correspond with 2/3 and 3/2 times the craft’s length.

Contrary to what might be expected, Fig. 7 shows that critical values of acceleration
are not to be expected at a relative wavelength of more than two. This means that vertical
acceleration decreases with increasing wavelength. In other words, the comfort of the small
craft will be much better on waves with a length of 20 m and a height of 1 m, than on waves
with a length of 10 m and a height of 0.5 m.

Similarly, the big craft will be more comfortable on waves of 100-m length and 5-m
height, than on waves of 50-m length and a height of 2.5 m, although the comfort in the
latter case is much better than the best that can be obtained with the small craft.

In reality the wave pattem is never found to be so regular as the theory supposes it to
be. On a seaway, wavelengths vary considerably, which tends to level off the extreme
accelerations. Furthermore small waves are always superimposed on the longer waves,
which will tend to raise the maximum accelerations to be found for values of \/I higher than
two. These two effects have been taken into account in the dashed curves shown in Fig. 7.

Hydrofoil Size and Seaworthiness 199

Fig. 6. Small hydrofoil craft

Sta 673m. Vadokn.

eae

MAX, VERT. ACCELERATION AT BOW }

WAVE LENGTH
LENGTH OF HYOROFOIL CRAFT

> AE

Fig. 7. Maximum vertical acceleration at the bow of a small hydrofoil craft and
a larger hydrofoil craft

200 A, Hadjidakis

As the behavior of a hydrofoil craft in the most unfavorable conditions is of decisive
importance for its seaworthiness and comfort, the critical accelerations at A/lJ = 2/3 and
d/l = 3/2 have been plotted in Fig. 8 as a function of craft length. This graph is valid for
one speed only, i.e. 30 knots, which makes it possible also to show the variation of the
Froude number with the craft length. The vertical acceleration at the bow for A// = lis
also shown.

a

ris
9
—__
=

=
9
So
Aa)
us
wz
Q
a
3
Ge
2

ce)
LENGTH OF CRAFT —=—

Fig. 8. Maximum vertical acceleration at the bow for critical values of A// as a function of
craft length / at a constant speed of 30 knots

Of these curves the one for A// = 3/2 is the most important, because the critical accel-
erations at A/I = 2/3 occur at relatively higher frequencies and can therefore be more
easily attenuated or even eliminated by means of foil suspension or similar improvements.

The influence of the speed of the craft is shown in Fig. 9, where the curve for \/l =
3/2 has been drawn for five different speeds. Evidently the vertical acceleration is approxi-
mately proportional to the square of the speed for a given craft length.

Thus it can be stated that vertical accelerations decrease rapidly with increasing
length of the hydrofoil craft, which means that the comfort, depending on the frequency of
variation of the accelerations (jerk), improves even more rapidly.

Hydrofoil Size and Seaworthiness 201

be
F]

VERT. ACCELERATION

° 20 0 60 1206.
LENGTH OF CRAFT) —=—

Fig. 9. Influence of the speed on the maximum vertical acceleration at the bow as a function of
craft length for the critical value A/J = 1.5

CONCLUSIONS

Assuming that the wavelength and height are proportional to the length of the hydrofoil
craft, in other words, that a craft of double length is running on waves which are twice as
long and high, it may be concluded that:

1. On a following sea, where maximum pitch is the critical factor, the seaworthiness is
nearly unaffected by size. Actually the seaworthiness improves slightly with decreasing

Froude number, i.e., increasing length.

2. When going against the waves, vertical accelerations are critical, which, with
increasing length of the craft, decrease more than proportionally with the Froude number.

Consequently the seaworthiness increases with the length of the craft, the more so
when the comfort is taken into consideration.

LIST OF SYMBOLS

a = angle of attack (effective)
b = maximum span of hydrofoils
C; = lift coefficient

c = _ speed of wave propagation

202

NNZ NYC HES SOR) cee Si rea Rees ayNAox

(1)

A. Hadjidakis

course angle relative to wave propagation
damping force

damping ratio

submerged foil area (effective)
Froude number

submerged foil area, contributing to damping
static roll amplitude (radians)
roll amplitude (radians)
acceleration of gravity
waveheight

index for heave

moment of inertia

design constant

lift force

craft’s length

wavelength

mass of the craft

damping

excitation frequency

natural frequency

index for pitch

static pitch amplitude (radians)
pitch amplitude (radians)
redressing moment (per radian)
density

steepness of waves

craft’s speed

relative vertical speed

static heave amplitude

heave amplitude.

REFERENCES

Buermann, T.M., Leehey, P., and Stillwell, J.J., “An Appraisal of Hydrofoil Supported
Craft,” Trans.of the Soc. of Nayal Arch. and Marine Eng., New York, 1953

[2] Crewe, P.R., “The Hydrofoil Boat; Its History and Future Prospects,” Trans. of the

Institution of Naval Arch., London, 1958

[3] Buller, K.}., “Neue und noch groszere Tragfligelboote,”Schiff und Hafen, 1959, p. 802

[4] Schertel, H. von, “Tragflachenboote,” Handbuch der Werften, Band II, Schiffahrts-
Verlag Hansa, Hamburg, 1952

[5] Schertel, H. von, “Tragfligelboote,” V.D.I. Zeitschrift, Band 98, No. 36, p. 1955,
Postverlagsort Essen, Dusseldorf, 1956

[6] Reinecke, H., “Tragfligelboote,” Schiffbautechnik, 8 Jahrgang, Heft 4, Berlin, 1958

Hydrofoil Size and Seaworthiness 203

[7] Berentzik, H., “Vergleich der theoretisch errechneten Beschleunigungen eines Trag-
fliigelbootes im Seegang mit den experimentell ermittelten Werten,” Schiffbautechnik, 10
Jahrgang, Heft 7, Berlin, 1960

[8] Heer, C.C., “Ergebnisse von Beschleunigungsmessungen an einem 10-m—Tragfliigelboot-
Groszmodell im natiirlichen Seegang der Ostsee,” Schiffbautechnik, 9 Jahrgang, Heft 11
and 12, Berlin, 1959

[9] Abkowitz, M.A., “The Effect of Antipitching Fins on Ship Motions,” Trans. of the Soc.
of Naval Arch. and Marine Eng., New York, 1959

DISCUSSION
P. Kaplan (Technical Research Group, Inc., Syosset, New York)

I want to know if the results just depend upon the assumption that the system is a
simple spring-mass system. It appears that hydrofoils alone as simple spring-mass systems
are overdamped in free motion as a single-degree-of-freedom. You have no such thing as a
natural frequency in many cases. Also, I would like to know if the result that vertical
accelerations are maximum at ratios of wavelength to craft length equal to two-thirds and
three-halves is true for all cases or just for the particular designs chosen for illustration.

A. Hadjidakis

I said at the beginning of my lecture that what I have done was done in a very general
way and in a very simple way, because, otherwise, one would need a computer and would
have to go into very difficult calculations. I have chosen this very simple spring and mass
system to get some quick results and to make it clear that in any case the seaworthiness
of bigger craft would be better than that of small craft. I meant, indeed, in all cases,
except fully submerged foil systems.

I might say now concerning what one should take for the characteristic length of the
craft; one had perhaps better take the distance of the extreme foils, or something like that,
to obtain better results. I do not pretend the diagrams I have shown are very accurate, but
the main thing I wanted to prove is the direction wherein we are going if we make craft
bigger. What you said about the damping ratios, that hydrofoil craft are overdamped, I do
not agree. Damping, as far as heave is concerned, is very high indeed, but damping on
pitch depends on what sort of hydrofoil system you have. With our one-foil system, on very
small craft we find damping ratios as small as 0.4, so in that case one can feel on different
wavelengths what is approximately the critical frequency of the craft. If you look at Fig.

4 it will be clear that this critical frequency is not a well-defined thing; you cannot measure
it very exactly.

P. Kaplan

You said that in following seas the important characteristic of seaworthiness is pitch-
ing. I think that heave is rather important, and that you get large possibilities of settling
in the water just due to that motion as well. -

204 A. Hadjidakis

A. Hadjidakis

I don’t agree, because although the heave is there, of course, the pitch will be more
important, making the craft nosedive. In that case the angle of attack of the foils decreases,
which means that lift decreases too (Fig. 5). Quite a big lift reserve on the forward foils
is then needed to take the craft up again for the next wave, because otherwise the nose will
touch that wave and then what we call seaworthiness in that particular case is finished and
one cannot go at full speed. That is what I meant; I insist on the pitch being the most
important factor on a following sea because, if you would go down horizontally, which is
the case when heave is the main factor, you still have the same lift-coefficient, and it would
be a very poor design if the craft wouldn’t go up then in time.

P. Kaplan

I want to know just how you get better performance from a small craft to a large one.
You did not just increase the size, you also changed the speed. Now with Froude scaling
we get the same relative values. Is there any particular way in which you did this or do
you see any particular optimum way of carrying out this change in size and speed to obtain
an improvement in the craft? .

A. Hadjidakis

The speed of 45 knots was chosen for the big craft with a capacity of about 400 passen-
gers because of economic and efficiency reasons, and in my opinion it is not necessary to
have higher speeds. The owner of the passenger-carrying craft will generally not gain
more money by a somewhat higher speed and I think the passengers will not even realize
the small time gains they will have in that case. Hence the number of 45 knots was not
chosen for some physical reason. There is, however, another aspect to this matter. We
have experience with all sorts of craft of different sizes and the biggest one could carry
about 70 passengers. We found weight is not going up as the cube of the length of the
craft, but somewhat less than that. It is logical to give a bigger craft a higher speed,
which helps a lot in keeping the foil dimensions and weight down. In our designs the foil
weight is about 5 to 8 percent of the total displacement weight.

DESIGN AND INITIAL TEST OF ONR
SUPERCAVITATING HYDROFOIL BOAT XCH-6

Glen J. Wennagel
Dynamic Developments, Inc.

Babylon, New York

INTRODUCTION

The Grumman Aircraft Engineering Corp. and its affiliate, Dynamic Developments, Inc.,
are engaged in research and development programs directed toward practical, high-speed
hydrofoil systems. Emphasis is placed on application to vehicles with water speeds
between 50 and 100 knots. Three of the test programs which assist in this effort are con-
sidered to be of special interest to the Third Symposium on Naval Hydrodynamics. The
first of these programs utilizes a whirling tank facility which enables hydrodynamic and
hydroelastic tests on waterborne or underwater devices up to speeds of 100 knots. The
second program utilizes a pendulum facility wherein hydrodynamic models swing through a
water tank of variable water temperature and, consequently, variable vapor pressure.

The third test program utilizes a research and test craft with supercavitating surface-
piercing hydrofoils, a supercavitating propeller and a gas turbine power plant. This vehicle
has been chosen as the subject for discussion herein. Construction and initial tests have
been performed under Contract Nonr 2695(00) with the Office Of Naval Research. The pur-
pose of this test program is to determine the hydrodynamic efficiencies of the propeller and
hydrofoil system over a speed range between 0 and 60 knots, and to investigate corrosion
preventive measures for gas turbine engines operating in a marine environment.

GENERAL CHARACTERISTICS

General characteristics of the craft are shown in Figs. 1, 2a, 2b, and 2c. Overall
length, with foils extended, is 23.3 feet. Clearance between the keel and the 60.0-knot
water line is 2.08 feet. Takeoff gross weight, with one pilot, is 2550 pounds. Table 1
gives a complete weight statement.

HULL

The hull has been constructed from an aluminum sport boat manufactured by Grumman
Boats, Inc. Three primary modifications have been made to the basic hull. The first is the
addition of Lockfoam covered with three layers of fiberglass cloth to the forebody bottom.
The new forebody lines were established through the addition of ribs installed external to

205
646551 O—62—_15

206 G. J. Wennagel

Fig. 1. The ONR supercavitating hydrofoil boat

the original hull. This addition increases forebody deadrise, so as to minimize wave impact
loads on the hull, and forms a step on the hull bottom. Vertical tubes are installed within
the boat and through the hull bottom aft of the step, to facilitate step ventilation. A second
modification is an aluminum box aft of the transom of the basic boat which gives the craft a
desired overall length and provides support for the tail hydrofoil and strut assembly. The
third modification is an opening for the engine exhaust in the starboard side-skin near the
aft end of the basic hull. Stainless steel sheet is used as a doubler around this cutout to
provide strength over a region affected by the high temperatures of the exhaust.

The basic hull skin is fabricated by stretch-forming two pieces of sheet which are
joined together at the keel. Riveted construction is employed to attach hull and box exten-
sion skins to ribs and longitudinal stiffeners. All the aluminum is 61S-T6 alloy.

270 240 210 180 150 120 90 60 30 fe}

LOA. 270
s

BASE LINE WL. 0.

25
--—__..__HIGH SPEED wt. b mal"
WATER SPEED INDICATOR PITOT _ Ser

12.95
28.9

POSITION |

Fig. 2a. Side view of the ONR supercavitating hydrofoil boat (all dimensions in inches)

ONR Hydrofoil Boat XCH-6 207

PILOT

DIAGONAL FOIL sag tt aad

99

ae YOKE
TAIL STRUT Noa ve
LOWER
TAIL STRUT
TAIL FOIL eee as
. DIA. ¢
ela FORWARD CRUISE

68 - FOIL ELEMENT

Fig. 2b. Front and rear views of the ONR supercavitating hydrofoil boat

Y RETRACTION AXIS
40

TURNING
ARC £ 10°

Te FLEXIBLE

COUPLING

-58 EXHAUST

Fig. 2c. Top view of the ONR supercavitating hydrofoil boat

208 G. J. Wennagel

Table 1
Weight Statement

Hull:
Basic Boat
Forebody Bottom Addition
Aft Box Extension
Step Vent Tubes
Pilot Seat and Supports

Forward Foil Assemblies and Supports:
Support Fittings and Structure
Upper Struts
Lower Struts
Diagonal Foils
Cruise Foils

Tail Foil Assembly:
Tail Strut
Yoke
Strut and Yoke Support Fittings
Propeller Pod

Upper Gear Box
Tail Foil

Transmission (All Gears, Shafting, Bearings, Seals, Couplings):

Powerplant:
General Electric T-58 and Reduction Box
Engine Mounts
Exhaust Duct and Hull Supports
Bellmouth and Screen
Cowl and Supports
Throttle Control and Supports
Tachometers and Throttle Slave
Engine Instruments and Panel

Lubrication and Fuel System:
Pumps
Fuel and Oil
Fuel Tank and Supports
Oil Tanks (2) and Supports
Fuel and Oil Lines
Fuel (1) and Oil (2) Filters

Steering System

Lear Actuator (For Tail Strut Incidence Adjustment)
Pilot and Equipment

Torquemeter

Air Bleed System

Battery

Propeller

Miscellaneous

ONR Hydrofoil Boat XCH-6 209

HYDROFOIL SYSTEM

Two surface-piercing hydrofoils (one on each side of the boat) are located forward of
the vehicle center of gravity and one fully submerged foil is located aft of the transom.
Approximately 77.0 percent of the vehicle weight is supported on the forward hydrofoils.

As seen in Fig. 3, each of the forward hydrofoil assemblies incorporates a vertical
strut supported to the hull in its upper region. A cruise foil element is attached to the
lower end of each strut. A portion of this element is cantilevered so as to extend inboard
of the strut bottom. In addition, it extends outboard and up from the lower end of the strut
with positive dihedral. The upper end of each cruise foil is joined to the upper region of
the strut by a diagonal foil element with negative dihedral.

Fig. 3. A forward hydrofoil assembly

Cross sections of the foil and strut elements of the forward foil assembly are shown in
Fig. 4. The cruise foil elements embody a basic supercavitating section which has a
circular-arc bottom shape, a flat upper contour, a sharp leading edge, and a blunt trailing

210 G. J. Wennagel

——— ane

Nemioean tint ARC LEADING EDGE RADIUS=0.0.15 IN.
FORWARD CRUISE FOIL SECTION
R=RADIUS OF UPPER SURFACE CONTOUR

LOCATION C IN. T HIN. R IN.
OUTBOARD END 11.940 0.811 177.733
DIHEDRAL BREAK 8.394 . 0.578 124 .688

CIRCULAR ARC LEADING EDGE RADIUS =0.015 IN.

FORWARD DIAGONAL FOIL SECTION
R=RADIUS OF UPPER SURFACE CONTOUR

LOCATION T IN.
OUTBOARD TIP 0.72
INBOARD END 1.20

PARABOLA

TRAILING EDGE RADIUS = 0.062 IN.
LOWER FORWARD STRUT SECTION

PERCENT CHORD T IN.
60. | 0.375
81.5 0.375
92.7 0.260

Fig. 4. Cross sections of the elements of a forward hydrofoil assembly

edge. Over the lower regions of this element, a tapered afterbody is added to the basic
section. At low speeds, prior to cavity formation, this afterbody reduces the drag which
would exist for the basic section. At higher speeds, after cavity formation, the afterbody

is essentially unwetted but still contributes to section strength and stiffness. The diago-
nal foil elements, which are wetted only during low-speed operation, employ a subcavitating
cross section with a vented base. This section employs a flat bottom, a circular-are upper

Ai

ONR Hydrofoil Boat XCH-6 211

contour, a sharp leading edge, and a blunt trailing edge. The strut cross section employs a
parabolic forebody and a tapered afterbody; the two regions of this section are separated by
a step.

The tail hydrofoil (Fig. 5) employs zero dihedral and is supported on the forward half
of the propeller pod; the pod is supported by a single strut. Figure 6 illustrates the strut
and foil cross sections. The tail hydrofoil section is of symmetrical, subcavitating design
and incorporates a low thickness ratio. A parabolic cross section with a blunt base is used
on the tail strut. Propeller pod fineness ratio is approximately 8.5.

Fig. 5. The tail strut, propeller, and hydrofoil

The forward, cruise foil elements and lower struts are fabricated from solid stainless
steel. Upper regions of the forward and tail struts are fabricated from heavy-gage aluminum
plate. Diagonal foils, tail foil, lower tail strut, and propeller pod are machined from solid
aluminum.

Each forward, cruise foil element is attached to its supporting strut by flush-head
bolts which are inserted from the bottom side of each foil element and which extend into
tapped holes of the struts. Diagonal to cruise element attachment is similar, with bolt

212 G. J. Wennagel

STRAIGHT LINE

ELEMENTS
ELLIPTICAL NOSE

T,
0.035c—

pl et oye set | 3

TAIL FOIL SECTION
ALL INTERSECTIONS OF STRAIGHT LINE ELEMENTS ARE ROUNDED

0.032 aa

LOCATION C IN.
ROOT 13.00 T=0.0375C
OUTBOARD TIP 3.66 T,=0.0254C

PARABOLA

TAIL STRUT SECTION

Fig. 6. Cross sections of the tail strut and hydrofoil

insertion from the upper side of each diagonal and with tapped holes in each cruise foil
element. These intersections are thus maintained hydrodynamically clean without resort to
bodies of revolution. Diagonal to upper strut connection on each forward foil assembly
involves one fitting and connecting bolts. Each of the two lower struts extends up into an
adjoining, hollow, upper strut; attaching bolts connect the mating segments. This design
enables easy replacement or modification of any of the four, separate components which
make up each forward hydrofoil assembly.

The propeller pod is split along a horizontal plane so as to form an upper half and a
lower half. Suitable machining allows for the insertion of a one-piece tail foil between the
upper and lower halves of the pod. Attaching bolts, which are externally flush, connect
these three structural elements. Thus, emphasis has been placed on obtaining a design
which allows easy replacement or modification of the tail hydrofoil.

A welded connection is employed between the upper half of the pod and the tail strut.
This welding extends around the entire perimeter of the strut cross section and has been
hand polished to obtain a smooth finish. Rigid and fixed connection between these two
elements was a design requirement so as to allow subsequent, internal machining of both
strut and pod in a manner that could guarantee proper alignment of transmission components.

An upper tail strut, of hollow construction, attaches to the top of the lower strut, and
provides support at its upper end for a right-angle gear box. Gear box and lower strut
attachment is by bolts to allow easy disassembly and inspection of internal transmission
components.

ONR Hydrofoil Boat XCH-6 213

All aluminum components are 61S-T6 alloy; all steel elements are 416 stainless alloy
with an ultimate tensile strength of 150,000 psi. The struts, which employ constant cross
section, were externally contoured by a single pass through a Whaley Machine of 14-inch
diameter. A Keller BL Model C Machine was used for obtaining the more complicated exter-
nal contours of the forward cruise hydrofoils.

CONTROLS AND RETRACTION

The boat is steered by rotation in yaw of the entire tail strut-pod-foil assembly. Tail
foil incidence with respect to the keel is adjustable while foilborne, and a trailing edge
trim tab is provided on the cruise foil element of one forward hydrofoil assembly. Each of
the three hydrofoil assemblies is separately retractable.

Tail assembly retraction, steering, and incidence adjustment in pitch are accomplished
by means of a unique, structural yoke. The yoke is an aluminum casting with machined sur-
faces. As shown in Fig. 2a, the yoke provides support to the tail strut at two, vertically
separated locations. Both of these supports are through sleeve bearings on cylindrical por-
tions of the yoke structure. The bearings, in turn, support an aluminum tube which is
rigidly bolted to the strut assembly at both support points. The sleeve bearings allow rota-
tion of the tail assembly in yaw, about the tube and steering axis. This axis, extended,
coincides with the 20.0-percent chord line of the lower strut. Each of the bearings can
transmit loads in a horizontal plane. Up and down loads are transferred from the tube to
the yoke at the lower and upper bearing supports, respectively. The upper end of the tube
and the top of the strut are joined by a fitting. This fitting supports the upper gear box and
has an arm extending out on its port side. Connected to the arm is a fore and aft push-pull
rod of the steering system. Rod loads establish equilibrium of the tail assembly in regard
to moments applied about the steering axis and motion of the rod provides steering action.
Strut displacement in yaw is limited by stops to plus or minus 6.0 degrees. The remainder
of the steering system provides for mechanical, irreversible control from the pilot’s wheel.
At high speed, only small strut rotations are required; a wheel-to-strut rotation ratio of

54.5 to 1.0 is employed.

Yoke support to the hull is provided at three pickup points. Two points at the upper
end of the yoke, separated athwartship, are hinged on a lateral axis. Rotation about this
axis allows for both retraction and for tail foil incidence adjustment. Each of the two
upper support points can transmit loads in all directions. The third support point is at the
bottom of the yoke and is attached to the hull through an electrically operated, linear actu-
ator manufactured by Lear, Inc., Instrument Division, Stamford, Connecticut (Model No.
434-AJ). Variable length of this support, in the fore and aft direction, allows incidence
change in pitch while foilborne. This attachment is designed to take only axial (fore and

aft) loads.

The vertical steering axis and the lateral retraction axis intersect at a point on the
vehicle vertical plane of symmetry. Also passing through this point is the centerline of
longitudinal, transmission shafting. A Rzeppa coupling in the transmission, at this three-
axis intersection, allows for simultaneous transmission of power, tail assembly incidence
change in pitch, and tail assembly steering. Manufacture of the flexible coupling is by

Dana Corp., Con-Vel Division, Detroit, Michigan (Model No. OR).

Three pickup points on the hull support each upper strut of the forward foil assemblies.
Two points on top of each strut, separated in the fore and aft direction, are hinged on a
longitudinal axis. Each of these supports can take loads in all directions. A third support

214 G. J. Wennagel

on each strut is located at the chine. It consists of a manually operated, locking handle
which can take loads in the lateral direction only. When this latter support is unlocked,
each forward foil assembly can be retracted by rotation outboard and up.

Provision has been made to avoid damage to the primary hull structure in case the
hydrofoils strike a solid object. On the forward foil assemblies, shear pins which attach
the support fittings, on top of the upper struts, to the upper struts are designed to fail
under specific loads. In the event of a failure, the affected forward assembly can leave the
vehicle in the aft direction; the support at the locking handle, near the chine, allows this
motion without restraint. On the tail assembly, retraction can occur, under a crash condi-
tion, by means of a shear pin incorporated in the lower yoke support and designed to fail
under a specified load. In the event of such failure, damage would be localized to the
Rzeppa coupling in the longitudinal transmission shafting.

A manually operated, trailing edge trim tab is installed on the cruise foil element of
the starboard forward foil assembly. It allows trim of the vehicle in roll.

TRANSMISSION

Line shafting runs aft from the engine to the upper gear box, down through the tail
strut to the propeller pod and then aft to the propeller. The transmission arrangement is
shown in Fig. 7. Precise design and manufacture has achieved a rigid design criteria of
minimum frontal area on the strut and pod. At 200 hp and at the maximum output speed of
the engine reduction box, 6000 rpm, gears and bearings are designed for a 50-hour life.
Design and manufacture was by the Medium Steam Turbine and Gear Division, General
Electric (Company, Lynn, Massachusetts.

Shafting between the engine and upper gear box incorporates a torque sensor, a torque
limiting shear pin and couplings with bearing supports. At the intersection of the transmis-
sion, retraction, and steering axes, a Rzeppa coupling allows both angular misalignment and
parallel offset of shaft centerlines.

Upper gear box and pod each contain a right-angle, spiral bevel gear set of 1.0 to 1.0
ratio wherein each gear incorporates a 35-degree spiral angle and a 20-degree pressure
angle. They are of 3.0-inch pitch diameter and 7.0 diametral pitch. All gears are keyed to
their respective shafts except for the pod pinion which has a major diameter fit fixed spline
and a rear face which shoulders against its drive shaft. Material is AISI-9310 aircraft
quality stock. Teeth were case-carburized to 60-63 RC after generating and then finish
ground. The gears were developed with proper contact patterns on both sides of the teeth
so that a single grinder set up could produce the gearing for both the upper and pod bevel
sets.

Connection between the upper gear box and the pod pinion drive shaft is made by a
free-floating, internally splined quill shaft in the upper strut. A ball bearing carries the
thrust reaction of the pod pinion, and this bearing has been located in an area well above
the high-speed water line where sufficient housing volume is available without detrimental
effect on drag. The length of the drive shaft between the thrust bearing and the pod pinion
is approximately 26.0 inches. Pod pinion radial reactions are carried by precision needle |
bearings. The pinion drive shaft is induction-hardened and ground in the locale of these
needle bearings to serve as integral bearing races; this feature acts to further reduce
frontal area. The pod pinion’s axial load component was selected to be of minimum value

215

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216 G. J. Wennagel

while the gear’s axial component is of maximum value. The hand of the spiral angle is such
that the gear’s axial reaction is out of mesh, or to the rear, in direct opposition to the pro-
peller thrust. This allowed for the use of relatively small propeller shaft thrust bearings,
since the net bearing thrust load is the difference of these forces. Two thrust bearings are
employed just forward of the propeller shaft seals. Two roller bearings straddle-mount the
gear to provide radial reactions.

Shafting in the upper gear box and the quill shaft are of 4340 stock with induction-
hardened splines. The pod pinion drive shaft is of 4350 stock while the propeller shaft is
of forged and age-hardened K-monel. The areas of seal contact on the transmission input
shaft and propeller shaft are built up with a copper flash followed by a nickel flash and a
hard chrome plate.

Seals are of the double lip type and made of Sirvene 7080, a material chosen for its
sea-water resistance. A single seal is used at the drive input with system lube oil
depended upon for lubrication. Tandem seals packed with a sea-water-resistant grease are
used on the propeller shaft. An interesting feature is that the pod structure is split horizon-
tally along its longitudinal axis and bolted together on assembly. The pod machinery area
is sealed by filling a machined groove, which runs along the periphery of each half, with a
room-temperature self-curing, silicone rubber compound.

Damage due to galvanic action has been controlled by careful selection of materials
exposed to sea water. Like the propeller shaft, the pod seal carrier is of K-monel. A lock-
ing ring, used to retain the seals, is cadmium plated. All lube and scavenge fittings and
lube oil supply lines and grease fittings are of aluminum. To date, there is no evidence of
erosion due to galvanic action.

The lubricant used conforms to specification MIL-0-6086 (light). This is a petroleum-
base aircraft gear oil containing extreme pressure additives. An electric pump in the lube
supply system delivers two quarts of oil per minute at 20 psig. A pressure sensing switch
has been incorporated in the supply line. If the pressure falls below 15 psig, this switch
opens a return line from the engine fuel control to the fuel tank and engine shutdown is
thereby automatic. A panel light then goes on to inform the pilot of the reason for engine
shutdown. Total oil flow is divided by a tee located at the boat transom so that one quart
per minute is supplied both to the upper gear set and to the pod gearset. Gear mesh and
input shaft thrust bearings are pressure-lubricated in the upper gear box. Oil mist and oil
flow by gravity lubricate the remaining gear box bearings, the spline couplings of the quill
shaft, and the thrust bearing of the pod pinion drive shaft. A single lip seal directly
beneath the pod pinion thrust bearing prevents leakage of oil to the pod. An oil scavenge
port is located at the level of this seal to allow the return of the lubricant to the lube sys-
tem. Needle bearings for radial load reaction on the pod pinion drive shaft are grease
packed on assembly, and a grease fitting is provided to the uppermost of these bearings.

A spray nozzle, to pressure lubricate the pod gearset, is located at the aft end of the pod-
strut juncture. Oil mist is depended upon to lubricate the propeller shaft roller bearings.
The pod is bored to give the effect of a shroud about the driven bevel gear. Dual channels
run axially from below the gear to a sump just forward of the foremost roller bearing. This
sump is shrouded to break up windage effects from a roller bearing locknut. The lubricant
is then scavenged through drilled passageways up through the strut where connection to the
lube scavenge system is made. Propeller shaft thrust bearings are grease packed and are
provided with ring seals. Wash out of the grease packing by lube oil is prevented by slinger
rings on the propeller shaft. Scavenge lines are joined at the transom into one line which
continues forward. An electric scavenge pump returns the lubricant to a reservoir. An oil
cooler has been found unnecessary and is not used.

ONR Hydrofoil Boat XCH-6 217

Magnetic drain plugs are employed in the strut and in the pod to provide indication of
gear or bearing wear.

ENGINE

Power is supplied by a General Electric YT-58-2 free turbine engine. Maximum turbine
output speed is 19,500 rpm. An engine gear box with a 3.25-to-1.0 ratio reduces the output
shaft speed to a maximum of 6000 rpm. At this speed, the engine has a guaranteed normal
continuous rating of 900 hp on a 59°F day and 765 hp at 80°. Although available power is
greatly in excess of vehicle requirements, its light weight and desirable output speed make
the T-58 particularly well suited for this application. Engine fuel and lubrication systems
are in accord with normal practice for this engine, with 5-micron and 25-micron filters
installed in the engine fuel and gear box oil supply lines, respectively.

A goal of this program has been to determine the location and degree of corrosion
effects on engine components and to test thereafter various coatings and changes in material
so as to systematically accomplish a final marinization of the YT-58-2 engine. Accordingly,
initial installation utilized a standard engine, No. SN-200-103, without any special compo-
nent protection, extemal cowling, or maintenance procedure.

External corrosion was first observed on small, unpainted areas of the engine’s magne-
sium front frame and of the magnesium reduction gear box. Further corrosion was then pre-
vented in these locations by a grease coating. Thereafter, stiffness became apparent-in the
linkage motion of the variable stator system. This stiffness was then corrected by periodic
application of oil. Subsequent corrosion occurred on other exterior components. Again,
after observation for a suitable period of time, further corrosion was halted in each locale by
regular application of grease or oil.

Engine operation was continued, with inlet exposure to direct salt-water spray, until the
turbine inlet temperature reading at idle had increased to approximately 100°C higher than
normal. Removal of the compressor casing revealed a significant amount of corrosion through-
out the entire compressor, with the effects of corrosion increasing toward the later stages.
Although the aerodynamic configuration of the higher stage blading had been materially
affected, engine starting difficulty had not been encountered. The compressor rotor was
cleaned by hand using large, gun barrel brushes; blades were removed and cleaned with an
aluminum oxide blast. The engine was reassembled and a cowling was installed to protect
the inlet from direct salt-water spray. In addition, engine washdowns were begun whereby
2.5 gallons of fresh water were sprayed into the inlet just prior to engine shutdown after each
test period. A second period of operation, equivalent to the first, was completed and internal
engine components were again inspected. It was found that the simple procedures then
adapted had essentially halted the spread of further corrosion. Engine temperatures and oper-
ation remained normal over the observation period.

A second engine, No. SN-200-102, has recently been installed and is currently being used
during test operations. It is a modified engine intended to test the effectiveness of certain
coatings in preventing corrosion of engine components. Table 2 gives a list of these compo-
nents and their coatings. Prior to installation, test-stand operation was completed by the Gen-
eral Electric Company to determine the effects of these modifications on aerodynamic charac-
teristics. They have been found small, and guaranteed performance ratings remain unchanged.
Vehicle operation will continue to use engine washdowns, exterior oil application and inlet
cowling. While no effects of corrosion have yet been observed with this engine, insufficient
operating time has been accumulated to allow proper conclusions.

218 G. J. Wennagel

Table 2
Marinization Changes to the YT-58-2 Engine

Engine Rees ee Material and Coating or
Dash No.* 8 Specification No. Treatment

102 and 101 | Front Frame Magnesium Nubelon S
AMS 4434
102 and 101 | Compressor Casing Steel, G. E. Nickel Plate
_ SAED B50T1131
102 and 101 | Compressor Rear Frame Steel, G. E. Nickel Plate
SAED B50T1131
102 Compressor Spool & Blade Assy Steel Nubelon S
AMS 6451
102 and 101 | Compressor Sta. Vanes & Rings Steel
AISI 405
Stages | thru 7 Nubelon S
Stages 8 and 9 Heresite
Stage 10 and Exit Vanes Silicone Alum.
102 and 101 | Combustion Chamber Casing Steel, G. E. Nickel Plate
SAED B50T1131
101 Variable Stator Actuating Rings Steel Chrome Plate
AISI 410
101 Variable Stator Actuator Levers Steel Nickel Plate
AMS 5120 '
101 Main Reduction Gear Case Magnesium Replace with
AMS 4434 Aluminum,
Alodine
101 All Magnesium Engine Components | Magnesium Replace with
AMS 4434 Aluminum,
Alodine
Compressor Spool Steel Nubelon S
AM-355

*This column identifies the changes as applying to the second engine, No. SN-200-102, or to the
third engine, No. SN-200-101.

A third engine, No. SN-200-101, is being modified for later installation and test. It
includes all of the coatings incorporated in the second engine, with additional changes as
given in Table 2. It is hoped that this engine, after test-stand operation and waterborne
service with suitable operating procedures, will represent a final marinization of the

YT-58-2 engine.

PROPELLER

The propeller is located on the aft end of the propeller pod behind the tail foil and
strut. It has been designed and manufactured by personnel of the David Taylor Model Basin
for zero cavitation number and in accordance with the DTMB three-bladed, supercavitating
propeller series. This series is fully described in Refs 1-3. Propeller diameter is 10.0
inches; pitch ratio at the 70.0 percent radius is 1.61; expanded area ratio is 0.457; blade
thickness fraction is 0.024; rotation is clockwise looking forward. Propeller material is

ONR Hydrofoil Boat XCH-6 219

stainless steel with an ultimate tensile strength of 140,000 psi. A design feature is a nylon
liner between the propeller hub and the propeller shaft. The shaft is tapered and propeller
attachment is by means of a locknut and a brass key. Friction between the hub and shaft is
reduced, by the nylon liner, to a very low value. This reduced friction, together with a
torque limit on the locknut, has enabled specification of key size such that key failure in
shear occurs at a predetermined value of propeller torque. Protection to transmission compo-
nents is thereby provided for a condition wherein the propeller might strike floating debris

or some solid object. Laboratory tests were undertaken to determine proper key size; failure
of the key occurs at a torque of 2500 inch-pounds.

INSTRUMENTATION

Instrumentation has been incorporated to provide cockpit indication of water speed, pro-
peller torque, propeller rotative speed, propeller thrust, keel trim in pitch, tail foil incidence
in pitch with respect to the keel, tail strut position in yaw, transmission and engine oil
temperatures and pressures, compressor and power turbine speeds and temperatures, and fuel
pressure.

The propeller torque measuring device is installed in the horizontal transmission shaft
aft of the engine gear box. It is a torque transducer, Model No. TG-5-3000A, manufactured
by the Crescent Engineering and Research Company, El Monte, California. The core or
sensor section of the transducer incorporates a solid shaft which acts as a torsional spring.
It is splined at its ends to mating sections of the drive shaft, and is enclosed by a cylindri-
cal, fixed pickup unit. Cylindrical crowns are attached to the shaft of the sensor section at
two locations which are separated along its longitudinal axis. The crowns have inter-
meshing teeth which extend around the center region of the solid shaft. Torque causes a
change in the size of air gaps which exist between the intermeshed teeth of the two crowns,
and this change modifies the magnetic reluctance. The variation in reluctance effects pro-
portional changes of impedance in two coils of the enclosing pickup unit. The coils perform
as two legs of an ac bridge to reflect electrical unbalance which is proportional to torque.
This unbalance is amplified, demodulated and applied to a microammeter which reads torque
directly.

Propeller thrust is calculated by employing strain gages affixed to the linear actuator
which attaches the lower yoke support point to the hull. Data is recorded on a portable
oscillograph. Measured compression loads in the actuator enable the calculation of propeller
thrust, as required for moment equilibrium about the retraction axis, based on estimated tail
foil lift and tail assembly drag components, together with associated centers of pressure.
The calculation of thrust is therefore approximate at the present time. It is planned to per-
form towing-tank tests of the tail assembly to accurately measure estimated quantities and to
thereby provide more accurate calculation of propeller thrust.

FLOW CONDITIONS

Flow conditions involve cavities over the upper surface and behind the blunt base of
each forward cruise foil, behind the blunt base of each forward diagonal foil, along the after-
body sides of the forward struts, and behind the blunt base of the tail strut. All other
regions of foils and struts experience subcavitating flow.

Hydrofoil and strut design provides several air paths to cavity regions. Forced ventila-
tion is provided through tubes to the blunt base of the tail strut and to the blunt trailing edge
regions of the diagonal and cruise foil elements of each forward hydrofoil assembly. Air
supply for this purpose is obtained from two bleed ports located aft of the compressor section

220 G. J. Wennagel

of the gas turbine power plant. Air flow utilized from each port is approximately 0.20 lb/
sec. Natural paths, for air flow from the atmosphere, are also provided. The blunt base of
the tail strut extends through the water surface. Similarly, air can flow from the atmosphere
to the sides of the afterbody on the forward two struts. At high speed, therefore, only the
parabolic forebodies of the forward struts are wetted. The cavities along the afterbody
sides can interconnect with cavities on the upper surface of the forward cruise foil elements.
A natural air path to the cavity regions of the forward cruise foil elements is also provided
along the blunt trailing edges of the diagonal and cruise foils. These blunt bases are con-
nected to each other at the intersection of the two foil elements. In addition, the cavity
regions of the cruise foils, at high speed, open directly to the atmosphere since these ele-
ments are surface piercing and this forms their primary source of air under cruise conditions.
A further possibility for air supply to these regions is through vortices from the foil inboard
tips which can extend aft and up to the water surface.

Air from the atmosphere flows to cavity regions with velocity and associated pressure
drop. Resultant cavity pressures are thus intermediate between vapor and atmospheric
pressures.

A goal of testing at Grumman Aircraft and Dynamic Developments, Inc. is to develop
hydrofoil systems which can make a smooth and stable load transition from subcavitating
conditions at low speeds to cavity flow conditions at high speeds. Each type of flow is
thereby utilized over the speed range where it best affords maximum lift-to-drag ratio. Both
surface-piercing and fully submerged foil systems are envisioned which can accomplish this
smooth flow transition, and it is expected that several foil arrangements will eventually be

tested on the XCH-6. -

PERFORMANCE

Curves of estimated vehicle drag and thrust available are plotted vs velocity in Fig. 8.
These are typical curves which pertain to current vehicle operation at a gross weight of
2550 pounds. In practice, both thrust available and drag curves can be altered.

THRUST AVAILABLE AT CONSTANT
PROPELLER TORQUE OF 2,100 IN. LB.

800 /
600
DRAG AND 4np
THRUST, LB.
si SMOOTH WATER DRAG AT
GROSS WEIGHT OF 2,550 LB.

0 10 20 30 40 50 60 70
FORWARD VELOCITY, KNOTS

Fig. 8. Estimated smooth water drag and thrust available

ONR Hydrofoil Boat XCH-6 221

Thrust available is dependent upon the size of the shear pin located in the horizontal
transmission shafting, aft of the engine gear box. These pins are hollow. The choice of
inside diameter and pin material alloy allows selection of maximum transmission and pro-
peller torque. This, in turn, determines available propeller thrust at each forward speed.
Normal vehicle operation utilizes a shear pin which is designed to fail when propeller
torque reaches 2100 inch-pounds.

Component drag prediction for operation in smooth water has been based upon methods
outlined in Refs. 4 and 5. Such prediction appears to be in good agreement with test results
obtained to date. The total smooth water drag values given in Fig. 8 can be considered to
correspond to normal operation, but the low-speed region can be altered depending upon pilot
adjustment of tail foil incidence in pitch with respect to the keel. Minimum takeoff speed,
the speed at which the keel just clears the smooth water surface, is approximately 20.0
knots at a gross weight of 2550 pounds.

The propeller was designed to develop 774.0 pounds of thrust at a speed of 62.0 knots
and at the maximum rotative speed of 6000 rpm. Predicted propeller efficiency, under these
conditions, is 73.9 percent. Associated propeller torque is 2100 inch-pounds.

For operation in smooth water at 60.0 knots, the total drag is estimated to be 660.0
pounds. Only 20.0 percent of this value corresponds to drag due to lift; 8.0 percent is
attributed to spray drag; 16.0 percent is due to air drag; the remaining 56.0 percent is due to
friction plus pressure drag of wetted elements. This distribution of drag is important when
considering vehicles of much larger size with the same cruise speed. Larger craft can be
expected to incorporate suitable aerodynamic streamlining, not employed on the XCH~, and
will benefit from considerably reduced friction drag coefficients associated with larger
chords and consequently higher Reynolds numbers.

INITIAL TRIALS
Several features of the present configuration have resulted from initial testing.

The nylon liner between the propeller shaft and the propeller hub was provided after an
early run wherein the propeller struck a piece of floating debris. The transmission was pro-
tected by a shear pin in the drive shaft for conditions of excessive torque from the engine.
When excessive torque was suddenly applied from the propeller, however, the inertia of
transmission components allowed the bevel gears within the pod to feel the overload prior to
shear-pin failure. The result was a tooth failure on one of the bevel gears. Damage to the
propeller, consisting of a few nicks on blade leading edges, was very slight and easily
corrected by filing. -

A reduction in drag, associated with forced air ventilation, was first established by
providing air to the blunt base of the tail strut. It allowed the maintenance of given forward
velocities with reduced propeller speeds. Provision for forced air supply was then extended
to include the blunt base regions of the forward hydrofoils.

During initial trails, a slight porpoising was encountered at approximately 40.0 knots.
The oscillation was of constant amplitude, without either build up or decay. It was theorized
that ventilation behind the blunt base of the tail strut extended over the sides of the pod
near its aft end so as to cause a small decrease in total lift of the tail assembly. This
change in lift caused a small increase in vehicle trim and a corresponding increase in pod

646551 O—62——_16

999 G. J. Wennagel

immersion. At the increased immersion, the flow again reseated on the affected region of
the pod, initial lift and trim were restored, and the process would then repeat itself, causing
a mild and steady oscillation. A small plate was added on top of the pod and behind the
strut so as to limit the extent of the air cavity. This change eliminated further oscillation.

Forward foil design has successfully provided a smooth and stable manner of cavity
formation over the cruise foils. It is significant that the vehicle has never experienced
adverse roll due to uneven extent of ventilation, one side to the other. Further, it has
never experienced abrupt changes in pitch as might be caused if ventilation or flow reseat-
ing occurred abruptly. This is true for both smooth and rough water operation, and for run-
ning at either a constant heading or during turning maneuvers.

CONCLUSIONS

1. Forced air ventilation, into regions behind the blunt base of a strut or foil, is a
practical means of drag reduction.

2. Within the limits of completed testing, predicted propeller and ventilated hydrofoil
efficiencies have been confirmed.

3. Ventilated hydrofoil systems can be designed to allow cavity formation in a smooth
and stable manner. They offer the elimination of abrupt loss in lift associated with some
subcavitating foil systems when ventilation occurs. Complete flow stability has been
demonstrated over a wide variety of sea conditions and tuming maneuvers.

4, Supercavitating foil sections, as compared to subcavitating types, exhibit a reduced
lift curve slope which minimizes incremental lift changes associated with orbital velocities
within waves. This feature, together with satisfactory hydrodynamic efficiencies, appears
to offer an important field for future development.

ACKNOWLEDGEMENTS

Appreciation is expressed to the Grumman Aircraft Engineering Corp. and to the General
Electric Company for freely-given technical assistance and component contribution, to Lear,
Incorporated for the donation of a linear actuator, and to the Kaman Aircraft Company forthe
donation of an oil reservoir and deairation device.

REFERENCES

[1] Tachmindji, A.J., Morgan, W.B., Miller, M.L., and Hecker, R., “The Design and Perform-
ance of Supercavitating Propellers,” David Taylor Model Basin Report C-807, Feb. 1957

[2] Tachmindji, A.J., and Morgan, W.B., “The Design and Estimated Performance of a
Series of Supercavitating Propellers,” p. 489 in “Second (1958) Symposium on Naval
Hydrodynamics,” Washington: Office of Naval Research, 1960

ONR Hydrofoil Boat XCH-6 223

[3] Caster, E.B., “TMB 3-Bladed Supercavitating Propeller Series,” David Taylor Model
Basin Report 1245, Aug. 1959

[4] “Study of Hydrofoil Seacraft,” Vol. I, Bethpage, New York: Grumman Aircraft Engineer-
ing Corp. and Dynamic Developments, Inc., Oct. 1958

[5] Johnson, V.E., Jr., “Theoretical and Experimental Investigation of Arbitrary Aspect
Ratio, Supercavitating Hydrofoils Operating Near the Free Water Surface,” NACA RM
L57116 (NASA, Washington), Dec. 1957

DISCUSSION

H. P. Rader (Vosper Limited, Portsmouth)

With my compliments to Mr. Wennagel and also to Mr. Tulin I should like to make just a
few remarks concerning the practical application of supercavitating sections. Recently we
have tested two propellers with supercavitating blade sections of wedge-shaped thickness
distribution. The efficiency of both propellers under noncavitating conditions was rather
poor. On analyzing the results according to the method of the equivalent polar curve as
devised by Prof. Lerbs we found that the drag coefficients of the equivalent blade sections
were much higher than those of supercavitating sections with quasi-elliptic or elliptic-
parabolic thickness distribution. In fact the drag coefficients agreed very closely with
values published in Dr. Hoerner’s excellent book “Fluid Dynamic Drag.” The drag coeffi-
cient ratios for various thickness-chord ratios are approximately as follows:

Thickness-chord Ratio Cy Wedge/Cp Ellipse
2.5

0.04

0.06 4.0
0.08 5.9
0.10 TE

When we design supercavitating hydrofoils and propellers we must not forget that the boat
fitted with the equipment has to pass through the noncavitating speed range. This is par-
ticularly important for hydrofoil boats which have a pronounced hump in their resistance
curve at a speed where supercavitating conditions may not be obtainable unless artificial
ventilation is used. Even with ventilation the drag coefficients of wedge-shaped sections
would be higher than those of quasi-elliptic or elliptic-parabolic sections under noncavitat-
ing conditions. I noticed in Mr. Wennagel’s interesting paper (or rather in Fig. 8) that at
the hump the resistance-displacement ratio of the craft described by him is approximately
0.26, which is rather high.

Marshall P. Tulin

With regard to Mr. Rader’s comments on the efficiency of supercavitating sections with
blunt bases operating in fully cavitating flow, it is, of course, certainly true that such sec-
tions are not particularly efficient because of the blunt base, which causes a drag that is
proportional to the square of the base thickness. In fact, it was with some surprise that the

294, G. J. Wennagel

David Taylor Model Basin tests of supercavitating propellers revealed that the efficiency of
such propellers at design advance ratio was essentially independent of cavitation number,
implying that the efficiency was a little bit less at high cavitation number, that is, under
fully wetted conditions, than it was in supercavitating operation, and thus indicating that
very little price was being paid because of the drag incurred by the blunt section operating
under fully cavitating conditions. With regard to the future, I would not care to speculate on
what shapes practical supercavitating foils, designed for high performance craft, will take.
Certainly in the case of fully submerged foils, flaps and perhaps other such devices will be
included; and I think that we must be patient to see how foil shapes will evolve.

Glen J. Wennagel

In regard to Mr. Rader’s comment on the ONR boat, the drag hump during takeoff is due
in good part to the foil and strut blunt trailing edges. The design gross weight as stated
was 2550 pounds. With a 2100-in./lb shear pin, which corresponds to 200 hp at 6000 rpm,
we reach a point where the boat will not take off when the gross weight reaches about 3000
pounds. This is without forced air. If the pilot turns on the air, the reduction in drag is
apparent and the boat will take off.

T. G. Lang (U.S. Naval Ordnance Test Station)
My comments are directed toward both Mr. Tulin’s paper and Mr. Wennagel’s paper.

The development of high-speed hydrofoil craft having surface-piercing hydrofoils or
struts requires a detailed knowledge of the effects of air ventilation on hydrodynamic forces.
The U.S. Naval Ordnance Test Station has conducted a series of experiments on hydrofoil
models in which air was forcibly exhausted through ports in the hydrofoil surface. These
experiments were conducted in the high-speed water tunnel at the California Institute of
Technology in 1958 and 1959. The objectives of the studies were to investigate the use of
forced ventilation for the purposes of control and of improving hydrofoil performance
characteristics.

Figure D1 shows a model of one hydrofoil used in the tests in which a spanwise series
of small holes can be seen just behind the leading edge. In this study, the chordwise loca-
tion of the holes was varied and the spanwise length changed.

Figure D2 is a top view of this model mounted in the water tunnel showing air being
exhausted through the series of holes, while Fig. D3 is a side view. The air did not spring
forward of the exhaust point unless extensive cavitation occurred behind the leading edge
or unless the hydrofoil was placed at an angle of attack above its fully-wetted stall angle.
Figure D4 is a top view of this model placed at the stall angle of attack. The air is seen
to ventilate forward of the holes.

Figure D5 is a top view of another model having the same contour in which a small
amount of air is exhausted through a single hole in the upper surface. The angle at which
the air diverged outward increased as the angle of attack increased. Figure D6 shows this
same model at the same angle of attack but with an increased air-flow rate. It is noted that
the sides of the air cavity are more distinct in the latter case and the thin, patchy type of
ventilation which was seen in the previous figure has disappeared.

ONR Hydrofoil Boat XCH-6 225

Fig. D3. Side view of the model

226 G. J. Wennagel

Fig. D4. Top view of the model at the stall angle of attack

— a

Fig. D5. Top view of a model with a single hole

Fig. D6. Same model as Fig. D5 with increased air flow

ONR Hydrofoil Boat XCH-6 227

Figure D7 is a plot of the lift coefficient versus angle of attack. The upper curve
refers to fully-wetted flow and the lower curve to vented flow. Note the large decrease in

YYY YY

“8 +6 -4 -2
ANGLE * ATTACK, DEG

Fig. D7. Lift coefficient of (top curve) fully wetted and (bottom curve) vented hydrofoil

lift when the hydrofoil is forcibly vented. This type of ventilation can be induced by
exhausting air through a port at any position on a hydrofoil and therefore may be used as a
means of controlling the hydrodynamic forces developed by the hydrofoil. It should also be
noted that the slope of the lift coefficient curve is reduced by ventilation. The references
cited at the end of this discussion include theoretical work which predicts the lift and drag
of two-dimensional hydrofoils of arbitrary shape vented at arbitrary chordwise locations for
the case of zero ventilation number. Good agreement is shown with experiment.

Figure D8 shows top and side views of a hydrofoil model having a cambered parabolic
cross section wherein air was exhausted through a hole in the trailing edge. The effective

kK: 0.100, Qz0.07/, @=8 5 R=1.$4¢

Fig. D8. Top and side views of a cambered parabolic
hydrofoil with air exhausted through a hole in the
trailing edge

228 G. J. Wennagel

aspect ratio in this photograph is 1.44. Note the shape of the air cavity at the tip. As in
the previous test series, the air was not seen to ventilate forward of the trailing edge unless
a region of separated water existed, such as that caused by stall, extensive cavitation, or
boundary layer separation. Figure D9 shows this same hydrofoil in two-dimensional flow
with a very small airflow rate. The vortex pattern seen behind the trailing edge disappears
when the airflow rate is increased.

Q20.02¢, R=4; A=c0

Fig. D9. Hydrofoil of Fig. D8, but
with a very small airflow rate

Figure D10 is a plot of the lift-to-drag ratio of the two-dimensional parabolic hydrofoil.
The lift-to-drag ratio approaches 25 as the airflow rate increases. Higher ratios could have
been obtained if tunnel blockage had not limited the minimum ventilation to K 2 0.14.

—_—_
pe 25. VENTED
; k~0.16
SYMBOL ,f7/SEC eal
° 30 L/o
3 20 16 VENTEO
SHADED SY-MBOL--ULLY WETTED L K~O.90

CLEAR SYMBOL ~ VENTED
(R= eo

&, OLCREES
of =6igsGe

Fig. D10. Experimental lift-to-drag ratios of the two-dimensional, cambered, parabolic hydrofoil

Figure D11 is a table showing an analytical comparison at noncavitating speeds of a
fully-wetted NACA 16-series hydrofoil and a base-vented parabolic hydrofoil. It is seen

ONR Hydrofoil Boat XCH-6 229

that the cavitation-free speed at the surface may be increased from 20 to over 50 percent by
use of the parabolic cross section. The lift-to-drag ratio, however, is decreased.

MCA /6-SERES —e — CAMBERED PARABOLA

PARAMETERS RESULTS ASSUMPTIONS -
Ze [eo Ya) | Scarmmae | BLN? HOW
Cawer) © Fe*2X10", "0039

3. EQUAL C, AMD Ye
4 CURCULARARC CAMBERLME
SK <<¢L0

18-SERIES.
PARABOLA

46 -SERVES.
PARABUA

-O10/ -67
i166 $0)
-WOf 25 )\.5/

15- SERIES 2099 53.55
PARABOLA OU? 43 6 &=0°
16- SERIES 7 R=08

PARABUA

4E-SERIES,
PARABOUA

16-SCRIES
PARABOLA

O08 27 '|-39
one 2
0086 58
0088 $7
25

ee 29 a
Fig. D11. Comparison of hydrofoils having equal lift coefficients and thickness-to-chord ratios

6 WOSWELPLACK

Figure D12 shows a vented hydrofoil which is designed for optimum efficiency. It is
constructed by first designing a strut having a thickness distribution and cutoff trailing edge
which provides zero cavity drag in accordance with Tulin’s linearized cavity theory for zero
cavitation number. This strut is then cambered in such a manner that the pressure on the

ASSUMPTIONS: N=0, 02=0° CIRCUIAR ARC CAMBER LINE, AR = 08,
a ee TURBULENT FLOW, Ge = 00089

CUT-OFF FOR

fill: iano

t=O \— CUP OUT TOREDUCE
FRICTIONAL DRAG

QOs3

Fig. D12. Base-vented hydrofoil having optimum efficiency

230 G. J. Wennagel

lower side is static depth pressure. Once this is accomplished, the lower surface can be
ventilated in order to reduce frictional drag. In this manner, all the lift is generated by the
upper surface and the lift-to-drag ratio is optimized.

Figure D13 is a theoretical plot of the lift-to-drag ratio versus cavitation number of
several hydrofoil shapes. The strength and lift of each of the hydrofoils was assumed to be
equal to that of an NACA 16-510 hydrofoil. As the cavitation number was reduced, the
chord length and thickness of each hydrofoil was varied to prevent cavitation. The assump-
tions were two-dimensional flow, a fully turbulent boundary layer, zero angle of attack,
circular-arc camber line, and cavity pressure equal to depth pressure.

90 iid Paka tio BASE VENTED HYOROFOIL HAVING
PO OPTIMOM EFFICIENCY
80 - a, a Ca
7O - ee = =
/
pe 7 -J BASE VENTED CUT-OFF 16- SERIES
/

Ve SO - -\ FULLY WETTED NACA Le- SERIES

\ BASE VENTED CAMBERED PARABOLA
% bette Gee ae

_ SUPERVENTILATING CIRCULAR ARC

Jo - Nea Le

Qo l | | | |

O O/ 0.2 03 OF AS O06 O.7

’ ! 1 oO t ! t

O Al 030 0.95 060 0.75 0.90 LOS
Opesen = 19 7

Fig. D13. Comparison of hydrofoils having equal strength and lift

The following reports published at the U.S. Naval Ordnance Test Station include more
detailed results of these studies:

1. Lang, T.G., “Base Vented Hydrofoils,” NavOrd Report 6606, Oct. 19, 1959

2. Fabula, A.G., “Theoretical Lift and Drag on Vented Hydrofoils for Zero Cavity Num-
ber and Steady Two-Dimensional Flow,” NavOrd Report 7005, Nov. 4, 1959

3. Lang, T.G., and Daybell, Dorothy A., Smith, K.E., “Water Tunnel Tests of Hydro-
foils With Forced Ventilation,” NavOrd Report 7008

4. Lang, T.G., and Daybell, Dorothy A., “Water Tunnel Tests of a Base-Vented Hydro-
foil Having a Cambered Parabolic Cross Section,” NavWeps Report 7584,
Oct 10, 1960

5. Fabula, A.G., “Application of Thin Airfoil Theory to Hydrofoils With Cut-Off,
Ventilated Trailing Edge,” NavWeps Report 7571, Sept. 13, 1960

ONR Hydrofoil Boat XCH-6 231

Marshall P. Tulin

We should all thank Mr. Lang of the Naval Ordnance Test Station for presenting us with
his recently obtained experimental information on base- and side-ventilated foils. Work of
this kind is most important to continue. I might perhaps mention at this time that the idea
of using a fully-wetted but base-ventilated hydrofoil for high-speed operation was, to the
best of my knowledge, originated at the Langley Laboratory of the NASA by Mr. Virgil E.
Johnson, Jr. These foils are very promising in certain respects, but are also probably very
sensitive, cavitation wise, to angle of attack. As was mentioned in the paper, we believe
that supercavitating foils intended for use on seagoing boats must be capable of operating
without cavitation under a reasonable range of design angles of attack; that range may be
larger than is possible with fully-wetted, base-ventilated, parabolic foils.

W. Graff (Versuchsanstalt fiir Binnenschiffbau, Duisburg, Germany)

The project of a full-cavitating hydrofoil is a proposal of mine made 24 years ago.
About 12 years ago J constructed and tested such a boat. The boat was driven by an air-
screw of 800 hp. The system of foil was somewhat similar to that shown by Mr. Wennagel
and had forward foils piercing the surface and a full-submerged foil aft. The trials of the
boat raised serious difficulties. Running on smooth water with full-cavitating foils we got
self-excited heaving and pitching oscillations, known as porpoising, such that it was
impossible to reach the full speed. Unfortunately, I could not finish these trials. The cal-
culations made on this matter seemed to show that the distance of the foils on high speed
was the most important factor in this problem. It was of interest to hear that Mr. Wennagel
also observed these phenomena and that he was able to eliminate them.

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DESIGN AND OPERATING PROBLEMS
OF COMMERCIAL HYDROFOILS

H. Von Schertel
Supramer A.G., Lucerne, Switzerland

Development of modem means of transportation undoubtedly trends to-
wards a progressive increase of speed in combination with an improved
riding comfort. Speed on the water is still far behind the speeds already
achieved with road or rail vehicles and aircraft.

Conventional types of watercraft cannot attain the speed of land
vehicles under economically acceptable conditions. The solution of the
problem by means of the hydrofoil boat is examined and the specific
qualities are discussed by which the requirements are fulfilled.

The paper then deals with the fundamental physical aspects of the foil
system as the most important design problem, and comparison is made be-
tween the surface-piercing foil and the fully submerged foil in relation to
drag-lift ratio and behavior in a seaway. The section ends with a review
of the practical adaptability of these two basic foil systems.

In the second part of the paper the commercial application of hydrofoil
boats is discussed. Technical data are given about the types which are
already being operated in regular passenger service. After a short review
about the introduction of the hydrofoil boat in public transportation,
follows an enumeration of the various existing hydrofoil passenger lines.

Finally the economic transportation problems are dealt with and data
are given about the profitableness of hydrofoil passenger services. The
field of application of commercially operated hydrofoil craft within the
framework of modern communications and their limitations as regards
speed is investigated.

INTRODUCTION

Development of modem means of transportation undoubtedly trends toward a progressive
increase of speed in combination with improved riding comfort. Considering the progress
achieved in passenger vehicles on road, on rail, and in the air from their first use in public
service until the present, one finds that their speeds have increased about tenfold. The
first railroad between Stockton and Darlington inaugurated in 1825 attained a speed of 10
miles per hour, while European express trains nowadays run at top speeds of 100 mph. The
first serviceable automobiles reached about 12 mph, which in the course of time was stepped
up in normal passenger cars to 120 mph. The passengers of the first commercial aircraft
were carried at speeds between 60 to 75 mph. Today we cross the Atlantic at nearly

600 mph. ©

Conditions in waterborne transportation are very different. Already in 1860 the “Great
Eastern,” the largest vessel of its time, attained a speed of 14.5 knots whereas the cruising
speed of the fastest passenger ship of today, the “United States,” surpasses this hardly
2-1/2 times. On inland waters the increase of speed has been by no means greater. The

233

234 H. von Schertel

most modern and fastest ships on European lakes surpass their 100 years older predecessors
only by twice their speed. The reason for this lagging behind of waterborne craft compared
with other means of transportation lies in the well-known large increase of drag with
augmented speed.

It is, therefore, not surprising that inventors, engineers, and scientists of several
nations have been endeavoring for many decades to find solutions which will result in com-
parable speeds on the water. Conviction that the shape of modern displacement bodies can
hardly be further improved, and that the solution can only be found in a reduction of the
immersed volume and of the wetted surface, led to the development of gliding or planing
vessels (V-bottom boats), in which dynamic lifting forces take the place of the static dis-
placement lift, and later on to the construction of hydrofoil boats in which the hull is finally
lifted completely out of the water.

The planing boats did not solve the problem of attaining the travelling times of land
vehicles under economically acceptable conditions and with good behavior in a seaway. The
reason for this must be attributed to the relatively high drag/weight ratio at the required
speed and to the extremely hard buffeting to which planing boats are exposed and which the
passengers cannot be expected to endure.

SPECIFIC CHARACTERISTICS OF HYDROFOIL BOATS IN COMPARISON WITH OTHER
FAST WATER CRAFT

Drag, Speed, and Economy

In Fig. 1 in which the specific power requirements of various water- and aircraft are
plotted against speed, displacement boats, planing vessels, and hydrofoil boats are compared

Displacement Commercial
boat Hydrotoil boat

Fig. 1. Power coefficient of various craft

Problems of Commercial Hydrofoils 235

with each other, all of them having—with regard to the Froude number—the same length of
160 feet, representing vessels of about 300-ton displacement.* It can be seen that hydrofoil
boats in the interesting speed range between 40 and 60 knots require only about 55 percent
of the propelling power of planing vessels, which shows that foil-supported craft can be
designed for comparatively high speeds and that they can operate at such speeds with rea-
sonable efficiency. This is to say that at high Froude numbers, hydrofoil systems are
known to function at drag/weight ratios below these of conventional motorboats. In conclu-
sion, hydrofoil boats must be expected to be comparatively economical within certain size
and speed ranges where other types of waterborne craft cannot operate effectively.

To illustrate this last statement we show in Fig. 2 the earning power of the hydrofoil
boat Type PT 20 in relation to speed. PT 20 represents a type of boat which today is
operating in regular passenger service in many parts of the world. This particular diagram
will be referred to in detail later on. It is shown that under the given service conditions an
interest of more than 50 percent on the invested capital can be achieved. The diagram,
which is based on actual experiences, reveals that the hydrofoil boat has succeeded in

|
|

= L

Effective power

|

1500 \yearly operat hours
60%\passeng. load

10 Ticket price acts/n.m. | ]

1
| |

|

|

|
t + }

30 325 35 375 40 425 454N

Fig. 2. Earning power of the PT 20 type
hydrofoil boat in relation to speed

*Some of the notations used in the text and figures are listed at the end of the paper.

236 H. von Schertel

attaining economically the present-day speeds of land vehicles, since the average speed of
fast trains in highly developed countries lies at 43 knots and at 32 for the whole world. But
in favor of the hydrofoil boat it should be mentioned that regular passenger services and
experimental trips along coastlines have demonstrated a further advantage, inasmuch as it
can reach certain destinations in an appreciably shorter time than land vehicles, which —
travelling with equal speed—must of necessity follow the more or less irregular coastline.
Such conditions for example prevail in Scandinavia.

Even better results can be obtained with a projected 300-ton Supramar hydrofoil boat.
In Fig. 3 is shown the obtainable profit for this type versus speed. Best results are
achieved at a speed of 50 knots yielding an interest on the invested capital of 57 percent.
When this project is realized, hydrofoil boats will show a clear superiority over land vehi-
cles with respect to travelling time whenever they operate between the same points on the
coast. This diagram also will be discussed in detail later on.

PS
S ~
S = 2000yearly operat hours
= = 60% passeng. load
= 6d Ticket price 7cts./nm.
110 4 Profi IN
rofrit
700 | 50! . N yi
IN
904 V,
go | 4C IA
INAS
a KE
60 | 30}. ZAIN
IS
50 | Ss
40-4 20! ese ole Bee day tL cae meres SAE INES
ie “a — iS
301 | | Effective power — XS
| | | Dit S
204 104 ce el N real
= wy
105 Sern ae i NN
| 1 Turbine 2Tunbines——3 Turb. IS
20 30 40 50 60 70 80 90 Vikn)

Fig. 3. Earning power of the PT 300 type hydrofoil boat (length = 160 feet)

Another quality distinguishes the hydrofoil boat from conventional craft. At higher
speeds, displacement and V-bottom boats cause the formation of a wave system of consider-
able amplitude in their wake which can interfere with the safe operation of other craft and
damage river banks. For this reason the speed of such boats is generally restricted in nar-
row waterways. Since hydrofoil boats develop only comparatively small wave-resistance
the disturbance of the water surface is so slight that the boats can maintain high speeds on
smaller lakes, rivers, and canals and in congested waters without endangering any other
craft or shore installations.

Considering now the airplane as the most serious competitor, we can derive from Fig. |
that the airplane can attain much higher speeds with lower specific propulsion power than
the hydrofoil boat. Therefore aircraft represents a very economical means of transportation.
However its commercial use requires an extensive and very expensive ground organization,

Problems of Commercial Hydrofoils 237

and it is well-known that aircraft operation over short distances does not pay very well.
Consequently hydrofoil craft may successfully compete with airplanes over short distances
because of their lower costs of operation and maintenance and of their point-to-point rather
than of airport-to-airport performance. Conditions may be such that a passenger choosing
the hydrofoil boat will arrive in less time at his destination than he would when choosing
the airplane. Although the airplane develops many times the speed of the hydrofoil boat the
journey by air involves the trip to and from the air terminals, which are often far away from
the original point of departure and destination. Over fairly short distances the helicopter
might be expected to be a serious competitor of hydrofoil craft. However, as shown in Fig. 1,
it requires rather high powered engines and its operation and maintenance costs are several
times those of the hydrofoil boat. Thus in many cases the prevailing conditions will favor
the use of hydrofoil craft instead of airbome means of transportation.

Sea-Riding Qualities and Passenger Comfort

Rapid connection between two points becomes meaningless unless the passenger can
rely on the timetable: in other words transportation must be regular and not be subject to
frequent interruptions on account of bad weather. Also the movements and accelerations of
a vessel caused by adverse weather conditions must be considered. If they assume propor-
tions which affect the passenger’s comfort, speed must be reduced.

It has already been stated that the problem of rapid transportation on the water cannot
be solved by planing boats on account of their buffeting in a seaway. At the cruising speeds
which are here under consideration accelerations of 6 g and more have been measured in
such vessels, accelerations which may be endured for a short time by the crew of a naval
craft but which exclude the use of such boats in commercial operation.

Principally it can be stated that hydrofoil boats are able to maintain a higher speed
level in a seaway than any other waterbome craft of similar size. Based on practical experi-
ences, Fig. 4 shows the approximate size of waves in which hydrofoil boats of the Schertel-
Sachsenberg system can still remain foilborne. The lower curve represents wave amplitudes
at which the boats can operate at full power; the upper curve shows wave sizes at which a
somewhat reduced cruising speed can be maintained while still keeping the vessel in foil-
borne condition. The diagram permits the estimation of the size of hull required for opera-
tion in waves of known sizes. With a new type of foil system presently being developed the
limits indicated in the diagram will be considerably widened.

Riding comfort in any watercraft is of course affected by the extent of its movements in
a seaway. Pitching, heaving, and rolling motions of hydrofoil boats are generally much
smaller than those of similar-sized conventional boats. It can also be stated that the ampli-
tude of these motions as well as the degree of submergence in the waves decrease with the
increase of speed and of the frequency with which waves are encountered. This phenomenon
in the performance of hydrofoil boats is due to a reduction of the influence of the waves on
the vessel’s inertia and of the orbital wave velocity on the foil’s angle of incidence with
the frequency of wave encounter. On the other hand, however, vertical accelerations
increase with speed. In accordance with the linear theory they increase also in proportion
to the slope of the waves.

Some indication of the extent of motion in a seaway may be gained from Fig. 5, which is
based on the latest tests undertaken with a 90-foot boat in waves estimated to have reached

a height of 5 to 6.5 feet and a length of 100 to 150 feet. The diagram depicts the average

646551 O—62—_17,

238 H. von Schertel

Boat not foilborne

| travel at full power

Height of Waves

0 20 71) 60 80 100 120
length of boat ft ——=—

Fig. 4. Limits of seaworthiness of Supramar boats

readings of the maxima taken during a period of 4 minutes. The roll angle is, of course, at
its maximum in a beam sea. Compared with displacement boats of similar size, however,
that angle is still small and the angle of pitch is very small indeed.

Figures 6 and 7 show the influence of speed or Froude number respectively and of
wavelength on the vertical accelerations of a surface-piercing front foil. These diagrams
are taken from a hitherto unpublished theory developed by Mr. de Witt of the scientific staff
of Supramar. For the accelerations shown in the figures the most unfavorable course of the
boat in relation to the direction of the waves was assumed in each case. The thick curve

ling one
sabe

<—ei——Wa ve——— > airection ——=— cea

Fig. 5. Mean values of pitching and rolling in rough water for the
PT 50 (wave height, 5 to 6-1/2 feet; wavelength, 100 to 150 feet)

Problems of Commercial Hydrofoils 239

Wave L/H = 15,7
Wave slope d= 11,30°
(= foil - distance

Fig. 6. Influence of the Froude number on vertical accelerations

Wg
08 lien
07 - |
|
06 7
05

04
° 7 |
a [

a

164-1 Pa
04+, —
ps
07 cai —+——
0
15 50 100 200 300 400 500600 ft

Fig. 7. Influence of the wavelength on vertical accelerations

240 H. von Schertel

in Fig. 6 corresponding to an extremely long wave of ten times the foil distance shows the
typical increase of acceleration with increasing Froude number. Accelerations in medium
waves, however, equaling the foil distance, are less dependent on the Froude number as
may be seen from the thin curve. The calculations represented in Fig. 7 are based on a foil
distance of 66 feet and a speed of 46 knots. The wave height for each respective wave-
length is shown by the dashed curve. Figure 7 shows that under the assumed conditions the
vertical accelerations increase only up to wavelengths approximately equal to the foil dis-
tance and decrease beyond that size, which is due to the decrease of the slope of the wave
with its increasing length.

Figure 8 shows the influence on vertical accelerations of the direction of travel with
regard to the wave direction for two different relations of wavelength to foil distance. Under

WAVE L/H= 15,7 |
WAVE SLOPE ¢ =/7,3
FROUDE-NUM BER F = 1,7

o° 30° 60° 90° 120° 150° 180°

WA meer Esra oi 7/ON— <a

Fig. 8. Influence of the wave direction on vertical accelerations

the given conditions (L/H = 15.7, F = 1.7) the maximum values are attained against the sea
and with a following sea, while at a course of about 80 degrees the boat remains at the same
place in the wave, which results in the disappearance of accelerations.

TECHNICAL CHARACTERISTICS OF THE TWO BASIC FOIL SYSTEMS

Having examined the suitability of hydrofoil craft for commercial transportation in gen-
eral, we shall now consider the main design problem in hydrofoil engineering, i.e., the foil
system itself, as it naturally has a decisive influence on the all-round performance of the
craft. Since it is not the purpose of this paper to present a survey of all the various designs

Problems of Commercial Hydrofoils 241

already in existence we shall limit the discussion to the two basic foil systems which
during recent years frequently formed the subject of discussions and led to differences of
opinions between the experts:

1. Surface-piercing hydrofoil systems—The surface-piercing systems consist either of
single V-shaped foils or of a combination of smaller foils arranged one on top of the other
like rungs of a ladder. A deviation from the equilibrium of the craft causes a change of the
wetted lift-producing area of the foil and automatically creates restoring forces. The sys-
tem therefore is automatically stable.

2. Fully-submerged hydrofoil systems—Fully-submerged foils have no inherent self-
stability. Depth of immergence must be maintained by means of mechanical, electrical, or
other controlling devices which—measuring either the distance between the hull and the
water surface or the foil submergence—give signals to contrivances which in turn affect the
lift by changing the angle of incidence of the foil or of flaps at its trailing edge. -

We shall now compare drag and compare behavior of the two basic foil systems, and
shall consider their possible future application.

Drag/Lift Ratio

We shall assume equal speed (45 knots), equal aspect ratio of A = b/c = 8.3, anda
suitable foil section for either type. An average submergence ratio of h = 1.2c is provided
for the dihedral surface-piercing foil (lowest point & = 2c) in accordance with the boats
which are at present in operation, and a lift coefficient of only Cy, = 0.22 with respect to
aeration.

The fully-submerged foil, however, requires a submergence ratio of not less than
h = 2c in order to avoid an excessive approach to wave-troughs but it is permissible to
apply a lift coefficient of 0.26, since this foil is not exposed to air-entrainment, and only
cavitation by influence of orbital motion has to be considered.

The total drag of a hydrofoil system may be expressed as
D=Cpsg

where S is the projected foil area and q is the dynamic pressure. The drag coefficient is
composed of four main components:

Oy Cp; + Cp, + Cp, + Cp

pa°*

The first component is Cp, = Cz,*/7A = minimum induced drag for infinite submergence.
When approaching the water surface the second component has to be considered:

Cp, edt CD, - = section drag.

The frictional component Cp, is strongly dependent on roughness of foil surface. For
smooth conditions and the applicable section thickness ratios of 0.05 to 0.10 skin friction
drag amounts to over 90 percent of section drag, so that the pressure drag Co, is very
small. For the determination of section drag mostly results of experiments are used.

242 H. von Schertel

Cp, of a V-shaped foil with an dihedral angle $ is increased to

Cp, = Cp, /cos t

due to the increased length of wetted surface. The third component

Cb, = wave drag

is negligibly small for high Froude number hydrofoil boats. The fourth component

CD, = parasitic drag

refers in this case to the foil struts, piercing the water surface. V-shaped foils with their
tips above the water surface during travel permit the use of relatively narrow struts since
the produced transverse forces can be taken up by structural elements which remain above
the water surface when the boat is travelling in foilborne condition. Because the struts are
only little immersed at cruising speed the parasitic drag of the foil is small. The fully-
submerged foil however requires very long struts which in view of the existing bending
moments must also be rather wide. Therefore a considerable parasitic drag is caused which
offsets the more favorable hydrodynamic qualities of the straight fully-submerged foil.

Calculation of the four drag components for the given example and on the assumptions
stated above yields a drag/lift ratio for the surface-piercing foil of 6.9 percent and for the
fully-submerged foil of 6.7 percent. Towing tank results obtained from two model foils of
the two systems confirm the theoretical analysis. They both produced drag/lift ratios of
approximately 7 percent. In conclusion, we can thus consider the two basic foil systems as
being equally favorable in regard to resistance at design cruising speed. In travelling
beyond cruising speed, however, the conditions change in favor of the surface-piercing foil
system. The area of the fully-submerged foil is determined by the capacity of takeoff at the
attainable Cy ax Value. The lift coefficient decreases then with the square of speed and
as a rule attains an unfavorably small value at top speed. On the other hand, when speed of
the surface-piercing foil exceeds cruising speed the wetted areas of the foil as well as that
of the appendages (struts emerge completely) are reduced to such an extent that their fric-
tional drag becomes less than that of the fully-submerged foil.

Experiments carried out recently with a 1-ton test boat originally fitted with surface-
piercing foils which were later replaced by a new system of automatically controlled sub-
merged foils also confirmed these inherent characteristics of the two types. Figure 9 shows
that cruising speed of the boat of about 50 km/h (27 knots) is reached with any of the two
foils at the same engine speed whereas the top speed attained with the surface-piercing foil
is 65 km/h (35 knots) as against 59 km/h (about 32 knots) with the fully-submerged foil.

Behavior in Sea Waves
The behavior of a hydrofoil vessel in sea waves is essentially the result of two func-
tions. The first one, called the “wing characteristic Z” indicates the stabilizing variation

(dL/dh); thus

Z = (dL/Lo) / (dh/bo)

Problems of Commercial Hydrofoils 243

where Lo = lift when travelling at normal waterline
by = normal wetted span of foil between surface-piercing points
h = depth of submergence.

V_

<

&

x

~

>

65

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50 ena
eee a
55
submerged
foil
50
45
y/
V
40
35 ol all

|
2000 2500 3000 3500
Motorspeed (r.p.m.)

Fig. 9. Comparison of the two foil systems on atest boat

The wing characteristic obtained with a surface-piercing foil in the Saunders-Roe
towing tank and measurements carried out on the 60-ton Supramar craft (dotted line) is
presented in Fig. 10. The hump in the curve of the 60-ton boat is produced by the stern-
heavy trim of the boat at low speed.

In designing a surface-piercing hydrofoil the value of the characteristic Z can be
influenced to a certain extent by selection of the dihedral angle, by suitable variation of
foil chord, foil camber, and angle of incidence in the vicinity of the piercing points. High
values of Z provide for efficacious stabilization but involve hard riding in a choppy sea.
Vice versa, reducing the slope of the curve shown in Fig. 10 leads to gentler behavior in a
seaway but low stability. In order to achieve better riding comfort it is obvious that the
wing characteristic should be adapted to wave heights and wavelengths; this, however,
presents difficulties with this type of foil. Nevertheless, trials which have been recently
undertaken in this respect with a smaller craft and an adjustable front foil of rather simple
design gave very encouraging results.

In order to make a vessel with fully-submerged foils stable, variations of lift as a
function of A must be provided by proper variation of the angle of incidence of the foil or

244 H. von Schertel

Lit ratio L Nt .
x=)

Rien aa wl, ;
b Wi

o=|span Wh

IL

S
n

“OS Ra? gus IQ”) Plas May = ais) get eas
Submergence (h- Yo,

Fig. 10. Wing characteristic of a surface-piercing V-foil

its flaps. The respective control device must be designed in such a way that it reacts to a
change of submergence, corresponding to the same value of Z as for surface-piercing foils.
The fully-submerged hydrofoil, however, has a significant advantage over the surface-
piercing type inasmuch as the value of Z can be changed at will whenever desirable and
inasmuch as a damping or filtering function can be introduced in such a manner that the foil
does not react to short waves and follows the contours of longer waves only insofar as to
keep the hull clear from the water surface. Furthermore, lift is being gradually modified in
conformity with the signal, whereas in the case of the surface-piercing foil, lift-variation
occurs more or less suddenly with an impact when travelling in waves. Consequently boats
with fully-submerged foils give better riding comfort. In rough sea conditions it is possible
to differentiate submergence against time and to modify the foil angle reaction accordingly.
The control device may also respond to accelerations; pitch and rolling angles can be
reduced by means of a gyroscope stabilizer. Such control is difficult to materialize for sur-
face-piercing foils because of their high autostability.

The second characteristic of a foil in a seaway is its reaction to orbital motion
indicated by

W = (dCL/da)/CL, =

: a-Ay

where dC,/dqa = lift-curve slope. A high value of VW means that a foil has a tendency to
react strongly to orbital velocites in waves. To make a foil less susceptible to these
motions, it would be desirable to make the lift-curve slope as small as possible and the
design-lift coefficient C7, as high as possible. Since the lift-curve slope is primarily a
function of aspect ratio, its reduction is bound to increase the drag, and since, on the other
hand, the lift coefficient C;, is limited by considerations of cavitation and ventilation, the
value of this parameter W cannot be reduced effectively.

Rigid hydrofoils are, therefore, always subjected to the influence of orbital wave
motions. This fact is without practical consequence when running against the sea, but the
influence of orbital motion is a very undesirable reality in a following sea. In such

245

Problems of Commercial Hydrofoils

conditions a maximum reduction of lift occurs when approaching a wave crest, and, on the
other hand, when approaching a wave trough, orbital motions have the tendency to increase
the foil’s lift. Consequently the phase of the foil travel is shifted under the effect of inertia
in relation to the wave contour (by about 120 degrees) and moves against it. Under unfavor-
able conditions the hull may be forced into the crest of the waves, thereby reducing the
speed of the craft considerably.

The amplitude of orbital motions reduces at a progressive rate, with increasing submer-
gence. A fully-submerged foil is therefore exposed to a smaller amplitude than the average
encountered by a comparable surface-piercing foil. Since, as already explained, the maxi-
mum lift coefficient will also be usually higher, a fully-submerged foil suffers less from
orbital motion. Apart from that, the control device can be adapted to compensate the orbital
influence by corresponding variation of lift.

Conclusion Regarding Practical Applications

After having discussed the characteristics of the two basic hydrofoil systems we can
make the following statements in conclusion of the first part of this paper:

Boats of the surface-piercing hydrofoil type have sufficiently low drag/lift ratios to
justify their use for high-speed commercial passenger service. Under equal conditions and
having equal cruising speed they can be expected to reach a higher top speed than vessels
provided with fully-submerged foils. Natural stability, simplicity of construction, opera-
tional reliability, ease of handling and maintenance, and, last but not least, a remarkable
invulnerability of the foil-system have contributed to the acceptance of hydrofoil boats as a
means of commercial passenger transportation. It can be taken for granted that the described
qualities will lead to a preference for this type for use on inland waters, in coastal regions,
and within protected sea areas.

Fully submerged foil-systems are in the same drag/lift bracket as the surface-piercing
type. They possess superior seariding qualities and offer higher riding comfort because of
their smaller and smoother heave and pitch response to sea waves. However, the complex-
ity of the height- and stability-control which is needed for this type of boat must be con-
sidered to constitute a serious drawback of this system. No doubt the electronically
controlled hydrofoil boats, which have been developed in the U.S., have shown excellent
riding qualities. In spite of this fact there remains for the traditional shipbuilder the
unusual conception that stability, naturally inherent to any properly built ship from histori-
cal times, should now be subjected to the faultless functioning of a number of complicated
gadgets. In case of a failure of such a control system, there may not remain sufficient time
to cut off the automatic and to “land” the boat by hand as is possible with an airplane.
Although an unexpected sudden tilting of a commercial vessel will not necessarily result in
any catastrophic situation, such an incident is liable to destroy the confidence of passen-
gers. In consequence the introduction of the fully-submerged hydrofoil system in public
service still presents problems and meets opposition of orthodox shipowners.

Therefore the requirement exists to maintain the stability of the fully-submerged foil
type directly by dynamic forces, similar to the surface-piercing foil, without inserting bulky
and vulnerable mechanical devices or electronic appliances. Based on ideas and experi-
ments carried out 15 years ago by the author, a new self-controlling system is in develop-
ment which —not being dependent on servo power sources and amplifiers etc. — can be
expected to offer sufficient reliability. In heavy seas the system can perform in the manner

246 H. von Schertel

described in the preceding subsection and there can also be introduced an additional simpli-
fied gyroscope control. A trial boat equipped with this new system is at present undergoing
tests conducted by the Supramar company, and has, so far, been successful.

Undoubtedly boats with fully-submerged foils will have a large field of application in
the future in less protected sea areas and as long-distance ferry boats on open sea. But
their range of operation will always be restricted by their faster competitor, the airplane, to
which preference will be given in all cases where its travelling time between two points,
including airport feeder service time, is considerably shorter.

PRESENT TYPES OF COMMERCIAL PASSENGER BOATS

In the second part of the paper we will deal with commercial application of hydrofoil
boats. We shall first regard the type of passenger boats which are at present in operation.
It is known that the first hydrofoil design which was used in scheduled commercial passen-
ger services was the Schertel-Sachsenberg type. The reason for the advanced technical
stage of this system lies in its structural simplicity and the reliability of the surface-
piercing foils used in this type. Another reason for the perfection of these boats is the fact _
that they are the result of organized development efforts which began with the author’s
experiments in the thirties and which were continued through the following years without
interruption. Design and construction have always been accompanied by extensive theo-
retical and experimental research as well as by trials with full-scaled craft. With the con-
struction of 16 different types and a total of nearly 60 hydrofoil boats to this date, of which
26 are passenger ferries, experience has accumulated to such an extent that Supramar’s
engineering staff is in the position to go ahead with the design of larger and faster economi-
cal passenger craft.

The predecessors of the present commercial boats were built during World War II. Out
of the six types which were designed for speeds up to 60 knots we shall mention the 80-ton
type VS 8 launched in 1943. This craft deserves attention because it was the largest hydro-
foil boat ever constructed and has not been surpassed even today. It was designed as a
high-speed long-range cargo carrier for operation between Sicily and Africa. The 105-foot-
long hull was constructed in light metal alloy. A maximum speed of 40 knots was obtained
using a twin arrangement of Mercedes-Benz Diesel engines with a combined output of
3600 hp. Although the corresponding ratio of 45 hp/ton is considered to be marginal for
satisfactory operation, the boat was nevertheless able to run at 37 knots against seawaves
up to 6 feet in height and 150 feet in length.

To date two prototypes find application in established passenger lines, the PT 20
(a 27-ton boat for 75 passengers) and the PT 50 (a 60-ton boat for 140 passengers). The
first PT 20 was built in 1955in the Rodriquez Shipyard at Messina. This craft (Fig. 11)
turned out to be the first of a series of very successful boats, 19 of which have been built
or are nearing completion.

The light-metal alloy materials used in building the PT 20 hulls are Al-Mg and
Al-Mg-Si. Watertight compartments are provided below the passenger decks and in other
parts of the hull. Several of these compartments are filled with foam-type plastic which
makes these boats practically unsinkable.

The engines installed in the PT 20 are supercharged 1350 Daimler-Benz Diesel engines
of the type MB 820 Db. The reversible gear, placed between the engine and drive shaft,
represents a special type developed by Zahnradfabrik Friedrichshafen. A 110-hp auxiliary

Problems of Commercial Hydrofoils 247

engine is located in the stern of the hull for emergency operation. Driving a small separate
propeller, this engine can be used to maneuver the boat in displacement condition.

Fig. 11. The PT 20 hydrofoil boat

The general arrangement of the PT 20 type is shown in Fig. 12. The boat is operated
and the machinery controlled entirely from the bridge which is located above the engine
room. Forty-five passengers can be accomodated in the forward cabin, while the rear cabin
is somewhat smaller and can take 30 passengers.

Together with the struts and a horizontal girder each foil forms a uniform framework
which, apart from giving static advantages, facilitates the exchange of the foil structure.
The foils themselves are made from medium steel as shells welded at the seams. The for-
ward foil can be tilted with narrow limits by means of a hydraulic ram acting on the girder
across the hull. It is therefore possible to adjust the angle of attack of that foil during
operation, thus counteracting the effect of larger variations in passenger loads and to ensure
optimum behavior in seawaves.

The PT 20 can be considered to be the smallest type of foilborne craft suitable for
passenger-carrying coastal service. In view of its novel and unconventional characteristics
the first boat of its class was originally licensed as “experimental” and restricted to opera-
tion in a 6-mile zone off the coast of Italy. After frequent inspections and supervision of
the service, the Registro Italiano Navale extended the license in 1958 to operation within a
20-mile coastal zone and in 1960 within a 50-mile limit. Today the PT 20 as well as the
PT 50 have been classed by the authorities of several countries.

Technical data which have an influence on the profitable operation of the type PT 20
are presented in Fig. 13. The shaft power of the engine was measured during operation by
means of a torque meter. The propeller was model-tested, thereby taking proper account of
the influence of shaft inclination. The drag/lift ratio was then derived from net power. This

248 H. von Schertel

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Fig. 12. The PT 20 hydrofoil boat

249

Problems of Commercial Hydrofoils

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ratio is D/ W = 8.6 percent at cruising speed (38 knots) and its minimum is 7.7 percent at a
speed of 34 knots. The corresponding maximum lift/drag is W/ D = 14.3.

A special type of PT 20 was designed for the transportation of engineers and workers
to offshore oil-drilling and pumping stations for oil-producing organizations in tropical
waters (Fig. 14). In this type, bridge and engine room have been arranged in the foreship in
order to obtain better vision since tropical waters have an occasional influx of driftwood.

Fig. 14. A special type of PT 20 for oil-workers in tropical waters

Tropical conditions were taken into consideration in the design and installment of the
engine plant, and the propeller has been placed in a specially protected position. Because
passengers board and leave the boat via the aftship this part has been fitted with particu-
larly sturdy metal guards. Because these boats are undergoing very rough handling by their
native crews the requirements with regard to their performance, sturdiness, and maintenance
are extremely high. Thus, for example, the boats must be able to remain foilborne even
when sudden full rudder is applied at full speed. Four boats of this special 29-ton type
have been constructed at the Gusto Shipyard in Schiedam (Holland) for the Shell Petroleum
Company in Maracaibo. »

Continued successful and profitable operation with the PT 20 stirred up sufficient
interest in a larger type of boat to be used in open waters farther away from the coast and
suitable for interisland services. The prototype of the PT 50 was constructed early in 1958,
again by the Rodriquez shipyard in Messina (Fig. 15).

In this 60-ton boat, passengers are placed below the main deck. Thus that deck con-
tributes considerably to the strength of a hull, allowing structural weight savings. The
machinery consists of two of the same type of Diesel engines tried out with great success
during a period of over 4 years in the PT 20 boats. Shafting is simplified as compared with
the smaller craft by eliminating the V-drive. Both rear and forward foil are rigidly attached
to the hull, but the lift of the forward foil can be modified by hydraulically operated flaps.

251

Problems of Commercial Hydrofoils

DEL CARIBE

riefunn

Fig. 15. The PT 50 hydrofoil boat

Figure 16 shows the general arrangement. The passenger room is divided into two com-
partments which are separated from each other by the engine room. The forward cabin con-
tains a bar, and a little saloon is provided on the upper deck, aft of which there are baggage
compartments. Two PT 50 vessels which have been delivered to Venezuela are fitted with
air-conditioning installations.

Figure 13 also shows the power curve Ne/2 of the PT 50 for comparison with the
PT 20. It reveals that the power is somewhat higher, especially at increased speeds, which
must be attributed to the twin shaft and twin rudder arrangement and to the not very favor-
able location of the propeller chosen in order to give better protection.

Technical Data for the two types are as follows:

PTE20 Bie SO
Length overall 68 feet 89 feet
Displacement fully loaded 28 tons 60 tons
Payload 6.8 tons 15.0 tons
Number of passengers 75 140
Max. power 1350 hp 2700 hp
Max. speed 42 knots 40 knots
Cruising speed 38 knots 36 knots
Power at cruising speed 920 hp 1000 hp
Range 300 naut mi 300 naut mi
Power/ton 48.2 hp/ton 45 hp/ton
Power at cruising/Payload 135 hp/ton 134 hp/ton
Power at cruising/Passengers 12.3 hp/pass. 14.3 hp/pass.

The Supramar company is working on the design of some larger and faster types of
hydrofoil boats intended for use under heavier, open-sea conditions and over longer dis-
tances. A 110 multiple-purpose type with a carrying capacity of at least 200 passengers
will be provided alternatively for Diesel and gas turbine propulsion. With two Mercedes-
Benz engines of 3000 hp each, a speed of 47 knots is expected, which will be

H. von Schertel

252

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Problems of Commercial Hydrofoils 253

increased to 53 knots if the Diesel engines are replaced by two 4200-hp Bristol-Proteus
turbines.

The next larger type was originally planned with a displacement of 120 tons but this
was later increased to 160 tons. With accomodations for 400 passengers over short dis-
tances or 320 baggage-carrying passengers and four motor cars, a top speed is expected of
50 to 60 knots, depending on the type of turbine.

The project of a 300-ton boat with supercavitating foils to carry about 550 passengers
will be discussed in a subsequent section with regard to its economical and attainable
speed and to the limits of its application. It is proposed to power this craft with two
Bristol Siddeley-Olympus turbines with a continuous output of 17,500 hp each. The cruising
speed is estimated to be 70 knots, thus reaching the technically possible limit, as we shall
see later on, but still representing from the commercial point of view a very profitable speed
under favorable conditions.

Depending on the area of application either the surface-piercing foil or the new type of
fully-submerged foil, now in development, will be employed in these future craft.

DEVELOPMENT OF COMMERCIAL PASSENGER SERVICE WITH HYDROFOIL BOATS

For the introduction of hydrofoil boats in public service, Supramar in 1952 constructed
a 9-ton boat (PT10) with a seating capacity of 28 passengers and powered by a 500-hp
Mercedes-Benz Diesel engine located in the stern. Soon after her first demonstration on
Lake Lucerne, Italian and Swiss navigation companies chartered this boat for passenger
services between the Swiss and Italian part of Lago Maggiore under the technical manage-
ment of Supramar. Thus the first scheduled hydrofoil boat service in the history of shipping
was inaugurated on May 16, 1953 (on the same lake, incidentally, on which Forlanini suc-
ceeded in getting his hydrofoil test-boat “foilbome” for the first time 55 years ago).

Interest in the new “flying boat” became widespread and in many cases took on enthu-
Siastic proportions. In a surprisingly short time travellers became accustomed to this
rather strange means of transportation. With 27,000 nautical miles covered during this test-
ing period, valuable experience was gained on the technical as well as on the commercial
side of the business. By the middle of 1956 the first 28-ton boat (PT20) had completed
several demonstration runs along the Italian coast and a round trip of 1600 nautical miles
from Italy to Greece. It had proved its seaworthiness on many occasions and in waves up to
13 feet high. On the initiative of Mr. Rodriquez a shipping company named Aliscafi was
established in Sicily and the first scheduled sea service inaugurated between Sicily and the
Italian mainland in August 1956. Cutting the port-to-port time from Messina to Reggio di
Calabria down to one-quarter of that of the conventional ferryboat and making 22 daily trips
the boat thus set an example for the operation of other hydrofoil services. The results of
this service after four years of operation are noteworthy: With a seating capacity of 75 pas-
sengers one boat alone has carried a record number of some 31,000 people in a single month.
The average daily number of passengers is today between 800 and 900. To date the boat
has carried a total number of approximately 1,000,000 passengers. The boats operating
around Sicily have covered a combined distance of approximately 465,000 nautical miles,
which is more than the round-trip distance to the moon.

After the establishment of the Messina-Reggio line other services have been organized
between the following localities:

646551 O—62——_18

254 H. von Schertel

1957 Messina-Taormina P20
1957 Messina-Liparian [slands-Palermo PT 20
1959 Venice-Trieste PT 20
1958 Lake Garda PT 20
1960 Naples-Capri-Ischia PT 50
1959 Maracaibo-Cabimas (Venezuela) 3 PT 20
1960 Stavanger-Bergen (Norway) PT 50
1960 Stockholm-Mariehamn (Sweden-Finland) PT 50

_ Buenos Aires-Montevideo 2 PT 50

The most prosperous services exist between Maracaibo and Cabimas and between
Naples and Capri. On the first mentioned line two PT 20 boats carry an average of nearly
1700 passengers daily and over 600,000 per year over a distance of approximately 20 miles.
The shipping company was able to amortize the two vessels within one year. On the
Naples-Capri line, operated with a PT 50, passenger fees had to be raised on account of an
excess of passengers. A second PT 50 has been ordered and two PT 20s have been sent
to Naples in the meantime to assist in operations during the season. A second PT 50 was
also requested for the Stavanger-Bergen line only two months after its inauguration.

On many occasions Supramar boats had the opportunity to demonstrate their seariding
qualities under very severe weather conditions. Soon after establishment of the PT 50 line
at Naples a heavy storm caused the 500-ton ferryboats to discontinue their trips between
Naples and Capri. The PT 50, however, maintained its service in an overloaded condition
carrying 170 passengers instead of the regular number of 140. On another occasion a PT 20,
passing through the Straits of Otranto was forced down by waves of an estimated height of
13 feet but short length. It was still possible to maintain an average speed of 15 knots. In
the Caribbean Sea another PT 20 got caught in the fringe of a hurricane and was able to
continue her journey in half-foilborne condition in long waves averaging 16 feet in height.

As regards maintenance, inspections are undertaken at regular intervals of about 2 to 3
months, including cleaning of hull, bottom, and foils. In a tropical climate, where intensi-
fied growth of barnacles affects the drag of the foils, more frequent inspection and cleaning
is indicated. If the foils are not retractable, this work is usually carried out by aqualung
divers. Disregarding major machinery overhaul, which is normally due after 5000 operating
hours, the maintenance of the foils requires about 25 percent of the maintenance work of the
entire boat.

Strength and reliability of the foil system was several times demonstrated when boats
ran aground and were still able to continue operation. A boat colliding with a pier caused
considerable damage to the latter while suffering only minor deformations of the plating
around the foil suspension point that did not interrupt service.

PROBLEMS OF ECONOMY IN COMMERCIAL HYDROFOIL OPERATION

Six years of experience in public passenger service with hydrofoil boats proved that
the commercial application of this type of craft is very profitable in areas with an adequate
passenger frequency. Similar to air transportation the comparatively high speed of hydrofoil
boats in relation to other waterborne craft results in a high earning power. Since the
present foilborne craft usually operate at speeds up to 3 times that of other boats, their
potential carrying capacity is up to 3 times greater. In other words, a hydrofoil boat can be
considered to be equal in capacity to a ship up to 3 times its size.

Problems of Commercial Hydrofoils 255

Figure 17 presents an analysis of actual commercial experience for the type PT 20
(assuming 2 boats in service) based on prevailing European conditions and a price of 6
cts/kg for Diesel oil and 53 cts/kg for lubricating oil. The graph indicates the yearly net
return to be expected over ticket price (cents per nautical mile) multiplied by the load fac-
tor (number of passengers per available seats) . As a function of three parameters the graph
thus enables anticipation of whether commercial application of hydrofoil boats would be
profitable under given local conditions. Assuming a fare of 8 cents per nautical mile, a
load factor of 0.5, and 1500 operating hours per year the graph shows a yearly net profit of
slightly more than 30 percent of the invested capital.

& 70 D

Pied th

NENe
a

Soe |

15 25 } ho GS del es ; 6 65 (ct)
ticket price [ct /nm] x passeng.load factor

Fig. 17. Commercial experience for the PT 20

The diagram is based on a cruising speed of 36 knots. In order to determine the influ-
ence of speed on profits and also the economical speed of the vessel at which the most
favorable relation between costs and revenues is obtained the curve depicted in Fig. 17 was
computed under the following suppositions:

Load factor: 60 percent
Fare: 8 cts/naut mi
specific weight: 4.3 kg/hp
Diesel engine < specific fuel consumption: 166 g/hp/hr
price of engine: $26/hp
specific weight: 1.3 kg/hp
Turbine specific fuel consumption: 320 g/hp/hr
price of engine: $44/hp.

The engine weights include gears and all auxiliaries as well as ducts, pipings, etc. The
highest profit is obtained with either Diesel or turbine propulsion at a cruising speed of 40
knots. Although the engine output at that speed is nearly 60 percent higher than at 30knots
the remarkable profit is obtained by 33 percent higher revenues in accordance with the equal
increase in passenger-miles. On account of the higher price of the engine and its higher

256 H. von Schertel

fuel consumption the interest on the invested capital earned with a turbine-driven boat is
only 85 percent of that of a Diesel-powered vessel. However, the fact that the modest
space requirements for turbines permit the accomodation of additional passengers has not
been taken into consideration. The conditions will be changed if prices and consumption
figures for turbines can be further reduced in the future.

We shall now consider the field of application for the commercial hydrofoil boat within
the framework of modern transportation and with regard to any competing craft. We will also
investigate its limitations as far as technical practicability, speed, and economy is
concerned.

Table 1 presents the principal data as they affect the economy of the hydrofoil boat
and its three competitors: the conventional displacement boat, the airplane, and the heli-
copter. The data are based on an assumed distance of 100 nautical miles. For an estimate

Table l

Comparison of Prices and Operating Costs

Block- Operat. costs
speed V | No. of

passeng. mi

(¢)

Passenger Boat | 26.5 tons : 700

Hydrofoil PT 20 2,800
Airplane Convair 18,000

Helicopter Sikorsky ~40,000
S-55

*V,= WV anes. boat «

of economy the two last columns are the most interesting. They indicate price per seat
divided by block-speed (port-to-port speed) and the direct operating costs per passenger
mile (expenditures less cost of ground organization). It appears that the foilborne craft is
very economical compared with the other means of transportation and under the assumed
conditions. In spite of the slightly lower “initial capital expenditure per passenger mile”
and operation costs of the displacement boat, the attainable revenues within a certain
operation period are higher for the hydrofoil boat because it travels three times as fast. It
must also be taken into consideration that the public is generally willing to pay higher
fares for faster transportation. The comparison with the commercial airplane reveals that
its initial costs are higher and its total operation costs very much higher if the expenditures
for the ground organization are taken into regard. With total operating expenses estimated
to be at least 7 times those of the hydrofoil craft, the helicopter cannot be considered to
represent an economical instrument of transportation at present.

In order to find the speed ranges within which the hydrofoil boat and its competitors
can be successfully employed we have already shown in Fig. | the power coefficient for
the craft in question over speed. First of all the eaming power for a hydrofoil boat with

Problems of Commercial Hydrofoils 257

550 passengers was calculated as a function of speed in order to determine the limit from
a commercial point of view. In doing this the following conditions were assumed:

Load factor: 60 percent (330 passengers)
Ticket price per nautical mile (in correspondence
with the average economy class fares for

airplanes): 7 cts
Operating hours per year: 2000
Range: 500 nautical miles
Total weight of turbine including gear and

accessories: 1.55 pounds/hp
Specific fuel consumption: 0.6 pounds/hp
Purchasing price of turbine including gear and

accessories: $42/hp.

For the fixed costs the same assumptions as in Figs. 2 and 17 were made. The
specific power requirements shown are based on measurements in various boats of the
Schertel-Sachsenberg system. It turned out that the power coefficient Ne/A uf given on a
logarithmic scale over the Froude number V/V2 (t= foil distance) lies on a nearly straight
line for all hydrofoil boats constructed, so that reliable figures are available. Values of a
much lower order given in other theoretical treatises, which neglect the fact that after a
relatively short operating time the foils are no longer hydraulically smooth, are not realistic.

In Fig. 3 the curves of profitableness and of the boat’s power in consideration of the
increasing displacement (turbine and fuel weight) are plotted against speed. The number of
turbines to be installed was based on the Bristol Siddeley Olympus turbine with 22,700 hp
maximum and 17,500 hp continuous output. Under the assumptions made, the best profit is
obtained at about 50 knots. With increasing speed earning power reduces and one may con-
sider 85 knots as the utmost limit at which the boat can still render a profit under otherwise
favorable conditions. The limit of technical practicability, however, can probably be
expected at 75 knots at the present stage of the art when observing the requirements of the
classification committees and the regulations of the London Ship Safety Convention, espe-
cially with reference to electrical installations, auxiliaries, and safety installations. Apart
from the fact that a larger engine plant can hardly be properly installed, the amount of fuel
necessary for greater speeds increases at such a rate that the number of passengers neces-
sary for obtaining a profit can no longer be maintained.

Figure 1 shows that 38 knots was chosen as the lower limit of application of the com-
mercial hydrofoil boat of the given size (160 feet) and 80 knots as the upper limit. Exceed-
ing this limit under economically acceptable conditions, for the case in question, seems
possible only if hydrofoil boats with better lift/drag ratios or lighter engines with less fuel
consumption are developed in future. In the speed range beyond 80 knots the superiority of
the airplane — as far as power requirement is concerned —increases continuously with the
speed.

The diagram also contains the dotted curve of the “flown” values of ground effect
machines (hovercraft) which operate at an altitude of about 5 percent of the craft’s diameter.
Because only scant results are available, this curve does not pretend to be correct. Data
of projects which have not yet been completed or power figures at altitudes of less than
0.05 diameter are not represented. A comparison between the hydrofoil boat and the hover-
craft is justified only if the flight altitude of the latter would be great enough that both
types can manage the same height of waves. However, it can be assumed that the respec-
tive curve will be considerably improved in the course of development.

258 : H. Von Schertel

The length of a route between two points which can still be operated advantageously in
competition with an airline is also dependent on the economic maximum speed of the hydro-
foil boat. Preference obviously will be given to the airplane if it requires much less time
for the trip and if both the airplane and the hydrofoil company charge the same prices.
Assuming a maximum speed for the hydrofoil boat of 75 knots and for the medium-distance
airplane a block speed of 360 knots, it can be safely predicted that for all distances beyond
300 nautical miles, for which the hydrofoil boat needs more than four hours, travellers will
prefer the airplane with its 50 minutes of flying time, even counting the time lost getting to
and from the airports.

It can be expected that under the existing circumstances the longest economically
favorable distance for the hydrofoil boat lies between 300 and 400 nautical miles. Although
quite a lot of publicity has been given to the idea of a future Atlantic hydrofoil service it
appears very improbable, if not impossible, that hydrofoil boats will ever be suitable for —
this task which can be much better accomplished and with better economic results by
modern airplanes.

NOTATIONS USED

= vertical acceleration
= span of hydrofoil
= chord of hydrofoil
=- submergence below water surface
= foil distance
= weight
= speed in ft/sec or in knots
= 0.5lpv” = dynamic pressure
b?/S =. aspect ratio
= fuel consumption

D/qS =: drag coefficient
L/S = lift coefficient
‘drag of hydrofoil

= lift of hydrofoil
w= wavelength

= Froude number

=- wave height

= motor power

=- projected foil area

= orbital sensitiveness

= wing characteristic

= displacement

= angle of incidence

= weight density of water

= dihedral angle

= wave slope

= y/q = mass density

ll

ipi~)
Il

PHY APNSMNVSRBHOHSVUAAWAOQVCSCPEwe™ ee

Problems of Commercial Hydrofoils 259

DISCUSSION

Christofer Hook (Atlantic Hydrofin Corportation, Villeneuve le Roi (S&O), France.

Baron von Schertel gave the example of phase shift in the case of following waves for
the surface-piercing system but he neglected to point out that the main advantage of the
submerged foil is the opportunity it provides for correcting this out-of-phase effect by
application of an advance or prediction signal to the incidence changes. Ideally, this would
lead us to a telescopic sensing device whose advance position with respect to the con-
trolled foil would be adjusted to suit different ratios of hull to wavelength. In practice, a
simplified compromise in the form of a fixed advance length is perfectly satisfactory since
only relatively short waves are significant.

It is also feasible to incorporate into such a sensing device a wave-height measuring
system, coupled to powerful damping, so that all waves up to a given maximum (correspond-
ing roughly to the height clearance) may be filtered out, resulting in practically level flight
over waves up to this size. For larger waves the aforementioned size may be subtracted as
a constant so that, for example, a wave 6 feet high may be dealt with as if it were only 3
feet high.

It is too often assumed that a given hydrofoil system must be restricted to a given
severity of sea and that beyond this some disaster must follow. The impression given is
not improved by the fact that the details of the type of disaster are left to our imagination!
This is, however, somewhat unfair since a similar restriction should also be placed on
normal boats, which should then also be sold with a “sea severity tag” attached — which is
not done. A good hydrofoil can perform in seas well beyond the capacity of a crash boat to
follow, as has been shown in U.S. Navy reports, and when failure to follow is eventually
arrived at it is not followed by any kind of disaster but merely by a sitdown, and the ship
can always continue in a half-foilborne condition; in fact, pitching is so severely damped
by foil action that a light hull fitted with foils will survive where the same hull with foils
removed will break up. The matter of wave filtering becomes so important at sea that fine
comparisons of relative lift/drag ratios of the two methods becomes pointless.

Finally, I think that the purely mechanical method cannot be lightly dismissed as
bulky and vulnerable since its reliability is unquestionable. Since an increase of C; is
obtained by incidence instead of by carrying reserve foil area we can offset the gain in
weight consequent in the elimination of this weight against the added hardware required to
manipulate the prediction method and we fill find that there is no added weight. On the
basis of cost, if I may be excused, I may perhaps point out that for a sport boat or
4-passenger size the cost of the hardware does not exceed $100. I think that all hydrofoil
men here today should realize that we are very much riding on the wave created 25 years
ago by the Schertel-Sachsenberg group and that, had it not been for the commercial suc-
cesses of this group, hydrofoils would never have got going again after the fizzle out that
followed the Bell experiments in about 1920.

H. Von Schertel

Mr. Hook is right when he points out that the description of phase shift for a hydrofoil
boat travelling in a following sea, given in the paper, refers to the surface-piercing foil

260 H. Von Schertel

system, and that the behavior of a boat fitted with fully-submerged foils controlled by a
predicting sensor would be different. I tried to give the advantages of fully-submerged
foils as far as | could in a paper of the present length.

L. S. Snell (The De Havilland Engine Company Limited)

Baron Von Schertel makes certain assumptions in Fig. 17 when comparing the perform-
ance of the Diesel engine and gas turbine. He goes on to say that “conditions will be
changed if prices and consumption figures for turbines can be further reduced in the future.”

It must be remembered that though the specific fuel consumption (S.F.C.) of the Diesel
engine can be taken as generally constant at about 0.4 lb/b.h.p.-hr, a wide variety of gas
turbines with equally widely differing performances can be produced. The S.F.C. quoted in
the paper has already been improved upon, though unfortunately at the expense of cost and
bulk. Figure Dl shows schematically three engine types recently evaluated for a small
high-performance craft. That on the right utilizes a simple open cycle and its design owed
much to aero engine practice. The efficiency of this simple cycle can be improved by rais-
ing the turbine inlet temperature and increasing the compression ratio, if this can be
associated with an increase in compressor efficiency. Modification to this cycle is desir-
able if good economy is the main aim. The center illustration shows the addition of a heat
exchanger which recovers some of the exhaust heat previously lost, while the remaining
diagram depicts a quite sophisticated arrangement using three stages of compression with
inter-coolers and exhaust heat regeneration.

Figure D2 shows the performance expected from each of these arrangements compared
with that obtained from the conventional Diesel engine. The component efficiencies and
maximum cycle temperatures used in these calculations are quite modest and are those
liable to be encountered in a marine engine rather than an aero engine. Examination of
these curves indicates a most significant point, that is whereas the Diesel engine has a
consistently good specific fuel consumption over a very wide load range, the simpler gas
turbines markedly worsen as the throttle is closed.

The author said that after using practically full power to get on the hydrofoils, cruise
then takes place at about 55 percent maximum power. Cruise economy at this point is,
therefore, the main consideration in the choice of prime mover. ‘Unfortunately the choice of
a simple open cycle working on high compression ratios and high turbine inlet temperatures
can be most misleading as, although the S.F'.C. is considerably improved at near 100 per-
cent full power, at 55 percent the S.F.C. is not materially better. This state of affairs
might well be more of a problem for the present generation of hydrofoil craft as it would
appear that as cruising speeds increase so the cruise power will proportionately increase.

A near ideal solution would be to evolve a variable-mass-flow engine fitted with an
exhaust heat regenerator. With decreasing throttle, mass flow would similarly decrease, and
if a near-constant pressure ratio and turbine inlet temperature was employed, the thermal
ratio of the heat exchanger would be markedly improved and, coupled with reduced ducting
losses, would give an increasingly better S.F'.C. with closure of the throttle.

During the course of the analysis previously referred to, account was taken of the
additional space available not only for the gas turbine compared with the Diesel engine but
for the varying amounts of space which could be saved with the different gas turbine con-
figurations and their associated fuel tanks. Surprisingly enough, the higher fuel consumption

261

Problems of Commercial Hydrofoils

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Fig. D2. Surface performance of air-breathing engines

of the simpler cycles was just about offset by the engine room volume and weight saved, so

that for ranges of up to 1000 nautical miles there was little to choose between the installa-

tions; above this figure the more sophisticated units showed to advantage. However, in this
design evaluation the installational size of the Diesel engine concerned cut the payload so

much that its use could not be considered.

This raises a further point, very much to the forefront of the engine designer's mind,
that above a certain size, Diesel engine specifics — both weightwise and volumewise —
greatly increase as high duty materials and associated workshop practices can no longer be
employed. One British engine, for instance, uses 36 pistons and 3 crankshafts in order to
get the largest engine size which can be made from the better class materials and precision
methods of manufacture. It would, therefore, seem that apart from small, highly specialized
or experimental craft, the gas turbine should really come into its own at shaft horsepowers
of, say, 4000 and above for the type of craft the author discussed.

Speaking very personally, I feel that no real progress will come with the adoption of
the gas turbine for high-speed marine craft until units are designed expressly for this pur-
pose, not merely rehashes of aero engines. The knowledge and facilities exist for design-
ing and developing these units; the engineer should be given the opportunity of doing this
at the earliest possible date.

R. N. Newton (Admiralty Experiment Works)

I would like to refer to two specific points arising from Baron Von Schertel’s excellent
paper which bear directly upon the general comments | made concerning hydrofoil craft and
GEMs in the discussion of Mr. Oakley’s paper on high-performance ships.

Problems of Commercial Hydrofoils 263

Near the end of the section titled “Present Types of Commercial Passenger Boats”
Baron Von Schertel listed particulars of the PT 20 and PT 50 Supramar craft. While the
figures quoted leave no doubt as to the economical performance of such craft in calm
weather, i.e., their relatively low horsepower for a given speed, it should be noted that the
maximum speeds quoted are not really high in comparison with modern planing forms and
the range, in particular, is very limited.

Now there can be little doubt, I think, that higher speeds will be achieved with the
larger craft which the Supramar Company are contemplating, but one is left to wonder
whether they will have the range for open sea voyages as claimed, or the ability to operate
in heavy sea conditions.

After listing the existing services the paper states that the craft have demonstrated
their sea-riding qualities and then goes on to quote three occasions upon which a craft was
forced down by the waves. In other words it is at least admitted that this type of craft is
limited by the sea condition. Unfortunately the paper does not say at what speed the craft
were able to proceed in waves, nor is there any proof of the heights of waves quoted —as
much as 16 feet in one case. One of the most difficult quantities to estimate, without the
use of apparatus only now coming into use, is the height and length of waves. I venture to
suggest that the heights quoted in the paper are heavily exaggerated.

To emphasize the point I would like to quote an extract from a recent report of David
P. Brown, President of the Board of Managers of the American Bureau of Shipping:

“After months of study of plans and design criteria as well as inspection during con-
struction, the first of several hydrofoil passenger craft is about to be submitted to the
committee for classification. Requests to class this type of craft were considered favorably
by the technical committee last autumn, subject to appropriate limitations as to the area of
operation and indication in the classification symbols that the craft is of special design
and not comparable to a normal vessel in respect of scantlings and certain machinery
details. Hydrofoil launches are now being built for use in Lake Maracaibo and in the
Caribbean.”

In conclusion may [| repeat the plea | made in the discussion on Mr. Oakley’s paper for
more research into the seaworthiness of this type of craft.

H. Von Schertel

Mr. Newton expressed the view that hydrofoil boats can get into sea conditions in
which they have to come down onto the sea. The behavior of hydrofoil boats in a seaway
depends, of course, on the length of the waves, and on the direction of travel of the craft
relative to the waves. Our experience over more than 10 years shows that the hydrofoil
craft can run in almost any sea condition, occasionally, of course, with reduced speed (as
is the case with any type of boat) and in a half-foilborne condition. - Also I would like to
add that in this condition hydrofoil boats have proved to have superior sea-riding qualities
over planing boats, owing to the damping action of the foils. This fact has also been
stressed by Mr. Hook. Conceming speed in waves, comparison runs which have been made
between our commercial boats and several Navy craft showed that the higher the waves
encountered the more the difference in speed was in favor of the hydrofoil boat.

264 H. Von Schertel

Mr. Newton emphasized that the range of 300 miles given in the paper for our hydrofoil
boats does not bear any comparison with fast planing craft. May I draw the attention Mr.
Newton to the fact that the paper only deals with rather short-distance commercial boats
with a high passenger load. In the case of a planing boat and a hydrofoil boat of equal
speed and displacement, the range of the hydrofoil craft will be considerably higher on
account of the much lower motor power and the associated low fuel consumption.

A. Hadjidakis (Aquavion Holland N.V.)

There are many things wherein I fully agree with Baron Von Schertel, for instance: that
the efficient speed for a hydrofoil craft lies between 40 and 50 knots; that the required
engine power per ton displacement is about 40 b.h.p.; and that seaworthiness increases with
size (it would be terrible if we disagreed on this point). Our thinking was very much in the
same direction. Compare for instance Baron Von Schertel’s Fig. 7 with Fig. 7 in my paper.

However, there are some points where we differ. In the first place, I wonder why the
foil incidence of your hydrofoils is such that a lift coefficient of only 0.22 is obtained. Is
this so chosen to prevent cavitation at full speed or perhaps to get better takeoff character-
istics, or is there some other reason? We use lift coefficients of up to 0.45 at top speed.
In the second place, it surprised me that a gas turbine engine gives a lower profit if used
in the PT 20 type, as is indicated in your Fig. 2. In my opinion it should be contrary. Let
the specific weights be 4.3 kg/hp for the Diesel engine and 1.3 kg/hp for the gas turbine,
which gives a difference of 3 kg/hp, or for 1375 hp a weight decrease of say 4 tons.
Because of double fuel consumption, fuel weight will increase say 1.5 tons. But there is
still 2.5 tons left, which means at least some 25 extra passengers; that is to say an
increase of income of one third, as the PT 20 carries 75 passengers normally. Furthermore,
only a small engine room would be required, so there would be more space left for other
things.

H. Von Schertel

The lift coefficient in the example given is only 0.22 because a speed of 45 knots was
assumed, so that the lift coefficient must be kept rather low to minimize or avoid ventila-
tion and cavitation. A certain margin must be left between the lift coefficient at which
ventilation occurs and the C;, value actually used in order to allow for the orbital motion
in the waves.

As to the question of operational economics with turbine propulsion for the type PT 20,
it has to be taken into consideration that the number of passengers is limited by the avail-
able space in the boat. The turbine-driven boat will be lighter and require a smaller engine
power, which however does not compensate for the high initial cost and consumption of
turbines. In my paper I remarked that economy would improve with the turbine boat if pas-
senger space could be gained by reducing the engine room size.

E. V. Telfer (Institute for Shipbuilding, Trondheim)

As a simple naval architect listening to your deliberations I have been a little puzzled
as to how to present your thoughts nondimensionally so far as these exciting craft are

Problems of Commercial Hydrofoils 265

concerned. At the moment it appears to me that you prefer to think entirely dimensionally
round each particular problem and in so doing do not necessarily see the picture as a whole.

For example, in Mr. Wennagel’s paper there is one very interesting diagram (here Fig. -

_D83B) given in dimensional form which can with advantage be re-presented in nondimensional
form. Normally in ship work we use a resistance coefficient based on a speed-squared
relation associated with a relative speed. One such combination is the Froude © and
presentation; and so long as resistance does vary approximately as speed squared, the cor-
responding value is substantially constant. When the performance of a fast vessel is so
presented, however, we see (Fig. D3A) that the speed-squared relation required for non-
dimensionality is no indication at all of the power of the speed with which the resistance is
actually varying. Actually, for speeds beyond the maximum , the resistance is more
nearly varying directly as the speed. In this case it is better to use a nondimensional
presentation which respects this fact, and this is easily obtained by plotting values of

©

72)
Lia
N
“Wve

e V/aé =(K)

Fig. D3. Comparisons of dimensional and
nondimensional presentations

266 H. Von Schertel

© . («) against «). This would convert Mr. Wennagel’s diagram from B to C. From the
latter we see most clearly that the conditions imply two regimes of broadly constant but
different ©) . (2) values. ‘What is the significance of these two regimes? Incidentally the
© . «® product reduces to (P/AV)/ «). ‘The term in parentheses, i.e., power per ton-
knot, is the transport coefficient and is a direct commercial measure of the design efficiency.

Our discussions include many references to Froude number, i.e., Froude speed-length
number. I doubt, however, so far as hydrofoil craft are concerned, whether we have a truly
representative length to use in the Froude number. [| suggest therefore, since for these craft
displacement or weight is constant, that the Froude displacement-speed factor, i.e.,

K = 0.5834 V/A”®, should always be used for their performance nondimensional presentation.

H. Von Schertel

The Froude number is based on the distance between the fore and aft hulls.

Douglas Hill (Grumman Aircraft Engineering Corporation)

We continue to be indebted to Baron Von Schertel for his reporting of the operating
experience of the craft of his design, as well as for his pioneering work in the development
of hydrofoils and their introduction into commercial service. Those of us interested in the
development of hydrofoils look to the experience with these craft as a milestone on the road
leading to the large-scale introduction of hydrofoils for commercial and naval use.

There can be little quarrel with Baron Von Schertel’s evaluation of the potential of
commercial hydrofoils on the basis of the evidence he presents and the scope of his presen-
tation. In the publicity which has attended the growing awareness of the potentialities of
hydrofoil craft, it is perhaps true that the feasibility of future vessels —a transatlantic pas-
senger liner, for example —has been overemphasized by the standards of today’s commercial
realities. Assuming that hydrofoil craft are better suited to coach than to stateroom accom-
modations, we can draw on the experience of other modes of coach transport to substantiate
Baron Von Schertel’s observations on the practical operating range of passenger-carrying
hydrofoils. A comparison of the length of journeys of travelers by air, railroad coach, and
intercity bus in the United States reveals that, while the average distance travelled varies
substantially between the modes of transport, the average duration of the journey is remark-
ably consistent:

Average Length of Average Average

BOT ETER ES Journey (statute miles) Speed (mph) Time (hr)
Airplane 577 220 2.6
Railroad Coach 110 40 Pa |
Intercity Bus 79 30 (est.) 2.6

It would seem reasonable to expect that the widespread use of hydrofoils for coach
transportation would likewise result in an average trip duration of about 2.6 hours.

Most journeys would be shorter than the average, some perhaps much longer. The peak
trip length can therefore be expected to be shorter than the length of the average trip. On
domestic United States airlines, for example, the most common trip length is of the order of

Problems of Commercial Hydrofoils 267

200 miles, corresponding to a flight of about 55 minutes. It is not surprising to find,
therefore, that the hydrofoil routes described as the most prosperous are also the shortest,
requiring runs of less than an hour. A journey of 300 to 400 miles, requiring 6 to 8 hours

of travel at 50 knots, would seem to be approaching the limit of coach-passenger endurance,
and we must agree with Baron Von Schertel that longer trips do not promise commercial
success.

If indeed the time required for a trip is the factor limiting the number of passengers
which may be attracted, it follows that faster boats will draw more passengers for a given
route. In the example of Fig. 3, it is indicated that the most profitable speed of the boat
under consideration is 50 knots, assuming a constant load factor. If the load factor itself
increases with velocity, however, the most profitable speed will be greater than 50 knots.

This only serves to point out that the best speed of a hydrofoil craft from a commercial
standpoint is influenced by market considerations as well as by hydrodynamic performance
and operating costs. To arrive at general conclusions as to the characteristics of a com-
mercial craft which will be most profitable, a more comprehensive analysis than the one
presented is required. We shall have to examine the entire range of feasible speeds, sizes,
route lengths, power plants, types of foil, and possible payloads. We shall have to examine
not only how variations in these basic parameters affect hydrodynamic performance but how
they may affect the operating costs and the value of the craft as transportation equipment. -
And all of these must be examined in the context of the environment — physical and
economic —in which the hydrofoil is expected to operate.

Such an analysis is not made in this paper, and we must read it with this caution. It
may appear to be more than it claims to be; it may seem to present conclusions where it
provides illustrations. -We have no reason to doubt, for example, that under the conditions
assumed the most profitable speed for the PT 300 is 50 knots, or that a Diesel-powered
PT 20 is more profitable than one which is powered by a gas turbine. We should not neces-
sarily infer, however, that 50 knots is thus the optimum speed for hydrofoil craft nor that
Diesel engines are generally more profitable than gas turbines.

The promise of hydrofoils would seem to justify a more searching evaluation: first, of
their economic characteristics and, second, of where these characteristics may pay off.
The Von Schertel boats have demonstrated that the hydrofoil may successfully perform the
role of a waterborne bus. - Why isn’t consideration being given to the analogous — and poten-
tially larger—role of ocean-going truck?

At present, the vast majority of waterborne commerce moves at speeds of 15 knots or
less. As Baron Von Schertel points out, no other waterborne vehicle has been developed to
provide higher speed service, as the motor vehicle and railroad do on land. - The need for
express cargo transport by water has been clearly demonstrated. Twenty-knot Mariner class
ships have been able to draw traffic away from slower ships. Shippers pay premium rates
to ship by fast passenger liner, at speeds of less than 30 knots. And at the far end of the
speed spectrum, transoceanic air cargo has been growing at the rate of 10 to 15 percent
per year.

In a great many respects, the hydrofoil craft seems to qualify as an “answer” to the
cargo aircraft for steamship lines:

1. Its speed range is more than double that of displacement ships of comparable size.

268 H. Von Schertel

2. Practical vessels can be built in much smaller sizes than displacement ships
without sacrificing sea-keeping capability. Smaller ships permit faster turnaround in port
and more frequent service. Delivery times of a day or two, rather than a week or two, are
possible.

3. As Baron Von Schertel points out, the hydrofoil can operate from existing facilities
and does not require extensive shore installations as airplanes do. It can provide a self-
contained express link in the transportation networks which now exist around the ports of
the world.

4. Economically, the hydrofoil is at its best at the shorter ranges where the aircraft —
present and projected —is at its poorest.

Within foreseeable advances in the state of the art, the ranges for economical operation
of hydrofoils will be substantially lower than those of aircraft. Carrying cargo instead of
passengers will not, of course, repeal these limits but it will enable the hydrofoil to service
routes where its own endurance, rather that that of its passengers, is the limiting factor.
While the hydrofoil cannot now be expected to provide economical transatlantic service,
there are major trade areas within its range capabilities. -These include the Caribbean area,
for example, and, it would seem, the seas adjacent to Europe. Furthermore, there are indica-
tions that the maritime industry, at least in the United States, is ripe for the introduction of
a high-performance, high-productivity vessel. Long a labor-intensive industry, the merchant
marine shows increasing signs of a transition to high-productivity capital equipment. Con-
tainerized cargo handling operations are now a reality and specially designed containerships
are now in service. Attention is being given to the automation of ship operation. The trend
toward faster displacement ships continues.

The possible role of the hydrofoil in cargo transport is indicated by the nature of the
cargo it might carry. Cargo aircraft are generally considered to be taking the “cream” of
the freight market —the highest valued commodities which can best afford the premium
freight charges. Of the commodities valued in excess of $40 per pound which are imported
to the U.S., for example, it is not uncommon for the entire amount to be shipped by air. On
the other hand, virtually no commodities valued at less than $1 per pound are imported
by air.

Hydrofoil cargo transports are not likely to recapture the very high valued cargo nor
will they be suitable for the low valued bulk cargoes. There is a middle ground, however,
consisting of general cargo —manufactured goods such as machinery, motor vehicles and
parts, electrical equipment, and textile manufactures — which constitute a substantial por-
tion of ocean-going commerce and for which the hydrofoil may be a suitable transport. -

The use of high-speed hydrofoil ocean transports would require that they be recognized
as an expensive, but valuable, piece of capital equipment. Their high cost would have to
be matched by high utilization to realize their productivity potential. -New concepts in
ocean shipping would be required, as well as the traditional ship operating virtues: minimum
crews which debark at the end of arun to be replaced by another crew in the manner of air-
line practice; turnaround times measured in hours or minutes rather than days; rapid engine
replacement in port so that major maintenance does not tie up the ship. »

The history of air transport is primarily the development of the passenger market. - The
passenger and his baggage have offered the highest revenues, and today’s airline passen-
gers are the heirs of the early travelers who paid premium fares for the novelty as well as

Problems of Commercial Hydrofoils 269

the speed of air travel. Only now are the airlines turning to large scale freight transport for
its own sake. ;

The initial commercial uses of hydrofoil craft have understandably followed the pattern
of early commercial aircraft. The virtues of the hydrofoil, however, are not those of the
airplane. As Baron Von Schertel points out, it will be well to recognize the pre-eminence
of the airplane for passenger transport beyond the range of a few hundred miles. If commer-
cial hydrofoils are to amount to more than a novelty, we shall have to look beyond the
operating experience of today’s boats. ‘It will be necessary to recognize the unique charac-
teristics of hydrofoils, to acknowledge their limitations, and to exploit their potentialities
with vigor and imagination.

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GROUND EFFECT MACHINE RESEARCH AND
DEVELOPMENT IN THE UNITED STATES

Harvey R. Chaplin
David Taylor Model Basin

Ground effect machine (GEM) investigations in the United States have
grown at an astonishing rate since early 1957. At present they involve
some 40 commercial firms, laboratories, etc., and represent an increas-
ingly significant investment of capital and engineering manpower.
These investigations are thus far largely exploratory in nature. They
represent a considerable variety of approaches, both as to types of
ground cushion phenomena employed, and vehicle applications en-
visioned. The various ground cushion concepts are reviewed, in terms
of elementary principles of fluid mechanics. All of the concepts are
shown to give direct relationships between vehicle performance and
vehicle size/height ratio. GEM developments are therefore expected to
tend ultimately toward large ocean-going vehicles operating very close
to the surface. Some of the outstanding advantages and problems of the
various ground cushion concepts are discussed, but present knowledge
does not support strong opinions asto which of the concepts will prevail.

Base area in square feet

Height above surface in feet

Perimeter of the base in feet

Vehicle length in feet

Vehicle beam in feet

Nozzle width in feet

Normal jet discharge angle in degrees (see Figs. 1 and 3)
Tangential jet deflection angle in degrees (see Fig. 3)
Effective cushion (base) pressure in pounds per square foot gage
Mass density of air in slugs per cubic foot

Mass density of water in slugs per cubic foot

Average jet velocity at discharge feet per second

Full-expansion velocity, in feet per second, of cushion air (velocity cushion air
would attain if allowed to expand to atmospheric pressure)

271

202 H. R. Chaplin

US Free-stream velocity in feet per second
% Free-stream dynamic pressure in pounds per square foot (q, = Fpl)
L Total lift in pounds
Ti Aerodynamic lift in pounds (lift contributed by pressure distribution associated
0 with external flow field)
Dy ee Ram drag (momentum drag) in pounds
D, Parasite drag in pounds
i Thrust component of jet reaction
D Net drag in pounds (D = D+ D, — T)
Cho Aerodynamic lift coefficient (C, 9 = L)/qS)
Cor Parasite drag coefficient (Cp, = D;/qy5 )
1. Cushion-system power in pound-feet per second
le Propulsion-system power in pound-feet per second
P Total power (P = P. + PF)
M Figure of merit M {=} V5)
2Vp
U Nondimensional velocity parameter |U = ————
Vives)
M, Ue Same as M, V, but with p,, Substituted for p
14 Augmentation efficiency
Na Duct efficiency
Ny Compressor efficiency
hen Internal efficiency
Np Propulsive efficiency
D, Discharge coefficient
E Energy ratio (ratio of maximum kinetic energy flux in system to power
dissipation)
k, Effective pressure recovery factor | k, = =P
= 2
; 9 pv.
D, Equivalent diameter D, = 4 ©
INTRODUCTION

Just over three years ago, in early 1957, a report [1] by the NACA of experiments with
a ground effect phenomenon sparked one of the most unique engineering and technical move-
ments of modern U.S. history, a movement which appears to be still gathering momentum. A
recent article in the technical press reported some forty firms now actively engaged in some
form of ground effect machine (GEM) investigations, financed by government support nearing
the ten-million-dollar mark plus an even larger investment of private funds.

GEM Research in the U.S. 273

Numerous economic and psychological reasons have been advanced to account for this
remarkable growth of GEM activity. Certainly a major factor is the fascinating simplicity of
the basic ground cushion concepts. The most casual technical training affords qualitative
understanding of the basic ground cushion principles, and innumerable arrangements of
readily available mechanical components which can produce a ground cushion come easily to
mind. It is not really surprising that so many individuals should have felt the urge to put
something together and putter around with it a bit. Even so, it may take something more to
explain the scope and vigor of GEM activity —and the financial investment which it repre-
sents — at a time when the GEM’s place in the transportation field is still highly uncertain.

This same simplicity of the ground cushion concept occasioned many a remark of sur-
prise, in those early days of 1957, that “someone hadn’t thought of it before.” Of course, it
turned out that someone had, notably, Kaario of Finland, Weiland of Switzerland, Cockerell
of England, and Frost of Canada. Even in the U.S., a rather astonishing number of GEM
models and vehicles of various types were soon found to be already in existence in widely
scattered back yards, basements, and garages. Nevertheless, nearly all of the many GEM
programs now in progress in the U.S. can be traced to the NACA report of 1957 and to the
subsequent government activity which it inspired.

Equally as interesting as the volume of GEM work now in progress is its diversity. At
least seven different ground cushion concepts, with almost countless combinations and
variations, are under serious study. Sizable independent programs are being devoted to
studies of engines, compressors, ducting problems, and structural design considerations.
While in itself one of the most interesting aspects of the U.S. GEM picture, this diversity
makes a comprehensive review of GEM research and development impractical. The present
paper will be confined to a review of seven of the more significant vehicle concepts, in
terms of simple fluid mechanics principles, with sketchy indications of the present stage of
development and prospects of each. Sources of more detailed information will be given
wherever possible.

THE GROUND CUSHION CONCEPTS

Seven ground cushion concepts have been chosen as representing, either directly or in
combination, most of the basic ideas under active consideration in the U.S. Several of these
concepts are at a rather primitive stage of development. In the analyses which follow, aa
attempt is made to reduce the various concepts to their lowest common denominator, in order
that their similarities and distinctions will be readily apparent. To this end, the analyses
follow what is termed, throughout this paper, “simplified ideal theory.” This entails assump-
tions of inviscid, incompressible flow, liberal application of the one-dimensional flow
approximations, and neglect of aerodynamic pressures and forces induced by the external
flow field. It might be argued with some justice that these analyses should be termed “over-
simplified ideal theory.” Certainly, there is a deplorable lack of rigor, and the results are
obviously not directly applicable to engineering problems. Nevertheless, it is believed that
the essential features of the basic principles involved are retained in the results; and that
the useful purposes of providing clear understanding and meaningful comparisons are served.
In the case of the most highly developed concept, the air curtain, an engineering analysis
(still highly simplified) will also be given, and compared to experimental results, to illus-
trate the relation between the simplified ideal theory and physical reality.

274 H. R. Chaplin

SIMPLE AIR CURTAIN

The air curtain (or “annular jet” or “peripheral jet”) concepts have received by far the
most attention and serious study. They are correspondingly the furthest advanced, in terms
of practical understanding and engineering data. The simple air curtain is illustrated in
Fig. 1. As in all the ground cushion concepts, the major source of lift force is the ground

C = Perimeter

Simplified Ideal Theorys
Cruise
L=ApS p V; GCV,
Aph= pV? G (1 - sin 6) vacate Yo

P.=V,GC(% pV? +% Ap) (1-3) VGC (% pV? +% Ap — q)

soa IE VV 2
pV s ps Vo. |
s

Fig. 1. Simple air curtain

cushion itself, a region of positive gage pressure trapped between the vehicle and the
ground. With the air curtain types, this ground cushion is generated and contained by a jet
of air exhausted downward and inward from a nozzle at the periphery of the base. With the
simple air curtain, the resultant jet reaction force is vertical, and propulsion must be pro-
vided by a separate device (shown schematically in Fig. 1 as a propeller).

Hovering

The jet reaction force contributes to the lift, but this contribution is small compared to
the cushion lift, so long as the vehicle is near the ground. The lift is approximated by the
product of cushion pressure times base area:

L=Aps. (1-1)

The cushion pressure also reacts against the air curtain with a force which must bal-
ance the momentum change within the curtain. This relationship is approximated by

Aph = pV? G(1 - sin @) (1-2)

GEM Research in the U.S. 275

where the jet discharge angle 0 is taken, by convention, to be negative for the inward-
inclined jet sketched in Fig. 1.

The power required is the product of jet volume flow rate times compressor pressure
rise, approximated by

P_ = VGC(>pV? + Ap) (1-3)

where the compressor pressure rise equals the jet total pressure and is the sum of mean
dynamic pressure (approximated by (1/2) p V2) and mean static pressure (approximated by
(1/2)A p).

Hovering performance is expressed by the dimensionless figure of merit,

Mie A= 1-4
Dip ee S (1-4)

which provides a direct indication of the important lift/power ratio L/P, and a direct com-
parison with the ideal shrouded propeller or helicopter (with fixed figures of merit, outside
ground effect, of 1.0 and \/1/2, respectively). Combining Eqs. (1-1) through (1-4), gives

nal - sin 0) S
Hil cin 7) or cgiaabheubl lias ankG
1 pe 1 - sin @)
which has a maximum value (when G/h = 1/2, 6 = —90°) of
Ss
Atty es (1-5)

A more thorough analysis (see Ref. 3, for example) gives slightly different values for
optimum nozzle width ratio G/h and jet angle 0, but almost exactly the same result for
optimum figure of merit M,,. Practical design limitations, internal losses, etc., will limit
actual vehicles to about M = 0.6 S/hC.

The ratio S/AC is called the size/height ratio. The initiate to the GEM field will find
that a few minutes devoted to firmly fixing the size/height ratio and its geometric meaning
in his mind will be well spent. This ratio is of predominant importance in virtually all con-
siderations of all types of ground effect machines. The size/height ratio may be thought of
as an area ratio, between the “cushion area” S of the vehicle base and the “curtain area” hC
of an imaginary peripheral screen sealing in the ground cushion. Or, it may be thought of as
a length ratio. Noting that, for a circular plan form, the quantity S/C is equivalent to one-
fourth the diameter, S/hC = D, /4h, where D, is the “equivalent diameter” of the plan form.

Cruise

When the simple air curtain vehicle moves horizontally in forward flight, two modifica-
tions to the hovering equations occur. First, a “ram” drag D,gm equal to the air mass flow
rate through the peripheral nozzle times the forward velocity, is experienced by the vehicle;

276 H. R. Chaplin

and the propulsion system must expend energy at the rate 7, = Dam Vo to overcome this drag
(Fig. 1, Eqs. (1-6), (1-7). Second, while the cushion power P. is still the product of air
volume flow rate times compressor pressure rise, the required pressure rise is now reduced
by the amount q, of the free-stream dynamic pressure recovered by the inlet (Hq. (1-8),

Fig. 1).

The cruise performance is expressed by the dimensionless “equivalent lift/drag ratio”
LV /P (Both the range of the GEM and its direct operating cost per ton-mile are directly
proportional to the equivalent lift/drag ratio.). Combining Eqs. (1-7) and (1-8) with (1-1) and
(1-2) and solving again for optimum G/h (= 1/2 (1 + U2)) and 6 ( = —90°) gives

=
0 S U
— = 2 nt (1-10)
es si hC Ne amiy2

where V is the dimensionless velocity parameter K)/ /L/ ps).

Since optimum nozzle width ratio G/h decreases with increasing forward velocity, Eq.
(1-10) represents an envelope curve for possible designs (or possible settings of an
adjustable-nozzle design).

Equation (1-10) describes a curve which rises asymptotically to the value 2 S/hC as
the forward velocity increases without limit. Consideration of the parasite drag will, of
course, cause the equivalent lift/drag ratio to fall below this simplified ideal solution and
to peak out at some value of V) corresponding to the “optimum cruise speed.” This, and
other practical considerations, will be discussed further when a simplified engineering
analysis of the air curtain vehicle is presented in a later section of this paper. It may be
said here that actual vehicles of the simple air curtain type will probably be limited to
equivalent lift/drag ratios of about 0.7 S/AC, at optimum cruise speeds corresponding
roughly to V = 1.0.

More detailed information on the simple air curtain is found in Refs. 2 to 5.

AIR CURTAIN WITH SKEGS

If the GEM is to operate exclusively over water, and at moderate speeds, a substantial
power saving is effected by the use of side plates, or skegs, which extend into the water as
sketched in Fig. 2. The equations are exactly the same as for the simple air curtain except
that the air curtain is furnished to only the portion 2b of the total perimeter 2b + 2/. Both
the figure of merit and the equivalent lift/drag ratio are hence improved in proportion to the

factor 1 + 1/b (Fig. 2, Eqs. (2-1), (2-2)).

In practice, of course, the drag on the submerged portion of the skeg becomes very sig-
nificant at high speeds. The exact breakeven point between the simple air curtain and the
air curtain with skegs depends on the roughness of the water surface (which determines the
minimum skeg submersion necessary to effect a seal). It is generally felt, however, that the
application of submerged skegs is confined to speeds below 50 knots.

A compromise between the simple air curtain and the air curtain with skegs is also
being studied wherein side plates extend below the base part way to the water, with air
curtains issuing from nozzles at the lower extremities of the side plates. The total curtain

GEM Research in the U.S. 277
b
[a Z ;

#))
AB

Simplified Ideal Theory:

Equations same as for air curtain, except that power is
furnished to only the portion (2 6) of the perimeter (27+2 6)

Fig. 2. Air curtain with skegs

area is 2 bh + 2 lh’ (where h’ is the clearance of the side plates above the water) as com-
pared to AC = 2 bh + 2 lh for the simple air curtain. The simplified ideal theory performance
is hence

l
Der GE
opt AC 1+ hs
bh
Fe) LindetSodent obDinaed. Ix ynction
P opt he v1 +2 1 +— pen

The proponents of this compromise believe that the side plates, because of their low frontal
area, can be designed to survive occasional high-speed impacts with higher-than-average
waves.

More detailed information on the air curtain with skegs is found in Refs. 6 and 7.

INTEGRATED AIR CURTAIN

The integrated air curtain provides for propulsion by directing the peripheral jet at a
rearward-inclined angle, so that the net resultant jet reaction force is inclined forward from
vertical. This system is illustrated in Fig. 3. Vanes in the peripheral nozzle deflect the
jet exhaust rearward through the tangential jet deflection angle 8. If the vehicle has a

278

H. R. Chaplin

= V, sin B (ram drag = thrust) (3-1)
ApS (3-2)

A ph= p VG (1 - sin @) cos? B 3-3)

(
P, =V,GC cos 8 (% py?+—! - 4) (3-4)
0 tan B

——— = (3-5)
VL/(p 8) y = (1 - sin 6)

(8)

Fig. 3. Integrated air curtain

GEM Research in the U.S. 279

slender plan form, with pointed bow and stern, so that the entire peripheral nozzle lies
nearly parallel to the direction of flight, then the rearward velocity component produced by
tangential deflection is, at all points along the nozzle exit, nearly equal to j sin 8B. The
thrust component of jet reaction is hence mass flow times Jj sin 8, and equals the ram drag
(mass flow times Vo) when

W=V,sin8. (31)

The base pressure and jet momentum relationship is modified from the simple air curtain
case (Eq. (1-2)) as follows: The effective nozzle area per unit nozzle length is reduced in
proportion to cos f, and only the fraction cos B of the jet momentum enters into reaction
against the ground cushion (since the tangential component of jet momentum is unchanged
as the jet curves outward). The equivalent relationship for the integrated air curtain is
therefore

Mph=- 0 ee G (1 - sin @) cos? B. (3-3)

The cushion power expression is the same for the simple air curtain (Eq. (1-8)) except for
the modified effective nozzle area:

P.=V,GC cos 8 (5p V,2 ++ -Ap - ay) | (3-4)

Combining, and solving for optimum nozzle width ratio G/h (= 1/2) and jet angle 6( = -90°)
gives for the equivalent lift/drag ratio

LV.
) S
ry SES olay Ge)
opt

Comparison of this equation with Eq. (1-10) for the simple air curtain shows that the
results are equivalent at very low speeds, but the integrated air curtain becomes vastly
superior at high speeds. Close examination of the integrated air curtain equations (Fig. 3)
will reveal that the power required, compressor pressure rise, internal air mass-flow rate,
and optimum nozzle width are all independent of speed. In practice, all of these advantages
are somewhat diluted (but not negated) by the effects of parasite drag and internal losses.
The best design practice might be to provide sufficient tangential jet deflection to counter-
act the ram drag, plus a separate propulsion system to counteract the parasite drag. This
and other practical considerations are discussed further in the simplified engineering analy-
sis of air cushion performance presented in a later section of this paper. It may be said
here that actual air cushion vehicles will probably be limited to equivalent lift/drag ratios
of the order of 0.9 S/AC, at optimum cruise speeds corresponding roughly to V = 1.0.

More detailed information on the integrated air curtain is found in Refs. 4 and 5.

WATER CURTAIN

In principle, a ground cushion can be contained by a peripheral jet of water in just the
same manner as by a jet of air. This concept is represented schematically in Fig. 4. Air is
pumped to the base of the vehicle until the ground cushion is established, at which point
equilibrium is reached between the change of momentum within the water curtain and the

280 H. R. Chaplin

Simplified Ideal Theory:

Equations same as for air curtain (neglecting gravity) except
that water sous appears instead of air sue

Simple Water Curtain

‘Note remarks in text!!

2,

a) ee eee

Fig. 4. Water curtain

reaction of the cushion pressure against the curtain. Gravity has a favorable effect on the
water curtain in that the jet gains momentum as it falls from the nozzle to the surface; and
it has an unfavorable effect on the pumping power, in that the water must be raised from the
surface to the level of the nozzle. The problem is greatly simplified, and the essential
character of the result is unchanged, if these effects are neglected. The equations for the
water curtain then become identical! with those for the air curtain (Figs. 1 through 3), except
that water density p, is substituted for air density p. The results, in terms of dimensionless
parameters referred to water density (Fig. 4, Eqs. (4-2), (4-4), (4-6)), are exactly analogous
to the air curtain results. However, when the parameters are referred to air density, it is
apparent that, in principle, the water curtain enjoys a tremendous advantage. In the hover-
ing case, and in the case of the integrated water curtain at cruise, the results (Eqs. (4-3)
and (4-7)) show the water curtain doing the same job as the air curtain with 1/29 as much
power required.

The trouble with this rosy picture becomes apparent when it is recalled that the opti-
mum nozzle width G on which these results are based, is half the operating height h. A
piping system large enough to supply such a nozzle, and filled with water, would weigh a

GEM Research in the U.S. 281

staggering amount. It is obviously necessary to compromise in favor of a much, much
thinner nozzle. In both of the cases cited, simplified ideal theory gives the power as pro-
portional to the quantity

1 A
Vipt-ae 25

h

The compromise to thinner nozzles thus destroys much of the theoretical advantage. (For
example, if G/h = 1/200 instead of 1/2, the power required is increased by a factor of five,
leaving the water curtain with roughly a 6:1 theoretical advantage over the air curtain com-
pared to the original 29:1 advantage.) Furthermore, experiments show that a thin water
curtain provides an imperfect seal, so that air must be pumped into the cushion at a quite
substantial rate to maintain the cushion pressure. This makes further serious inroads into
the theoretical advantage. Just how much advantage is left, if any, is not quite clear at
present. Certainly, the water curtain concept should not be discounted without careful
study, and a vigorous study program is, in fact, underway.

More detailed information on the water curtain concept is found in Ref. 8.

PLENUM
The plenum is by far the simplest of the ground cushion concepts, both in principle and
in physical embodiment. The plenum concept is represented schematically in Fig. 5. The

vehicle has a recessed base. Air is simply pumped into the recess and allowed to leak out
along the ground.

Hover

Neglecting internal velocities in comparison to the full-expansion velocity, V_, gives

Lift EL = ApS (5-1)
Cushion pressure Ap = + p v2 (5-2)
Cushion power cE = VD he Ap (5-3)
Ao Ss
Fj f merit MzVWaas 5-4)
igure of meri 2 2D, hC (

where D, is the discharge coefficient of the leakage flow, D, AC being the flow area at full
expansion. Using a typical value of 0.61 for D, gives

S

M=0.58 Fe: (5-5)

282 H. R. Chaplin

S

|

C = perimeter

vi ’

Ap

Area

rea
AC { Wt 9, AC

A
V. — fA :

Vv

e

Simplified Ideal Theory:

Cruise

pV, D,AOV,

=D,_V,

VD, hC (Ap — Go)

Fig. 5. Plenum

Cruise

ram 0O

The ram drag (mass flow times flight velocity), propulsive power, and cushion power

(same as in hovering, except for effect of ram pressure recovery in inlet) are given by

Dram = PVeDe hCV,
P, = D, am Vo
P, = V,D,hC(Ap

Introducing the dimensionless velocity parameter

v = ¥/ VE - y2

and again using the value 0.61 for D, gives

= 90):

Vo
V,

(5-6)
(5-7)
(5-8)

(5-9)

GEM Research in the U.S. 283

S U

—s— = 1.16 —~—: ———_. (5-10)
m 1+50?

Compared with the corresponding air curtain results, the plenum concept falls consid-
erably short, both as to figure of merit and (especially) equivalent lift/drag ratio. Never-
theless, it has its supporters among GEM investigators, because of its extreme simplicity.
Of the GEM concepts, the plenum is the most compatible with cheap, lightweight, rugged
construction.

More detailed information on the plenum concept is found in Refs. 9 and 10.

RAM WING

The ram wing is believed to be the oldest of the ground cushion concepts, having been
introduced in Finland by Kaario as early as 1935. The ram wing concept is represented
schematically in Fig. 6. The vehicle takes the form of a box with the bottom and front side
removed. The basic ram wing has no hovering capability, but if it moves forward very close
to the ground (so that velocities within the ground cushion are negligible), the ram pressure
pV7 /2 builds up beneath the base, giving a lift force

L= 5 pves. (6-1)

Simplified Ideal Theory:

Cruise

= %pV2 1b

= pV, (21AD,) Vo

Fig. 6. Ram wing

Only the air which leaks out at the tips contributes to the ram drag, since the air leaked out

the rear recovers it original rearward momentum. Assuming the discharge coefficient D, to
be 0.61,

Dram = PVo (21hD,) VY (6-2)

284 H. R. Chaplin

P = DranVo (6-3)
“.
LVy_\., See
p -1AGe —- - (6-4)

This elementary solution gives equilibrium flight at only one speed, corresponding to

Vo

= —————.- / 2. =
i YL7 (eS) 2 (6-5)

In practice, one of the other ground cushion concepts must be combined with the ram wing to
maintain the ground cushion during acceleration to and deceleration from this equilibrium
speed. For equilibrium at speeds higher than given by (6-5), the height 4 must increase
until the cushion pressure falls to something less than the full ram pressure. Even at V = 2,
the lift contributed by suction pressure induced on the upper surface of the wing (neglected
by the simplified ideal theory) is significant, but it does not dominate the problem. At
higher speeds this suction-pressure lift does begin to predominate, and the simplified ideal
theory must be abandoned in favor of approaches along the line of conventional wing theory.

It is interesting to note, from Eq. (6-4), that the ram wing, like the conventional aircraft
wing, gives much better performance with high aspect ratio, b/I.

There has been practically no development effort devoted to the ram wing in the U.S.
However, several U.S. groups have recently begun study programs, and a rapid expansion of
activity in this area is not unlikely.

Kaario’s discussion of the ram wing concept is found in Ref. 11.

DIFFUSER-RECIRCULATION SYSTEM

From an academic point of view, the diffuser-recirculation system is perhaps the most
interesting of all the ground cushion concepts. It is the only one which can, in principle,
sustain a vehicle at a finite height above the ground without dissipating any power.

The diffuser-recirculation system is represented schematically in Fig. 7. A recirculat-
ing flow, rather like a standing ring vortex, is maintained within and under the vehicle. At
the periphery of the base, the flow passes through a nozzle G, and is then exposed to atmos-
pheric pressure until it passes through the gap & between the underside of the base and the
rim. The static pressure at the gap A is thus essentially atmospheric. The geometry of the
base and internal passages is so arranged that the highest velocity (and thus the lowest
pressure) of the flow occurs at the periphery, where the flow is exposed to atmospheric
pressure. The average static pressure under the vehicle is therefore higher than atmospheric,
giving a net lift force on the vehicle. Under the assumption of inviscid flow, this is a
closed system with no energy dissipation. Simplified ideal theory thus gives

M = 1 Lyk ..
oor i apie taal Slows

GEM Research in the U.S. 285

Ve
Ca opt oe
EA WSS WN
IS Seat OI Y Cr Test

Nozzle @-—— Sr WY ‘

TITITTTIRFIV IVI TIT IVIFITITIT?. Sa Section
Gap h
Diffuser-Recirculation Open-Jet
Vehicle Wind Tunnel
Simplified Idea] Theory: Wind Tunnel Analogy:

Fig. 7. Diffuser-recirculation system

Most of the research and development effort in the diffuser-recirculation area is con-
cerned with modifications and variations of the basic system described. The practical out-
look for the basic diffuser-recirculation system is not encouraging, primarily because of the
following considerations:

1. The geometric relationship between the nozzle G and gap 4 is rather critical. This
relationship can be established properly only for a single operating height; and variations in
h from wind and surface disturbances cannot be avoided. Particularly disconcerting is the
occurrence of an unstable condition, wherein lift force decreases with decreasing height, at
heights below the optimum.

2. In practice, as in theory, the performance is totally dependent upon efficient diffu-
sion (deceleration and static pressure recovery) of the flow as it moves inward between the
vehicle base and the ground. Irregularities in the surface over which the vehicle moves
would make efficient diffusion impossible.

An interesting analogy can be drawn between the diffuser-recirculation vehicle and the
open-jet wind tunnel, also sketched in Fig. 7. The wind tunnel is also a closed system in
which the maximum velocity occurs at a section (the test section) where the flow is exposed
to the atmosphere. The power dissipation of the wind tunnel would be zero with inviscid
flow; and, like the vehicle, its actual performance depends upon an efficient diffusion proc-
ess. Tunnel performance is often expressed by the “energy ratio” E, defined as the ratio of
the kinetic energy flux through the test section to the power dissipation. Applying this
terminology to the vehicle, taking the gap area AC as analogous to the test section area,
gives

646551 O—62——20

286 H. R. Chaplin

1 C
PF. = D) pV ih

L

ApS = ky 5 pV2S

where kp is the effective pressure recovery factor. This gives

M= SL/L aeleeieaeae
2 fo Pe Si) 2h) |) Res

Open-jet wind tunnels typically have energy ratios around 3.0. Using this value, and
assigning (arbitrarily, and probably optimistically) the value of 0.7 to k, would give

- S
M2 0.6 Te

as a rough estimate of actual hovering performance, under favorable conditions. This is the
same as the estimated actual hovering performance of well-designed air curtain vehicles.

This numerical result is questionable, due to the assumptions employed. Of greater
significance is the demonstration that, for the diffuser-récirculation concept, just as for all

of the other ground cushion concepts, the performance is directly dependent on the size/
height ratio, S/AC.

More detailed information on the diffuser-recirculation concept is found in Ref. 12.

SIMPLIFIED ENGINEERING ANALYSIS OF THE AIR CURTAIN VEHICLE

The foregoing “simplified ideal theory” analyses serve only to give mathematical expression
to the basic ideas involved in the various ground cushion concepts. Comparisons between
the various concepts on this basis are still only comparisons of ideas. Realistic compari-
sons of engineering merit will require not only more complete analysis, but a considerable
background of systematic empirical information. Such comparisons are not possible at the
present state of the art. More complete analyses of several of the concepts will be found in
the references, but only in the case of the air curtain concepts is there a sufficient body of
systematic experimental data to support a realistic assessment of the practical performance.

It would be superfluous to repeat here any detailed analysis of air curtain vehicle per-
formance. It may be useful, however, to follow a simplified development, along the lines of
the simplified ideal theory, but accounting, in an elementary way, for the most significant of
the effects previously neglected. These are:

1. Internal losses, accounted for by a duct efficiency 74, compressor efficiency 7p, and
internal efficiency 7,_, = Np Na:

2. Base pressure loss, accounted for by an augmentation efficiency 7, .

3. Aerodynamic lift coefficient Cz), and parasite drag coefficient Cpz, produced by
the external flow field.

GEM Research in the U.S. 287

4. Propulsion system losses, accounted for by the propulsive efficiency 7;,.

Both tangential jet deflection B (Fig. 3) and separate-propulsion-system power will be
accounted for. This is appropriate to a semi-integrated air curtain concept, wherein part,
but not all, of the thrust is derived from the air curtain itself. The resulting equations will
be reducible to the simple or integrated air curtain cases simply by setting B equal to zero,

or setting air-curtain thrust equal to net drag, respectively.

The lift is approximated by
L = ApS j+ Cy ay Si (8-1)
ih

(The box will be used to indicate modifications to the simplified ideal theory.)
The base pressure is approximated by

G- 8
: qe fe BP ale
Np = pve, , (1 - sin 0) cos* 8 Nal: (8-2)

The cushion power is approximated by

ips ag 1 1

a 2 Bek x Casen a0
cm 1V; GC cos Al p Ver 4 poP Mg do). (8-3)
pee

|
c |
|

The ram drag, parasite drag, and air curtain thrust are given by

Diam 7 PV; GC cos B Vy (8-4)
D,-=\Cn- gq, S ' -
pat
iF (ee Ine 6 V.2 GC cos B sin B aaa (8-6)
J J Cy

where the factor 2//C accounts for-the fact that the tangential force component of each
element of the air curtain lies tangential to the local periphery, not, in general, precisely
parallel to the direction of flight.

The propulsion power is net drag times flight velocity, corrected for propulsive
efficiency:

a! St err Vy vo

Combining, and forming dimensionless ratios:

288 H. R. Chaplin

G
P. 1 (1 - sin AM ett 1

L3/2//' 6S int

4

; 3/2 Cr %
(1 ee v7 - nq? (1 — v4 (3-8)

' S : G
2 (1 =*sin sane (1 -- sin 2 lee. cos? B

D
. Vv (8-9)

a ae - sin @) = Ce.
2

(1 - sin @) 74

aS
(Ss
i}
Ss
+

tan “L 21
an
sg ae ie ac Sate (1 = 2 = (8-10)
(ths "Sai G3) = e
Ac A
LV P_.P
Svan peed SePeele (8-11)

P L3/2// 0S

These last four equations form a system from which realistic estimates of air curtain per-
formance (whether simple, integrated, or semi-integrated air curtain) can be made — if esti-
mates of the parameters 74, Cy 9, and Cp¢ can be obtained. The problem of providing a
reliable basis for estimating these three parameters occupies much of the systematic GEM
research now underway in the U.S. (The internal-, duct-, and propulsive-efficiency problems
are, of course, age-old problems, not peculiar to GEM’s, and highly developed techniques for
dealing with them are available in the literature.) Some progress has been reported in the
references. A good deal of additional progress can be expected in the immediate future.

It will suit our present purposes to side-step the question of how these estimates can
be made, and to use, for our discussion, a source of data for which the necessary param-
eters have been evaluated to some reasonable degree of accuracy, experimentally. Such a
source is Ref. 5, which presents data from wind-tunnel tests of the David Taylor Model
Basin’s Gem Model 448. This model represents the nearest approach to a realistic,
practical GEM configuration for which laboratory-controlled performance test data are
presently available.

Photographs of the model are presented in Fig. 8. The sketch in Fig. 3, used to illus-
trate the integrated air curtain concept, is also representative of the geometric arrangement

GEM Research in the U.S. 289

Fig. 8. DTMB GEM Model 448

and airflow path of Model 448. The four compressor nacelles are powered by calibrated
electric motors. Several interchangeable sets of nozzle control vanes were employed to
produce various tangential deflection angles B (see Fig. 3). The model is not equipped
with an actual propulsion system, but propulsive power was accounted for in the test data
reduction by arbitrarily assuming it to equal 1.25 DVp (i.e., 7, = 0.8). (It should be noted
that, as explained in Ref. 3, the assumption P, = 1.25 DV becomes unrealistic at low
forward speeds. Nevertheless, it provides a convenient and reasonable basis for quick
examination of the overall performance characteristics.) The performance parameters
necessary for evaluation of Kgs. (8-8) through (8-11) vary slightly with changing test condi-
tions, but it will suffice for our purposes to use the fixed set:

Tint = 0.57 N40 = 0.73
Wet 1.0.00 Cho = 0.42
Nr = 0.80 Cor¢ = 0.074.

290 H. R. Chaplin

(These performance parameters reflect design features dictated by scale and by research
utility considerations. A highly developed vehicle should have substantially improved
internal and duct efficiency and parasite drag coefficient, perhaps 0.8, 0.9, and 0.04,
respectively.) The pertinent geometric data for the model are:

S = 20.50 square feet
C = 18.85 feet

lL = 8.45 feet

G =. 0.058 foot

9 = ~45°.

Performance calculations from Eqs. (8-8) through (8-11) are compared with experimental
results in Fig. 9. The consistent agreement, over wide ranges of test conditions, strongly
suggests that the very elementary considerations employed in the simplified engineering
analysis correctly represent the major physical phenomena involved.

(It is not our purpose here to consider the detailed performance of the air curtain GEM.
However, in passing, one should note: (a) the strong, almost linear dependence of the
equivalent lift/drag ratio, LVo/P, on size-height ratio, S/hC, shown in Fig. 9a, (b) the
significant performance advantage indicated in Fig. 9b for the vehicle with controlled tan-
gential deflection B of the air curtain, and (c) the further advantage indicated for such a
vehicle, in Fig. 9c, in terms of the power-required distribution between cushion power and
propulsive power. The total installed power required, if separate power sources are used for
the cushion system and propulsion system, will be the sum of the maximum cushion power
required and the maximum propulsion power required. This sum, especially for the simple
air curtain (8 = 0) is very much larger than the maximum total power required at any given
instant.)

Having confirmed that the simplified engineering analysis gives a reasonable represen-
tation of the actual facts, we are in a position to examine the relationship between the
simplified ideal theory and physical reality. It will be easiest to consider the simple air
curtain (8 = 0) for this purpose. For direct comparison, the appropriate form of simplified
ideal theory is obtained from Eqs. (8-8) through (8-11) by setting

URE eR my ee

The most important information one may hope to obtain from simplified ideal theory is a
qualitative prediction of equivalent lift/drag ratio. The calculation of optimum equivalent
lift/drag ratios from Eqs. (8-8) through (8-11) is rather laborious. However, as suggested by
Ref. 4, and confirmed by Fig. 9, an excellent indication of effects on optimum performance
is obtained by considering performance at V = 1. Figure 10 gives the result of such a cal-
culation, starting with simplified ideal theory, and with successive curves which result from
setting the performance parameters, in steps, to the values appropriate to Model 448. Each
of the “real” effects gives a successively less optimistic performance prediction compared
to the simplified ideal theory. At the end, the actual equivalent lift/drag ratio of Model 448
is very much lower than predicted by simplified ideal theory (about one-half), but the essen-
tial conclusion of the simplified ideal theory as to the nearly linear relationship between
equivalent lift/drag ratio and size /height ratio is borne out.

291

GEM Research in the U.S.

——— Calculated

OOA Experimental

0 1 2
v

(a) Equivalent lift/drag versus speed

Ly
P

Equivalent Lift/Drag

Caiculated
Aoa Experimental

(b) Effect of 8 on equivalent lift/drag

Fig. 9. Comparison of “simplified engineering analysis” calculations with experiment

(P,1U07;)) JueuITIedxe YIM SUOTJe[NOTeO ,stsA[eue Sutssoursua peryttduis, jo uostseduior) *6 “31 ad

6 = 3D4/S ‘suiaisks uorsjndoid pue uorysno useMieqg Jomod jo uolNngqINsiIq (2)

(b ()
Gat Celene 90e mai0 C0 0 Ga Oil i020) eri 20 0

jejuawiiadxjo =
payejnojeq —

jequawiadxjo a
payejnojej—

H. R. Chaplin

-18M0qg Uols|ndoig

292

Jamog uorysny

293

GEM Research in the U.S.

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—

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294 H. R. Chaplin

Similar demonstrations have been or could be made, from data available in the refer-
ences, that the hovering performance (figure of merit) is also, experimentally, almost
directly proportional to the size/height ratio in the cases of all the air curtain types, the
water curtain type, and the plenum type. The hovering performance is less significant, in
terms of vehicle usefulness; but this lends additional support to the essential conclusions
of the simplified ideal theory.

RESUME

The simplified ideal analyses furnish valuable insight into the fundamental ideas of the
air cushion concepts. They furnish, further, the invaluable information that all of the air
cushion concepts have in common the property of rapidly improving performance with increas-
ing size/height ratio. Unfortunately, they leave unresolved —as does also experience to
date —the essential question: Which concept is best for which application?

The following remarks are in large part merely the author’s opinions:

Air Curtain — Most thoroughly studied, and correspondingly best understood of the air
cushion concepts; most logical choice for early applications.

Plenum — Simplest and cheapest to build but ranks low in performance; very poor pros-
pects for favorable high-speed performance unless combined with another concept, such as
ram wing.

Ram Wing — Interesting only in combination with some other concept to provide low-
speed flight capability; requires much additional research to evaluate practical problems and
merits; appears to have great potential, if stability and control problems prove tractable.

Diffuser-Recirculation System — Intriguing idea, but unlikely to prove practical.

Water Curtain — Falls far short of theoretical expectations, but may eventually prove
advantageous for moderate-speed over-water applications; not amphibious

Skegs — (Previously considered in combination with air curtain, but equally suitable for
combination with any of the other concepts.) Likely to prove advantageous for moderate-
speed over-water applications; not amphibious.

THE GEM’S PLACE IN TRANSPORTATION

In every one of the ground cushion concepts considered, elementary analysis indicates
a direct dependence of the performance on the size/height ratio. Except for the air cushion
types, the available experimental evidence is rather limited; but every bit of data which is
available tends to confirm this direct dependence. This alone is enough to identify the
natural form and habitat of ground effect machines: very large vehicles operating very close
to the earth’s surface. Since land areas are, in general, topographically unsuitable for the
operation of large, high-speed vehicles at low heights, this means ocean-going GEM’s. A
more specific idea of the sizes and heights involved is afforded by Fig. 11. For a GEM
with LV, /P = 0.9 S/AC (appropriate to a highly developed air curtain vehicle) the equivalent
lift/drag ratio is plotted in comparison with existing aircraft, and the size is given for each
of several operating heights. The argument implied in Fig. 11 is, of course, somewhat
superficial. There is the possibility that one of the other ground cushion concepts will

GEM Research in the U.S. 295

S
Size/Height —
ize/Heig 7

1000
8
25
500 _s Es
Fa
=
oo o
) uw
a
> :
F Typical
~ Airplane
=
= Typical
s Helicopter
lw

S
Size/Height —
ize/Heig AC

Fig. 11. The relationship between performance
and size for a GEM

provide better performance than the air cushion. There is the possibility that a GEM can
compete successfully, even with somewhat inferior lift/drag ratio, because of its more
flexible load-handling abilities and terminal facility requirements. And there is the possi-
bility that the GEM’s apparent compatibility with nuclear power will, in future years, loom
larger than the lift/drag consideration. Nevertheless, it would be necessary to stretch any
or all of these possibilities rather far to avoid the conclusion that ocean-going GEM’s will
be very large vehicles indeed. As such, their development represents a very imposing
economic undertaking, and, while a great deal of current GEM research and study is directed
toward the ocean-going machine, it may be assumed that serious development efforts will
await clarification of the present confused state of knowledge.

In the meantime, GEM applications will be made in “fringe” areas where special cir-
cumstances or special requirements make moderate-size GEM’s attractive. The two main
categories of such fringe areas are:

1. The limited application possibilities over flat inland areas and protected waters,
where extremely low operating heights afford good performance for moderate-size vehicles.
These include passenger and cargo ferries, emergency craft, sport craft, and swamp buggies.

2. Limited military application possibilities where requirements for speed, amphibious
capability, and load-carrying ability outweigh the conventional economic performance
criteria.

Early applications are likely to center around the air curtain concepts. Only in this
area is the state of the art sufficiently advanced to permit rational design. Where the future
will lead is anybody’s guess.

Whatever form the GEM of the future takes, this much is certain: If energetic and

enthusiastic activity count for much, the GEM’s future in the U.S. should be very bright
indeed.

296 H. R. Chaplin

REFERENCES

Air Curtain

[1] Von Glahn, U.H., “Exploratory Study of Ground Proximity Effects on Thrust of Annular
and Circular Nozzles,” Washington, National Advisory Committee for Aeronautics

TN 3982, Apr. 1957

[2] Chaplin, H.R., “Theory of the Annular Nozzle in Proximity to the Ground,” David Taylor
Model Basin Aero Report 923, July 1957

[3] Crewe, P.R., and Eggington, W.J., “The Hovercraft —A New Concept in Maritime Trans-
port,” London, The Royal Institution of Naval Architects, Nov. 1959

[4] Chaplin, H.R., “Ground Cushion Research at the David Taylor Model Basin —A Brief
Summary of Progress to Date,” Presented at Princeton Univ. Symposium on Ground

Effect Phenomena, Oct. 21-23, 1959

[5] Johnson, A.E., “Wind Tunnel Investigation of DTMB GEM Model 448,” David Taylor
Model Basin (In preparation)

[6] Cathers, D., Hirsch, A.A., and Walker, W., “Air Pressure Levitation,” Paper presented
at the Feb. 1960 meeting of the Chesapeake Section of the Society of Naval Architects _
and Marine Engineers, Washington, 1960

[7] Hirsch, A.A., “The Hovering Performance of a Two-Dimensional Ground Effect Machine

Over Water,” Presented at Princeton Univ. Symposium on Ground Effect Phenomena,

Oct. 21-23, 1959
Water Curtain

[8] DeVault, R.T., “Introduction to the Hughes Hydrostreak Concept,” Culver City, Calif.,
Hughes Aircraft Co. Report X-424, Nov. 1959

Plenum
[9] Dobson, F.A., “The Airlifter,” Dobson Aircraft Corp., Whittier Calif., Dec. 1957

[10] Wright, E., “The Effect of Configuration on the Lift of a Two-Dimensional Open Plenum
Ground Effect Machine,” Princeton Univ. Report 516, May 1960

Ram Wing

[11] Kaario, T.J., “The Principles of Ground Effect Vehicles,” Presented at Princeton
Univ. Symposium on Ground Effect Phenomena, Oct. 21-23, 1959

Diffuser-Recirculation

[12] Gates, M.F., and Sargent, E.R., “Development of a Unique GEM Concept With Potential
for Achieving Efficient Forward Flight,” Presented at Princeton Univ. Symposium on
Ground Effect Phenomena, Oct. 21-23, 1959

GEM Research in the U.S. 297

BIBLIOGRAPHY

Liberatore, E.K., “GEM Activities and Bibliography,” Buffalo, Bell Aerosystems Co.,
Mar. 1960

ADDENDUM

ILLUSTRATIVE COMPARISON OF THE GROUND CUSHION CONCEPTS

Direct comparisons between the various ground cushion concepts are very risky at the
present state of the art. The simplified theory presented in the body of this paper neglects
many factors, which may affect one concept more than another, and the degree of experi-
mental understanding is by no means the same for all of the concepts. For this reason it
was originally decided to avoid direct comparisons.

However, in retrospect it became apparent that direct comparisons would go far toward
clarifying many of the points made. It was decided to present the comparisons shown in
Fig. 12 after all. The reader is implored to accept Fig. 12 as merely illustrative of some of
the distinctive characteristics of the different concepts, and to use it cautiously or not at
all as a criterion of their relative practical usefulness.

100
| % | of
oo
\p b = es G@) Simple Air Curtain
=. Ss Q@) Air Curtain with Skegs
: 4 (2/b= 4)

4 @ Integrated Water Curtain
o© RF (G/h = 0.005)
4 ae el ©) Plenu=

‘gn | jaca
=

| = Cruise Performance

| <—! Per simplified ideal theory;
Parasite drag limits shown:

5 | Comparisons are illustrative only.
| he =10

S Pg %
= a ay | = 4 (}) Integrated Air Cu-tain

'

'

’

Equivalent Lift/Drag

6.1 0.5 1.0 5
Vv
10 30 100 500
L
Vapn, ee =a

298 H. R. Chaplin

The cruise performance curves for the six concepts shown are plotted directly from the
simplified ideal theory. Special assumptions, such as length/beam ratio of 4 for the air
curtain with skegs, and nozzle-width/operating-height ratio of 0.005 for the water curtain,
are noted in the legend. The seventh concept, the diffuser-recirculation system, is omitted,
since the dearth of practical experience makes it impractical to plot a curve of even illus-
trative value.

Also shown in Fig. 12 are performance limits imposed by parasite drag, under assumed
parasite drag coefficients:

Dy
Cor = il
= 2
5 pV" Ss
De
Cee : era aS
9 Pw Vana S

The “simplified ideal theory” curves account for cushion power and momentum drag but
ignore parasite drag. The “limit” curves account for parasite drag but ignore all other power

~

requirements, as follows:

limit ~ 0
2
Te pS Vo
\y2
LV,
0 " 2 hi 2 Epi
P e

The concepts which do not involve contact with the water (curves 1, 3, 5, 6) are assumed,
for illustrative comparison purposes, to have parasite drag coefficients of Cp = 0.04; the
concepts which do involve water contact are assumed to have Cp, » = 0.0017. (A calcula-
tion which included cushion power, momentum drag, and parasite drag would give a curve
which followed closely below the “simplified ideal” curve at low speed, then peaked over
and followed closely below the “limit” curve at high speed.)

The relative performance picture afforded by Fig. 12 should be interpreted as follows:

1. Within the no-water-contact family, the relationship between curves 1, 3, and 5 is
probably in reasonable qualitative harmony with the facts. If lower S/hC or lower Cp¢ were
assumed, the integrated air curtain would appear slightly better, compared to the simple air
curtain; the plenum, slightly poorer. Curve 6, for the ram wing, is only a single point, since
under the approximations used in theory, equilibrium flight occurs at only one speed coeffi-
cient, U = V2. If lower S/hC or lower Cp were assumed, the ram wing could be made to
appear superior to the other members of the no-water-contact family, by assigning it a higher
aspect ratio b/I.

GEM Research in the U.S. 299

2. Within the water-contact family, the positions of curves 2 and 4 relative to each
other are not meaningful, since they can be substantially changed or reversed by assigning a
different nozzle-width ratio G/h to the water curtain and/or a different length/beam ratio //b
to the air curtain with skegs.

3. Between the two families, absolute performance comparisons are meaningless for the
same reasons. However, the significant and valid point that the water-contact concepts are
superior at low speeds, while the no-water-contact concepts are superior at high speeds, is
very well illustrated. The exact degree of superiority, and speed range to which it applies,
depends very heavily on the respective parasite drag coefficients. The values of Cp; and
Cp, w used in Fig. 12 were selected arbitrarily, for illustrative purposes only.

DISCUSSION

R. L. Weigel (University of California)

I do not want this to be construed as a criticism of Mr. Chaplin’s paper: the subject he
was given was extremely broad to cover. However, there is one important aspect that he did
not touch upon and that is the wave resistance of this type of vehicle moving over the water
surface. I have chosen this one aspect because I think it of general interest to naval archi-
tects and furthermore, as this meeting is dedicated to Sir Thomas Havelock, I think it is
interesting that the technique used in predicting the wave resistance is due to Sir Thomas
Havelock. Rather than considering a true ship, he considered a pressure disturbance moving
over the surface of the water, and this is precisely the problem of the ground effect machine.
Sir Thomas Havelock’s main advance is that he considered a pressure area rather than a
pressure point, and he formulated his problem in such a manner that numerical results could
be calculated from the equations rather than just looking at the integrals and so forth. Fur-
thermore, he even went so far as to calculate the resistance in almost the exact form that is
needed to obtain the information for designing a ground effect machine. Sir Thomas Havelock
considered a pressure disturbance something like that shown in Fig. D1 where pressure is
measured vertically and the radius of the disturbance is measured along the abscissa, with
axial symmetry assumed. Now, he chose one particular shape; however, tests that we have
made have indicated that we can have quite irregular shapes such as a double hump shape

and the resistance is practically the same as one obtains from this shape that Havelock
assumed.*

Havelock’st numerical results are shown in Fig. D2. Very often in naval architecture
we are dealing with the deep-water aspects and a Froude number based upon the length of
the ship. If we deal in inland waterways we talk about a Froude number based upon the
water depth.. It turns out for ground effect machines we must consider both of these simul-
taneously. So we have a parameter which is the diameter of the pressure disturbance
divided by the water depth. This is a dimensionless resistance which is a specific weight
of water (pg) times the resistance, and this is the wave resistance (R) divided by 27 times
the diameter of the ground effect machine times the maximum pressure squared (that exists

* R.L. Wiegel, C.M. Snyder, and J.B. Williams, “Water Gravity Waves Generated by a Moving Low
Pressure Area,” Trans. Amer. Geophys. Union 39 (No. 2):224-236 (Apr.. 1958).

+ T.H. Havelock, “The Effect of Shallow Water on Wave Resistance,” Proc. Roy. Soc. (London)
A100 (No. A705):409-505 (Feb. 1, 1922).

300 H. R. Chaplin

p
Prax

0 r
D = Effective diameter of pressure area, feet.
d = Water depth, feet.
p = Mass density of water, slugs/ft.”
g = Gravity, ft./sec>
R = Wave resistance, pounds.
Pmax Maximum pressure of pressure area, |bs/ft?

Fig. D1. A pressure disturbance on the water surface

I
Theoretical D/d = 2.7; vA /% gD=0.86
ia
curves obtained by
Havelock.

Deep water
max.

0.4 0.6 0.8 1.0 1.2 1.4 16

es Y29D

Fig. D2. Theoretical curves obtained by Havelock

in the base cavity). Then, for various values of the ratio of the diameter of the disturbance
to the water depth we can get a series of curves as shown (Fig. D2); where the diameter is
great compared with the water depth we have a shallow water resistance peak, where the
diameter is small compared with the water depth we have the typical deep-water peak, and
where we have intermediate values we have double peaks, the deep-water and the shallow-
water peak, or shoulders. I think the thing of primary interest is just how great is the wave
resistance due to a high-speed vessel of this sort. Using the results of Havelock I made
several calculations just briefly to give you an idea. If we choose a craft that is large,
that’s the type the Navy is considering, the type that Mr. Chaplin mentioned, with the diam-
eter of 300 feet, flying over the ocean at some relatively high speed (I have had to choose a

GEM Research in the U.S. 301

speed of only 65 knots, which is low, but if I chose higher it would have been off these
curves) the resistance will be lower than what I show here. If it is moving over water deep
enough so that D/d = 0 (where d is the water depth), then V/y gD/2 = 1.6, and pgR/2nD pmax
= 0.085. In very deep water it turns out that the horsepower that is utilized in wave resist-
ance is only 12-1/2 hp, for a base pressure of 5 pounds per square foot, which is practically
negligible, as we are talking about a craft with a 40,000- to 50,000-hp motor. It is almost
unbelievable. The horsepower loss is dependent upon the square of the base pressure, so if
you get up to the more modern concepts, which have a higher base pressure loading, and you
go up to say 30 pounds per square foot the horsepower goes up to 440, and even this is only
of the order of 1 percent. It is becoming appreciable but is still relatively small. In operat-
ing the GEM, it will move from deep water into shallow water as it is coming ashore and
here is where we have to be careful. We can hit some of these “hump” speeds and for one
combination which I picked at semirandom (I chose a water depth of 100 feet and a forward
speed of 55 knots) the wave resistance can rise to over 1500 to 1600 hp. In general when we
are operating over deep water at high speeds, the wave resistance will be negligible. The
only time it will tend to become important is when one is bringing it in over relatively shal-
low water. The stopping of one of these vehicles is a major problem. Consequently this
high resistance that one gets as one comes into shallow water probably will turn out to be
very useful because I think that maybe for slowing a GEM we can make use of this shallow
wave resistance.

In practice the peak pressure will be higher than the average base pressure on a GEM,
and the power necessary to overcome wave resistance will be higher than the values men-
tioned, perhaps by a factor of almost two.

L. W.. Rosenthal (Folland Aircraft Limited)

I would like to thank Mr. Chaplin for the very pleasant paper he presented, but I am
going to be so ungracious as to suggest that it had the wrong title! I think perhaps it might
well have been called “A Simplified Method of Comparing the Performance of Ground Effect
Machines.” During the whole of the Symposium, I have hoped that the hydrodynamicists
would have discussed the position of the ground effect machine in the light of the conclu-
sions that Mr. Chaplin had arrived at. When Mr. Oakley spoke, he did in fact suggest that
attention had been given to the position of the ground effect machine, with the limitations
that we know, in the marine transport pattern. Unfortunately, he did not develop this point.
Mr. Van Manen, discussing the Crewe Eggington hypothetical project, quietly but firmly took
it to pieces on the structure weight grounds, but he gave no constructive comment on what
he would suggest instead. Mr. Newton followed this up by saying that the structural problem
concerned with a large ground effect machine in a seaway needed investigation, but here
again he gave no lead at all. Mr. Chaplin broadly touches on the place of the ground effect
machine but his comments are based on a general parameter and not necessarily on the
specialized uses to which we might be able to put the vehicle at the moment.

What I would like to suggest, if this is not ungracious again, is that possibly the hydro-
dynamicist and the naval architect are dragging their feet on this particular problem, or is it
that no conclusions have yet been reached and we are looking for something or some informa-
tion which does not yet exist.

Another point to which Mr. Chaplin made no reference in his assessment of the small
machines, and this could hardly have been considered a secondary one, is the means of
obtaining pitch and roll stability and the penalties associated therewith. I would like to

646551 O—62——_21

302 H. R. Chaplin

ask if Mr. Chaplin would like to discuss these points in the light of the title of his paper
“Research and Development of Ground Effect Machines in the United States.”

J. L. Wosser (ONR, Washington)

After listening to all the papers on hydrofoils and associated subjects that have been
delivered to date at this meeting, let me state that I am very impressed with the level of the
work that has been done. There has been just this one paper presented in the area of ground
effect machines (GEMs). Most of this was Mr. Chaplin’s original work. It included a brief
summary of some of the other things that have been done, but very brief. I would like to
point out that as far as we in the United States are concerned, we consider the GEM state of
the art stands about where that of the airplane did in 1904, or where you gentlemen with the
hydrofoil boat were in 1930. We are just getting started in this field, research wise. How-
ever, there is a considerable research effort underway.

Within the Office of Naval Research we are supporting some 16 research tasks, all
coordinated into an overall GEM research program. Our objectives in this area are not the
far-out aims that Mr. Chaplin mentioned at the conclusion of his paper: the 600-foot trans-
oceanic GEM. In order to get this type of research effort underway, we have more immediate
aims. We are looking for the areas where the GEM’s unique capabilities to travel over all
types of surfaces (water, land, ice, snow, and mud) with equal facility can be utilized for
military purposes. Now, this military approach is not the direction of the effort that is going
on in England, and I am sure that Mr. Shaw will tell you about their commercial interests in
a moment.

In addition to the work of the Office of Naval Research, extensive programs are being
sponsored by Bureau of Naval Weapons, Bureau of Ships, and the Army Transportation
Corps —all looking at their own areas of interest where the GEM has unique capabilities.
The Army is particularly interested in overland vehicles that can travel over ice fields, mud
flats, river deltas, etc., to open up these areas for future exploration or military operations.
To give you a complete outline of the work that is being done would take too long. A thumb-
nail sketch would show that we have programs underway that will utilize more sophisticated
approaches to some of the simplified theories you have heard presented today: viscous
analyses of flow inside and external.to the machines, propulsion and structural loads
analyses, stability and control criteria, and ditching and flotation tests at the Seakeeping
Basin here at Wageningen.

In conclusion, let me tell you a few of the things that have been achieved in the GEM
area to date. During his talk Mr. Chaplin showed you some of the machines now in existence.
Remember that we consider this thing to be in the period 1904 aircraft wise, or just starting
out hydrofoil wise. In England, they have flown a GEM at 60 mph over the water on the
Solent and successfully operated in eight-foot waves. In Switzerland, we watched a demon-
stration of a 30 x 30 foot-machine move over the water at 51 knots. The Curtiss-Wright Air-
car has been officially clocked at 65 mph. At the present time, Mr. Carl Weiland, now
working for the Reynold’s Metals Company, is building a 90-mph machine at Louisville,
Kentucky. This is supposed to be flying by Christmas (1960).

Yesterday I received a letter which I have already shown to Mr. Chaplin, so this will
not come as any surprise. About two years ago at an Institute of the Aeronautical Sciences
meeting in New York I met a Dr. Bertelsen. We discussed theoretical aerodynamics and
GEMs for an hour and a half before I found out that he was an M.D., an obstetrician from

GEM Research in the U.S. 303

Illinois. He delivers babies and dabbles in aerodynamics on the side. So, with that as a
background, these are abstracts from the letter I received from him yesterday: “Since I saw
you in January 1960 at the IAS meeting, considerable work on Ground Effect Machines has
obviously been done all over the world. We too have been busy. We now have what we con-
sider to be the world’s best Ground Effect Machine from the standpoint of control, hill
climbing, stability and all around utility. We have a 200-hp, four-passenger flying machine
in the chassis stage and have gotten well along in testing it. We get 12-inch altitude at
1600 Ibs. gross weight, climb a 10-degree slope and go 50 mph over land or water. I feel
that with all the theory and experimentation, we lead the world and challenge all comers in
this category of small personal Ground Effect Vehicles.”

R. A. Shaw (Ministry of Aviation, London)

It seems to me that you are mostly ship men here and therefore at a disadvantage in
assessing the possibilities of what we call hovercraft in England and what you call ground
effect machines in America. (What you call them on the continent, I have not yet discovered.)
You are at a disadvantage because you look at them and you think, Reynolds’ number,
Froude’s number — must explain it all this way.. It isn’t really very helpful in working out an
understanding of hovercraft or ground effect machines because, perhaps fortunately for
everyone, the Froude number is not very important as far as ground effect machines are con-
cerned, at least you soon get out of the regime in which it is important. Perhaps the particu-
lar case in which it will continue to be important is when your machine enters shallow water
and does the transition from water to land. Apart from this, as ship people I feel that you
are at a disadvantage because the problems of ground effect machines are largely the prob-
lems which have been studied by the aircraft industry in the past. Hovercraft certainly
stand astride the two fields but a proper appreciation of ground effect machines and their
future, I think, is easier for the aircraft people who rather rub their hands and say “jolly
good, this is easy stuff,” by comparison with the ship people who say “what a troublesome
problem.” It isn’t really easy stuff because there are a lot of very curious things in it.
What I wanted to bring home to you was that you couldn’t assess it simply by ship stand-
ards. There are many subtleties in ground effect machines, and to give you an example, in a
machine like the SRN-1, the British hovercraft, there was more power lost between the fan
and the ducts than was used in the lifting and propulsion of the machine. This is just part
of the problem of design, how to use your power effectively and efficiently in a novel way.
People haven’t got habits of thinking about hovercraft so they don’t naturally choose a good
solution. We have got to go through that process of finding good solutions and that’s why
people like Dr. Bertleson, who are following the example of the Wright Brothers and starting
at the beginning, are just as likely at this stage to hit on a good solution as well-informed
gentlemen at national establishments. In fact, we are in the pioneering stage and have to
assess our progress in terms of that stage; it is 1904 as Col. Wosser said. But despite the
fact that it is only 1904, or perhaps to be precise 1905, we are in England at the moment,
building a 25-ton hovercraft capable of cruising at about 70 knots and carrying about 50 or
60 people several hundred miles. This is being achieved within a few years of the concept
taking hold and within a year or 18 months of our first demonstration. By contrast, in the
hydrofoil business, hydrofoils have been talked about for a generation and although they are
now at the 80-ton stage, it has taken a generation to get there. With a background of air-
craft experience it does look possible to get into the ground effect business in a matter of
five years. I want to redress any influence which Mr. Chaplin’s final statement might have
on this audience in thinking that unless you got into the 600-foot or 10,000-ton class,
ground effect machines have no place, by reinforcing what Col. Wosser said and that is you
must not consider them in relation to conventional forms of transport on conventional routes.

304 H. R. Chaplin

You have got to think of the places you can’t get to any other way, the marshes, the bogs,
the ice flows, the out of the way places where you can’t land an aeroplane. These are the
places where I believe the hovercraft will really take root and where we shall gain the
experience we need to allow us ultimately to build big ocean craft.

L. Landweber (Iowa Institute of Hydraulic Research)

The following comments were written before the contents of the present paper were
known. That can be justified, however, by noting that these comments will serve as a
bulwark to the title of Mr. Chaplin’s presentation which they supplement by describing the
work in this field that is being done at the Iowa Institute of Hydraulic Research. These
comments were written by Lawrence R. Mack and Joachim Malsy:

Increasing interest in the possibilities of ground effect machines hasled many organiza-
tions in several countries to undertake basic studies of ground effect phenomena or studies
of the means of practical utilization of these phenomena or both. During the past two years
the Iowa Institute of Hydraulic Research, with the financial support of the Office of Naval
Research, Contract Nonr 1509(03), has conducted both analytical and experimental investiga-
tions of the behavior of an annular-jet nozzle in proximity to solid and liquid surfaces. The
results of the first year’s work were presented at the Princeton Symposium on Ground Effect
Phenomena in October 1959.* Since then a thesis describing an experimental study of an
annular jet moving over water has been completed;' the results of this study, together with
certain results for a stationary jet over land obtained preparatory to the over water experi-
ments, are contained in a forthcoming report to the Office of Naval Research.* It is the
intent of this discussion to summarize briefly these results.

A rigidly supported 7-inch-diameter annular nozzle discharging air vertically at different
rates through a 1/8-inch gap was towed at different altitudes and speeds (including zero
velocity) over initially quiescent water. Chosen combinations of these three variables led
to 18 runs, for each of which the configuration of the water surface in the vicinity of the
nozzle and the pressure distribution on the base of the nozzle were determined and plotted
in the form of contour drawings.

The measurements of water-surface configuration were obtained by means of a capaci-
tance wire and a point gage and were supplemented by stereo photographs as an aid in pre-
paring contour maps. All stationary cases show a deep annular depression, caused by the
impinging jet sheet, about 1.2 nozzle radii from the center line of the jet. The water surface
under the nozzle base was considerably higher than what would, under the assumption of
hydrostatic pressure distribution in the water, correspond to the pressure within the base
cavity, a ridge at about 0.7 nozzle radius actually projecting above the still water surface.
Forward speeds produced a considerable change in surface configuration from that of the
static case. A forward ridge, or bow wave, was clearly discernible in all runs. With higher
jet momenta and especially with higher speed, this bow wave seemed to split into two
separate ridges, each situated in front of the nozzle about 45 degrees from the direction of

* Lawrence R. Mack and Ben-Chie Yen, “Theoretical and Experimental Research on Annular Jets
over Land and Water,” Proc. Symposium on Ground Effect Phenomena, Princeton, Oct. 1959, pp. 263-
284. Also available as Reprint No. 164, State University of lowa Reprints in Engineering.

t Joachim K. Malsy, “Experimental Investigation of an Annular Jet Traveling over Water,” M.S.
Thesis, State University of Iowa, Aug. 1960.

¢ Lawrence R. Mack and Joachim Malsy, “Experimental Studies of an Annular Jet,” Iowa Institute
of Hydraulic Research Report to the Office of Naval Research, Sept. 1960.

GEM Research in the U.S. 305

forward motion. At low velocities a half-moon-shaped depression, concave rearward, was
beneath the front part of the base plate, the deepest parts being situated on either side of
the nozzle center. This deep, very clearly visible depression moved to the rear with
increasing speed and decreasing jet momentum. A high wave directly to the rear, which at
times almost touched the nozzle, behaved similarly. In general, it can be said that all
observed surface phenomena moved to the rear and became less pronounced in magnitude
with increasing forward speed and decreasing jet momentum.

Base-plate pressures were measured at 14 piezometer holes, the number of locations
being tripled for some runs by making two 45-degree rotations of the base plate, and isopi-
estic lines were drawn. In the stationary cases, especially for the higher altitude, a dip in
pressure was noticed at about 0.7 nozzle radius, the same location as an elevation in the
water surface. For all cases of forward speed the pressures were larger in the rear than in
front, pressures at the center and extreme rear tending to be the highest. A roughly circular
pressure valley at about 0.7 nozzle radius was clearly discernible, culminating in a saddle
point between the two peak pressures. The high pressures to the rear and low pressures to
the front combined to give nose-down pitching moments acting on the base plate for all
cases of forward motion.

The total base-plate lift, and hence the lift-augmentation factor, for each run was
obtained by integration of the pressure distribution. As expected, the augmentation factor
was found to increase with decreasing altitude for all conditions. For the stationary runs
the augmentation increased with decreasing jet momentum for a given altitude, in qualitative
agreement with the theoretical predictions of Mack and Yen (first footnote) even though the
surface-configuration data did not support certain of their assumptions. The numerical
magnitudes of augmentation, however, were less than their idealized predictions. For con-
stant altitude and jet momentum, forward speed, within the experimental range, improved the
augmentation initially, then caused a decrease at still higher speed. This feature, particu-
larly the increase of augmentation with increasing speed at low speeds, was more pronounced
for the lower altitude tested. It thus appears that there is an optimum forward speed for
each combination of altitude and jet momentum.

The same basic 7-inch annular nozzle, but with interchangeable mouthpieces, was also
tested statically over a ground board. Discharge angles of 0, 15, and 30 degrees inward
with a 1/8-inch annulus width were used, as were 0- and 15-degree angles with gaps of
5/16 and 1/2 inch. As expected, the 30-degree discharge angle gave the highest augmenta-
tion; no significant distinction in augmentation was noted, however, between the 0- and
15-degree angles. It was found that, within the gap range tested, the augmentation factor
and the radial uniformity of pressure both increased with increasing gap width. This
behavior of the former is in qualitative agreement with the theoretical prediction of Mack
and Yen (first footnote, Fig. 1) within its range of validity (altitude less than the nozzle
radius).

The results summarized herein of tests conducted at the Iowa Institute of Hydraulic

Research on annular jets moving over water and stationary over land are described more
fully in the report to the Office of Naval Research already referred to.

E. C. Tupper (Admiralty Experiment Works)

Mr. Chaplin did a very good job in condensing this research work into such a relatively
short paper and I hate to suggest more work for him, but I would like to make two points.

306 H. R. Chaplin

First of all, I should like to emphasize how important it is that we should not progress
too far with detailed experimental and theoretical work into the optimum form for resistance
and propulsion alone. It may well be that many of the configurations so studied will prove
quite unacceptable from the point of view of stability or seaworthiness. In my opinion, a
broad, less detailed, study is first required into all aspects of design to determine the
ranges of the principal dimensions which are likely to prove suitable for a balanced design.

Second, I would like to support Mr. Rosenthal’s plea for some comments upon stability
for the various configurations which have been discussed. This obviously is of prime
importance in operation over waves as well as over calm water, and I would like the
author’s comments on whether passive stability may be adequate in some cases, or whether
active stabilizing will always be necessary.

Harvey R. Chaplin

I will not reply at great length to the various comments, but will touch just a couple of
the specific questions. About the optimum plan form for waves, certainly we don’t have the
answers on this yet but there is a program at the Netherlands Ship Model Basin underway
now which will give answers having some bearing on this question and other programs within
the United States too which are touching on this question. The plan form on which most of
the recent air curtain research has been concentrated has been the plan form of the model
which the photograph has shown, which is not too far distant from the ship forms and may
not be totally unsuitable from the standpoint of possible wave impact. The stability ques-
tion of course, is a very important question and it would be inexcusable not to cover the
stability if we knew what to say about it. Again on the air curtain, a good bit is known
about the stability; we know how to stabilize the machine, how to give it some natural
stability, and we are close to knowing an answer to the question of whether it will be pos-
sible to have inherent stability or whether it will be necessary to have artificial stability.

I personally feel that for the air cushion at speeds up to 100 knots it will be possible to
have inherent stability without the black box. Beyond those two specific points, if I may
abstract other comments en mass, they would add up to the fact that I have left an awful lot
out and this is certainly true. I am afraid I cannot undertake to fill in very many of the gaps
in the limited time that we have. Fortunately, some of the gaps were filled in by the com-
menters themselves and | thank them for that. Certainly the one thing that I shouldn’t have
left out is what was pointed out by Professor Wiegel, and this is the wave drag problem
which bears on the person to whom the Symposium is dedicated. I certainly owe him grati-
tude for pointing out this inexcusable oversight. As to the title of the paper, I have to take
my excuse from the fact that, as you know, in the scheme of things, the title and abstract of
the paper were submitted some months in advance of the paper itself and as is often the
case, the paper at the time it was submitted was quite a bit different from the way it was
envisioned at the time the title was submitted.

HYDRODYNAMIC ASPECTS OF A DEEP-DIVING
OCEANOGRAPHIC SUBMARINE

P. Mandel
Massachusetts Institute of Technology

INTRODUCTION

There is little question that the major problems that arise in the design of very deep
diving submarines are in the field of structures and not in the field of hydrodynamics. How-
ever, several hydrodynamic problems exist that are peculiar to the very deep diving oceano-
graphic submarines that are of little or no importance in the design of the most advanced
military submarines. These problems arise in connection with: (a) the ballast systems
needed for operation at great depths, (b) the oscillations that may be excited by vortex
shedding during vertical ascent or descent, (c) the effects on depth and trim angle control
of the compressibility of the submarine hull and of sea water, (d) the precise control in
both the horizontal and vertical planes needed to perform the mission of the boat, and
(e) directional stability when under tow on the surface. This paper will concern itself
largely with these five questions plus some introductory discussion of how the configura-
‘tion of a deep-diving submarine compares to military submarines and bodies of revolution
of elementary shape. -

This paper will draw on experience gained during the design of the oceanographic sub-
marine Aluminaut which is described in Ref. 1. The design of this unique vehicle was
based on general concepts initiated by Dr. Edward Wenk, formerly of Southwest Research
Institute, San Antonio, Texas. The design was developed to its current state under a
project initiated by Mr. J. Louis Reynolds, Vice President, Reynolds Metals Company.
Negotiations for construction of the boat are currently nearing completion.

The Aluminaut differs in basic concept from the previous generation of deep-diving
vehicles marked by the bathyscaphes FRNS and Trieste which were developed by Auguste
Piccard. It will be recalled that it was the Trieste which made the record-breaking dive on
January 22, 1960, to a depth of 37,800 feet. These vehicles depend on a buoyant liquid,
gasoline, to support about 90 percent of their total weight while their small pressure hulls
support only about 5 percent of their total weight. The remainder of their weight is sup-
ported by the buoyancy of structure. In contrast the buoyancy of the pressure hull of the
Aluminaut supports over 80 percent of its total weight. The marked effect of this change of
concept can be seen by study of Table 1. With a total submerged displacement of slightly
more than half of the Trieste, the Aluminaut has about 9-1/2 times the useful volume and
possesses mobility and endurance that are vastly superior to the Trieste. It is these defi-
ciencies of the Trieste plus the difficulties in handling large quantities of volatile gaso-
line that prevent full appreciation in oceanographic research work of her superlative depth
capability. Even the retrogress of the Aluminaut from the remarkable depth capability of
the Trieste is more apparent than real since only about 40 percent of the world’s ocean area
is greater than 15,000 feet deep. Thus, the Aluminaut has great potentiality as an oceano-
graphic research vehicle. -

307

308 P. Mandel

Table 1

Comparison of Characteristics of the Trieste and the Aluminaut

[ee [me | Te | Si

Length overall, ft
Max. Beam, ft

Draft, before dive, ft
Inside diameter of pressure hull, ft

Inside length of pressure hull, ft
Thickness of pressure hull material, inches
Total submerged volume at depth
Internal volume of pressure hull
Buoyant volume of pressure hull
Pressure hull structural weight
Other structural weight

Volumes

(cu ft)

Weights
(1b)

21,520
33,600

Ballast, permanent and jettisonable 24,6 20
Propulsion, batteries, pumps, controls ?
Flotation fluid 157,000

Scientific payload ~500
Total submerged displacement

5 252,000

Materials Pressure hull material Steel Aluminum
Flotation fluid Gasoline Silicone

Performance Operating depth, ft 15,000

Characteristics | Test 17,000

Collapse depth, ft 21,000
Max. horizontal speed, knots 4.7
Horizontal endurance, miles 96
Endurance, hours 36 normal

72 emergency
Vertical speed, fps 3 normal

11 max
Power available for instruments, kwh 64
Crew Number 3

*1/2-4 hours at depth depending on time for vertical traverse.

THE ALUMINAUT CONFIGURATION

The configuration of a deep-diving submarine is largely determined on the basis of
structural and arrangement requirements. For the Aluminaut a cylindrical pressure hull with
hemispherical ends was selected as a good compromise between an optimum structural con-
figuration and a configuration most suitable for housing people and equipment. The forward
pressure hull hemisphere and the cylindrical portion of the hull form the basic external hull
of the submarine since these are not entirely unsuitable from a hydrodynamic point of view.
For hydrodynamic reasons, however, the aft pressure hemisphere is enclosed within a non-
pressureproof, almost conical, stern capsule that provides some hydrodynamic fairing and
also serves to house the electric propulsion motor and control mechanism. This obviates
the need for large mechanical penetrations of the pressure hull. This capsule is filled with
a buoyant silicon fluid (specific gravity—0.76) that provides a suitable environment for the
motor in addition to providing a modest amount of buoyancy (~13 percent). The stern

Deep-Diving Submarine Hydrodynamics 309

capsule is pressure equalized to the open sea at all times so that it may be constructed of
relatively light structure. Figures 1 and 2 show the general configuration of the submarine.

The configuration, thus determined, has somewhat different proportions than the usual
military submarine. The length/diameter, L/D, ratio is lower and the prismatic coefficient,
Cy, (ratio of volume of submarine to volume of circumscribing cylinder) is higher than the
corresponding proportions of military submarines. These properties of the Aluminaut are
conducive to minimizing wetted surface and hence frictional drag at the expense of an
increase in separation and form drag. The differences in wetted surface between the
Aluminaut and military submarines and the wetted surfaces of elementary body of revolution
shapes similar to submarines are shown in Fig. 3. ‘The ordinate of this figure is the wetted
surface of solids of unit volume. (For example, a cubic prism of unit volume has a wetted
surface of 6.) To compute the wetted surface of geometrically similar volumes, it is only
necessary to multiply the value of the ordinates in Fig. 3 by the volume raised to the 2/3
power. It is seen that not only does the Aluminaut have substantially less wetted surface
than military submarines because of her low L/D and high C,, but with those proportions
her hemispherical nose and essentially conical stern offer almost as low wetted surface per
unit volume as the less hydrodynamically satisfactory combination of conical nose and
stern. The comparison is as follows at the Aluminaut L/D and C,:

Con fi guration Wetted Surface for Unit Volume
Hemi-nose; conical tail 6.90
Conical nose; conical tail 6.89

The additional useful information shown in Fig. 3, computed on the basis of information
given in Appendix A and taken from Fig. 7 of Ref. 2, is in the nature of a postscript to the
main purpose of this paper. Primarily, the figure shows the large penalty in wetted surface
associated with high length/diameter ratios, which is well known, and the variation of
wetted surface with prismatic coefficient and end shape, which is not so well known. Within
the range of L/D’s suitable for streamlined bodies (L/D>3) and at low values of Cp, the
bodies with conically shaped nose as well as tail, have the least wetted surface, the bodies
with hemispherical noses and conical tails have the next greatest, while bodies with sub-
marinelike noses and tails have the most wetted surface. This latter fact is consistent with
the frequent observation that achieving reasonable form drag usually involves increased
wetted surface and frictional drag. At high prismatic coefficients the effect of end shape
on wetted surface is not nearly as decisive as at low values of C,. It might also be
observed that to achieve the complete continuity at the intersections of the parallel middle
body and the end shapes that is needed to minimize form drag, and that is so obviously
missing with pure conical or even hemispherical ends, much higher prismatic coefficients
are associated with a fixed percentage of parallel middle body L,,’ for the submarinelike
bodies than for the simpler shapes. All in all it is evident from Fig. 3 that the L/D ratio is
the most important parameter influencing wetted surface and that prismatic coefficients in
the practical range from 0.50 to 0.90 are also conducive to minimizing wetted surface for
any fixed L/D ratio. For a more thorough discussion of this particular question, the reader
is referred to Section 3 of Ref. 2.

Figures 1 and 2 indicate that the Aluminaut is encumbered with many appendages.

Most of these are needed to house equipment or ballasting functions (described subsequently)
that are advantageously mounted external to the main pressure hull. As shown in Fig. 2, and
also subsequently in Fig. 6, some of these appendages are not suitably faired, but this prob-
lem is fully appreciated by all concerned and it has been decided that the building plans

will incorporate suitable fairings. Even faired, it is estimated that the appendages will more
than double the bare hull-drag of the submarine in horizontal motion. Drag estimates will

be shown in Fig. 13.

P. Mandel

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Deep-Diving Submarine Hydrodynamics 313

BALLAST SYSTEMS

Although water is ordinarily employed for variable ballast on submarines, the energy
and pump weights required to eject water at great depths render almost essential the use of
alternate ballast systems at such depths. Actually four ballast systems are employed on
the Aluminaut, three of which are variable and one of which is fixed:

1. The ballast intended for routine ascent at great depths is 4,000 pounds of iron shot,
contained in two amidship external saddle tanks shown in Figs. 1 and 2. These tanks are
loaded through filling tubes in the superstructure when the boat is on the surface and
emptied as desired through hollow solonoids at the bottom of both tanks. When energized
(they draw only 36 watts each) the solonoids magnetically solidify the shot so as to plug
the shot-tank aperture. When current to the solonoids is cut, either deliberately or by acci-
dent, gravity will cause the shot to flow out at any depth of operation. Thus, this system
which was devised by Piccard for the bathyscaphes is relatively fail safe. To prevent
electrolytic action with the iron shot, the aluminum saddle tanks are lined and the filling
tubes molded of plastic. Special measures such as the prevention of exposure of wet shot
to the air must also be taken to prevent the iron shot from corroding and consolidating. The
iron shot described here has been successfully used with the bathyscaphes.

2. To augment the preceding system and also to permit the submergence of the boat
from the surface to be under the complete control of the pilot within the boat a conventional
water-ballast system is also provided. The tanks of this system which are also external to
the pressure hull have a capacity of 3,000 pounds of sea water and are always open to the
sea. Like conventional submarines, water is excluded from these tanks by means of a
closed air valve; it can be admitted by opening the air valve and can be ejected from these
tanks by compressed air stored in flasks located above the pressure hull as shown in Fig.
1. Because these flasks are initially charged for practical reasons to a pressure of only
2,200 psi, the water tanks can only be blown at depths less than about 4,800 feet. It is
this fact that necessitates the provision of the iron shot system for use at greater depths.

3. For faster (and more reliable in the event of inadvertent flooding) emergency ascent,
provision is made for jettisoning a 7,000-pound chunk of aluminum-wrapped lead stored in
the lower keel structure. This ballast can be quickly released by the pilot by a simple
mechanism that cuts two 1/4-inch-diameter supporting cables. The penetrations for these
cables are the only mechanical penetrations of the pressure hull.

4. The remaining ballast is fixed and is divided between nonwastable ballast needed
for stability purposes and wastable ballast margin available for design, construction, or
future growth. Two thousand pounds of nonwastable ballast for stability purposes is
located very low in the boat in the free-flooding keel in the form of solid aluminum bars.
Five thousand pounds of wastable growth margin (on paper until the boat is completed) is
allowed for at the location of the center of ‘gravity of the entire boat. Upon completion of
the boat, the remnant of this margin would actually have to be installed as ballast in the
keel with the nonwastable ballast where it would further improve stability. The distinction
between these two kinds of fixed ballast is further clarified in Section 5 of Ref. 2.

The three variable ballast systems permit the submarine to be operated at a great
variety of equilibrium and nonequilibrium conditions. The more important of these condi-
tions for the Aluminaut are tabulated in Table 2, which also shows the freeboard to the
deck for the surface conditions and the metacentric heights for all conditions. Some of the
same information is illustrated in Fig. 4.

P. Mandel

314

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316 P. Mandel

Normal diving and ascending procedures will be able to be carried out as follows on
the Aluminaut. Condition 2 (see Table 2 and Fig. 4) would obtain when the Aluminaut is
alongside an auxiliary vessel preparing for a dive. All solid variable as well as fixed
ballast is on board, but it may be considered that the iron shot and water ballast had been
jettisoned on a previous dive. When the boat has been made ready for submergence in all
other respects, iron shot would be added from the mother ship to produce condition 3. In
this condition the water tanks are still empty so that the boat is still afloat with over a
foot of freeboard to the deck. The final submergence of the boat is now under the complete
control of the pilot and when he is ready, he can flood the ballast tank, bringing the boat to
condition 4 with a negative buoyancy of about 500 pounds. In all probability, the pilot
would choose at this point 'to speed his descent by propelling the boat to the desired depth
of operation. It would take roughly 50 minutes to reach 15,000 feet with a 30-degree down
angle utilizing full power on the main propeller (40 minutes with the vertical propeller work-
ing also). Otherwise, it would take about 4-1/2 hours.

With no further change in ballasting the boat would be at neutral equilibrium at 15,000
feet since due to compressibility effects (which will be discussed subsequently) the initial
500 pounds of negative buoyancy would disappear at that depth. This corresponds to
condition 5.

Ascent can be achieved by any of the ballast arrangements shown in Table 2 and
Fig. 4. The fastest panic condition ascent, which involves jettisoning the solid lead
ballast and iron shot at 15,000-foot depth and the ballast water at 4,800 feet, utilizing full
power on both the horizontal and vertical propellers, and a 30-degree up angle would take
22 minutes from 15,000 feet with no leakage. With no power available (and no leakage)
the panic ascent would consume about 44 minutes. Consideration was given earlier in the
design to provision of an overboard discharge pump capable of handling a very modest
amount of leakage at 15,000 feet. It was concluded that the weight saved by not installing
the pump was greater than the weight of water which the pump could eject during an ascent.

MOTION CHARACTERISTICS DURING VERTICAL ASCENT—PRELIMINARY
CONSIDERATIONS

Piccard reported in Ref. 3 that on several occasions the bathyscaphes experienced
rather violent oscillations while rising freely under the influence of positive buoyancy.
Since it was known that the Aluminaut would possess substantially less metacentric sta-
bility than the Trieste, which would tend to permit larger oscillations, it was decided to
study this problem thoroughly during the feasibility and design study of the Aluminaut.

Bodies which are streamlined in planes through their longitudinal axis of symmetry,
are usually very blunt in planes normal to their axis of symmetry. It is well known that
such blunt bodies will shed Karman vortices. For cylindrical bodies moving normal to their
axis, which corresponds fairly closely to the case of submarines rising vertically, it can be
shown that the dependence of the period of the vortex shedding upon the various charac-
teristics of the flow is expressed by a single curve of the Strouhal number D/T,V against
the Reynolds number VD/v, where T, is the period of the vortex shedding (period of excita-
tion), D is the diameter of the cylinder, V is the velocity, and v is the kinematic viscosity
of the fluid. This curve was obtained experimentally in Ref. 4 for Reynolds numbers up to
about 10° and plotted in such forms in Ref. 5 that extrapolation to slightly higher Reynolds
was not too difficult. It is shown in Fig. 5 for Reynolds numbers up to about 4 x 10®.

Also shown in Fig. 5 is cylinder-drag data as a function of Reynolds number taken from

Deep-Diving Submarine Hydrodynamics 317

Ref. 6. It will be noted that the critical Reynolds number reflecting an abrupt change in
the flow regime takes place between the model- and full-scale range. These data were
used to compute both the vertical velocities of ascent as well as the approximate
periods of excitations that could cause oscillations of the Aluminaut during a rapid
ascent.

If the excitation periods computed in the preceding correspond to the natural period
of oscillation of the submarine during any particular ascent, severe oscillations would

Limit of experimental
Hata with primes | Be scale range l|to7 ft/sec

model

0.8

Cylinder Drag Strouhal
Coefficient

=

aay

o O06

‘

°o

Ss)

mo

=

ra)

a
S)

o
‘p

0.2

48 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 656
l0gig (Reynolds Number)

Fig. 5. Effect of the Reynolds number on the vortex shedding frequency and the drag of cylinders

646551 O—62

22

Num ber

Strouhal

318 P. Mandel

doubtlessly ensue. The natural rolling period of a submerged submarine is conveniently
expressed as follows:

Tn = 271kx/V gBG
where = the natural roll period
=- radius of gyration of weight about the longitudinal axis of
symmetry (varies from about 3.3 feet for condition 9 in
Table 2 to 3.6 feet for conditions 4 and 5)

g = the acceleration due to gravity
BG = the metacentric height (tabulated in Table 2).

=~
eae

Utilizing all of the preceding information, it would be possible to predict which of the
nonequilibrium conditions shown in Table 2 was likely to result in severe oscillations. How-
ever, this information would be insufficient to permit an estimate of the amplitude of oscilla-
tion. Furthermore, a freely rising submarine could obviously partake of coupled motions that
might render the elementary analysis discussed up to this point inadequate. For these
reasons the decision was made to conduct an experimental model study of the ascent of the
Aluminaut.

MODEL STUDY
The Model and Facility
On the basis of a model-scale study conducted by the author, Southwest Research Insti-

tute constructed the 1/12-scale, 49-inch-long hollow aluminum model shown in Figs. 6 and
7. The unfairnesses evident in Fig. 6 were ameliorated by the use of modeling clay, and as

Fig. 6. Assembled aluminaut model

Deep-Diving Submarine Hydrodynamics

319

Fig. 7. Internals removed from aluminaut model, showing adjustable

weights for vertical ascent tests

indicated earlier these changes will presumably be incorporated into the full-scale. The
internal weight arrangement evident in Fig. 7 permitted considerable adjustment not only of
the net buoyancy of the model (by means of adding or subtracting weights), but also of its
natural period of roll for any fixed net buoyancy (by means of altering the vertical position
of the weights as well as the radius of gyration). The amplitude of these adjustments is
evident from Table 3 which describes the conditions that apply to each vertical ascent test

conducted with the model.

Table 3

Vertical Ascent Tests; Model Variables Tested

TestNo. | Date of Test

i 6/10-12/59*
li 6/10-12/59
lil 6/10-12/59
iv 6/10-12/59
Vv 6/10-12/59
vi 6/10-12/59
Vii 6/10-12/59
viii 6/10-12/59
1x 6/10-12/59
x 6/10-12/59

Meta-

centric

Height

(in.)*
64.2 33.2 0.79 4.33
64.2 Boe 0.79 4.33
82.5 14.9 1.09 4.02
86.1 Ting 1.14 3.97
89.6 7.8 1.19 3.92
79.4 18.0 1.05 4.07
70.3 ial 0.90 4.23
82.5 14.9 1.09 4.02
86.1 les 1.14 3.97
79.4 18.0 1.05 4.07

Free
Roll
Period
(sec.)*

1.57
1.57
1.23
1.20
1.15
1.27
1.41
1.23
1.20
1.27

Bilge | Stern
Keels§ | Fins§
Narrow | Small
Narrow | Small
Narrow | Small
Narrow | Small
Narrow | Small
Narrow | Small
Narrow | Small
Wide Small
Wide Small
Wide Small

*Net buoyancies of the 6/10-12/59 tests were computed on the basis of 97.4 pounds total buoyancy.
+Weights, metacentric heights, and gyradii do not include the effects of inadvertant flooding water.

§See Table 4,

320 P. Mandel

Table 3 (Continued)
Vertical Ascent Tests; Model Variables Tested

Meta-

centric Bilge
Height i i Keels$
(in.)+#

TestNo. | Date of Test

6/10-12/59* gen) or. 9 Wide
7/27-29/59t : ; 3 8 ; Narrow
7/27-29/59 al: ‘ a2 : : Narrow
7/27-29/59 ail : A : 58 Narrow
7/27-29/59 oll : 5 6 = Narrow
7/27-29/59 : : : : i Narrow
7/27-29/59 3 ot é J ; Narrow
7/27-29/59 Ee ; 2 ; : Narrow
7/27-29/59 5 £5 ; 5 : Narrow
7/27-29/59 7 3 alk : all Narrow
7/27-290/59 3 : ? ‘ ‘ Narrow
7/27-29/59 : ; oll é al Narrow
7/27-29/59 ; : : ; : Narrow
7/27-29/59 . f ; : ; Narrow
7/27-29/59 ‘ p : ; .3 Narrow
7/27-29/59 ‘ ; : : : Narrow
7/27-29/59 : F : alt ; Narrow
7/27-29/59 5 Wide
7/27-29/50 7 Wide
7/27-29 /'59 Swill Wide
7/27-29/59 5 ; : 4 al Wide
7/27-29/59 5 BD : ; al Wide
7/27-29/59 ; 9 56 | Whee

*Net buoyancies of the 6/10-12/59 tests were computed on the basis of 97.4 pounds total buoyancy.
tNet buoyancies of the 7/27-29/59 tests computed on the basis of 102.0 pounds total buoyancy.

+Weights, metacentric heights, and gyradii do not include the effects of inadvertent flooding water.
§See Table 4.

Since it was known that the shape and size of external appendages would strongly influ-
ence the damping of any oscillations, two different sets of bilge keels and stern fins were
made to be fitted to the model. These are described in Table 4 and identified in Table 3.

Table 4

Narrow Bilge Keels 29 x 3/4 in. Xs

Wide Bilge Keels 29 x 1-1/2 in. «x re:
Small Stern Planes 2.6 x 3.7 in. x 2
Large Stern Planes 5.8 x 3.5 in. x 2

Deep-Diving Submarine Hydrodynamics 321

The vertical ascent tests were conducted at the Underwater Weapons Tank Facility of
the Naval Ordnance Laboratory at White Oak, Maryland. This facility proved to be well
suited to the purposes of the tests. Figure 8 shows the facility setup schematically. The
fact that the launching platform could be freely raised and lowered permitted great flexi-
bility in conducting the tests. Moving pictures of the vertical ascent tests were obtained
by underwater photographers from the staff of the U.S. Naval Underwater Photographic Unit
in Washington.

U.S. Naval Underwater Photographic Unit in Washington.

The instrumentation used to measure the results obtained in this facility was, however,
quite crude. The speeds of ascent were measured with stop watches while the roll oscillations

~— FREE WATER SURFACE

SWIMMER WITH
MOVIE CAMERA
(USNUPU)

PLATFORM HOISTING
CABLES

SRE | > VIEWING PORTS

RELEASE FIXTURE,
REMOTELY CONTROLLED
FROM FREE WATER SURFACE

Fig. 8. Vertical ascent test setup at NOL Underwater Weapons Tank Facility

322 P. Mandel

had to be based on visual observations through the viewing ports in the sides of the tank. —
It might be noted, however, that in spite of the limitations imposed by both the crudity of
the measurements and inadvertent leakage of the model, the tests provided important guid-
ance to the design of the Aluminaut.

Model Drag

Results showing the relationship between ascent velocity and the net buoyancy and
drag for the model conditions and configurations tabulated in Table 3 are shown in Fig. 9.

100

90

Points plotted indicate experiment results as follows:

NARROW KEELS, SMALL STERN FINS, 6/10-I2 TESTS
80 NARROW KEELS, LARGE STERN FINS, 7/ 27-29 TESTS
WIDE KEELS, SMALL STERN FINS, 6/10-l2 TESTS
WIDE KEELS, LARGE STERN FINS, 7/27-29TESTS

LINES [INDICATE COMPUTED RESULTS
70

Pounds

oO
(o)

Drag or Net Bouyancy,
aS oa
(@) (e)

30

20

O 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Ascent Velocity, Ft./Sec.

Fig. 9. Model vertical-drag data, experimental and computed

Deep-Diving Submarine Hydrodynamics 323

Also shown are computed data for the same configurations utilizing data given in Fig. 5 and
Table 4. The scatter of experimental results is large, but considering the uncertainty intro-
duced by the presence of an unaccountable amount of flooding water during some of the
tests, the experimental results were considered as roughly confirming the computed data.

In particular, the wide keel tests of 6/10-12/59 showed consistent serious deviation from
the computed data. It will be noted that in none of the tests did the velocities approach
those associated with the sharp discontinuity in the drag curve.

Model Oscillations

Test results of roll oscillation are plotted in Fig. 10 as a function of both ascent
velocity, determined experimentally, and natural roll period, as tabulated in Table 3. Maxi-
mum double amplitude rolls observed during each test are shown symbolically for each point
plotted. Also shown in Fig. 10 are the approximate periods of the roll excitation caused by
the periodic vortex shedding computed on the basis of the Strouhal number curve shown in
Fig. 5. With the introduction of bilge keels located at the maximum beam of the model, the
maximum width to be used in the computation of the excitation period could be either 8
inches, the diameter of the cylinder, or 9-1/2 inches, the maximum tip to tip beam. This
accounts for the range of values of excitation period shown in F'ig. 10.

In general, as would be expected, the closer the correspondence between the natural
roll period and the period of excitation, the larger the roll amplitude that resulted. It might
be noted that it is very likely that the actual natural roll periods of the model during some
of these tests were longer than the computed periods because of leakage. A moderate
increase in the natural roll periods of some of the test spots would have improved the corre-
lation between maximum roll amplitude and resonance.

While the proximity to resonance explains several of the large observed roll amplitudes
in Fig. 10, in most instances the test conditions with low metacentric heights (large natural
periods) showed larger roll amplitudes than conditions with large metacentric heights (small
natural period). Although not directly apparent from Fig. 10, the preceding statement applies
at constant tuning factor (ratio of natural period to excitation period). For example, the
comparison of results of test 14 to tests 10 and 7 or test 11 to test 4 shows this tendency.
The only exception to this pattern are tests 13 and 17. This effect of metacentric height is
consistent with vibration theory (e.g., Ref. 8).

Only data for the narrow bilge keel tests of July 27-29 are shown in Fig. 10 since none
of the wide keel tests showed strong oscillations and, in fact, most of those tests showed
no oscillations at all. Therefore, it may be concluded that a substantial reduction in roll
amplitude can be achieved with wider bilge keels. However, this is achieved at the expense
of a considerable increase in vertical drag (see Fig. 11), particularly in the range of full-
scale Reynolds number where the bare cylinder drag is a much smaller fraction of the total
than it is in the model range.

In addition to quantitative data on roll amplitude and vertical velocity, the free ascent
tests revealed strong cross-coupling between vertical forces and horizontal motion. As a
result, the model usually achieved some ahead velocity while solely under the influence of
a vertical force. This cross-coupling is probably enhanced by the strong fore and aft asym-
metry introduced by the presence of large stern fins. Usually, only a slight pitch or yaw
developed during the tests. However, during test 16 a strong pitch developed. Nevertheless,
in no case was there evidence that the coupled motions prevented the predicted occurrence
of roll oscillations.

324 P. Mandel

2.6 GF
4 Narrow : Bilge Keels
(
2.2 GY Approximate range of

roll and sway excitation
periods

91/2 "MAX. MODEL WIDTH

G
2.0 | Z
VY

w
= 1.8
fo)
1S)
®
a 7&10
o 1.6
<2
a 4 3
1.4
3)
1.2 ©
Oe S
T 92 a
Ke) he)
Numbers of circles correspond to 7/2/7-29 test numbers. Vy
Location of circles correspond to experimentally determined 4)
08 ascent velocity and computed natural period of roll oscillation. mW,
Smallest angle between lines in circles corresponds to observed
maximum amplitude of roll.
0.6
O 0.5 1.0 1.5 2.0 2.5 3.0 B.S) 4.0

Ascent Velocity, Ft./Sec.

Fig. 10. Model oscillation data, experimental and computed

PREDICTION OF FULL-SCALE DATA

While the model results are of great importance in confirming design philosophy, they
cannot be applied directly to predicting full-scale performance. Examination of Fig. 5 shows
that both the nondimensional drag and roll excitation period change radically between the
model range and the full scale. However, with the assumption that the limited experimental
data modestly confirm design calculations within the model range, one can utilize the full-
scale data in Fig. 5 with more confidence.

Deep-Diving Submarine Hydrodynamics 325

Full-Scale Drag

Full-scale drag vs ascent velocity curves are shown in Fig. 11. ‘The uppermost curve
corresponds to the Aluminaut design, as it is contemplated at this time with 12-inch bilge
keels and large stern fins. It is seen that this condition results in more than double the
bare cylinder drag. Also shown in Fig. 11 are the intercepts corresponding to actual buoy-
ancy conditions of the Aluminaut shown in Table 2. Comments have already been made con-
cerning the speed of descent in condition 4 and the speed of ascent in the panic conditions
8 and 9. In the normal ascent if all of the shot ballast were dropped and water ballast

18,000

ALUMINAUT HULL I2" BILGE
KEELS & NEW STERN

POSITIVE BOUYANCY OR DRAG, POUNDS

5
VELOCITY OF ASCENT, fps

Fig. 11. Full-scale computed vertical-drag data

326 P. Mandel

pumped out at moderately shallow depths (condition 7), it would take approximately 62.5
minutes to reach the surface. In fact, better time would be made if no ballast were dropped,
but full power were preserved and utilized for the ascent using a 30-degree up angle. With
this assumption it would take only 49 minutes to reach the surface. It appears very likely
that this method of ascending will be preferred in the normal situation to dropping and losing
expensive shot ballast.

Full-Scale Oscillation

Predicted excitation and oscillation periods for the full-scale Aluminaut are shown in
Fig. 12. Bearing in mind the results of the model tests, it might be surmised that normal
ascent conditions 6 and 7A might suffer the most severe oscillations of any of the condi-
tions. This conclusion is tempered by the fact that, as noted in the previous section, it is
very likely that propulsion would be used to speed the ascent in these conditions. With the
boat driven to the surface with, say, a 30-degree up angle, it is not likely that severe oscil-
lations would ensue even if the shot ballast was also dropped.

While it is likely that propulsion will be utilized for normal ascents, in the panic condi-
tions, 8 and 9, propulsion may not be available. Therefore, it is of particular significance
that these two conditions are, in fact, very far removed from resonance (tuning factor = 5
and 6 respectively). Therefore, in spite of their low metacentric heights, these conditions
should not suffer large roll amplitudes.

Although not noted in Fig. 12, the range of predicted excitation periods on that figure
is bounded by two different water temperatures. For the full scale, the influence of the
range of possible values of water kinematic viscosity on ascent velocity and frequency of
excitation is much more pronounced than the influence of assumed maximum body width.
Since water temperatures between 40° and 80°F might be encountered, kinematic viscosities
associated with these temperatures were used to form the boundary conditions.

CONCLUSIONS FROM VERTICAL ASCENT TESTS AND COMPUTATIONS

1. Despite large scatter, the experimental drag and oscillation data reasonably conform
to analytically predicted results.

2. At resonance a substantial reduction in roll amplitudes can be achieved with wide
bilge keels at the expense of a considerable increase in drag.

3. Strong cross-coupling caused the model to attain some ahead velocity. ‘Pitch and
yaw also were evident during many of the tests. These did not appear to interfere with the
predicted roll oscillations.

4. For the normal ascent, the vertical velocity achievable by dropping ballast is so
low that power preserved and utilized for propulsion would provide larger ascent velocities.

5. The only two buoyancy conditions liable to experience large roll oscillations are the
normal ascent conditions 6 and 7A. This situation is vitiated by the fact that in actual
operation it will probably be preferable to utilize propulsion for the normal ascent rather
than drop ballast.

PERIOD- SECONDS

Deep-Diving Submarine Hydrodynamics 327

NATURAL PERIODS OF ROLL

OSCILLATION FOR DIFFERENT
BUOYANCY CONDITIONS

Oo I 2 3 4 5 6
ASCENT VELOCITY, fps

Fig. 12. Full-scale computed excitation and oscillation periods

6. The two panic ascent conditions, 8 and 9, are so far removed from resonance that

strong oscillations are extremely unlikely.

EFFECTS OF COMPRESSIBILITY

Pressure hulls of either shallow- or deep-diving submarines deform elastically under
the external pressure of submergence. The associated loss in volume is usually small and

328 P. Mandel

is to some extent compensated for by the compressibility of water. In relatively-thin-hulled
shallow-depth submarines, however, the loss in volume represents a hull compressibility
which is greater than the compressibility of the surrounding sea water. As a consequence
the hull becomes less buoyant as it dives, which is an inherently unstable condition. Thus,
a thin-hulled submarine that is neutrally buoyant near the surface must by some means dis-
charge ballast at deeper depth in order to attain neutral buoyancy again at that depth.

Contrary to this situation, the thick hull of the Aluminaut is less compressible than
water. As a result the boat becomes lighter as it sinks. Thus theoretically it should be
possible when it is desired to dive the Aluminaut to its operating depth to take in a precise
amount of excess ballast so that the Aluminaut has negative buoyancy near the surface.
Without further intake or discharge of ballast the boat should then sink slowly to its operat-
ing depth where it will be in perfect equilibrium if the proper amount of excess ballast was
initially taken aboard.

Particulars are shown in Table 5. The density of water at 15,000-foot submergence is
about 2 percent greater than its density near the surface. Thus an incompressible hull would
gain about 2 percent of its near surface buoyancy at a depth of 15,000 feet. On the other
hand, if a thin-hulled submarine such as are now designed for shallow depths could be
designed somehow to accept 15,000-foot submergence, it would lose about 5-1/2 percent of
its near surface volume at that submergence and thus lose about 3-1/2 percent of its near
surface buoyancy. The Aluminaut pressure hull is closer to the incompressible hull in this
respect and loses only about 0.5 percent of its near surface volume at 15,000-foot submer-
gence. Thus, it gains 1-1/2 percent of its near surface buoyancy at 15,000 feet. However,
the overall compressibility of the entire Aluminaut configuration is considerably greater
than 0.5 percent, primarily because the pressure-equalized silicone fluid in the stern cap-
sule is itself much more compressible than water. It is estimated that the compressibility
of the entire Aluminaut configuration is about 1-1/2 percent; thus it gains only about 0.5
percent of its near surface buoyancy at 15,000 feet. However, this calculation is believed
to be too sensitive to subtle influence to be entirely depended upon, and must be carefully
checked during the initial dive of the boat itself.

Table 5
Effects of Compressibility

Ratio of Water | Ratio of Hull

eee Density at Volume at Gain
Compressibility of Hull Depth and Depth and or loss
oN Water Density | Hull Volume | Buoyancy
at Surface at Surface

Incompressible hull 1% gain

Incompressible hull 2% gain
Hull as compressible as water 0

Hull as compressible as water 0
Thin hull submarine 1.5% loss
Thin hull submarine 3.5% loss
Aluminaut, pressure hull only 1.5% gain
Aluminaut, entire configuration 0.5% gain

Deep-Diving Submarine Hydrodynamics 329

The statements that have been made concerning the compressibility of thin-hull sub-
marines apply with equal verity to vehicles like the bathyscaphes that use light-density
fluids for flotation. Although the pressure hull of the Trieste may also be less compres-
sible than sea water, the fact that the Trieste derives 90 percent of its buoyancy from light-
density gasoline makes its overall compressibility almost as large as that of gasoline. Since
gasoline compresses at a rate about twice that of water, the bathyscaphes, like the thin-
hulled shallow-depth submarines, are unstable in depth.

There is yet another effect of compressibility with the Aluminaut hull in addition to the
effect on net buoyancy at depth. The compressibility of the silicone fluid in the stern cap-
sule of the Aluminaut introduces a significant shift forward in the longitudinal position of
the center of buoyancy between near surface operation and operation at depth. This fact in
conjunction with personnel movements on board was used to help determine the capacity of
the trim tanks that are installed inside the pressure hull.

Because of the contraction of the pressure hull with depth, allowance has had to be
made with attachment of all internal and external mountings to make sure that neither they
are damaged nor the hull is locally restrained. Calculations indicate that at the test depth
of 17,000, the radial displacement of the Aluminaut pressure hull is about 0.172 inch.

MANEUVERING IN 80TH PLANES

Most of the oceanographers who would be the potential users of the Aluminaut empha-
sized to the designers the necessity for precise control in both the horizontal and vertical
planes. Because of the low speed of the Aluminaut, it was apparent that the desired degree
of control could not be achieved by control surfaces alone as it is with higher speed sub-
marines. lor that reason two techniques are employed that are not usually used with sub-
marines in addition to several other more conventional techniques. One is a propeller to
provide thrust in the vertical direction and the other is a swiveling main propulsion pro-
peller to permit directing the thrust in the horizontal plane. These are described in the
subsequent sections.

Control in the Vertical Plane

To permit positive control in the vertical plane at all speeds a four-foot-diameter pro-
peller driven by a 5-horsepower motor is mounted on top of a small superstructure about 15
percent of the length forward of amidship as shown in Figs. 1 and 2. This superstructure,
which is also oil-filled, houses the motor, which is reversible and has a stepped speed con-
trol. With 5 horsepower the propeller can develop about 400 pounds of thrust as shown
subsequently in Fig. 13. This thrust can be directed either upward or downward by revers-
ing the motor. Since the magnitude of the thrust is not dependent on the speed of advance
of the boat as it would be if it were developed by control surfaces, it should provide posi-
tive vertical depth control at any speed of advance.

In addition to the vertical propeller, two additional systems are provided for control in
the vertical plane. The most effective of these, particularly in the speed range of the
Aluminaut, is a conventional hydrostatic trim system that can be used to control pitch angle.
The system as presently designed employs a pump driven by a 1-1/2-horsepower motor that
can transfer 1,000 pounds of water between the two trim tanks shown in Fig. 1 in a period
of two minutes. The moment produced of 26,500 foot-pounds is sufficient to trim the

330 P. Mandel

submerged Aluminaut about 20 degrees against her metacentric stability. As mentioned
earlier, the trim system is currently sized primarily to accommodate the trim unbalance
imposed by the loss of buoyancy in the stern capsule at depth and the movement of person-
nel within the boat. However this system could be readily enlarged after completion of the
boat if it is evident that additional capacity is needed to augment pitch angle control.

Stern control surfaces are also employed partially. to improve directional stability in the
vertical plane, but also to serve as trim tabs to assist in maintaining horizontal flight with-
out constant use of the vertical propeller. It is possible that they may not be able to fulfill
even the latter limited mission in the event that the critical speed (see Section 7 of Ref. 2)
falls within the operating speeds of the Aluminaut. At this speed no combination of hull
pitch angle and stern plane angle can simultaneously balance out both the hydrodynamic
forces and the combination of hydrodynamic and hydrostatic moments that act on any sub-
marine. However, at speeds just slightly removed from the critical value the stern planes
will be effective as trim tabs even though they may have to be used in a sense opposite to
that which intuition would dictate.

Even with the preceding systems for controlling pitch angle and depth, difficulty may
be encountered in maneuvering along an irregular bottom. For this purpose use of a trail
rope such as is used by balloonists for maintaining constant distance above an irregular
land terrain may be helpful. However, there has been little or no experience with this
device with submarines and the forces involved may be too small to be effective. In the
event that difficulties are experienced after the boat is in operation, the alternatives remain
of refining the controls of the vertical propeller or increasing the capacity and refining the
controls of the hydrostatic trim system.

Main Propulsion and Control in the Horizontal Plane

A 15-horsepower dc motor located in the stern capsule and connected via bevel gears
to a four-foot-diameter propeller is used for main propulsion. This motor (as well as the
topside motor for the vertical propeller) has hydrodynamic problems of its own since it must
operate in an environment of very high pressure oil. To improve motor efficiency, both the
rotating as well as the fixed elements must be especially streamlined to minimize hydro-
dynamic drag. Even with these measures a motor efficiency of only about 50 percent is
assumed. It is expected, however, that the motor can be substantially overloaded because
of the excellent heat dissipation into the surrounding medium. As shown in Fig. 13, a maxi-
mum horizontal speed of about 4.7 knots is expected with 15 horsepower.

Stability and control in the horizontal plane is effected by large fixed fins, balanced
rudders (each 6 square feet in area) and provision to swivel the main propulsion propeller
through a total arc of 120 degrees. Both the rudders and propeller pivot on a common verti-
cal shaft as indicated in Figs. 1 and 2. The swiveling propeller by itself should assure
excellent control at any speed in the horizontal plane.

DIRECTIONAL STABILITY IN TOW TESTS

Full-scale operations with the Aluminaut doubtlessly, will at some time require that
she be towed on the surface to the scene of diving operations. Experience in the past with
vessels of similar shape to the Aluminaut has revealed that in some conditions, the behavior

D, DRAG IN POUNDS

1200

1000

800

600

400

200

Deep-Diving Submarine Hydrodynamics

NOTE: CURVES OF CONSTANT SHP ARE
DISCONTINUOUS BETWEEN HORIZONTAL
AND VERTICAL MOTION, BECAUSE OF
LARGE DIFFERENCE IN HULL INTERFERENCE
BETWEEN THESE TWO CONDITIONS

SHIP SPEED, KNOTS

Fig. 13. Resistance and power, ship and shaft

331

332 BeiMandel

of the towed vessel is so erratic that towing becomes difficult and hazardous. It was for
these reasons that tests were carried out with the 1/12-scale Aluminaut model in order to
observe its behavior while being towed on the surface.

Test Arrangements

The Aluminaut model was towed about two model lengths behind a model of an Oceano-
graphic Research ship at the M.I.T. Ship Model Towing Tank. The research ship model was
in turn towed by the erdinary gravity towing arrangement employed at the M.I.T. tank. Thus,
the towing vessel was restrained to straight-ahead motion while the towed vessel was unre-
strained except for the tension on the towing line.

Several towing arrangements were employed:

(a) Ahead tow - single tow line attached to a bracket on the centerline of the forward
deck of the Aluminaut.

(b) Ahead tow - bridle tow lines attached to the extremeties of a one-foot-wide spanner
fixed to the forward deck of the Aluminaut.

(c) Ahead tow - bridle tow line attached to the leading edges of the Aluminaut bilge
keels.

(d) Astern tow - single tow line attached to upper rudder of the Aluminaut.

Tests were conducted with both the small and the large stern planes and rudder
described in Table 4. During all tests the model was equipped with the wide bilge keels.
Limited tests were also conducted with the stern fins removed.

Test Results

Distinct differences in the behavior of the towed Aluminaut model were observed with
the different towing arrangements. The Aluminaut model was initially set directly behind
and in line with the towing model. Upon release and movement of the towing model, the
following situations could be observed:

Characterization

ipti Behavio i
Jess EE Eee of Behavior

1. The Aluminaut model would veer off Unsatisfactory
course and collide with nearest tank
wall.

2. The model would initially veer off Unsatisfactory

course, but return and cross its initial
position and collide with the opposite
tank wall.
3. The model would initially veer off Satisfactory
slightly to one side, but within the
length and width of the tank, it would
tend to return to a stable equilibrium
position.

Deep-Diving Submarine Hydrodynamics 333

Characterization

Description of Behavior :
a as Pai ld of Behavior

4. The model would remain in a stable Satisfactory
equilibrium position throughout the
run.

Tests with the small stern fins yielded the following results:

Towing Characterization
: Comments
Arrangement of Behavior aan
(a) Unsatisfactory
(b) Satisfactory Equilibrium position

slightly to port of
initial position

(c) Satisfactory Equilibrium position
slightly to port of
initial position

(d) Satisfactory

Similar results were obtained with the large stern fin tests except that with towing
arrangements (b) and (c) the model veered more to port and the equilibrium position was
further to port. In order to help ascertain the cause of this distinct bias, configurations (b)
and (c) were tried with the model set initially heading to starboard. The model still turned
to port and established a stable equilibrium position on the port side of its initial position.
Subsequently, the model was also tested with the stern fins removed entirely. In this condi-
tion, towing arrangements (b) and (c) became unstable, although no bias either to port or
starboard was evident. It was concluded from these tests that both the original and enlarged
rudders were not set exactly parallel to the center line of the model, and that the bias to
port which was even more evident with the enlarged rudders was due to the inaccurate initial
setting of the rudders.

Application to Full-Scale Towing

The model results indicate that towing arrangements (b) and (c), or (d), would be satis-
factory for the full scale. From a practical point of view, (c) or (d) appear preferable to (b)
which involves the fitting of a large spanner subjected to very heavy cantilevered loads.

The bias that the model possessed for the port side is considered of little consequence

since the zero position of the full-scale rudders can be easily adjusted. It was concluded
from the tests that satisfactory towing of the Aluminaut on the surface could be accomplished
with at least two simple towing arrangements.

SUMMARY OF PAPER

The contrasts between the Aluminaut and both the previous generations of deep-diving
vehicles and conventional military submarines are emphasized in this paper. In the field
of hydrodynamics, special problems arise with deep-diving submarines that require different
ballasting systems as well as different devices for effecting superior control at low speeds.

646551 O—62—_23

334 P. Mandel

In addition, the oscillations arising from rapid vertical ascent or descent and the effects of
compressibility become nonnegligible for deep-diving submarines. It is believed that the
compromises made in the design of the Aluminaut effect reasonable solutions to each of
these problems.

In particular, the use of iron shot as ballast appears to be a satisfactory substitute for
water ballast and pumps at very deep depths. The possibility still exists, nevertheless,
that the iron shot may prove to be redundant for normal ascents and it may prove feasible to
abandon its use. In that event its equivalent weight would be incorporated in solid ballast
for use in emergency ascents.

Model tests have reasonably confirmed theoretical predictions of the measures needed
to avoid large oscillations in vertical ascents. In particular, the tried and true measures of
avoiding resonance and incorporating damping devices such as bilge keels promise to yield
satisfactory solutions. It has been demonstrated that in the panic ascents where it would
be particularly desirable to avoid large amplitudes of roll the tuning factors are in fact far
removed from resonance.

The fact that the overall configuration of the Aluminaut is somewhat less compressible
than sea water yields the desirable characteristic of stability in depth. This is in contrast
to thin-hull submarines and earlier bathyscaphes that must drop ballast to achieve neutral
buoyancy at any depth below the surface.

The use of a vertical propeller augmented by a hydrostatic trimming system as well as
stern planes is expected to provide satisfactory depth and pitch angle control at all speeds.
The possibility of using a trail rope for navigating over the bottom remains to be explored.
Excellent control in the horizontal plane should be achieved by the ability of the main pro-
pulsion propeller in conjunction with the rudders to swivel through an arc of 120 degrees.

Model tests indicated that satisfactory towing of the Aluminaut on the surface could
probably be accomplished either by towing it stern first or by towing it bow first with a
bridle arrangement attached to the leading edges of the bilge keels.

Taken as a whole, the Aluminaut promises to open a new era in ocean exploration and
ocean utilization. While the primary problems associated with its development lie in the
field of structures, this paper shows that careful consideration has also been given to its
potential hydrodynamic problems.

ACKNOWLEDGMENTS

The author is indebted to the Reynolds Metals Company of Richmond, Virginia, and the
Southwest Research Institute of San Antonio, Texas, for sponsoring his work in connection
with the Aluminaut and for giving him permission to use the results contained in this paper.
In addition, the majority of the figures contained in this paper were prepared at Southwest
Research Institute.

In particular, the efforts of Mr. Ernest Brunauer of Southwest Research Institute are
acknowledged. It was largely he who shouldered the burden of carrying out the vertical
ascent tests at the Naval Ordnance Laboratory. In this connection the efforts of the staffs
of the Underwater Weapons Tank Facility, the Office of Naval Research, and the U.S. Naval
Underwater Photographic Unit are acknowledged. Mr. Ernst Frankel, Research Assistant at
M.I.T., conducted the towed directional stability tests.

Deep-Diving Submarine Hydrodynamics 335

But, it is to Dr. Edward Wenk, Jr., formerly of Southwest Research Institute, that the
author is principally indebted. Dr. Wenk enlisted his work on this project and it is largely
his enthusiasm, initiative, and drive that have carried the project through the difficult
formative stages.

REFERENCES

[1] Wenk, Dehart, Kissinger, and Mandel, “An Oceanographic Research Submarine of Alumi-
num for Operation to 15,000 feet,” Royal Institute of Naval Architects, Mar. 23, 1960

[2] Arentzen, E.S., and Mandel, P., “Naval Architectural Aspects of Submarine Design,”
Trans. Soc. Naval Architects Marine Engrs., Vol. 68, 1960

[3] Piccard, A., “Earth, Sky, and Sea,” New York: Oxford University Press, 1956

[4] Relf, E.F., and Simmons, L.F.G., “The Frequency of Eddies Generated by the Motion
of Circular Cylinders through a Fluid,” British Advisory Committee for Aeronautics,
R. and M. 917, 1924

[5] Landweber, L., “Flow About a Pair of Adjacent Parallel Cylinders Normal to a Stream,”
David Taylor Model Basin Report 485, July 1942

[6] Marks, L.S., “Mechanical Engineers Handbook,” 6th edition, New York: McGraw-Hill,
1958, pp. 11-84, Fig. 21

[7] Hoerner, S.F., “Aerodynamic Drag,” Midland Park, N.J., 1951, pp. 21-25 (The 2nd edition,
y 8 P
published in 1958, is titled “Fluid-dynamic Drag”)

[8] McGoldrick, R.T., “A Vibration Manual for Engineers,” 2nd edition, David Taylor Model
Basin Report 189, Dec. 1957 (obtainable from Office of Technical Services, Dept of
Commerce, PB131785)

APPENDIX

The data plotted in Fig. 3 for the elementary body of revolution shapes are based upon
the expressions for the dimensional and nondimensional volume and wetted surface of
simple cylinders, cones, and hemispheres as follows:

Cylinder Hemisphere Cone
Dimensional Volume, V yale = iD ke
Nondimensional Volume, C, 1.0 2/3 1/3
2
Dimensional Wetted Surface*, S 7DL 5D? GDL: art 4
2
Nondimensional Wetted Surface, C, 1.0 1.0 - > + 4
t

*Excluding area of flat ends in these cases.

336 P. Mandel

Where: L = length

L, = length of cylinder (parallel middle body)
L, = length of single conical end
D
V

= diameter of cylinder, hemisphere, or base of cone

= volume
Cp = prismatic coefficient =
2p 2p,
4
S = wetted surface
S
= tted f ffici ——————
Ge wetted surface coefficient ADL

When these basic shapes are combined together with either the cones or the hemispheres
forming the ends, the sums of the wetted areas as expressed above form the total wetted
surface including that of the ends. The total Length L is then the sum of L, plus the
length of both ends. It then becomes convenient to introduce the nondimensional expressions

L,’ = L,/L and L,’ = L,/L. With these definitions the following relationships apply:

Hemispherical Ends:

aL ing) Li y3

rae ree 1

Bin miganaiad fy « eo
q Pel

Rit

TDL, + 7D2 |
Lege are rae

Conical Ends:

| OIE) BPS) DONE So) 2
TT TT
qdLx +@ DL: 4
ej
— D2,

4

7 D2
MDL, + > DL, Ee 4 1./D2 Fs
C, Se) ae =l- 2L 4’ + Dy L2 + 4L,

Hemispherical nose, Conical Tail:

D 1 D
L,=L-L,->; L,! = 1-Le -aF

TT TT

—D2L +—~ D3 +s Dats!

ihe Beteaiss al EA eum dani a jbo 2/3! =
4 D?L

2
nDL, +% D2 +4 DL, You + 4 -
cn = z 2 be =1-L,’ +~>5+ 4L,'?
age FR EDL STI Om ane: TCM POT IONe whet

ale
Sls

Deep-Diving Submarine Hydrodynamics 337

Flat Ends: PLS Le, = Eine ea 0
DL, + yD? an
Cat, a nDied aot yRikae

For all configurations the wetted surface can be expressed in terms of the volume

V, Cp, Cs, L/D as follows:

e

1/3 2/3
5)

S.C. fin

Cursory inspection of all of the preceding relationships will indicate the wisdom of
leaving them in the parametric form in which they are expressed. Since both Cp and Cs
are expressed in terms of L,’, it could be eliminated as a parameter and the wetted surface
could then be expressed directly as a function of C, and L/D for any given volume. The
resulting expressions would, however, be extremely ponderous.

DISCUSSION
E. C. Tupper (Admiralty Experiment Works)

My remarks are not so much a question directed to the author, but rather a request to
him that he might study certain problems associated with the control of this submarine when
it is in operation. I think it is desirable to go a little into the background of the problem to
bring out the points I wish to make. There are two points of great interest in the control
of an underwater body in the vertical plane. So as not to become involved in any problems
of nomenclature, | will call these points A and 3.

Point A is a fixed point in the submarine which would typically be about a third or a
quarter of the length from the bow. Ignoring transient effects, a vertical force applied at
point A would cause the submarine to rise or dive while still maintaining level trim. Point
B moves with changing speed; at very high speed it is almost coincident with point A, but,
with decreasing speed, it moves aft until, at zero speed, it is infinitely aft of the sub-
marine. Again, ignoring transient effects, a force applied at point B will cause a submarine
to change trim but not to change depth. The author did mention that the action of these
after hydroplanes might be reversed at low speed. This will occur at the speed at which
point B coincides with the position of the after hydroplanes. At that speed, the after hydro-
planes can only change the trim and not depth. A downward force applied ahead of point B
would cause an increase in depth and a downward force applied after point B would cause a
decrease in depth so that, in passing through the speed at which point B coincides with
these after hydroplanes, there will be a reversal in the effect on depth of using the planes.

I would like to suggest to the author that it would be interesting to study, when he has
the submarine in operation, whether the vertical propeller, which would seem to be almost
at point A, does in fact control depth without significant effect upon trim, and also to study
the actual speed, which may well be about 2-1/2 knots, at which the reversal of the after
hydroplanes’ influence on depth occurs.

338 P. Mandel

Serge G. Bindel (Bassin d’Essais des Carenes, Paris)

I was very interested in the results given by Prof. Mandel, and particularly in those
concerning the rolling oscillation during vertical ascent; they confirm the results obtained
in the Paris Model Tank when studying the bathyscaph of the French Navy.

I should like, however, to make one remark regarding the dynamic stability of such a
ship during vertical motion. There is a problem for ascent, but there is generally no problem
when diving. This fact may be due to the presence of the bridge fairwater, which plays a
nonsymmetrical role in the two directions of the motion, even if, as in the present case, its
height is not too large. The effect of the fairwater is, in some manner, like this of a longi-
tudinal fin on the stability on straight course for a surface ship; when diving, the fairwater
is aft and its effect is favorable; in the contrary, when ascending, the fairwater is fore and
its effect is unfavorable. The problem is, of course, more complicated than for a surface
ship, because of the existence of a static positive stability. Therefore, if Karman’s vor-
tices are responsible for the excitation, my opinion is that it is necessary to take into
account the dynamic stability of the motion and, when designing such a submarine, to
avoid, if possible, a too developed bridge fairwater.

Owen H. Oakley (U.S. Bureau of Ships)

I couldn’t resist commenting on Mr. Mandel’s excellent paper because he and I were
involved together in the business of submarine design for a number of years. I would like
to make only one comment and that with respect to the comparison of the Aluminaut with the
Trieste. Mr. Mandel noted a ratio of one to ten in the internal volumes of the two craft in
favor of the Aluminaut, but that is really not quite fair. One thing should have been empha-
sized more, and that is that the Aluminaut is good for about 15,000 feet of submergence,
where the Trieste is good for some 37,000 feet. ‘The problem of density of structure required
for the deeper submergence comes into the matter very strongly, as does the basic difference
in the means of providing buoyancy. Mr. Tupper covered a point very nicely that I wanted to
touch upon, namely, what is the critical speed, and I would like to inquire whether an attempt
was made to estimate this.

P. Mandel

Mr. Oakley has pointed out an oversight in the oral presentation of the paper. The fact
that the Trieste is capable of diving to more than twice the depth of the Aluminaut places
some very severe constraints on her design that are not nearly as limiting for the Aluminaut.
I have more amply covered this point in the written text of the paper.

I have not attempted to predict the critical speed for the Aluminaut. The critical speed
is extremely sensitive to the magnitude of the hydrodynamic vertical force and moment due
to asymmetry, designated Z* and M* in Ref. 2. There is no known method of analytically
predicting this force and moment and attempts to rationalize experimental values have been
unsuccessful. In any event there is little necessity for accurately predicting the critical
speed in the case of the Aluminaut since it is intended to place primary reliance on the
vertical propeller and on the hydrostatic trim system for control in the vertical plane.

With respect to the oscillations in dive as well as in ascent, the Aluminaut can only
descend by means of overweight at a very modest speed. As shown in Table 2, the

Deep-Diving Submarine Hydrodynamics 339

Aluminaut can only be ballasted to be some 500 pounds overweight and with that overweight
its descent to 15,000 feet would take some four and one-half hours. Therefore, the vertical
descent shown in Fig. 12 corresponds to a very modest speed where oscillations do not
occur, and where the conditions are well removed from resonance. This is why we were
primarily concerned with the ascent condition in the investigation.

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SUBMARINE CARGO SHIPS AND TANKERS

F. H. Todd
National Physical Laboratory

INTRODUCTION

The naval architect has always had to battle with the twin elements of wind and water,
and these have provided very powerful barriers to any great increase in the speed of ships.
Because a ship is floating at the boundary between two elements of very different densities,
it creates waves on the surface as it moves and any increase beyond a certain speed leads
to an excessive increase in the wavemaking resistance and consequently to an uneconomic
rise in costs of construction and operation. This situation may be thought of in some
respects as being analogous to the “sonic barrier” in aircraft design. Much research has
been carried out into the reduction of wavemaking resistance, both by experiment and theory,
and this has led to the adoption of very fine waterlines forward, U shaped sections and bulb-
ous bows, which has enabled the speed of surface ships to be raised appreciably in the
course of the years.

The effect of the wind is not only to increase the drag of the ship by its direct effect,
but by creating rough seas or storm conditions forces the ship to slow down to avoid the
excessive pitching and heaving motions and the damage to the structure which would even-
tually ensue. Such slowing down means loss of time or subsequent expenditure of extra
power to make up schedules, since in general no owner wishes to do other than make full
use of a very expensive investment. Again, research has done much to help the naval
architect combat such conditions. The adoption of fine entrances, high freeboards, and good
flare above water have, for example, enabled designers to improve the seagoing qualities of
trawlers almost beyond recognition, while the perfecting of antirolling fins has reduced this
motion to almost negligible amount in those ships fitted with them. There remains the prob-
lem of reducing pitching by similar means, and the bulbous bow does present some advan-
tages in this way also.

The naval architect may well have envied his aeronautical colleague who, faced with
similar weather problems, has taken the modern aircraft to very high altitudes where atmos-
pheric conditions are calmand stable. He has long been aware of the possibilities of lifting
a ship above the water surface on foils to reduce its own wavemaking and avoid the surface
wind-generated waves, and of the advantages of taking a ship below the surface to escape
both rough weather and wavemaking resistance.

The perfecting of the internal combustion engine, which enabled the aeronautical
engineer to achieve his desires by giving him a source of power having a high power/weight
ratio and a relatively high efficiency, also allowed the naval designer to build hydrofoil
boats which can attain the same speed as a high-speed conventional motor boat for about
half the power. These are not new in conception, for the first such hydrofoil craft “flew” in
Italy as long ago as 1906, attaining a speed of some 38 knots [1]. Of recent years the inter-
est in hydrofoil boats has greatly increased, many new foil configurations have been evolved,

341

342 F. H. Todd

and passenger-carrying hydrofoil craft are now in use in a number of sheltered waters. It is
natural that efforts should be made to extend this principle to larger ships, but the difficul-
ties in the way are considerable —the foils and their supports become more massive and
heavy, detracting from the deadweight carrying capacity, they are a source of trouble due to
the large draft they cause when the boat is on the surface coming into port, and since they
are essentially high speed craft the large powers necessary must be transmitted from the
hull to propellers a long way below the hull. Studies of such designs are going on, and we
shall doubtless see much larger craft of this type built for specialised services, but their
use for carrying the bulk cargoes of the world seems unlikely. Moreover, such boats still
have to contend with the ocean waves on the surface of the sea, and therefore for any given
size have limitations as to the sea state in which they can operate. By going below the
surface, both wavemaking resistance and bad weather can be avoided altogether, and with
the advent of nuclear propulsion the naval architect for the first time is in a position to
seriously contemplate such a procedure. The use of nuclear reactors as a source of heat
has freed the ship from dependence on the atmosphere and made possible the true submersi-
ble, which would spend all its seagoing time deeply submerged and only surface near the
ends of its voyages. The hull can therefore be shaped for minimum resistance under deep-
submerged conditions, with little concern as to its surface performance. The benefits to be
gained are, of course, the elimination of surface wavemaking resistance, the reduction of
form resistance to a minimum by streamlining the hull, and escape from the effects of bad
weather.

The availability of nuclear propulsion for ships, as in the case of other new inventions,
has led to a search for useful and practical applications. So far as the navies of the world
are concerned, there is no doubt as to the value of nuclear propulsion in submarines to
enable them to remain submerged almost indefinitely and in other ships in order to maintain
fleets at sea for long periods without refueling. Nuclear-propelled submarines, apart from
their military use, could be of extreme value for the delivery of fuel and supplies to com-
batant ships or to beleaguered islands or ports. From the national safety point of view,
also, submarine tankers and cargo ships would be of inestimable value because of their
relative immunity from enemy attack. A considerable number of military nuclear-propelled
submarines are now in commission and it may well be that the first nuclear-propelled sub-
marine supply ship will also be built by one of the navies of the world. The Russians,
because of their particular geographical problems, have made the first application of nuclear
propulsion to a surface ice-breaker which can, as a result, stay at sea for long periods to
keep the lanes open throughout the winter.

The spectacular voyages of the USS NAUTILUS and SKATE under the North Pole in
1958 have also shown that submarines can be navigated for long distances under water
without having to surface for sights of the sun or stars. This fact has led to much specula-
tion regarding the possible use of such ships on routes which are closed by ice for part or
all of the year.

As we shall see later, the submarine ship must operate at fairly high speeds to show
material advantages over the surface ship, perhaps 30 knots or above. This fact may have
some attraction to passengers who suffer from travel sickness but enjoy the restful atmos-
phere of a ship, and limited passenger accommodation may become a feature of such vessels.
True, the passengers would not see the sea, but neither do many travellers today, who
divide their time between dining saloon, bar, and cabin, and there is certainly no less to see
than in the upper atmosphere!

When we come to consider such ships for commercial use, however, a number of operat-
ing problems are introduced, and it is the purpose of this paper to review these and, it is

Submarine Cargo Ships and Tankers 343

hoped, to stimulate discussion and draw forth new ideas. One of the main advantages of
nuclear-propelled ships is the saving in space no longer required for bunkers, and the con-
sequent increase in carrying capacity. The weight of the shielding around the reactors
cancels out much of the saving in weight by the elimination of fuel, and this would suggest
that such machinery is most suitable for long-haul voyages. The average dry-cargo ships
have been developed largely along the requirements of certain trades, and these are not
likely to alter materially in type. Moreover, the oil fuel is very often carried in the double
bottom, and such space is not of use for additional cargo. These considerations lead to the
general conclusion that the ships most suited to nuclear propulsion are those carrying bulk
cargoes such as ore, grain, and oil. In the case of the latter, with a liquid cargo the ease of
handling is such that the ships could just as easily be submarines as surface ships, and it
is this aspect of nuclear propulsion which seems to have caught the public imagination.

Economic nuclear propulsion in the strictly commercial sense is as yet some time away,
but if it is eventually to be applied to submarine cargo ships, then there are without doubt
many difficult problems to solve, and it is certainly not too early to begin research into them.
if we are to be ready to design and build such ships.

HYDRODYNAMIC ADVANTAGES OF SUBMARINE SHIPS

Although this symposium is supposed to deal with the field of naval hydrodynamics, it
would not be realistic to treat this particular problem purely from that standpoint. There are
so many practical limitations which would come into the design of a submarine cargo ship or
tanker that they must be given consideration, as only in this way can we give due weight to
all the conflicting claims. In this section the question of powering will be considered and
estimates made for both surface and submarine ships. These will be used later in the paper
as a basis for discussion when we come to deal with the problems of operating and main-
taining such craft.

From the hydrodynamic point of view, the two greatest savings that we can expect by
going under water are the elimination of the wavemaking resistance and the escape from the
effects of rough weather. Owing to the good hull form which one can design for a real sub-
mersible, it is reasonable to expect that the form resistance would also be reduced.

The wavemaking resistance is a relatively small part of the total resistance for surface
ships at low speeds, and therefore one would not expect any great savings in power by its
elimination at low speeds of operation. Indeed, the submarine under such conditions begins
with a considerable handicap in that the wetted surface will be considerably greater than
that of the equivalent surface ship and therefore the skin-friction resistance of the sub-
marine will be considerably higher. Only if this can be offset by a corresponding reduction
in form resistance will the submarine be able to break even with her surface counterpart. At
higher speeds the wavemaking resistance of the surface ship begins to increase very rapidly
(in the order of V°) and to attain these higher speeds a continual fining of the hull is neces-
sary together with an increase in ship dimensions in order to carry a given amount of dead-
weight. Eventually we reach a speed at which it is no longer economic to drive the surface
ship, and it is here that the submarine would show a very real advantage in terms of power.
We can thus conclude on purely general grounds that the elimination of wavemaking resist-

ance by using a submarine ship will only begin to give substantial returns when used at
comparatively high speeds.

344 F. H. Todd

In order to have some actual figures on which to base a discussion, calculations have
been made of the horsepower required for both surface and submarine ships of displacements
from 25,000 tons up to 150,000 tons. From these figures estimates have been made of the
powers required for surface and submarine ships of the same deadweight, as this is the only
fair way of assessing their respective merits. This particular range of displacement was
chosen because the lowest figure represents the smallest ship for which marine nuclear
installations at present envisaged would be suitable and the highest figure is comparable
with the largest tankers at present being built and having deadweights in the neighbourhood
of 100,000 tons.

For the surface ships the results have been taken from the David Taylor Model Basin
Series 60 [2]. For a number of displacements within the above range the EHP has been cal-
culated for Series 60 models having block coefficients of 0.60, 0.70, and 0.80. The ships
all have a length to beam ratio of 6.5 and a beam to draft ratio of 3.0. The ship powers have
been estimated from the model results using the A.T.T.C. 1947 line with a correlation allow-
ance AC of +0.0004. In designing ships of different speeds, it would be natural to use a
finer block coefficient for the higher speeds, and the comparisons have therefore been made
at a speed appropriate to each fulness. These speeds have been determined from the
modified Alexander formula

V
MV. 2 (1.06 = Cp).

In order to convert these EHP values into DHP at the propeller, it is necessary to make
certain assumptions regarding the propulsive efficiency and appendage allowances. In the
first place, it has been assumed that where the DHP is less than 40,000, the ship will be
propelled by a single screw, where it is between 40,000 and 80,000, it will have twin
screws, and above this higher figure, will have four propellers. For the single screw ships
a quasi-propulsive coefficient of 0.72 has been used, in accordance with the experiments
with Series 60, and no allowance has been made for appendages. It is assumed, in effect,
that with modern large ships of all-welded construction, the allowance of AC; = +0.0004
will be sufficient to take account of any ordinary appendage resistance as well as any hull
roughness effects. In such cases, therefore, the DHP has been taken as equal to the EHP
divided by the QPC. For the twin and quadruple screw ships, the QPC has been taken as
0.68 and an appendage allowance of 10 percent and 20 percent respectively has been added
to take account of the bossings or shafts and A-brackets in these arrangements.

For the submarines, estimates have been made for two different prismatic coefficients,
namely 0.60 and 0.65, using various published works by Weinblum, Amtsberg, Crago, etc.
These apply to streamlined bodies of revolution having their maximum diameter at a point
40 percent from the nose. Again, the skin-friction resistance of the ship has been estimated
by using the A.T.T.C. 1947 line and including an allowance of AC = +0.0004. The
length/diameter ratio has been taken as 7.0, which is approximately the optimum for vessels
of this form having the appropriate tail surfaces to give adequate directional stability.

The dimensions of the surface ships and submarines for the displacement ranges
covered are shown in Tables 1 and 2 and Fig. 1, and charts of EHP/V® in Figs. 2 through 6.
The EHP values for submarines of different displacements are also listed in Table 3, and
those for surface tankers in Table 4. This latter table also includes estimates from NPL
data, which show close agreement with those made from the Series 60. As will be seen in
making the comparison for vessels of equal deadweight, the use of a circular submarine form
leads to very unrealistic dimensions as regards draft. In order to be able to assess the

Submarine Cargo Ships and Tankers 345

Table 1
Dimensions of Surface Tankers

L/B = 6.5; B/d =.3.0; V/\/Lpp = 2(1.06 —:Cg)

EHP in-
Lpp (ft) B (ft) d (ft) V/V L pp V/V Ly ie cluding
+ 0.0004

= 0.60; A/(L/100)® = 135.3; L = 100 #A/5.14

25,000 : : 58,000 : : : 17,800
50,000 5 3 92,000 : : : 39,200
75,000 g : 120,600 : ; , 62,200
100,000 ‘ 146,200 5 : d 87,150
125,000 . 166,000 5 5 ; 110,000

Cp = 0.70; A/(L/100)? =. 160; L = 100 $A /5.43

57,300
91,000
119,500
144,800
168,000

Cz = 0.80; A/(L/100)? = 180.4; L = 100 ¥A/5.65

25,000 : : 57,000
50,000 5 : 91,200
75,000 : 120,000
100,000 : ; 144,800
125,000 ; 168,000

346 F, H. Todd

Table 2

Dimensions of Submarines with Circular Cross Section

Wetted Surface (sq ft)

25,000 : 65,200
50,000 : 104,000
75,000 : 136, 000
100,000 164,600
125,000 191,200
150,000 ‘ 216,000

25,000 4 65,400
50,000 : 104,000
75,000 ; 136,200
100,000 i 164,800
125,000 : 192,100
150,000 : 216,000

IN FEET

LENGTH

1,000

300

800

700

500

400

Submarine Cargo Ships and Tankers

25000 50000 75000 100000 125000
DISPLACEMENT IN TONS, A

Fig. 1. Dimensions of surface ships and submarines

159.9000

YSL3aWVIG

1333 Ni

347

348

F. H. Todd

idk
ak

dod

L \
=

30
SPEED IN KNOTS

Fig. 2. EHP values for surface ships relative to speed

Submarine Cargo Ships and Tankers 349

A IN
TONS

150,000

(25,000

—

100,000

V KNOTS

EHP VALUES FOR 0-60 Cp SUBMARINE
EHP VALUES FOR 0-65 Cp SUBMARINE --------

Fig. 3. EHP values for submarines relative to speed

646551 O—62——24

350 F. H. Todd

4-0

anu

aoe
e
sy

=

=
BN

sume”

xy
mee

ie) 25,000 50,000 75000 {100,000 {25,000 150000
& TONS

Fig. 4. EHP values for surface ships relative to displacement

Submarine Cargo Ships and Tankers 353

Table 3
EHP Values for Submarines with Circular Cross Section

(based on A.T.T.C. 1947 line + 0.0004)

EHP (for displacement in tons)*

ht 25,000 50,000 75,000 100,000 | 125,000 | —_ 150,000
= 0.60

1,194 1,870 2,407 2,885 3,335 3,748
8,960 14,000 18,120 21,700 25,080 28,180
29,040 45,500 58,920 70,530 81,480 91,540
67,400 105,100 135,900 163,500 188,900 212,200
128,800 201,800 261,500 312,200 360,500 407 ,300
220,100 342,800 443 ,200 533,246 615,800 691,600

1,236 1,920 2,498 2,985 3,450 3,865
9,260 14,430 18,750 22,400 26,000 29,130
30,200 47,000 61,000 73,300 85,000 95,120
69,800 109,000 141,500 169,000 196,000 219,500
134,000 208,700 271,000 325,000 377,000 422,000
227,600 356,500 462,000 552,000 640,500 717,000

* Figures below lines require DHP of more than 200,000.

354 F. H. Todd

Table 4
Comparison Between EHP Values for Surface Ships
Derived from NPL Data and Series 60

25,000 17,820 17,800
50,000 39,360 39,200
75,000 62,500 62,200
100,000 87,440 87,150
125,000 110,710 110,000

25,000
50,000
75,000
100,000
125,000

25,000
50,000
75,000
100,000
125,000

* Model results extrapolated by Froude frictional coefficients, no allowance.
t Model results extrapolated by A.T.T.C. 1947 line with an allowance ACp = +0.0004.

Submarine Cargo Ships and Tankers 355

effects of this, estimates have also been made for submarines having elliptical cross sec-
tions in which the maximum beam was four times the maximum draft. For a circular-section
submarine the draft could also be decreased by an increase in the length/diameter ratio, but
this would involve a progressively greater increase in drag and also a considerable penalty
in the form of extra hull weight.

In order to convert the EHP values for the submarines to DHP, the same assumption as
for surface ships has been made regarding the range of powers for one-, two-, and four-screw
arrangements. The QPC has been assumed to be 0.80 for the single-screw designs and 0.67
for the twin- and quadruple-screw. The propulsive efficiency for the single-screw arrange-
ment has been taken somewhat higher than that for the corresponding surface ship because
of the better wake conditions attained behind the body of revolution form. For appendage
resistance, 20 percent has been allowed for the conning tower, rudders, stern and bow diving
planes, flooding holes, and similar items not present on a surface ship, a further 10 percent
for bossings or shaft brackets on the twin-screw ships, and 20 percent for the quadruple-
screw ships. These figures give total allowances of 20 percent, 30 percent, and 40 percent
for the one-, two-, and four-screw arrangements respectively.

The above figures for the submarine are based on the assumption that it is sufficiently
deeply immersed below the surface that there is no residual wavemaking. This means as a
rough guide that it is immersed to a depth of at least half its length or some 4 or 5 diam-
eters. The effects of depth of submersion have been investigated by model experiments in
the Saunders-Roe tank. Models representing an 80,000-ton-displacement submarine were
run at various depths below the surface from 100 to 300 feet and over a corresponding ship
speed range of 20 to 50 knots [3]. The values of the resistance for a length/diameter ratio
of 7 are shown in Table 5 and plotted in Fig. 7. There is certainly some depth effect still
in evidence at the deepest depth of submersion, namely 300 feet, as is shown by the in-
crease in resistance with speed at that depth. As the depth is decreased it is seen that the
increase in resistance for the lower speeds does not increase very rapidly until the depth
reaches something approaching 100 feet. For the higher speeds, however, the increase is
much more rapid and for 50 knots, for example, the increase between 300- and 100-foot sub-
mersion is no less than 200 percent. It is therefore evident that if we are to obtain the full
benefit from the elimination of wavemaking resistance any large submarine of this type must

Table 5
Effect of Depth of Submersion on Resistance
for a Submarine of 80,000-Ton Displacement, L/D = 7, and Maximum
Section 40 percent L from Nose (from Ref. 3)

Speed (knots)

Saicesteheta cle DRL MM AS wl 50k, |

Values* of R/V? x 10-4

(ft)

*R = resistance in pounds; V = speed in knots.

356 F. H. Todd

0-5

R IN POUNDS
VIN KNOTS

0-4

ae 10

300 209 fete) (eo)

DEPTH OF SUBMERSION IN FEET,
MEASURED TO CENTRE LINE OF SUBMARINE

Fig. 7. Effect of depth of submergence on a
submarine of 80,000-ton displacement, L/D
= 7.0 (from Ref. 3)

run at a considerable depth below the surface. From these same model experiments Mr. Crago
has shown that if the submarine is running in shallow water there can be a serious bottom
effect also. For the same submarine of 80,000-ton displacement, for example, at 50 knots in
water 300 feet deep, the resistance is some four times as great as in deep water when deeply
immersed. For a ship of this size this figure is somewhat academic, as no one could ever
contemplate driving such a ship at 50 knots in such a depth of water! But the results serve
to indicate the problems that are likely to be met with when a large submarine is approaching
shallow water on the continental shelf or when coming into estuaries. In fact, for a vessel of
this size we must revise our ideas of what we mean by shallow water.

Before leaving this question of resistance, it is worth while pointing out that in a sub-
marine of this type the resistance would be almost wholly frictional, and to obtain the full
benefit in power reduction it would be essential to keep the hull clean at all times, since
any penalty due to roughness and fouling would be considerably greater than in the corre-
sponding surface ship. This points to the need for frequent dockings, and as we shall see
when discussing the operation of such a ship, this poses considerable problems. For a ves-
sel of this type running deeply submerged in the comparatively calm conditions a long way
below the surface of the sea, it is interesting to consider the possibility of maintaining
laminar flow over the hull to a greater or lesser extent. The benefits to be derived from
such a possibility are, of course, great. For a submarine 650 feet in length with a dis-
placement of 75,000 tons running at a speed of 20 knots, the corresponding Reynolds number
is 1.7 x 10°. The values of Cp for turbulent and laminar flow are 0.00144 and 0.00010
respectively, and if we assume that the laminar flow hull is perfectly smooth and omit the AC,
allowance of +0.0004 in this case, but add 20 percent in each for conning tower, etc., then
the ratio of total resistance for laminar and turbulent flow is 0.25. This is an ideal condi-
tion, of course, for a number of reasons. To encourage laminar flow over the hull, the shape

Submarine Cargo Ships and Tankers 357

would have to be modified, in particular by moving the maximum diameter further aft, and
this would increase the form drag. The hull would have to have certain fittings like conning
tower, bow planes, flooding holes, anchors, etc., and no matter how these might be housed
in recesses with covering doors, etc., they would inevitably act to some extent as turbulence
stimulators, and the actual resistance would in fact be much nearer to the turbulent figure.
While the idea of maintaining laminar flow is therefore extremely attractive, the practical
difficulties of maintaining the requisite smoothness over the whole surface of a steel struc-
ture of this kind immersed in salt water and subject to all the effects of corrosion and foul-
ing present almost insuperable difficulties. Of recent months other devices to maintain
laminar flow have been suggested, such as the use of a soft skin of rubber with liquid back-
ing which would damp out the onset of turbulence and so prevent transition. Not enough is
yet known of these ideas to be able to express any real opinion about them but they would
certainly be expensive both to fit and to maintain and, unless they were immune to surface
deterioration and fouling, it seems would soon lose their efficiency.

COMPARISON OF SURFACE AND SUBMARINE SHIPS OF THE SAME DEADWEIGHT

A considerable number of papers has been written giving comparisons between conven-
tionally propelled and nuclear propelled surface tankers. These have shown in general that
the nuclear propelled tanker cannot yet compete economically with one having conventional
machinery unless the capital cost of the nuclear plant and the cost of nuclear fuel become
considerably less than their present values. This paper is not concerned with the relative
merits of conventional and nuclear machinery, but since the submarine cargo ship or tanker
can only be contemplated on the basis of nuclear propulsion, any comparisons made are on
the basis of nuclear machinery for both ships. In this way the weight of the machinery will
be comparable in both cases and will therefore affect the deadweight available equally in
each case.

Several authors have given values of the deadweight/displacement ratio for nuclear
propelled surface and submarine tankers, and values of this ratio are shown in Figure 8.
Although there is considerable scatter in this diagram, it is considered that if we take a

+++ SURFACE TANKERS
© ©© SUBMARINE TANKERS

ASSUMED RATIO FOR
SURFACE TANKERS

ASSUMED RATIO FOR
SUBMARINE TANKERS

) 20,000 40.000 60,000 80.000 {00,000 120000
DISPLACEMENT IN TONS (A)

Fig. 8. Deadweight /displacement ratios

358 F. H. Todd

value of deadweight to displacement ratio equal to 0.60 for submarine tankers and 0.75 for
surface tankers, a fair comparison between the two types would be obtained. A comparison
between surface and submarine tankers is shown in Table 6 for ships having deadweights of
18,750, 56,250, and 93,750 tons. Using the appropriate deadweight /displacement coeffi-
cients from Fig. 8, the displacements have been calculated and the ship dimensions, EHPs,
and DHPs have been taken from the curves obtained as described in the preceding section.
The results for three of these cases are shown in Figs. 9, 10, and 11, where curves are
given of DHP to a base of speed in knots for a surface tanker and two submarine tankers
having respectively circular and elliptical sections. The submarine tanker throughout has a
prismatic coefficient of 0.60. On the other hand, the block coefficient of the surface tanker
has been varied to suit the speed/length ratio, being 0.80 at the lower end of the speed
range and 0.60 at the top end. The figures for the surface tanker do not include any allow-
ance for the effect of rough weather, and the following remarks therefore apply to smooth
water conditions for the surface ships, and are favourable to them in this respect. Making
the comparison in this way, however, enables certain other points to be brought out.

Looking first at the results for the circular section submarine and the surface tanker,
we see that at the lowest speeds in question, namely 10 to 15 knots, there is practically no
difference in the power requirements between these two types of ship. Since there is very
little wavemaking in the case of the surface ship at these speeds we can infer from this
that the reduction in form drag for the streamlined submarine as compared with that for the
surface ship is sufficient to compensate for the increased frictional resistance of the sub-
marine due to its augmented wetted surface. At higher speeds the circular section submarine
tanker always requires less power. For vessels having a deadweight of 18,750 tons the dif-
ference in power is some 5000 on 25,000, and this is about the smallest ship for which a
nuclear propulsion plant could be designed at present. For the largest ship having a dead-
weight of 93,750, which is comparable with the largest tankers built today, the power for the
surface ship at 27 knots is some 190,000 and this is reduced to about 140,000 in the case of
the circular submarine. It is also worth noting that there is no appreciable difference in
power for these two designs up to a speed of 23 knots and to get the benefit from the sub-
marine design one needs to go to higher speeds; however, at 27 knots the figures just quoted
show that we are faced with a power plant comparable with those of the “Queens,” and this
of course would involve many problems regarding the housing of nuclear reactors, heat
exchangers, etc., within reasonably sized containers for fitting in a submarine. When allow-
ance is made for the effects of weather, the comparison would be much more favourable to
the submarine, either from the point of view of the greater power and fuel consumption to
maintain speed on the surface ship or from the loss in speed of the surface ship at the same
power. Opinions as to the weather allowances necessary on different trade routes vary, but
the tables and curves given here will enable anyone interested in this problem to make his
own estimates in these matters for any desired weather conditions.

Curves are also shown in these three figures for a submarine of the same deadweight
but having a beam equal to four times the draft. The only factor which has been taken into
account in making the estimates for these elliptical section ships is the increase in wetted
surface necessary to obtain the same volume within the elliptical section. Doubtless in
such a design the displacement would have to be increased to a considerable extent to
allow for the effect of the different shape of hull upon steel weight, and it is probable that
the residuary resistance coefficient would also be considerably increased as compared with
that for the circular section. For both these reasons, therefore, the power curves shown are
likely to be an underestimate.

Teasdale has published similar curves for surface and submarine tankers, both nuclear
propelled, having a deadweight of 45,600 tons including the weight of the reactor [4].

Submarine Cargo Ships and Tankers 359

Table 6

Comparison of Tankers of Constant Deadweight*

ae B d V :
or °
(ft) (ft) (ft) (knots) PAS

Deadweight = 18,750 tons

569 87.5 21.93
538 82.8 16.73
518 79.7 11.83

Surface
Surface
Surface

Submarine
Circulart
Circular
Circular
Elliptical?
Elliptical
Elliptical
Circulart
Circular
Circular
Elliptical?

Elliptical

Elliptical

eee eo

Deadweight = 56,250 tons

820 | 126.2 42.1 26.3
777 | 119.6 39.9 20.1
747 | 115.0 38.3 14.22

Surface
Surface
Surface

Submarine
Circulart
Circular
Circular
Elliptical+
Elliptical
Elliptical
Circulart
Circular
Circular
Elliptical?

Elliptical

Elliptical

PNOeRK PRK RK PNK PH RK

145,800

Deadweight = 93,750 tons

Surface 125,000 - 962 | 148 49.3 28.5 110,000
Surface 125,000 921 | 141.7 47.2 21.9 45,300 73,300
Surface 125,000 : 885 | 136.0 45.3 15.5 15,180 21,080

Submarine
Circulart 156,250 t 13,780
Circular 156,250 - : p 37,800

20,680
73,400

Circular 156,250 ‘ 81,050 169,300
Elliptical+t | 156,250 | 0. : 20,100 30,200
Elliptical | 156,250 | 0. 55,150 115,300
Elliptical | 156,250 | 0. 118,100 247,200
Circular? 156,250 i : 14,230 21,380

Circular 156,250 5 ‘ 38,900 75,500
Circular 156,250 MN a 83,600 175,000
Elliptical? | 156,250 | 0. : 20,800 31,200
Elliptical | 156,250 | 0. f 56,800 118,700

PPE PNY Ph Pe Rte

Elliptical 156,250 E 122,000 255,200
* Deadweight /displacement ratio taken as 0.75 for surface tankers and 0.60 for submarine tankers.

¢ Streamlined body of revolution, circular sections, L/D = 7.0.

$ Streamlined body, elliptical sections, B/d= 4.0.

§ EHP based on A.T.T.C. 1947 line plus 0.0004.

360 F. H. Todd

SURFACE TANKERS ARE DESIGNED FOR A SPEED GIVEN BY

V = . -
Fz? (' 06 Ce)
ee SECTION SUBMARINE TANKERS, USING OEADWEIGHT

— DISPLACEMENT RATIOS FROM REFERENCE A.
20,000

20000

O.H.P.

{0,000

V IN KNOTS

Fig. 9. Comparison of surface and submarine tankers of 18,750
tons deadweight

Submarine Cargo Ships and Tankers

SURFACE TANKERS ARE DESIGNED FOR A SPEED GIVEN BY

2 (108- )

@ CIRCULAR SECTION SUBMARINE TANKERS, USING DEADWEIGHT
— DISPLACEMENT RATIOS FROM REFERENCE A.
160,000

{40,000

190,000
80,000
D.H.P.

40,000

20,000

V IN KNOTS

Fig. 10. Comparison of surface and submarine tankers of 56,250
tons deadweight

361

F. H. Todd

SURFACE TANKERS ARE DESIGNEO
FOR A SPEED GIVEN BY

z =2 (1-06 - Cg)

CIRCULAR SECTION SUBMARINE
TANKERS USING DEADWEIGHT —
DISPLACE MENT RATIOS FROM

{50,000

D.H.P.

100,000

$0,000

[~)
Vv IN KNOTS

Fig. 11. Comparison of surface and submarine tankers of 93,750

tons deadweight

Submarine Cargo Ships and Tankers 363

His power curves are reproduced in Fig. 12 and show the same general relationship between
the circular and elliptical section submarines although the estimates are based on quite
independent data. The principal difference between his curves and those shown in the
present paper is that the power curve for the surface ship is considerably higher and much
nearer to the elliptical submarine than to the circular one. This difference can be accounted
for by the fact that Teasdale included a “normal” allowance for service conditions, in other
words, for the effect of wind and rough seas.

DWT = 45600 TONS ((NCLUDING REACTOR WEIGHT)

NO WEATHER ALLOWANCE ON
SUBMARINES AC p= +0-0004

{2 14 16 {8 20 22 24
V KNOTS

RFACE SUBMARINES

SHAPE |HULL FINED TO SUIT HIGHER SPEEDS JPT

300
pose | ost

45

so

644290 |67860 | 629860
74650 | 76850
WS SQ FT 110,000 AVERAGE {25070 | 204120

Fig. 12. Comparison of nuclear Saleen surface and submarine
tankers (from Ref. 4)

Q
a
Q
Cc
cr
>
D
a)
c

OPERATING PROBLEMS

The strength of submarine cargo ships and problems of manoeuvrability are both
extremely important factors in the design of such ships. The first is not strictly “hydro-
dynamic” and the second is to be discussed in another paper, so the present remarks will
be confined to very general aspects of these two problems.

It has been said that in the case of submarine tankers carrying liquid cargoes of one
sort or another, they need only have a central pressure hull containing the machinery,

364 F. H. Todd

living spaces, and buoyancy tanks, the external cargo spaces being of very light construc-
tion since the liquid pressures inside and outside could be equalised. It has been claimed
in fact in a study conducted in Japan that this could result in an actual reduction in steel
weight. But can we indeed accept this conception? There is always the possibility of a
power failure or other mishap which would force the ship to surface at sea and possibly in
bad weather, and it would seem that under these conditions the submarine would need to
possess longitudinal strength comparable with that of any surface ship. This would seema
hazard particularly to be catered for in any pioneer submarine tankers, because in these
early ships there will doubtless be many precautions taken to ensure a shutdown of the
reactor in the event of any unforeseen or unacceptable faults in the equipment. It may even
be that a shutdown could follow a mere fault in the controlling instrumentation. Moreover,
with the very large draft of such a vessel, she may in many cases have to load or unload at
a deep water terminal in a relatively exposed position where the conditions of the sea may
be quite rough as compared with those normally existing in harbour. Also in approaching
port when the water depth becomes less than some 400 or 500 feet such a submarine would
undoubtedly have to surface and do the last part of the journey as a surface ship and
therefore subject to the effects of rough weather. Taking all these possibilities into
account, it seems extremely doubtful whether the conception of a very light structural hull
for submarine tankers would be acceptable, at least until considerable experience had been
gained in their operation. Fora cargo ship there is no question but that a pressure hull
would be necessary throughout. The Mitchell Engineering Company have carried out an
investigation into such a cargo submarine and have come to the conclusion that the hull
would need to be made of mild steel up to 3 1,/2 inches in thickness amidships in way of
the machinery and compensating tanks, and vary down to 1 inch in thickness toward the
end. On this basis they estimated the deadweight/displacement ratio to be 0.56 for an
ore-carrying submarine of 50,000-ton displacement [5]. There are many other problems
involved in the construction of a hull of such dimensions. As has already been pointed out
when discussing the resistance qualities of different designs, the circular hull submarine
leads to very large drafts such as could not be catered for by any existing docks or terminal
ports. On the other hand, the circular hull is the only economical one from a structural
point of view and any departure from exact circularity will mean the provision of extra stiff-
ness and greater thickness of plating [6]. Although liquid cargoes could be handled very
easily from a submarine, other types of cargo could be very awkward in their demands for
large hatches which would have to be pressure resisting and which would, in fact, form
another weakness in the strength qualities of the hull. It is probable from a strength point
of view that the only reasonable cargoes would be bulk ones such as iron ore or similar
high-density materials.

Two of the most important qualities in such a ship would be its directional stability
and controllability. If we consider a submarine some 800 feet in length travelling at an
immersion of 400 feet and built to resist pressures up to twice this depth, we are asking
the crew virtually to fly the vessel in an atmosphere whose depth is only equal to the
length of the submarine. This would probably give any air pilot nightmares! Since in
order to compete commercially on a power basis with the surface ship the submarine would
also have to travel at some 30 knots or more, it is easy to see how by the accidental
occurrence of even a moderate trim by the bow the ship could very soon exceed her safe
diving depth. The question of providing adequate directional stability and controllability
is therefore a paramount one in the whole design.

In order to obtain adequate transverse and longitudinal stability, it will be necessary
for the submarine to carry permanent ballast, as is done in military craft. It is also impera-
tive to have some surplus buoyancy when on the surface, and in any commercially operated
craft the amount of such reserve buoyancy would almost certainly be controlled by

Submarine Cargo Ships and Tankers 365

legislation. It is probable that these two items together would amount to not less than 25
percent of the displacement. Although such a vessel would not have to carry bunkers, there
would be little if any saving of weight in this respect since the shielding for the reactor
would be of the same order of magnitude. All these considerations add up to the fact that a
submarine cargo ship or tanker would require considerably greater displacement than a sur-
face ship to carry the same deadweight.

Any nuclear propelled ship will call for a much greater skill in the engine room crew
than does a conventionally powered ship. If in addition the nuclear propelled ship is also a
submarine, then the navigating crew must also be very highly skilled and undergo long peri-
ods of training comparable with those of similar navy crews. The additional cost of this
skill could be considerable and may be an important factor in the overall comparison.

At least in the single-screw designs it will be necessary to have some means of auxil-
iary propulsion such as motors and batteries or diesel engines for use on the surface in an
emergency. If a single-screw submarine were designed for use under ice, it would seem
almost essential to have electrical auxiliary propulsion since in these circumstances it
would be impossible to bring the ship to the surface until she was clear of the ice. In the
case of a twin-screw ship with two entirely separate propulsion plants, auxiliary propulsion

could probably be dispensed with.

Considering in more detail the operation of the circular and elliptical submarines and
the surface ships for which the estimates of power are shown in Figs. 9, 10, and 11, we see
that the circular section submarine requires a lower power than either the surface ship or
the elliptical submarine. However, we see that the advantage is not large unless we con-
template speeds of 25 or 30 knots and it is a question of the economics of the oil industry
whether it is justifiable to carry a commodity such as oil or indeed any other nonperishable
bulk cargo at such high speeds. It is not proposed here to enter into these economic ques-
tions, but it may be sufficient to point out that these will vary from time to time, as wit-
nessed by the fact that during the Suez crisis the cost of transporting oil via the Cape rose
to £19 per ton as compared with the normal freight rate of £2 per ton via Suez. In addition,
it has been estimated that the capital cost of a submarine tanker would be of the order of
twice that of the equivalent surface ship [7,8]. It would appear, therefore, that the advan-
tages from a commercial point of view would not be large enough at commercial speeds to
counteract the increased capital cost of the submarine and the increased costs of mainten-
ance, crew, insurance, and the large expenditure which would be necessary for research and
development before such a ship could be built. Beyond these operational problems is the
further one that a circular hull submarine large enough to justify nuclear propulsion and
competition with similar surface ships would have a very large draft and would require new
building facilities, new dry-docking facilities, and new terminals. It would also be very
limited in respect of the world ports in which it could be accommodated and would certainly
not be able to pass through the Suez or Panama Canals for many years to come. If sucha
ship to carry oil from the Middle East to Europe had to go via the Cape, then this would far
outweigh any advantages it might have in regard to lower power requirements.

If we restrict the draft of the submarine to a figure comparable with that of the equiva-
lent surface tanker, then we are forced into accepting a submarine having elliptical sec-
tions. As an extreme case the figures have been calculated for such submarines of different
deadweights having a beam/draft ratio of 4:1 and the power curves for these have been given
in Figs. 9,10, and 11. It is at once seen that from a power point of view, even after making
some weather allowance for the surface ship, there is no longer any overall benefit to be
derived on the basis of power. If the thesis is accepted that for a submarine tanker the
structure carrying the oil can be of light scantlings, then there would not be a very great

646551 O—62——25

366 F. H. Todd

penalty on weight in going to the elliptical section. However, in a cargo-carrying submarine
or a tanker in which the whole hull was built to resist pressure, then there would be a very
substantial penalty in weight through such a change in shape. The comparison between the
submarine and the surface ship is therefore seen to be more unfavourable to the submarine

' when we consider a hull with elliptical sections.

The outcome of this whole discussion points to the fact that from a commercial point of
view there is little attraction in submarine cargo ships or tankers so long as we restrict our
thoughts to speeds within the limits for which economic surface vessels can be designed.
However, if we wish to go beyond these speeds, then the submarine would be the only
answer for the carriage of bulk cargoes. For lighter specialised cargoes, it may well be that
the answer is to go the other way and use hydrofoil craft. If submarine ships were justified
by a sufficient demand for the transport of bulk cargoes at such high speeds, the figures that
have been given for the powers required, even when we are not compelled to make provision
for surface wavemaking and rough weather, still mount rapidly and very soon attain astro-
nomical proportions. In Table 3 a line has been drawn showing where the power required
exceeds 200,000 DHP, which means a propelling installation of the same order of magnitude
as that of the QUEEN ELIZABETH, and of course to house any such nuclear plant in a sub-
marine would be quite a task. The containing vessel for the 20,000-shaft-horsepower nuclear
plant for the American cargo ship SAVANNAH is 50 feet long and 35 feet in diameter, and
one can imagine that with 200,000 horsepower together with the space required for perma-
nent ballast, trimming tanks, buoyancy tanks, living quarters, and so on, the deadweight/
displacement ratio would be rather small.

From time to time it has been suggested that a smaller tanker of the nuclear propelled
submarine type might be competitive with a much larger surface tanker. Such a comparison
has been made by Teasdale [4] between a 47,000-ton-deadweight surface tanker and a
26 ,000-ton-deadweight submarine tanker of circular cross section and having dimensions
which do not exceed those of the surface ship. He estimated that if the submarine were
travelling at a high enough speed to deliver the same quantity of oil per year as the surface
tanker, the fuel consumption would be about four times as great. From his analysis he
doubted whether the capital cost of the submarine would be sufficiently less than that of
the surface ship to offset such a penalty.

TERMINAL PROBLEMS

The draft of water in many ports and harbours today does not exceed 35 feet and in
very few does it exceed 50 feet. This would impose many problems on the building and sub-
sequent dry-docking of submarines of the size we have been discussing. The draft problem
could be overcome by going to an elliptical section shape for the hull, but this would give a
submarine of excessive beam for most dry docks and would also bring in the other problems
which we have mentioned in the previous section. For the ordinary routine loading and
unloading processes, it would be necessary to build offshore loading terminals or use feeder
services to bring the oil in to the refineries. This would mean that the ship might be sub-
ject to rough weather and therefore would have to be designed with normal longitudinal
strength and not be able to have a light scantling outer structure. Another point of some
interest for such large ships approaching the shore is the fact that once the depth of water
was reduced much below 600 feet they would have to surface and do the last part of their
journey as surface ships. To achieve the high efficiency of underwater propulsion which
has been the aim of all these designs it is necessary to go to a streamlined body of revolu-
tion with the very minimum of appendages. However, when such a vessel has to come to

Submarine Cargo Ships and Tankers 367

the surface it has about the worst possible shape for good performance. The maximum speed
under these conditions would probably be quite low, and if the approach to the terminal
ports runs for long distances over a continental shelf or similar shallow water, this could
add appreciably to the time of passage and so offset to some extent the better submerged
performance. Such a large craft would be exceedingly difficult to manoeuvre in shallow
water, and would be quite unwieldy at the terminals, and especially if made of light
scantlings would be liable to serious damage and during berthing operations would need the
assistance of tugs. In common with nuclear propelled surface ships, it would also be nec-
essary for the port authorities to monitor continually for radioactivity and to maintain the
equivalent of a fire-fighting service to give warning and to cope with any such accident.
Even voyages under ice can be fraught with considerable dangers other than those of navi-
gation. According to information resulting from the voyage of the NAUTILUS, the ice over
the North Pole is some 9 to 13 feet in thickness but with occasional ridges penetrating some
100 feet below this. In addition, from time to time icebergs were met with which were
imprisoned in the surface ice and reached no less than 1000 feet below the surface. The
use of a polar route into the Pacific is also subject to the limitation that water in the
Bering Straits is only some 150 feet in depth. It is obvious that extremely careful naviga-
tion would be called for to cope with such hazards.

CONCLUSIONS

The general conclusion which can be drawn from all the above evidence is that sub-
marine cargo ships and tankers of circular cross section could be designed to compete with
surface ships of the same deadweight as regards their power requirements, especially when
one takes into consideration the effects of rough weather, from which the submarine would
be immune. Such submarine ships would have excessive drafts, however, and if this is
avoided by using elliptical sections, then the submarine’s superiority soon disappears.
However, up to such speeds as those for which an economical surface ship can be designed,
say of the order of 25 to 30 knots, the capital cost of the submarine, of the necessary dock-
ing facilities, and of the provision of offshore terminals could not at present be justified on
economic grounds. At the very least it would be necessary for a consortium of owners to
combine in the building of terminals and dry docks for common use before a fleet of such
submarines was put into service. At higher speeds, above 30 knots, the submarine begins
to have very appreciable advantages as regards power, but of course at these high speeds
the price to be paid in terms of fuel consumption is also high and the propelling machinery
would begin to take up more and more of the volume of the ship, thus reducing her dead-
weight. Whether such ships would be a good financial proposition or not would depend
on the interplay of capital cost, insurance, maintenance, fuel and operating costs, the
freight rate obtainable for oil or other bulk cargoes, and the route on which the ship is
employed. If this were between the Middle East and the United Kingdom, for example, a
submarine tanker of circular section of anything greater than about 25,000 tons dead-
weight would be unable to use the Suez Canal in its present form. The economic advan-
tages of large size in the carriage of bulk cargoes could therefore only be realised in
this trade so far as submarine tankers are concerned by routing them around the Cape, with
all the attendant disadvantages. These would not, of course, enter into other routes, such
as Venezuela to Europe. The first marine nuclear propulsion plants will certainly be
expensive, but the ingenuity of the engineer will undoubtedly reduce these as time goes on.
Nuclear fuel costs, which at present are perhaps twice those of conventional fuels will also
come down in the course of time and with the more widespread use of atomic energy in

other fields.

le Ee re

368 F. H. Todd

So far we have restricted ourselves to a comparison between nuclear propelled surface
and submarine ships. The case for the submarine would be even less favourable at this
time if we were to compare it with a conventionally propelled surface tanker. The British
Shipbuilding Research Association has compared surface tankers having respectively con-
ventional and nuclear propulsion plants and has shown that even on the assumption of
economic parity between capital and fuel costs there is no commercial attraction in
installing in a surface tanker a nuclear plant of higher power than the conventional ones
being installed at present [9]. It has also been shown, in the case of a tanker of 65,000
tons deadweight with 25,000 shaft horsepower employed on the Middle East-United Kingdom
route, that while the cost per ton of transporting oil via the nuclear ship might be within
15 percent of that for the conventional ship, the cost for the nuclear ship begins to rise
again at speeds above 17 knots [10]. This means that from the point of view of economy
there is really no inducement to fit higher powers in nuclear ships than are at present fitted
in conventional ships. The greatest commercial incentive for submarine ships at the
moment would appear to be their use on special routes where surface ships cannot be used
and on which there might be the attraction of making special profits. One such route which
has been mentioned on many occasions is that from Canadian ports carrying out ore in
winter underneath the ice, and another has been the suggestion put forward in the London
Times that by going under the polar ice it is possible to halve the distance between London

and Tokyo [8].

Leaving aside economic questions, there is no doubt as to the extreme advantage of
having such craft for military use and for the transport of valuable cargoes in wartime. It
may well be that some government will build a craft of this type very soon, both for its
military potential and national prestige, and to gain experience in the operation of such
ships. Something of the magnitude of the tasks facing the designer of such submarine cargo
carriers as we have been discussing may be realised when it is recalled that the largest
submarine built to date is the USS TRITON, which has a length of 447 feet, a displacement
of 5900 tons, and two nuclear reactors. This is the longest and most powerful submarine
yet built and it is a very long way to go from this to tankers of the order of 60,000 or
80,000 tons. Although the tasks are formidable, there is no doubt as to the challenge they
present and the extreme interest which they engender in the mind of the designer. Using a
completely submerged hull of ideal form, the propeller designer is presented with wonderful
opportunities of realising his ideal propeller—a circumferentially uniform wake, the chance
to apply impeller theory to marine problems, and the knowledge that at such deep depths
cavitation will no longer be a paramount inhibitor always looking over his shoulder. To the
hull designer, the challenge will be to eliminate all possible sources of parasite drag by
attention to hull form, appendages, and smoothness of surface and to ensure adequate direc-
tional stability and control, while tantalisingly in the offing will be the lure of someday,
somehow, achieving laminar flow over some of the hull surface. Inevitably progress will be
made only by small steps, and the development of a successful submarine cargo ship or
tanker will involve many years of research and development and much money. From the
scientific and technical point of view, such a vessel presents a great challenge to the naval
architect and marine engineer, who will doubtless continue to dream of the possible solu-
tions until the day comes when they are given the opportunity to translate their ideas into
reality. ;

Submarine Cargo Ships and Tankers 369

ACKNOWLEDGMENTS

The work described in this paper forms part of the Research Programme of The National

Physical Laboratory and is published by kind permission of the Director.

[1]

[2]

[3]

[4]

[5]
[6]

[7]
[8]
[9]

[10]

REFERENCES

Crewe, P.R., “The Hydrofoil Boat; Its History and Future Prospects,” Instn. Naval
Architects Trans. 100:329 (1958)

Todd, F.H., Stuntz, G.R., and Pien, P.C., “Series 60—The Effect upon Resistance
and Power of Variation in Ship Proportions,” Soc. Naval Architects Marine Engrs.
Trans. 65:445 (1957)

Crago, W.A., “Test Results on Submarine Tankers,” Impulse, No. 4, Mar. 1958

Teasdale, J.A., “Characteristics and Performance of Nuclear-Powered Submarine

Cargo Vessels,” North East Coast Instn. of Engrs. and Shipbldrs. 57:461 (1958-59)
Mitchell Engineering Co., “Nuclear-Powered Submarine ‘Moby Dick’,” 1960

McKee, A.I., “Recent Submarine Design Practices and Problems,” Soc. Naval Archi-
tects Marine Engrs., paper 11 for meeting Nov. 12-13, 1959

Article on submarine tankers, Journal of Commerce, Jan. 17, 1957
Shipping Correspondent, “Undersea Voyages,” The London Times, Aug. 13, 1958

Smith, S.L., and Richards, J.E., “Nuclear Power for the Propulsion of Merchant
Ships,” Instn. Engrs. Shipbldrs. in Scotland 100(pt7):698 (1957)

Norton, E., “Future Trends in Marine Engineering,” Shipbuilding and Shipping Record,
International Design and Equipment Number, 1958

DISCUSSION

R. N. Newton (Admiralty Experiment Works)

We should all be grateful to Dr. Todd for presenting what seems to me to constitute the

_ most comprehensive review of this popular topic, to date. He has dealt with almost every
aspect of the general problem and I would like merely to place more emphasis on some of
the points which he has made and which militate against the opinion, frequently expressed,
that submarine tankers offer great advantages in the commercial field.

370 F. H. Todd

In that part of his paper which compares the two types of tanker on the basis of the
same deadweight, Dr. Todd assumes, from data provided by other authors, a ratio of dead-
weight to displacement in a nuclear propelled submarine of 0.6 (see Fig. 8). With this
value of 0.6 the DHP curves of the circular sectioned submarine tanker in Figs. 9, 10, and
11 lie below those of the surface tanker at all speeds worth considering. The crossover
points beyond which the submarine tanker has the advantage over the surface tanker are
11, 14, and 22 knots for equal deadweights of 18,750, 56,250, and 93,750 tons respectively.
This paints a rather rosy picture for the submarine, but if the ratio of deadweight to dis-
placement for the submarine tanker is less than 0.6, a very different picture would be
obtained. It is not clear from the paper how the authors concerned arrived at their figures
for this ratio, but it is significant to remark that the values indicated in Fig. 8 show a much
wider scatter than those for surface tankers. Immediately I notice that Dr. Van Manen gave
in his paper a figure of 0.41 for a 100,000-ton tanker and 0.50 for a 43,000-ton tanker. There
is only one way to arrive at a realistic figure, viz., by carrying out a detailed design, and,
speaking with some experience in the design of military submarines, I venture to suggest
that a more realistic figure would be nearer to 0.5 than 0.6, for several reasons. The most
important of these is the inherently unsuitable shape of the pressure hull to accommodate
machinery and equipment with a degree of compactness consistent with good maintenance
such as is possible in a surface ship. Others include the weight and space necessary for
additional ship services such as trimming and compensating systems, special air-condition-
ing apparatus, three sets of control gear (two hydroplane gears and one rudder gear), addi-
tional navigational and operational aids, and the fact that many of the ship services must
be designed to much greater pressures than in the case of a surface ship. The effect of a
lower deadweight/displacement ratio would be to raise the submarine curves in Figs. 9, 10,
and 11 and to greatly increase the speeds at which the submarine gains the advantage over
the surface tanker to such values as to raise grave doubts as to whether the machinery to
drive the vessel at such speeds could be accommodated. In the case of the 56,250-ton
tanker in Fig. 10, for instance, if a deadweight/displacement ratio of 0.5 were used the
crossover point would rise to the region of 27 knots and the DHP required to 118,000.

Immediately the question arises “could the machinery and propellers to provide this
power be accommodated in any reasonable size of pressure hull to suit other requirements?”
I am strongly of the opinion that it could not. If this opinion is correct, it follows that, if
for any reason a submerged tanker of such size became necessary, then with the presentmeans ~
of propulsion and types of fuel it would be necessary to accept a very heavy loss in dead-
weight ratio compared with a surface tanker.

Against this, however, must be weighed the distinct advantage of being able to
operate with impunity in any weather except when approaching the terminal points of the
voyage, as pointed out by Dr. Todd.

In the section of the paper dealing with operating problems, Dr. Todd has called
briefly, but very effectively, to the need to take account of dynamic stability and control
when submerged. Even with the comparatively small military submarines now being built,
the problems involved are both complex and numerous. Compared with the submarines of
the last war, underwater speeds have doubled and trebled. The speed beyond which the
submerged tanker begins to gain over the surface ship is in this region. Consequently the
time available to the commanding officer, and to whatever control apparatus with which the
submarine is equipped, to take corrective action is very much shorter —not more than one-
third of that in an orthodox submarine of the last war.

Fortunately much has been learned regarding the dynamic stability and control of
normal sized submarines. By the application of theory and model experiments correlated to

Submarine Cargo Ships and Tankers 371

the results of full scale trials, it is now possible to determine stability criteria and to
design and use automatic control systems to suit the stability and the response of the sub-
marine. For such large submerged tankers as those under discussion similar investigations
would have to be carried out and would, in fact, be vital since the margin of safety involved
in terms of time and the depth range is obviously much smaller.

On the question of structural design it is fair to state that the unpressurized structure
which may contain liquid cargo offers no insuperable design or constructional difficulties,
but, as Dr. Todd points out, it is indeed a grave assumption that it could be of light scant-
ling. On the other hand the problem of designing the pressure hull and the consequences
which could result from the application of incorrect principles are much more serious.
Sudden collapse of the hull can arise from elastic failure of the structure or plastic failure
of the material, so that the problem involves not only the theory to avoid one or both of
these but also the development of special steels. This development work becomes more
and more difficult as the thickness of the steel increases, i.e.,as the size of the pressure
hull increases. Construction of the pressure hull is a highly specialized technique. Not
only must the hull be built to a small degree of ovality, or “out of circularity,” but it is
necessary to employ highly specialized welding techniques. Local weaknesses brought
about by poor design or bad workmanship introduce a danger factor since they could lead
to early local failure and sudden collapse of the hull.

Merely to emphasize the vital nature of these two problems of stability and strength let
us take Dr. Todd’s example of a submerged tanker 800 feet long with a pressure hull say
600 feet long and a collapse depth 800 feet. Let us also suppose, as he does, that the
tanker is proceeding at about 30 knots at 400 feet depth in order to gain the advantage of
not generating waves, and then let us suppose that for some unforeseen reason it takes on
a bow-down angle of 10 degrees for just 40 seconds. In that short space of time the fore
end of the pressure hull would have exceeded the collapse depth and complete disaster
would follow. It is as well to stress that this is by no means a hypothetical case but a
real possibility.

It is for such reasons as those which I have attempted to emphasize, and many others
which Dr. Todd has mentioned, that I incline to the opinion that, although the possibilities
of submarine tankers as cargo vessels for commercial use present a great challenge to the
naval architect and marine engineer, there are many equally challenging and important
aspects of surface ship propulsion which require our immediate and undivided attention in
the near future. Not the least of these is the design and construction of hull forms capable
of maintaining higher speeds in rough weather.

F.H. Todd

Mr. Newton rightly points out that the relative merits of the surface and submarine
tankers in the matter of power depend very greatly upon the values assumed for the ratio of
deadweight to displacement, and stresses that only detailed design calculations can give
reliable guidance on this point. At the time of writing this paper, no such data appeared to
be extant, but recently the results of such detailed weight estimates have been published
in the U.S. by Russo, Turner, and Wood.* Detailed designs have been made for a number of
submarine tankers with nuclear machinery having streamlined, body-of-+revolution hulls, with

* Vito L. Russo, H. Turner, and Frank W. Wood, “Submarine Tankers,” Society of Naval Architects
and Marine Engineers, New York, Nov. 1960.

372 F. H. Todd

and without parallel body, and rectangular hulls having rounded corners. The deadweight/
displacement ratios depend upon the size of the ship and its speed, and some representative
figures taken from the reference are given in Table D1, as they are of general interest.

Table D1
Deadweight/Displacement Ratios (from Russo, Turner, and Wood)

Speed 2 Parallel | Length | Beam | Depth

. Deadweight = 20,000 tons
20 Circle 0 525 75.5 | 75.5

Power Increase
for Rect.
Section (%)

20 Rect. 59.5 555 80 40 62
30 Circle 0 560 79.0| 79.0 -
30 Rect. 58 570 90 40 40

ao = 30,000 tons

20 Circle 85.5] 85.5

20 Rect. . 8 oe 100 40

30 Circle 625 89.5} 89.5

30 Rect. a 5 625 120 40 :

Deadweight = 40,000 tons

660 93.5 | 93.5
. 6 710 120 40 : 48,400
oe 98.0} 98.0 : 104,400
120 : 152,000

Circle
Rect.
Circle
Rect.

For the tankers having circular sections the ratio at 20 knots varies from 0.546 at
20,000 tons deadweight to 0.566 for 40,000 tons. At 30 knots the corresponding figures are
0.467 and 0.500. These ratios are referred to surface displacement. If we convert them to
submerged displacement, which is the one used in my paper, the values become 0.496,
0.515, 0.425, and 0.455 respectively. These values are all appreciably less than the
figure of 0.60 used in the original power estimates. The latter have therefore been
revised, using the new deadweight/displacement figures as derived above, but otherwise
they are upon exactly the same basis as the earlier estimates. The calculated powers are
shown in Table D2 and the appropriate spots have been added to Figs. 9, 10, and 11. For
the smallest tanker, of 18,750 tons deadweight, the circular-hulled submarine now requires
exactly the same power as the surface tanker at 20 knots, so that the crossover point (for
smooth water operation) has been raised from about 11 knots to 20 knots. The same
applies to the intermediate 56,250-ton-deadweight tanker, the crossover rising from 14 knots
to 20 knots. For the largest ship, of 93,750-ton deadweight, the submarine tanker does not
become less resistful than the surface ship (in smooth water) until the speed is some
28 knots.

These figures substantially agree with those suggested by Mr. Newton. | was aware at
the time of writing the paper that the assumed value of 0.6 for the deadweight/displacement
ratio was probably high, especially as many of the examples originated from the protagonists

Submarine Cargo Ships and Tankers 373

of the submarine cargo ship. Indeed, in an earlier paper* I had used a value of 0.5, but felt
it necessary to err on the side of optimism rather than to run the risk of being charged with
pessimism and lack of foresight! This comparison is for circular-hulled submarines, which,
of course, suffer from the extreme drafts noted in the paper. In the American paper referred
to, power requirements are also given for rectangular section submarines, some of which are
shown in Table D1. While this type of section enables the draft to be kept within reason-
able bounds (not greater than 40 feet for 40,000 tons deadweight) the price to be paid in
power is very considerable, varying from 60 to 80 percent increase at 20 knots and 40 to 50
percent at 30 knots. These powers are higher than those given in my paper for elliptical
sections, but, of course, they have the advantage of less beam. With limitations on beam,
draft, or both, forcing us to adopt elliptical or rectangular sections, it is obvious that the
commercial merits of the submarine tanker as compared with those of her surface sister will
be hard to find, even when allowance is made for rough sea effects upon the surface ship,
as long as there is no demand for the high speed (and therefore high cost) transport of bulk
cargoes across the oceans.

In view of Mr. Newton’s experience in submarine design problems, his remarks on the
difficulties inherent in providing adequate strength in such large submarines will be of
great value to all interested in this subject, and we are indebted to him for his penetrating
analysis of the problem.

Table D2
Power Calculations for Submarine Tankers

(Body of Revolution Hull, No Parallel Body, G, = 0.60; L/D =°7.0)

Deadweight (tons) 18,750 56,250 93,750
a
0.530

H. Lackenby (British Shipbuilding Research Association)

I should like to make a few general remarks on the strength of the author’s conclusions,
especially as I have since ascertained that they have a bearing on remarks I made on Dr.
van Manen’s paper on Monday which touched on the same subject. As I said at that time,

*F, H. Todd, “Submarine Tankers,” Shipbuilding and Shipping Record, Aug. 21, 1958.

~ 2S + =

374 F. H. Todd

when comparing the performance of surface and submarine vessels one has to be careful
about the basis of comparison. Although for a given deadweight of say 25,000 tons the
break-even point as regards power for certain designs may be as high as 25 or 30 knots, one
has to bear in mind that a surface tanker of 25 knots would not be really economical com-
pared with a commercial tanker which would have a speed of about 15 or 16 knots. As I
said before, I am speaking here of economy on the basis of cost per ton-mile of cargo
carried under present conditions. This was one of the findings arising from freighting cal-
culations carried out by the British Shipbuilding Research Association in conjunction with
ship-owners in connection with extensive nuclear energy studies, reference to which the
author has made in the paper. This point is really covered by the second paragraph of the
author’s conclusions with which I certainly agree. Incidentally it might be of interest to
mention that I understand that freighting calculations have also shown that, if anything, the
speed of some recently built conventional tankers is perhaps a little on the high side as far
as economic running is concerned. From the military point of view, of course, the situation
could be quite different as the author has clearly explained.

What I am really saying here is that in my remarks on Dr. van Manen’s paper on Monday
I anticipated these particular aspects of the author’s conclusions, although I was not aware
of this at the time.

I should also like to say that I agree with the author’s conclusions generally, in addi-
tion to those to which I have referred already. There is just one point of detail I should
like to raise, namely, the limit of power for a single screw surface ship which the author
has taken as 40,000 SHP. Some years ago we looked into this at BSRA in connection with
the nuclear studies for tankers referred to earlier and we were advised that for a number of
reasons and in particular the possible incidence of excessive cavitation erosion, 20,000 to
25,000 SHP was about the practical limit at the time. Perhaps the author may care to com-
ment on this point. .

F. H. Todd

The British Shipbuilding Research Association has carried out a great deal of research
into the subject of nuclear propulsion, and it is therefore gratifying to know that Mr.
Lackenby is in general agreement with my conclusions as to the commercial merits of its
application to submarine cargo ships and tankers. With reference to the use of an upper
limit of 40,000 horsepower for a single-screw ship, such power absorption is, of course,
fairly common on multi-screw ships. In single-screw surface ships the wake variation con-
ditions are considerably more onerous, and as Mr. Lackenby has said this introduces the
danger of cavitation erosion. In the submarine this is not nearly such a serious problem
since the circumferential variation can be kept quite small by paying good attention to the
shape and location of appendages. Moreover, we are here talking of the future, and with
the advance in knowledge about propeller-excited vibration and the necessary stern shapes
and clearances to reduce the forces to a minimum, we may look forward to a steady increase
in the maximum power we can put through a single screw.

J. M. Ferguson (John Brown and Co., Limited, Glasgow)

I do not wish to discuss at all the technical side of Dr. Todd’s paper but I wish to
express a thought which has caused a great amount of discussion between my colleagues
and myself, mainly during lunch periods. We have been discussing the possibilities of

Submarine Cargo Ships and Tankers 375

these submarine tankers because at the moment at Clydebank we have been building such a
number of present day modern large tankers. Now, as you are all aware, in these present-
day tankers a great deal of attention is being paid to the amenities for the crew. The very
rapid turn round of these tankers, both at their loading port and at their delivery port means
that the crew generally get very few advantages of leave or of getting home and the condi-
tions on board the tankers are made all the more pleasant and congenial for these men.
When you come to the submarine tanker, I would like you to think of the psychological
aspect of the environmental conditions. These people are to be living in artificial condi-
tions, with no daylight for days and weeks at a time, and the financial cost of making these
conditions congenial and tolerable would add very considerably to the economic aspect of
running these ships. In the case of the American submarine NAUTILUS, which did that
epoch-making trip under the ice in the Polar regions, the crew, when you read the descrip-
tion of the voyage, had quite extensive arrangements made for their entertainment and to
keep them happy; but they were doing a special job, they were being filled with a feeling of
prestige attached to the job. To maintain, as would you call it, an equally high standard of
“esprit de crew,” then, quite a lot of money is involved. I would like that thought to be
kept in mind because I do feel that when such craft do come along that aspect of it which
apparently has not been considered so far will require quite an amount of attention.

F. H. Todd

Mr. Ferguson has called attention to a very important point in the operating of sub-
marine cargo ships and tankers —the maintenance of the health and happiness of the crews.
This would without question involve owners in a considerable increase in running costs.
As mentioned in the paper, the operation of such craft will call for very highly skilled per-
sonnel, both for navigational and engineering purposes, and this also will add still further
to the costs, and these two factors together may be of importance in the overall economic
comparison.

M. F. Gunning (Netherlands United Shipbuilding Bureau)

We have made some fairly detailed investigations inte a 20,000-ton tanker with a
moderate eccentricity of the ellipse of the midship section with proper diving depth, reason-
able strength of outer hull, etc., and we find that we come well above 0.6, closer to 0.65,
deadweight ratio. Here, I am sorry, I am very much at loggerheads with my very good friend
Mr. Newton.

F.H. Todd

The question of deadweight/displacement ratio has been discussed at length, and Mr.
Gunning’s figure of 0.65 seems to be very much higher than those quoted by people with
experience in the design of naval submarines and those given as the result of a very
thorough study of commercial submarines by the Maritime Administration in the U.S., which
I have referred to. Mr. Gunning is himself a submarine designer, and I can only leave this
point to the experts in this field to sort out!

376 F. H. Todd

J. A. Teasdale (Furness Shipbuilding Company, Limited, Billingham)

The largest known military submarine has a displacement of only one-fifth that of the
smallest vessel considered in this comprehensive paper and so far no fully commercial sub-
marine has been in service. The author, therefore, is to be congratulated upon his approach
to the comparison between surface and submarine merchant vessels and the interesting
presentation of the results of his work.

In order to properly compare these results with other works on the same subject a little
more information is required on the data used in the study. Firstly, is the displacement
referred to in Tables 2, 3, and 6 and Fig. 8 one of surface or submerged condition? Sec-
ondly, what form drag allowance has been assumed for the submarines? Various sources
put this allowance as high as 15 percent of the skin friction value. Thirdly, the method of
determining the deadweight of the submarines seems to be very approximate and perhaps
misleading. The deadweight/displacement ratio is unlikely to be constant for a range of
displacement of over 100,000 tons and a range of horsepower of more than 200,000 DHP.

Since these designs of submarine tanker are intended to be of relatively high speed
compared with conventional submarines and accurate depth control at low speeds is not so
important, there should be no necessity for bow diving planes. Even without these it is
doubtful whether the appendage allowance would be less than 25 percent of the EHP of the
bare hull.

The author states, “If we restrict the draught of the submarine to a figure comparable
with that of the equivalent surface tanker, then we are forced into accepting a submarine
having elliptical sections.” The writer would venture to suggest that if ever a submarine
tanker project becomes fact, then the practical design is likely to have a substantial
parallel body of rectangular section with radiused corners. The ends, of course, would be
as streamlined as possible.

R. Brard (Bassin d’Essais des Carénes, Paris)

Undoubtedly, the question that Dr. Todd has chosen for this Symposium is an interesting
one. Numerous papers have already been written on this subject, that is, a consequence of
the promises of atomic propulsion. But the problem of submarines, cargo ships, and tankers
is quite different from that of navy ships and there is no evidence that the excellent navy
solution is excellent for merchant marines. Dr. Todd made a very acute analysis of the
various parts of the problem. He examined them not only from a hydrodynamical point of
view, but also from others that are to be considered, and particularly those of the operating
problems, terminal problems, and economical problems. I think that the values of the main
parameters on which Dr. Todd grounds his analysis, should not give rise to discussion, but
in a final synthesis, the weights of each aspect could be somewhat subjective. The con-
clusion of Dr. Todd seems, however, prudent and full of sense. Many studies are necessary
before submarine cargo ships and tankers become practical, but those concerned with this
question will find in Dr. Todd’s reflections many very useful materials.

F. H. Todd

I am grateful to Admiral Brard for his remarks, and agree completely with his opinion
that military and commercial submarines are two quite different propositions. It was my

Submarine Cargo Ships and Tankers 377

principal aim to give sufficient hydrodynamic data covering a variety of sizes of submarines
for anyone studying the subject to make his own power estimates and draw his own conclu-
sions. As power is only one of many factors, however, I did endeavour to call attention to

other features in such ships, and, as Admiral Brard says, the relative importance of each of
these will vary from ship to ship according to the purpose it is to serve.

In conclusion, I would like to make a few remarks on the philosophy of the theme of
this symposium —high performance ships. I have tried to show in the paper that the naval
architect and naval scientist have long been aware of the possibilities —and problems — of
using submarines for commercial purposes, and how they have to a large extent been waiting
for a propulsion plant independent of atmospheric air. From time to time —and indeed in the
introductory proceedings of this conference —comments are directed against us to the effect
that we have only increased the speed of ships twofold in a hundred years, whereas our
aerodynamic friends have done it tenfold in fifty years. I would like to suggest, neverthe-
less, that we have made some progress in that time — from the old SAVANNAH to the new
SAVANNAH in a hundred years seems quite a step, as does that from the old Cunard
BRITANNIA of 1840 to the QUEEN ELIZABETH of 1940. As a naval architect and scien-

tist, I feel that we have not altogether failed in developing our science.

One of the great problems from which we have always suffered, and which I have tried
to bring out in the paper, is the continual insistence in our particular art or science, which-
ever you like to call it, that ships must always pay for themselves in the economic sense.
This, I think, is where our aeronautical friends have had a terrific advantage. We have
striven for years to get just one research ship of our own on which to carry out the many
full-scale experiments necessary for the progress of our science, but to no avail. The
aeronautical engineer designs an aircraft, and builds one, two, three, or more prototypes for
development testing, and is undeterred if some of these, and many millions of pounds, go up
in smoke in the process. I believe that if we were given similar opportunities in the way of
government support for research and development, and if our steps were not continually
dogged by this attitude that every ship built must be a commercial success, we would make
a great deal faster progress.

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EXPERIMENTAL TECHNIQUES AND METHODS OF ANALYSIS
USED IN SUBMERGED BODY RESEARCH

Alex Goodman

David Taylor Model Basin

This paper deals with the various experimental techniques and methods
of analysis which are either presently being used or are contemplated in
the near future at the David Taylor Model Basin in the field of dynamic
stability and control of submerged bodies. The advantages and disad-
vantages of the various techniques are discussed. Primary emphasis is
placed on the principles and operation of the DTMB Planar-Motion-
Mechanism System, since this device is unique and is being used exten-
sively to provide stability and contro] data for submerged bodies in six
degrees of freedom. Emphasis is also placed on the concepts, tech-
niques, and philosophies used in simulator studies and motion analyses.
A brief description of the new rotating arm facility, and the free-running
model technique are also presented.

NOTATION

The nomenclature defined in DTMB Report 1319 is used herein where applicable. The
positive direction of axes, angles, forces, moments, and velocities are shown in the accom-
panying sketch. The coefficients and symbols are defined as follows:*

Symbol Dimensionless Form Definition
A,
A, A, = =a Projected area of bow planes
A,
A, Sa Projected area of rudders
12
' os
A. A= oy Projected area of stern planes
AP After perpendicular
AR Aspect ratio

*All derivatives with respect to angular quantities are given as “per radian.”

379

Alex Goodman

380

JON3Y3I53549
TVLNOZIYOH

Analysis Used in Submerged Body Research 381

Symbol Dimensionless Form
B
U
; ry oa
9 PL U
CB
CG
c
ce
jet
d qd =7
FP
h
' I,
Oy Ea Los = i
gel
K ——
9 pleU?
“5
K Ko =
P Pp 5 pltU
Ks
K; a ea ae
get
K K ‘ K,
r he 5 el
’ K;
K. Ki ae
r ip 1 ois
2
; K
K, K, = =
5elsu

646551 O—62——.26

Definition

Buoyancy force

Center of buoyancy of submarine
Center of mass of submarine
Damping constant

Critical damping constant
Diameter of body
Forward perpendicular

Depth of submergence to center of mass

Moment of inertia of the body about x-, y-, z-axis

Hydrodynamic moment about x-axis through center of
gravity

Derivative of moment component with respect to angu-
lar velocity component p

Derivative of moment component with respect to angu-
lar acceleration component p

Derivative of moment component with respect to angu-
lar velocity component r

Derivative of moment component with respect to angu-
lar acceleration component 7

Derivative of moment component with respect to ve-
locity component v

382

Symbol

Dimensionless Form

Alex Goodman

Definition

Derivative of moment component with respect to ac-
celeration component v

Derivative of moment component with respect to rudder
angle component 6.

Radius of gyration of ship and added mass of ship
about y-axis

Characteristic length of submarine

Hydrodynamic moment about y-axis through center of
gravity

Derivative of moment component with respect to angu-
lar velocity component q

Derivative of moment component with respect to angu-
lar acceleration component

Derivative of moment component with respect to ve-
locity component w

Derivative of moment component with respect to ac-
celeration component w

Derivative of moment component with respect to con-
trol surface angle component 6

Derivative of moment component with respect to bow
plane angle component 6,

Derivative of moment component with respect to stern
plane angle component 6.

Symbol

Mo

Ma’

I

i}

Analysis Used in Submerged Body Research 383

Dimensionless Form
raha de ret Sd oe) UN)

Me

1 1392
9?

M,

Lal 3772

2

m
es
= 7)
9?

N
1 43,72
=o) 0)
5?

Definition

Derivative of moment component with respect to pitch
angle component 0

Hydrodynamic moment at zero angle of attack

Mass of submarine, including water in free-flooding
spaces

Hydrodynamic moment about z-axis through center of
gravity

Derivative of moment Component with respect to angu-
lar velocity component r

Derivative of moment Component with respect to angu-
lar acceleration component r

Derivative of moment component with respect to ve-
locity component v

Derivative of moment component with respect to ac-
celeration component 0

Derivative of moment with respect to rudder angle
component 5.

Angular velocity component relative to x-axis

Angular acceleration component relative to x-axis

Angular velocity component relative to y-axis

ae ee a

384

Symbol Dimensionless Form
$72
: a ql
q q ~ ye
: es
Ty)
p on rl?
r r ge
fi /2
U U'=1
V,
v gee
U
5 ie a
U2
w was
U
a eeu
U2
X apenas
5 pet?
x
iB
Xp» Zp <p =,
ee “B
Bigat wal
LEG
AE tog = 1

Alex Goodman

Definition

Angular acceleration component relative to y-axis

Angular velocity component relative to z-axis

Angular acceleration component relative to z-axis

Time for oscillatory motion to damp to one-half initial
amplitude

Velocity of origin of body axes relative to fluid in
feet per second

Velocity of origin of body axes relative to fluid in
knots

Component along y-axis of velocity of origin of body
relative to fluid

Component along y-axis of acceleration of origin of
body relative to fluid

Component along z-axis of velocity of origin of body
axes relative to fluid

Component along z-axis of acceleration of origin of
body axes relative to fluid

Hydrodynamic longitudinal force, positive forward
The longitudinal axis, directed from the after to the

forward end of the submarine with origin taken at the
center of gravity

Coordinates of center of buoyancy with respect to
body axes

Distance from reference point to center of gravity of
model.

Symbol

¥

Analysis Used in Submerged Body Research 385

Dimensionless Form

a
5 Pl?U?
V7
ape
=pl°U
’ Y.
Y. = ; Le
= pl4
aes
ene Y
“dooksia
9?
ee ee
DONS hte
_—pl
3?
Y
Y,;'= §
= pl?0?
Ys
Ys ‘= if
r 5Pl?U?

ip it ath! Stig

Qe

Definition

Hydrodynamic lateral force, positive to starboard

Derivative of lateral force component with respect to
angular velocity component r

Derivative of lateral force component with respect to
angular acceleration component r

Derivative of lateral force component with respect to
velocity component v

Derivative of lateral force component with respect to
acceleration component 0

Derivative of force component with respect to control
surface angle component 5

Derivative of lateral force component with respect to
rudder angle component 6.

Distance along the transverse axis, directed to star-
board with origin taken at center of gravity

Hydrodynamic normal force, positive downward

Derivative of normal force component with respect to
angular velocity component q

Derivative of normal force component with respect to
angular acceleration component 4

Derivative of normal force component with respect to
velocity component w

386

Symbol Dimensionless Form
peer
Z; x
gel
ae
Zs Z, ‘=
b b 1 512y?
2
Z, fine =
$ Seay
s s 1 72772
= pl-U
3?
Z,
Liz Z,'= ;
=e
3?
z
a
B
)
5,
6.
a)
s
6
p p'=1

Alex Goodman

Definition

Derivative of normal force component with respect to
* .- =
acceleration component w

Derivative of normal force component with respect to
bow plane angle 6,

Derivative of normal force component with respect to
stern-plane angle component 6,

Normal force at zero angle of attack

Distance along the normal axis, directed from top to
bottom (deck to keel), with origin taken at center of

gravity

The angle of attack; the angle to the longitudinal body
axis from the projection into the principal plane of
symmetry of the velocity of the origin of the body axes
relative to the fluid, positive in positive sense of ro-
tation about the y-axis |

The drift or sideslip angle; the angle to the principal
plane of symmetry from the velocity of the origin of
the body axes relative to the fluid, positive in the
positive sense of rotation about the z-axis

Angular displacement of a control surface

Angular displacement of bow planes, positive trailing
edge down

Angular displacement of rudders, positive trailing
edge port

' Angular displacement of stern planes, positive trail-

ing edge down

The angle of pitch; the angle of elevation of the x-
axis positive bow up

Mass density of water

Roots of stability equation, i=1, 2,...

Analysis Used in Submerged Body Research 387

Symbol Dimensionless Form Definition
wy The angle of yaw
ms w' = 4h Circular frequency of oscillation
ol
o, a 7 Natural frequency of undamped oscillation
Subscripts:
in In-phase component of force or moment
out Out-of-phase or quadrature component of force or
moment
o Maximum amplitude
1 Associated with forward strut
2 Associated with aft strut
m Model.
INTRODUCTION

The problems associated with the dynamic behavior of the submarine and other sub-
merged bodies, that is, stability, performance, and ease of handling, have become increas-
ingly more and more important with each new increase in submerged speed. This has been
particularly true for motions in the vertical plane since the submarine must be operated,
strictly on instruments, within the confines of a layer of water usually no greater than a few
boat lengths. Also, the increase in submerged speed has given rise to some serious dy-
namic problems for motions in the horizontal plane and made the problem of emergency re-
covery increasingly more acute.

In the early stages of development in the field of submarine stability and control the
designer was faced with the problem of providing a combination of characteristics and means
for controlling the submarine which would result in satisfactory dynamic behavior. A major
difficulty in providing for this was the lack of sufficient information to guide him in the
choice of a combination of physical characteristics which would result in adequate stability,
performance, and ease of handling. Another major difficulty was the lack of straightforward
design methods and experimental or theoretical techniques for obtaining a desired combina-
tion of characteristics. In those cases where the results of model tests in the form of the
various hydrodynamic coefficients were available, the interpretations that could readily be
made relating them to adequate dynamic behavior were rather limited. The designer was
further handicapped in that he had no standards by which he could evaluate the dynamic be-
havior of the submarine.

The mission of the Stability and Control Division at the David Taylor Model Basin has
been, therefore, to remedy this situation. This has been partially accomplished by the

388 Alex Goodman

development of a set of desirable handling qualities which could be used in optimizing the
submerged body design [1], development of methods which give an objective evaluation of
the submerged body’s dynamic behavior, and improvements in, and development of, new pre-
diction and experimental techniques which could be used by the designer to evaluate the
submerged body performance.

This paper presents and describes the various experimental techniques and methods of
analysis used at the David Taylor Model Basin in the field of dynamic stability and control
of submerged bodies. The advantages and disadvantages of the various techniques are pre-
sented. Primary emphasis is placed on the principles and operation of the DTMB Planar-
Motion-Mechanism System [2] as well as on the concepts, techniques, and philosophies used
in simulator and motion analysis studies. Throughout the presentation, it will be noted that
frequent references will be made to the use of the various techniques as applied to the sub-
marine stability and control problem. Most of the data acquired by these techniques have
been for submarines because of the urgency that submarine problems have assumed in recent
years. It should be noted, however, that these techniques are also applicable to other types
of marine vehicles.

COMPARISON OF EXPERIMENTAL TECHNIQUES

The various approaches to the solution of problems in the field of dynamic stability and
control of a submerged body moving through a fluid, shown in Fig. 1, have been employed by
naval architects and aerodynamicists for many years. Many investigators have used the free-
running or flying-model techniques since a direct evaluation of the performance of the design
is provided [3,4]. However, this method does not provide data which can be related to the
physical characteristics of the design or to support the design changes required for improved
performance. The full-scale technique suffers from the same shortcomings. However, this
technique is used mainly for providing data for model-full-scale correlation.

In recent years, the development of general-purpose analog and high-speed digital com-
puters has resulted in the extensive use of the mathematical-model technique. This tech-
nique, which is based on a thorough analysis of the differential equations which govern the
motions, provides a solution and basic understanding of the dynamic stability and control
problems of a submerged body. However, these differential equations of motion are
comprised of numerous coefficients or derivatives which are of hydrodynamic origin.

et FULL-SCALE TESTS Slate

PROBLEMS THEORETICAL PREDICTIONS

IN a asa

PIRICA
STABILITY pags Me

AND ify
CONTROL | CAPTIVE _MODEL [>| MOTION ANALYSIS|C>

—— FREE FLYING Bie
MODEL TESTS

Fig. 1. Approaches to problems in stability and control

SOLUTIONS

Analysis Used in Submerged Body Research 389

Consequently, to obtain solutions for any given configuration by means of this technique, it

is necessary to know these coefficients with reasonable accuracy. Many attempts have been
made in the past to fulfill this requirement by utilizing various experimental (captive model)

and theoretical techniques, or combinations of both.

Among the various captive-model techniques used, fairly refined methods have been de-
veloped by model basins and wind tunnels for measuring forces and moments due to body
orientation and control deflection; the so-called static stability and control coefficients.
However, the various experimental methods used to determine forces and moments associ-
ated with variations in linear acceleration, angular velocity, and angular acceleration have
been successful in only a limited number of cases. The techniques that have been tried in
this respect have required the use of facilities such as the rotating arm, free oscillator,
forced oscillator, curved-flow and rolling-flow tunnels, and curved models in a straight flow
facility [3]. Each of these techniques has certain limitations and problems associated with
either accuracy of instrumentation, model support, friction, accuracy of flow (curved and
rolling flow) [5,6], and accuracy of model construction (curved model). Also, none of these
techniques provide a direct measure of all the hydrodynamic coefficients required in the equa-
tions of motion for six degrees of freedom. The DTMB Planar-Motion-Mechanism System re-
cently developed at the David Taylor Model Basin, however, incorporates in one device a
means for experimentally determining all of the hydrodynamic-stability coefficients required
in the equations of motion for a submerged body in six degrees of freedom.

A comparison of the main advantages and disadvantages of several of the basic experi-
mental techniques that have been discussed is presented in Table 1. Based on this compar-
ison, it can be seen that the mathematical model technique, in conjunction with accurately
determined hydrodynamic coefficients, provides the most powerful and versatile design and
research tool for the study and analysis of stability and control problems.

PLANAR-MOTION-MECHANISM SYSTEM

The DTMB Planar-Motion-Mechanism System incorporates in one device a means for ex-
perimentally determining all of the hydrodynamic-stability and control coefficients required
in the equations of motion for a submerged body in six degrees of freedom. These include
the static-stability and control, rotary-stability, and acceleration coefficients. The unique
features of the system are the methods used to impart hydrodynamically pure pitching, heav-
ing, and rolling motions to a given submerged body, as well as the dynamometry and data
analysis equipment employed. These enable the explicit and accurate determination of indi-
vidual derivatives without resort to the solution of simultaneous equations as is necessary
when other types of oscillation devices are used. Although the system was designed prima-
rily for submerged body research it can also be used to determine the hydrodynamic coef-
ficients for other types of marine vehicles such as hydrofoil boats and ground effect machines.

General Considerations

The derivatives and composition of the equations of motion have formed the subject of
numerous text books and papers [7-9]. For the purpose of this paper, therefore, only the
general nature of these equations are considered. This is done to give some insight into
the problems which must be faced in the design of experimental facilities for the evaluation
of the equations.

390 Alex Goodman

Table 1

Comparison of Experimental Techniques

(a) Free-Running Model Technique

1. Direct evaluation of performance of 3. Can be used to study extreme maneu-

specific designs vers which may be too complicated to
handle by analytical or computer
2. Can be economical and expedient methods
1. Does not provide data which can be 3. Froude or dynamic scaling necessary
directly related to the design of body for mass, moment of inertia, and meta-
and individual appendages centric stability

2. Does not provide data to directly sup-
port design changes to effect improve-
ments in performance

(b) Mathematical Model Techniques

Advantages

1. Provides means of evaluating han- 4. Provides measures of capabilities of
dling qualities or ahaa charac- the design tempered by existence of
teristics of proposed design well in ad- human operator or automatic control
vance of construction device in control loop

2. Provides simple means of studying ef- 5. Can be used in human engineering
fects of practical variations of basic studies
design which may be necessary to im-
prove performance - 6. Can be used to study overall perform-

3. Can be used to determine effects of ance of various combined systems

various environmental conditions on

peeeneanee 7. Can be used to train personnel

Disadvantages

1. Requires an accurate set of hydrody- 2. Requires expensive facilities
namic coefficients of the design for the

equations of motion

(c) Captive Model Techniques

1. Provides basic data on which to sup- 4. Results can be utilized to study ef-
port design of individual body and fects of proposed design changes

appendages without additional model tests

2. Provides basic data which can be used 5. Data are perpetuated so that further
with the equations of motion in com- studies can be made at a later time
puter and simulator studies to evaluate without the need for additional model
inherent and closed-loop performance tests

3. Provides data which can be directly 6. Provides powerful tool for doing re-
utilized in design of automatic control search or systematic series work on
systems stability and control

7. Dynamic scaling not necessary

Disadvantages

1. Data are one step removed from directly 3. Requires expensive facilities
indicating all the handling qualities

2. Method ae become uneconomical for eval-
uating performance of one specific design

Analysis Used in Submerged Body Research 391

The hydrodynamic forces and moments which enter into the equations of motion as coef-
ficients are usually classified into three categories: static, rotary, and acceleration. The
static coefficients are due to components of linear velocity of the body relative to the fluid,
the rotary coefficients are due to angular velocity, and the acceleration coefficients are due
to either linear or angular acceleration. Within limited ranges the coefficients are linear
with respect to the appropriate variables and thus may be utilized as static, rotary, and ac-
celeration derivatives in linearized equations of motion.

It may be concluded from the foregoing classification, that the experimental determina-
tion of the coefficients of the equations of motion requires facilities which will impart linear
and angular velocities and accelerations to a given body with respect to a fluid. For exam-
ple, the usual basin facilities have carriages designed to tow models in a straight line at
constant speed. Such facilities can be equipped to orient models in either pitch or yaw to
obtain the static coefficients. However, more specialized types of facilities, such as rotat-
ing arm or oscillator, are required to impart the angular velocities that are necessary to ob-
tain rotary coefficients. The oscillator type of facility provides also linear and angular ac-
celerations so that the acceleration coefficients may be determined experimentally.

The choice of a suitable facility for determining hydrodynamic coefficients involves
many considerations pertaining to accuracy, expediency, and ease of data analysis. A de-
tailed treatment of these problems is beyond the scope of this paper. However, of primary
concern is the degree to which the experimental technique involves explicit relationships
and avoids the need for solutions of matrices. Also techniques which involve extrapolations
should be avoided. To illustrate, a carriage which tows a model at uniform velocity in
straight-line pitched or yawed flight is a direct and explicit means of determining static
coefficients. Similarly, a rotating arm which:tows a model at uniform angular velocity and
tangential to the circular path at each of several different radii is a means for determining
rotary coefficient explicitly. On the other hand, the use of the rotating arm to obtain static
coefficients should be considered as an indirect procedure since the data must be extrap-
olated to infinite radius. The usual oscillator techniques are even more indirect and, at
best, require solutions of simultaneous equations to.obtain rotary and acceleration derivatives.

Each of the techniques mentioned can be used most advantageously for obtaining one
category of hydrodynamic coefficients. The straight-line towing carriage supplies only the
static coefficients. The rotating arm supplies rotary coefficients directly and static coef-
ficients indirectly. The oscillator supplies all three categories of coefficients, but all in-
directly.

The foregoing considerations suggest the desirability of having a single system to de-
termine explicitly all of the coefficients required in the equations of motion for six degrees
of freedom. To accomplish this objective, it is necessary to develop a facility which can
move a body through water with “hydrodynamically pure” linear velocities, angular veloci-
ties, linear accelerations, and angular accelerations in all degrees of freedom. This con-
cept forms the basis of the DTMB Planar-Motion-Mechanism System.

Principles of Operation

The DTMB Planar-Motion-Mechanism System as it physically exists is described briefly
in the next section. It is desirable, however, to consider first the principles underlying the
operation of the mechanism so that the design concept can be generally understood. In the
interest of simplicity the mode of operation applicable only to submerged bodies in the

ae ee en ee a

392 Alex Goodman

vertical plane will be used to describe the principles of the system. The system of axes as
well as the symbols and coefficients used in this section have been defined in the notation
at the beginning of this paper.

The kind of motion for static coefficients is commonly used by wind tunnel and model
basin facilities and, therefore, dces not need to be explained in detail. Figure 2 schemati-
cally represents this type of motion. The components are given with respect to a body-axis
system with the origin at the center of gravity, CG.

0 (@=a)

Fig. 2. Straight-line pitched motion for steady-state tests

The system produces this motion by using a towing carriage to tow the model in a
straight path at constant velocity. Discrete pitch angles for each run are set by a tilt table
which supports the model through a pair of twin struts. Control surface angles are also set
discretely for each run. Forces are measured by internal balances at each of the two struts
to obtain static forces and moments.

The unique feature of the DTMB Planar Motion Mechanism is the kinds of motions pro-
duced to enable the explicit determination of the rotary and acceleration coefficients. Si-
nusoidal motions are imparted to the model at the point of attachment of each of the two
towing struts while the model is being towed through the water by the carriage. The mo-
tions are phased in such a manner as to produce the desired conditions of hydrodynamically
“pure heaving” and “pure pitching.” It is possible also, if required for any reason, to pro-
duce various combinations of pitching and heaving. Figure 3 illustrates various types of
motions including (a) the type of motion usually associated with oscillators, (b) pure heav-
ing, and (c) pure pitching. The latter two are the basic motions associated with the DTMB
Planar Motion Mechanism. The pure rolling motion is not illustrated but consists simply of
oscillations about the longitudinal body axis.

The oscillator motion depicted by Fig. 3a is actually a combination of pure pitching and
heaving motions. Since the CG is constrained to move in a straight path while the model,
which oscillates in a see-saw fashion, assumes sinusoidally varying angles of attack and
pitch angles. As a result a mixture of static, rotary, and acceleration forces and moments
is produced. It becomes necessary, therefore, to perform a similar oscillation about a second
reference point. The two oscillation conditions together with the static tests provide data
which can be used to separate the hydrodynamic coefficients. The solution of simultaneous
equations involved in this process, however, could lead to errors because of the wide differ-
ences in magnitude between the various individual coefficients. The oscillator type of

Analysis Used in Submerged Body Research 393

2 »

(a) Combined pitching and heaving

(b) Pure heaving

Ny C2a-b

(c) Pure pitching

Fig. 3. Oscillation types of motion

394 Alex Goodman

motion is produced by the Planar Motion Mechanism when the two struts move sinusoidally
180 degrees out of phase with each other.

The pure heaving motion shown in Fig. 3b is obtained when both struts move sinusoi-
dally in phase with each other. This results in a motion whereby the model CG moves in a
sinusoidal path while the pitch angle @ is invariant with time.

The pure pitching motion shown in Fig. 3c is obtained by moving both struts out of
phase with each other; the phase angle between struts, ¢,, is dependent upon frequency of
oscillation, forward speed, and distance of each strut from the CG. This relationship can
be expressed as

and is derived as Eq. (A20) in Appendix A. The resulting motion is one in which the model
CG moves in a sinusoidal path with the model axis always tangent to the path (angle of at-
tack a = 0).

The process for obtaining translatory acceleration derivatives from pure heaving tests
is represented diagrammatically in Fig. 4. The diagrams across the top of the figure show
the motions of the aft and forward struts with respect to each other. Corresponding posi-
tions of a component resolver (which has been replaced by a sine-cosine potentiometer in
the new system), provided with the electrical system to rectify the sinusoidal signals from
the force balances, are also shown. At the left is a column of graphs showing the resulting
motions and forces at the CG. The right-hand column contains the mathematical relationships
represented by each graph. Descending from the top of Fig. 4, there is the vertical displace-
ment z curve, the associated velocity 2 curve, the associated acceleration z curve, and the
vertical force Zp curve. It may be noted that the Zp curve is displaced in point of time
from the z curve by phase angle ¢. Thus Zr can be considered as being made up of two
components, one in phase with the motion at the CC, Z;,, and the other in quadrature with
the motion at the CG, Z,,,. The shaded area per cycle under each curve represents the mag-
nitudes of Z;, and Z,,,, respectively.

The process for obtaining rotary and angular acceleration derivatives from pure pitching
tests.is represented diagrammatically in Fig. 5. The order followed is similar to that shown
in Fig. 4. In this case, the pitch angle traces (0, 0, and 6) are of primary interest. The Zp
curve is displaced in point of time from the 0 curve by phase angle ¢. The procedure for
resolving the resultant force into in-phase and quadrature components is similar to that for
the pure heaving case. The shaded area per cycle under each curve represents the magni-
tudes of Z;, and Z,,,, respectively.

In the pure heaving case the in-phase component of force is directly related to the linear
acceleration and, therefore, can be used to compute explicitly the associated acceleration
derivatives. Similarly, in the pure pitching case the in-phase component of force is directly
related to the angular acceleration and the quadrature component is directly related to the
angular acceleration and the quadrature component is directly related to the angular velocity.
Thus both the angular acceleration and rotary derivatives can be computed explicitly. The
relationships between the various rotary and acceleration derivatives and the respective

Analysis Used in Submerged Body Research 395

Pure Heaving Synchronous Switch
F'wd 270°
SZ

%=0

Z=W= wa COS wt=W, cos wt

oo e 2 . 2 °
Z=w=- W404 Sin wt =— Wo SIN wt

2,72. Sin Wt-%) =(Z, cos ¢) sinut

-(Z_ sind)cos wt ='Z,, sinwt+ Z,,, cosut

wT

Zin? S° z| | % sin(wt-¢) d(wt)—
em
f Zosin(ot—sacr) | = Z, cos >
Tv
1,

Zo Eo | [2 sin(ut—¢) d(wt)—

3/57
J sinwt+d)d(wt) +

27

f Z,sin(wt- ¢) d ot | -Z,sing
37

Fig. 4. Analysis of pure heaving motion

396 Alex Goodman

Pure Pitchin Synchronous Switch

CT
gc car Fae: te
_ 5 2 z | = J
9 “rats | Kr Xo, Xj +x =b

wt Yw, w, Z
Zz j oN =
z Las . Las wt Zz, =a sinwt
/2~ |" en 37
72 Noe Zo =a sin (wt -%.)

60 /+\ Q= sak = - 49 sin [costut-5)]
ds
wonepe|

Bo % cos(wt-—)

= Q=WO sin (wt- 5)

BY

ds

6 =4=w"@, cos(wt-—)
lens ,
Zp=Zo cos [(wt-) - 4]
: ¢
=Z, Cos dcos(wt-S2)+Z, sin g sin(wt- 3)
yore m+ $8
Zin? ae ZRd(wt)- zagiet] zoe
$5 wns
5 ome m+ 8

is yaaa $

dyes

zou" F-2s| | Zp att) -[ ZR don) + | ze gt
ec re

\ ar Ss a

N
>

N
2
eM
Wi

=Z_ sing

Fig. 5. Analysis of pure pitching: motion

Analysis Used in Submerged Body Research 397

in-phase and quadrature components of force are presented in Table 2. The derivation of
these reduction equations is presented in Appendix B.

Description of Facility

The DTMB Planar-Motion-Mechanism System is a complete system for obtaining hydrody-
namic coefficients from model tests. It embraces all mechanical, electrical, and electronic
components necessary to carry out all functions from the delivery of the model to finalized
processing of data preparatory to analysis. This includes preparation of the model for test-
ing, conduct of static and oscillation tests, sensing and recording of test data, and process-
ing data digitally in tabulated form. The main features of the system such as model support
and positioning equipment, forced-motion mechanism, dynamometry, and Instrumentation
Penthouse containing recording and control equipment are shown in Figs. 6 to 23. A descrip-
tion of each of these components has been presented in great detail in Ref. 2. However, the
roll-oscillation mechanism and new instrumentation system shown in Figures 15, 20, and 22
were developed and placed into operation after the publication of Ref. 2 and have not been
described heretofore.

Roll-Oscillation System — Briefly, the roll-oscillation system produces an oscillation of
the body about its longitudinal axes. Modifications and additions to the drive system and
roll gage assembly, shown in Figs. 13 and 17, respectively, were required to incorporate
this additional mode of motion.

\emee
TOWING

OSCILLATOR MOTOR
PHASE CHANGER

fi)
CARRIAGE | SUPPORT BRACKET
1 XK VARIABLE-SPEED BELT DRIVE

Cm
Le.< PISTON
WATERLINE

APPROX.
10'

- —

GAGE ASSEMBLY

Fig. 6. Schematic arrangement of DTMB Planar Motion Mechanism with model attached

646551 O—62——27

398 Alex Goodman

Table 2
Reduction Equations for Rotary and Acceleration Derivatives*

Vertical Plane

_ OZ "Din + (22dinl O02, ')in + Za" dinl
wae ae a

aes ol(Z,'), = (Z,")i nl x el(2, din > (Z,')i 41
t ow, 4 ow,

°

UZ" dour + aJoued — UZ "dour + Fr'Jourl
2q,' ; 2q,'

x Ol(Z 2" dour ry (2; Douel CZ a" dour ~ (21 Jour!
iv eM ae -

Hees
°q,’ 4 aq,’

MCE Di CAS OZ int ain
2g,’ 0g," 7

x (2Z,')in = i Jinl es 2002 4"Jin ~ 1'Dinl
4 oq," w? 4 0q,' w?
Horizontal Plane
Safar Bo

atC¥ "din + Pa ial Din * Pa"dinl
r ov,’ ov,”

'
Me

x CK, 7 sind ie sin 7 M1 ind
a ov,’ q ov,’

= Os Dour ¥ a out! i OLCK dour y CADE!

‘ ,
or, or,

OLY a" Jout ot CY ourd ah OK a Dour i Cia Dane

or,’ 4 ae”
Cle Suet vy Y,')inl Cle Fae Sh (Ye dinl
“ or,’ or,'

xe OlLOXa ite Olan inl ri MOG Oy SOO) ioe
a [o Waco) mua eat Stop or,’

LXKD CK" )in
3, Ww,"

°

OK") out + O(K') out

2 chen’ or,
2 ie OK'd in
oe” or.’

o

= AK dour S OK") out
ep,’ op,’

* mys (Mod m Ny) m> and (K,),, determined from standstill tests

Analysis Used in Submerged Body Research 399

Fig. 7. Tilt table with model attached
mounted on storage stand

—— ls

j

'-

te ian
i,
al
|
if
*

Fig. 8. Tilt table with model attached being moved
by overhead crane

Alex Goodman

400

gales io

—

Si

Fig. 9. Tilt table support bracket attached to towing carriage

Fig. 10. Tilt table

Analysis Used in Submerged Body Research 401

Fig. 11. Tilting mechanism

é

Fig. 12. Counterbalancing device

The lower pulley shaft of the drive system, shown in Fig. 13, was modified so that it
would operate in two modes; as a power transmitting shaft for the pure heaving and pitching
modes, and as a power transmitting shaft for the rolling mode. This is accomplished by
mounting the lower set of Gilmore pulleys on a shaft which is support by ball bearings to an
inner shaft which is directly coupled to the main drive shaft. For pitching and heaving, the
outer and inner pulley shafts are connected by means of a tapered pin. For the rolling mode,
the taper pin is removed, and the strut-pistons are locked. The pulley at the end of the
idler shaft, shown in Fig. 15a, is connected by means of a Gilmore belt to a pulley-shaft
system attached to the aft strut-piston. This pulley can be positioned along a keyed shaft
so that a fixed relationship is maintained between the pulley and the aft strut. Power is
transmitted to the slider crank mechanism attached to the aft strut, by means of another

402 Alex Goodman

(a) Top view (b) End view
Fig. 14. Phase-changer and synchronous switch

pulley-Gilmore belt system. The slider crank mechanism used in this case has an eccen-
tricity of 0.125 inches and a connecting rod length of 5.000 inches; or a length of connecting
rod to eccentricity of 40.0. The sinusoidal motion produced by the slider crank mechanism
is transmitted to a 1.0-inch-diameter push-rod, shown in Fig. 15b, which is guided by linear
bearings attached to the aft strut. The end of the push-rod is attached to the modified roll
gage by means of the 3.000-inch crank arm assembly shown in Fig. 15c. The maximum am-
plitude of the roll oscillation is therefore, +2.38 degrees

403

Analysis Used in Submerged Body Research

goueleq [[O1 03
quawuyoRye pol-ysng (2)

UWISIUBYOOUI UOTII[IOSO [[OY “GT “sty

por ysnd [Joy (q)

USTUBYIOU BATIP UOTIe[[LOSo [[OY (B)

404 Alex Goodman

Fig. 16. Modular force gage

(a) Assembled with gimbal

(b) Individual components

(c) Modified roll gage

GIMBAL BLOCK

PUSH-ROD ATTACHMENT

ME Beil

Fig. 17(a), (b), (c). Roll gage

Analysis Used in Submerged Body Research 405

(a) Top view

Ce

(b) Fore and aft view

Fig. 18. Gage assembly

Alex Goodman

406

. Penthouse mounted on top of carriage

19

Fig.

Fig. 20. Inside view of penthouse showing instrumentation

Analysis Used in Submerged Body Research 407

SERVO DIGITAL ER OSRAMMEK
AMPLIFIER INDICATOR RGAE

IBM
ELECTRIC
TYPEWRITER

SELECTOR
SWITCH

STATIC

BALANCE &
SENSITIVITY
CONTROL

DYNAMIC

SYNCHRO E FORCE INEST ae BROWN
SWITCH OMPONENT (LONG- RECORDER
SEPARATOR CONSTANT FILTER)

Fig. 21. Instrumentation system for Planar-Motion-Mechanism System

400
SUPPLY

GAGE} -CONTROU_ISTATIC on

A400 CYCLE] } --- doyrrare-o| SCANHER
SUPPLY ee bg ro

Ee er
ae Ear lke
R
AMPUFIER” —TFUNCTION RELAYS IBIER
CEROMICALY HVTAERD 40
MECHANISM DRIVE SHAFT | 3 77777 TH ee TEGRATE|
SWE- COSINE
POTENTIOMETER
ELECTRICALLY ACTUATED
BY MECHANISM
ORIVE SHAFT

Fig. 22. New instrumentation system for static and dynamic stability tests

The roll gage assembly, shown in Figs. 17a and 17b, was modified as sketched in Fig.
17c. The longer shaft is now supported in the gimbal block by two ball bearings. For the
static, heaving, and pitching tests, the shaft is attached to the gimbal block by a large
taper pin. In this mode the balances and gimbal block assembly function as described in

Ref. 2.

For the rolling tests, the taper pin is removed and the push-rod attachment is installed.
The body can now be oscillated in roll through the roll gage located at the aft strut and the
three-degree-of-freedom gimbal located at the forward strut.

The range of oscillation frequencies and frequency changing procedures for the rolling
mode are the same as for the pitching and heaving modes [2].

408 Alex Goodman

Fig. 23. Digital indicator

Instrumentation for Static and Dynamic Stability Tests — A comparison of the old and
new measuring and recording systems, presented in Figs. 21 and 22, respectively, shows
that both systems are essentially the same for the static stability tests. The major differ-
ence is in the method and equipment used in the dynamic tests. For completeness, how-
ever, the measuring and recording equipment used in the static and dynamic tests, shown by
the block diagram in Fig. 22, will be described.

The recording equipment for the static stability tests is a digital system designed to
display and read out the unique steady-state value of each force and moment sensed by the
transducers for any given test condition. The system is made up of seven channels to con-
form to the number of gages in the model. Each channel is separate in all respects except
for the power supply that it shares in common.

Briefly, each channel is essentially an automatic null-balancing system; the transducer
in the gage and the digital indicator combine together in a servo system. ‘The transducer
output is balanced by a potentiometer. When a gage in the model is deflected, the resulting
error signal from the transducer is amplified and drives a servo motor which positions a
potentiometer to restore electrical balance, or null, to the system. The amount that the
potentiometer is moved is then a measure of the force or moment applied at the gage. The
various components and circuitry which constitute the recording systems are shown by the
block diagram in Fig. 22.

The upper path of Fig. 22 applies to the digital system used for static stability tests.
A Brown recorder could be used in place of the digital system. The term gage is used to
denote the variable reluctance transducer whether it be the magnigage used with the modular
force gages or the magnitorque used with the roll gages. The 400-cycle power source sup-
plies a 4.5-volt carrier to the gage in such a manner that the current divides into two paths,
one about each coil. If the core of the gage is electrically centered, the impedances of the
gage halves are equal, and consequently the voltages are equal. As the core is displaced,
the impedance of one gage half increases and that of the other decreases with corresponding
voltage changes.

Analysis Used in Submerged Body Research 409

Alternating voltage from the gage passes to the balance and sensitivity control box,
which contains a silicon diode bridge as well as other adjustments and refinements that are
described in more detail later. The voltage is rectified by the diode bridge to produce full-
wave rectified direct-current voltage. The total rectified voltage obtained across both coils
is constant and is used as a reference voltage. Polarity is established by making one side
of the line positive and the other negative. The voltage measured between each coil
changes, however, when the gage core is displaced; the voltage across one coil increases
while the other decreases an equal amount so that the reference voltage always remains con-
stant. This is analogous to a three-wire system in which the voltage across the outside
lines remains constant but the voltage from one side to the common is made variable. The
feedback potentiometer, which is on the shaft of the digital indicator, is wired similarly; the
voltage across the end terminals is the reference voltage and the common is attached to the
potentiometer slider. When the gage core is at electrical center, the potentiometer slider is
at midposition. When the gage core is displaced, an error signal results.

The error signal is fed through a mode selector switch (static or dynamic; digital or
recorder) to a chopper servo-amplifier similar to that contained in the Brown recorder manu-
factured by Minneapolis-Honeywell Company. The chopper converts the direct-current error
signal into 60-cycle alternating current. The resulting signal is amplified to drive a servo
motor which in turn drives the potentiometer slider until the error signal is reduced to zero
and a null-balance established.

The digital indicator, shown in Fig. 23, is the active part of the feedback loop. The
assembly is made up largely of commercially obtainable components. Beginning from the
left, it may be seen that there is a digitizer with a detent unit, servo motor, speed reducer,
and potentiometer. The components are aligned axially and are connected together with
Oldham couplings to minimize binding. The Metron speed reducer, is an antibacklash plane-
tary gear box with a reduction of 21 to 1. It is inserted between the servo motor and 10-turn
Heliopot (+0.05 linearity) so that the range of the system is +5 turns on the potentiometer,
with a little to spare. The digitizer is connected directly to the through-shaft of the servo
motor. The digitizer is a unit manufactured by the Dayton Instrument Company. It is essen-
tially a four-digit mechanical counter and plus-minus wheel, equipped with electrical con-
tacts. Eleven wires are brought out of each decade, one for each unit and one common.
These wires lead to the programmer and scanner and then to the readout equipment. The
detent unit, which is integral with the digitizer, consists of a solenoid-operated star wheel.
The star wheel is used to center the unit decade on a contact, when command for readout is
given the solenoid is energized and the digit wheels are placed in contact with the electrical
brushes.

The digital indicator operates in two modes: balancing and readout. In the balancing
mode, it is part of the feedback loop, as explained earlier. In the readout mode, the servo
motor is automatically stopped and the digitizer serves as a memory which stores the last
reading obtained.

The scanner and programmer unit is the brain of the readout system. It serves two func-
tions: first, introducing predetermined data such as run number, body angle, and control
surface angle and secondly, scanning and sequencing the data actively obtained during the
test. When a test run is made, the digital indicators are allowed to settle out at an approxi-
mately fixed reading. At command, the servo motors are automatically stopped and the scan-
ner unit scans each channel, decomplimenting if necessary, and feeds the readings in cor-
rect sequence, one digit at a time, to the solenoid-operated IBM electric typewriter. The
typewriter tabulates the data on a form specially prepared for the purpose, as shown by the
reduced sample given in Fig. 24.

410 Alex Goodman

SAY STABILITY ARO CONTROL DATA SHEET

pRac-TR. .13 (2-58) SHEET No, 23

ENGINEERS) Young-Schwarting MOTESStrip tests top rudder ‘and prop. off
Sens 5:1 all cares except ¥'s

0000 0000 0000 0000
-0203 -0021 0016|-0070
-0212 0232 -0058 -0084
-0213 0482 -0120 -0088

-0207 -0012 0019 -0066

-0195 -0295 0110 -0056

-0187 -0536 0158 -0032

-0190 -0793 0173 -0008

-0188 -1041 0155 -0019

-0200 -1060# 0061 0062

* after nO means hit limit

-0212 -0060 0111 -0063 0011 -000
=0216 0186 0045 -00/5 OOK -0001
=0218 0427 -0004 -0085 0157 -0001
-0214 -0057 0111 -0062 -0000 -0001
-0200 -0335 0193 -0041 0056 -0002
-0198 -0580 0243 -0014 0194 -0002

0211 0014 -0061 -0066
-0211 0268 -0139 -0073
=0210 0520 -0196 -0075
-0206 0018 -0062 -0057
20194 -0254 0016 -0050
0027 -0188 -0499 0059 -0026

2026
20:
2028
20
20
20
20
20
20
20
20
20
2038
20
20
20
20
20
20

Fig. 24. Typical data sheet for static and dynamic tests

As mentioned earlier, the balance and sensitivity control box contains features that are
provided for the purpose of maintaining accuracy and increasing versatility of the system.
Among these features are the means of checking zero, adjusting and checking sensitivity,
changing zero reference, and filtering to smooth out the data.

A zero-check switch is provided to separate any change of reading due to causes other
than actual gage displacement. These changes could be due to causes such as changes in
value of circuit resistors or diodes. When the switch is closed, the primaries of two input
transformers of the control circuit are connected in parallel so that their voltages must be
equal regardless of gage core position. If the zero-check reading differs from the original
value, the difference is due to changes in the control unit circuitry rather than the gage.
Thus, the reading obtained on the digital indicator may be corrected by this amount. If the
gage is not balanced at the time of testing due to preload or core offset, it is desirable to
balance it directly. This is accomplished by a “gage zero” potentiometer which is adjusted
to make the impedances across the two gage halves equal.

The “pen position” adjustment is provided to set the initial reading of the digital indi-
cator or recorder to any desired value while the model is at rest. The usual practice for
steady-state tests is to adjust the digital indicator to read zero when there are no hydrody-
namic loads on the system. The setting is periodically checked before each run or group of
runs to maintain the zero. The advantages of this procedure are that it provides a means for

Analysis Used in Submerged Body Research 411

determining easily whether any changes other than hydrodynamic have occurred in the total
system and it eliminates the need for subtracting arbitrary readings on each channel to ob-
tain the net readings. The pen position adjustment is accomplished by a potentiometer
which is connected in parallel with the feedback potentiometer.

A span or sensitivity adjustment is provided to establish the calibration of the digital
indicator in terms of the load on the gage. The span control is a potentiometer which is
placed in series with the part of the circuit that goes with the feedback potentiometer slider.
Thus the unbalanced voltage resulting from displacement of the gage appears across the
span potentiometer as well as any other resistors placed in series with it. The range of
sensitivity varies from nearly zero to an amount somewhat in excess of that required to ac-
commodate the maximum sensitivity of all the types of transducers used in the tests. The
span potentiometer has a calibrated index and can be locked into place. However, since the
control units and gages can be interchanged, the span potentiometer setting is not suffi-
ciently accurate. Therefore a system for setting sensitivity which is independent of the
transducer movement is provided. This “span check” is made by applying a step signal to
simulate an actual transducer change. To do this, a precision resistor in the control box is
shunted across one gage coil. The resulting span check reading on the digital corresponds
to the given sensitivity and to the nominal setting on the span potentiometer.

Span control settings and span checks are usually established with the modular force
gage or roll balance mounted on a calibration stand. Since all components of the measure-
ment system are linear, the settings are determined on the basis of that required to give a
reading of exactly 1000 counts on each channel for some predetermined load. As mentioned
earlier, the usual sensitivity is 200 pounds for 1000 counts; however, sensitivities of twice
or one-half of this amount are used from time to time, depending on the range of loads en-
countered in the test. The span checks for each calibration are recorded in a log book.
These values have been found to hold true for any particular gage and control box combina-
tion over a period of years.

The signal coming from the gages is a fluctuating one even in steady-state tests. This
is due largely to carriage vibrations which are transmitted to the model through the rigid at-
tachment. A filter is provided in the control box to smooth this signal to obtain one steady
value at the digital indicators. The filters are made up in steps so that only the amount
needed for smoothing is used without needlessly sacrificing speed of response. The filter
switch connects successive values of capacitance between the span potentiometer slider
and one side of the feedback potentiometer. The polarity between these two points is al-
ways the same, so that electrolytic capacitors of reasonable size can be used. Since this
capacitance is outside the servo feedback loop, it introduces no instability.

The recording and measuring equipment for the dynamic tests uses the same control
unit, 400-cycle power supply, and digital readout or Brown recorder system. The distin-
guishing features are: the introduction of the signal amplifier, the force component separa-
tor, the integrating amplifier, the multiganged sine-cosine potentiometer, and the integrate
timer and function relays. These components are shown by the lower leg of the block dia-
gram of Fig. 22. They become part of the measuring and readout system when the selector
switch is thrown from static to dynamic. The selector switch also selects the type of read-
out; that is, digital or Brown recorder. The digital system, however, is represented in

Fig. 22.

The signal amplifier shown by the block diagram is a high-gain transistorized amplifier
that provides accurate amplification of signals in millivoltages over the frequency range from
direct current to 20 kilocycles. Some of the principle features of this amplifier are: high

412 Alex Goodman

input impedance, and therefore negligible loading effect on transducer initiating signals,
drift of less than 2 microvolts per 10°F change in ambient temperature for the 0.1- to 30-
inillivolt input ranges, and for frequencies of 0 to 10 cycles per second, an equivalent input
noise of about 4 microvolts peak to peak. This amplifier, the Accudata III, is manufactured
by Minneapolis-Honeywell Company. Seven of these amplifiers are used in the system; one
for each gage.

The multiganged sine-cosine potentiometer (eight potentiometers mounted on one shaft)
is mechanically attached to the Planar Motion Mechanism driveshaft in the same manner as
that used for synchronous switch, shown in Fig. 14, and rotates at the displacement fre-
quency w. Each sine-cosine potentiometer has a function conformity of +0.25 percent from
0 to 360 degrees. These potentiometers are manufactured by the Beckman Heliopot Company.

The output of the signal amplifier is applied to the input of the sine-cosine potentiome-
ter within the circuitry of the force component separator. The force component separator
consists of seven channels, one for each gage. The major components of the force compo-
nent separator unit are the function switch, normal-reverse switch, and function relays.

The function switch has four settings which are identified as quadrature, reset, in-phase,
and calibrate. The quadrature position electrically multiplies the amplified signal from each
signal amplifier with the cos wt generated by a corresponding cosine wiper of the sine-
cosine potentiometers. The in-phase position performs a similar multiplication with sin ot.
In the reset position, the feedback capacitor of the integrating amplifier is shorted out and
the output. voltage of the integrating amplifier is returned to its zero level. In the calibrate
position, the sine-cosine wipers are disconnected and the signal is multiplied by unity in-
stead of either sin wt or cos wt. The significance of this will be discussed later. The
normal-reverse switch changes the polarity of the signal feeding into the sine-cosine poten-
tiometer. This feature eliminates the requirement for knowing the static zero precisely; that
is, it eliminates the need for having the transducer balanced exactly.

The product of the signal with either the sin wt or cos wt is fed into the integrating am-
plifier. This amplifier is another Accudata III operated in the open-loop mode and having a
precision capacitor in the feedback loop. Again, one integrator is used for each gage in the
system. The feedback capacitor has a value of 0.5 microfarad (accurate to within +0.1 per-
cent) and is manufactured by Arco Electronics Company. The output voltage of the integrat-
ing amplifier is fed through the contacts of the function relays; identified as “integrate”
and “hold.” These relays are controlled by a precision stepping switch located in the in-
tegrate timer unit. The operation of the stepping switch is controlled by a microswitch
which is actuated by a sweeper mounted on the drive shaft. The number of pulses to be
counted, corresponding to revolutions of the drive shaft, can be selected by a switch located
on the integrate-timer unit. The home position of the stepping switch and function relays
corresponds to the reset position mentioned previously. Upon being pulsed, the stepping
switch operates the integrate and hold relays and then rotates an amount corresponding to
the selected number of cycles. At this point the integrate relay is deenergized, opening the
input to the integrating amplifier. At the same time the hold relay maintains the charge on
the feedback capacitor of the integrator. The integral of the signal is therefore, the output
voltage of the integrating amplifier divided by the selected time of integration. This voltage
can be recorded and read out by either the digital servo system or Brown recorder.

The operation performed by this system is equivalent to determining the Fourier coef-
ficients of the fundamental of the gage signal [10,11], as illustrated in Appendix C.

Analysis Used in Submerged Body Research 413

The validity of the force-component-separator and integrator system, shown in Fig. 22,
has been established on the basis of controlled laboratory tests. A sinusoidal load of known
amplitude and frequency (0.1 to 1.0 cycle per second) was applied to a gage. The phase
angle between the gage signal and the sine-cosine: potentiometer was varied known amounts
over a range of +90 degrees. The study demonstrated that the amplitude of any individual
component is determined to an accuracy of better than | percent.

One of the main advantage of this system, over the one described in Ref. 2, is the re-
duction in test time for the dynamic mode (about 50 percent) and the use of the digital-readout
system for recording purposes.

Typical Test Results Obtained with the System

All the hydrodynamic coefficients required in the equations of motion for submerged
bodies in six degrees of freedom can be obtained with the DTMB Planar-Motion-Mechanism
System. This is accomplished by appropriate orientation of the model and mode of operation
of the system.

Both the linear and nonlinear coefficients associated with static stability and control
are determined using the system. In the case of the rotary and acceleration derivatives,
however, only the linear coefficients are determined.

The various static, rotary, and acceleration coefficients that can be evaluated using
this system are presented in Appendix B and Table 2. It is believed to be pertinent, how-
ever, to include representative samples of test results obtained for each of the three classes
of coefficients. Examination of these samples should provide insight not only into the nature
of these coefficients but also the quality with which they can be determined by the system.
Before proceeding, it is reemphasized that all of the coefficients are obtained from the ex-
plicit relationships given in Table 2.

Typical test results for coefficients of the “static” variety are shown in Figs. 25, 26,
and 27. The variation of normal force and pitching moment with body angle is shown in Fig.
25 and the variation of normal force and pitching moment with stern plane angle is shown for
various body angles in Figs. 26, and 27, respectively. The body-angle and plane-angle
range covered in this case is only +6 degrees. Data are, in general, obtained over a body-
angle range of +15 degrees and a control-surface angle range of +20 to +45 degrees, depend-
ing on the particular control surface being investigated. As indicated in the figures, the
derivatives are obtained from the slopes of the appropriate curves faired through the data
points. The slopes of the body-angle curves are taken through a body angle of zero and be-
come the static stability derivatives. The slopes of the control-surface curves for zero body
angle are taken through a control-surface angle of zero and become the control derivatives.

The kind of results obtained from pure heaving tests is shown in Fig. 28. The slopes
of the separate in-phase force component curves are used with the formulas given to obtain
the linear acceleration derivatives, the added mass Z and associated moment M,.

Typical results of pure pitching tests which are used to obtain (damping) force and mo-
ment derivatives are shown in Fig. 29. It may be noted that the quadrature components of
force measured at each of the two struts are plotted separately. It has been found desirable
to do so since the slopes of the two curves can then be substituted directly into the two for-
mulas shown in the figure to obtain the damping force derivative Z , and damping moment
derivative M,.

646551 O—62

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Alex Goodman

<4 “4
10 x20 Zw is slope of Z’curve -taz0 By
Mw is slope of M curve at ox 20
W/U =W’ for small angles
6 4
12 3
N 8 2s
s 5
pie alls a pee =
8 fo] (ors
er re ae
_ -4 ‘ll
Pcs abet
A, Shes
-(2 -3
-16'
-6 4 2 fo) 2 4 6
Angle of Attack a in degrees

Fig. 25. Typical curves of static force and
moment versus body angle

at Mg, is slope of neat

M’ curve

=
=
a
7
=
i
o
=
re
=
a
CEE aac)
ESSE b boli,
“24
-8 420 2 4 6 8 Sioule4ne
ar Plane Angle §, in degrees Stern-Plane Angle 6, in degrees
Fig. 26. Typical curves of static force ver- Fig. 27. Typical curves of static moment
sus control-surface angle for various body versus control-surface angle for various

angles. body angles

Analysis Used in Submerged Body Research 415

In-Phase Component of
Normal Force Coefficient

, OZ) in + (Z2 Vind

Ma x 8[(Za)in-(Z) in]
wit rs
eo.

fo) 002 004 O06 O08 O10 Ol2 O14 O16
Linear Acceleration Parameter Wo

Fig. 28. Typical curves of forces versus linear acceleration
amplitude from pure heaving tests used to obtain added mass

(and associated moment)

Fig. 29. '‘l'ypical curves of forces versus
angular velocity amplitude from pure pitching
tests used to obtain damping force and damp-
ing moment derivatives

=
3,
Py

fe}

8 (Zour +(Za' out J
849

(2g +™m )*

&
GB

Quadrature Component of Normal Force Coefficient

O 002 004 O06 O08 O10 Ol2 O14
Angular Velocity Parameter q,

The curves of in-phase force components shown in Fig. 30 are also typical of the results
obtained from pure pitching tests. Here again, the force components measured at each strut
are plotted separately. The angular acceleration derivatives, the added moment of inertia
M; and associated force Z;, are obtained from the slopes of the curves using the formulas

given in the figure.

Results are obtained in a similar manner for all the other hydrodynamic coefficients.

ROTATING ARM FACILITY

Another captive-model technique that has been used extensively in submerged body re-
search, to determine explicitly the rotary coefficients for the differential equations of motion,

416 Alex Goodman

x 2UEBig i 2g]

In-Phase Component of
Normal Force Coefficient.

O.l 0.2 0.3 04 06 O7 08

Angular Acceleration Parameter q,

Fig. 30. Typical curves of forces versus angular acceleration am-
plitude from pure pitching tests used to obtain added moment of
inertia (and associated force)

is the rotating arm technique [12]. In this case, the body is towed at uniform angular veloc-
ity in a circular path at different radii and the resultant forces and moments are determined
for various body angles. The rotating arm technique has been used also to obtain the static-
stability derivatives. In this case, the data for the various body angles are extrapolated to
infinite radius and cross-plotted against the body angle. This is an indirect procedure and
is generally not recommended.

The DTMB Planar-Motion-Mechanism System determines only the linear rotary deriva-
tives. The DTMB rotating arm, using the same models, will supplement these results. and
provide a measure of the nonlinearities which are presently being estimated theoretically.

A detailed description of the DTMB rotating arm facility is presented in Ref. 13. For
the purposes of the present paper a brief description of the main components, shown in Figs.
31 to 33, is presented herein.

The DTMB rotating arm basin is 260 feet in diameter and 21 feet deep. The arm pivots
in the center on tapered roller bearings designed for centrifugal forces of 145,000 pounds.
The drive system is located at the extreme end of the arm and consists of two 250-horsepower
direct-current electric motors, directly coupled to two steel wheels which support the arm and
run on an outer peripheral track. Kach wheel is preloaded against the track by nested com-
pression springs which provide a normal force of 61,000 pounds. A maximum steady-state
speed of 30 knots can be achieved at the 120-foot radius for runs restricted to one turn. The
arm, shown in Fig. 31, is a tabular aluminum structure weighing about 37,500 pounds and
having natural frequencies in the vertical, horizontal, and torsional modes of greater than 3
cycles per second.

Submerged models are attached by a pair of towing struts to a model positioning appa-
ratus, shown in Fig. 32. The positioning apparatus is attached to a carriage (Fig. 31) which
can be remotely positioned along the arm to any radius from 12.5 feet to 120 feet from the

Analysis Used in Submerged Body Research 417

Direction Walkway Aluminum fairing is to be installed
of fl between motors of assembly

.\ Carriage Drive

\ “earings Plan View “Carriage Position Indicator -
“Drive Speed Control
Assembly
= SS Slip Ring
Js . y dn i Support
ah : / | Post
yi i} ut DY
i Slee aa Bre Arm Speed

Rotating Arm == rriage a : —s— Model Support
Protective orice t in ee ae Center Island © er oe a\ N Struts
de Centrifugal Force of Soa nas Pivot Bearing ASsently 650 Towing Beam Model

Assembly is to be Transmitted Side Elevation Rear Elevation
to the Upper Rear Intersections

of Outer Bay Truss and Main

Chords by Additional Structure

Fig. 31. Rotating arm general assembly

—___—_———— |0'-0" MAX.
5'-0"

PITCH YAW BEARING
DRIVE

2 tia)
STRUT BEAM — m T&D So,

=H

iSS//5=
[aaa (gaa SROLL DRIVE

STRUT STRUT.

FORWARD

= |||

MODEL =

Fig. 32. ‘Towing struts and model positioning arrangement for the rotating arm

Alex Goodman

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Analysis Used in Submerged Body Research 419

center of the arm. The positioning apparatus is operated remotely and permits model atti-
tudes of: yaw, plus or minus 30 degrees; pitch, plus or minus 15 degrees; and roll, 10-
degree outboard, 40-degree inboard.

For yaw- and pitch-angle changes, the test location of the submerged body is unchanged.
However, for roll angles, the center of roll positioning is high about the CG of the body, and
an adjustment in radial position on the arm is necessary to maintain the proper location of
the body. The positioning apparatus was made external to avoid large cutouts in the body
with respect to the towing struts. Individual motor drives and position readouts are used to
position the body at any attitude. Also, the support struts can be oriented with respect to
the flow to minimize strut-body interference. The strut spacing is adjustable between 3.5
feet to 10 feet to accommodate various length bodies. The struts shown in Fig. 32 are simi-
lar to those used with the Planar-Motion-Mechanism System [2]. The main difference is that
the upper end of the rotating arm strut has a larger chord (2.5 feet as compared to a 1.0-foot

chord).

The internal balance system used with the rotating arm system is identical to that used
with the Planar-Motion-Mechanism System. Likewise, the measuring, recording, and pro-
gramming system, shown in Fig. 33, is essentially the same as the static-stability measur-
ing system used with the Planar-Motion-Mechanism System.

Test runs are generally made within one turn of the rotating arm. The tangential veloc-
ity for submerged models is held constant for all radii at about 10 feet per second. For each
of several discrete radii, the yaw angles (or pitch angles of a submarine model mounted on
the side): is varied incrementally to investigate planar forces and moments.

MOTION ANALYSIS SIMULATOR AND FACILITY

The main purpose of any motion simulation facility is to represent, in the laboratory,
the characteristics of proposed specific designs of surface ships, submarines, missiles, and
other vehicles as well as the effects of the surrounding environment. Such a facility enables
the designer to evaluate and improve the handling qualities or operational characteristics of
a design, on the basis of established figures of merit, well in advance of construction. The
facility may be used for parametric studies to investigate the importance of the various hy-
drodynamic coefficients and other parameters in the equations of motion. For simulating
manned vehicles, the facility is used to study the responses of the human in the control loop
as affected by indicators, displays, control linkage design, and environment. In addition,
operating personnel can be trained well in advance of commissioning.

A simulation facility consists of a number of general-purpose electronic analog com-
puters which are used to solve linear and nonlinear differential equations of motion such as
presented in Appendix D. A general-purpose analog computer is an assembly of electronic
and electromechanical units, which uses direct-current voltages as variables and can per-
form specific mathematical operations. When these units are connected together properly
they can be used to solve mathematical equations. The independent variable is represented
by time in the computer. The suitability of the computer for solving differential equations
arises, therefore, from the ease with which an integration of voltage with respect to time can
be achieved using a high gain direct-current amplifier having capacitance feedback [14,15].
The following basic mathematical operations, required to solve differential equations, are
performed by an electronic analog computer: inversion, algebraic summation of a number of
variables, multiplication by a constant, integration of a variable with respect to time,

420 Alex Goodman

multiplication or division of one variable by another, generation of trigonometrical functions,
generation of nonlinear functions, and discontinuities.

The various components that comprise the DTMB Motion Analysis and Simulator Facility
are presented in Figs. 34 to 39.

A typical computer circuit diagram for a submarine in six degrees of freedom is presented
in Fig. 34. The various symbols used to represent the various computer components are pre-
sented in Fig. 34d. These components are interconnected by means of a plug-in patchboard.
The patchboard, therefore, represents the “mathematical” model of the system. A computer
is equipment with several patchboards, and therefore one mathematical model can be stored
while another study is being conducted.

The DTMB analog computer, shown in Fig. 36, consists of four general purpose com-
puters, manufactured by the Midcentury Instrumatic Corporation, and one analog computer
manufactured by the Reeves Instrument Company. This facility presently consists of 168
operational amplifiers. (The number of operational amplifiers is usually used to express the
size of a facility.) In addition, auxiliary equipment such as diode function generators, white-
noise generators, X-Y plotting boards, two 8-channel Sanborn recorders, strip-chart recorders,
and various instrument displays are available. The present facility will be expanded shortly
to about 300 operational amplifiers when several new computers are procured. With this ad-
dition, the facility will be capable of handling, simultaneously, two six-degree-of-freedom
submarine studies as well as other small problems.

Studies which involve the human operator as part of the closed loop are performed using
the submarine simulator facility shown in Figs. 37 and 38. The simulator consists of cube-
shaped cab (about 7 feet x 7 feet x 7 feet) capable of turning on a single axis equivalent to
the pitch axis of a submarine. It is hoped that an additional degree of freedom, to simulate
the rolling motion of the submarine, will be added in the near future. The cab contains two
control stations, each equipped with an aircraft-type control stick (stick-wheel) and the nec-
essary display instruments. The motion of the cab is governed by an electrohydraulic servo
system receiving inputs from the analog computer. A block diagram of this circuit is pre-
sented in Figs. 34c and 39.

A comparison of the submarine control cycle with the simulator control cycle, presented
in Fig. 35, shows that the cab constitutes one link in a complete submarine simulator con-
trol loop which involves the human operator. The equations of motion set up on the analog
computer (mathematical model) constitute another link in the control loop. Movement of the
control stick in the cab produces inputs to the computer which then calculates the resultant
path of the submarine. The submarine attitude angle, depth, depth error, and plane angles
are displayed by the instruments and the cab rotates to the computed pitch angle. Simulta-
neously, a graphical record of the submarine trajectory and other pertinent information is
recorded by a multichannel recorder.

The Motion Analysis and Simulator Facility in combination with the concept of the
definitive maneuver provides a powerful tool for studying the handling qualities of subma-
rines. Several of the typical definitive maneuvers, shown in Figs. 40 to 43, are performed
using this facility to provide numerical measures of the inherent characteristics of sub-
marines.

The meander maneuver, shown in Fig. 40 provides numerical measures of the dynamic
stability in the vertical plane; such as, time to damp to one-half amplitude ¢, ,», damping
ratio c/c,, and damped period. The overshoot maneuver also shown in Fig. 40 provides

Analysis Used in Submerged Body Research 421

(a) Vertical Plane

(a) Vertical plane

Fig. 34— Analog computer diagram of submarine equations

422 Alex Goodman

(b) Horizontal Plane

(b) Horizontal plane

Fig. 34— Analog computer diagram of submarine equations (continued)

+840

Analysis Used in Submerged Body Research

CONTROL CIRCUIT

+6r¢

+55 ~5s¢ tos

by = + (5p + 5s)
br = (Bp - 65)
where
bac, Ore are effective diving planes and rudder ordered angles
Spe» Sse are ordered angles to port and starboard shafts
5p , 5 are actual angles of port and starboard shafts
64, or are effective diving plane and rudder angles which

would produce forces and moments equivalent to
deflection angles of stocks

(c) Thrust representation, axial force and control circuit diagram

Fig. 34— Analog computer diagram of submarine equations (continued)

423

424 Alex Goodman

SYMBOLS DESCRIPTION SCHEMATICS

Summing Amplifier

=|>— Integrating Amplifier

Hi Gain Amplifier

~-—{ }--—--- Diode Limiter

Non-Linear Function Generator

—C)——— Grounded Potent iometer :

(d) Description of symbols used in analog computer diagrams

Fig. 34— Analog computer diagram of submarine equations (continued)

Analysis Used in Submerged Body Research 425

INSTRUMENTS

7
SUBMARINE -7 SUBMARINE
4

CONTROL | “CYCLE ——o
7
ie

OPERATOR Z

CONTROLS

(a) Submarine control cycle

we INSTRUMENTS \

DIVING STATION

SUBMARINE TILT. TABLE
SIMULATOR [2 a)
CONTROL CYCLE aN
OPERATOR
>
S

ANALOG COMPUTER

(b) Simulator control cycle

Fig. 35. Comparison between submarine control cycle and simulator control cycle

Fig. 36. DTMB analog computer facility

numerical measures of control effectiveness, such as time to reach execute, overshoot angle
in the vertical plane, and overshoot depth in the vertical plane. A similar maneuver, such
as shown in Fig. 42, can be used to define the ability to initiate and check a course change
in the horizontal plane. The results of the Dieudonne spiral maneuver shown in Fig. 41 pro-
vides some measure of the inherent dynamic stability in the horizontal plane. These results
are also indicative of the course-keeping characteristics of the submarine. Steady-turn
studies such as shown in Fig. 43 provide numerical measures such as tactical diameter, ad-
vance, transfer, time to change heading 90 to 180 degrees, and loss of speed in the turn.

426 Alex Goodman

Fig. 37. DTMB submarine simulator facility

Definitive maneuvers which provide measures of the capabilities of the submarine tem-
pered by the existence of a human operator or automatic control in the control loop can also
be performed using the facility. Maneuvers such as depth-keeping and course-keeping, under
various environmental conditions, provide numerical measures such as rms depth or course
error, percent time on target, degrees traveled by control surfaces, and maximum values of
these variables. Various other maneuvers such as limit-dives and emergency-recovery maneu-
vers also can be performed in'the laboratory and evaluated before the submarine is built.

Instrumentation studies, which are in the nature of human engineering, can be carried out
using the submarine simulator. The effect of various displays, indicators, lighting, and other
devices on the performance of the submarine can be evaluated.

Thus, it can be seen that the DTMB Motion Analysis and Simulator Facility is a power-
ful and versatile design and research tool for the study and analysis of stability and control
problems.

FREE-RUNNING MODEL TECHNIQUE

The free-running model technique has been used extensively in the past in studying the
turning and maneuvering characteristics of surface ships [4]. With the emphasis that has been
placed the performance of high-submerged-speed submarines, the necessity for turning data
from submerged model tests has become more important. Karly submerged maneuvering tests,
in the vertical and horizontal planes, conducted at.the David Taylor Model Basin employed a
specially constructed 5-foot model provided by NACA. The model was powered electrically
and controls were operated by compressed air through a flexible tubing which trailed astern
of the model. Model trajectories were obtained, by photographing the movements of the model

Analysis Used in Submerged Body Research 427

(b) Port and starboard control sticks and displays

Fig. 38. Interior views of the submarine simulator facility

Alex Goodman

428

yino1to 1oye| UIs ay} JO WeiseIp yoolg “6g “Sty

fetdstq

Jeqnduog queuma4sut

pareoquey4s

YS [01}U0D psvoqieys

ketdstq

queumi4sut

440d

YONS [O1}JUOD WOd
dumd

oF Tneapsy

Analysis Used in Submerged Body Research 429

MEANDER

Fig. 40. Typical meander and
vertical overshoot maneuvers
(definitive maneuvers)

Rate of Change

®
©
o
ie]
re
ole
Se
@
Cir
@
slo
a

Rudder Angle
Rudder Angle

(a) Stable ship (b) Unstable ship

Fig. 41. Typical curves from spiral maneuvers

relative to a grid attached to the towing carriage, using cameras mounted inside the model.
The main objections to this system are that it required an expensive model and that the ex-
cessive drag resistance of the tubing affected the maneuvering characteristics of the model.

With recent advances in electronic and electromechanical instrumentation, and water-
proofing techniques, it became possible to perfect the free-running model technique using
the standard 20-foot (nominal) submerged models used for both resistance and propulsion
tests and stability and control tests.

The free-running submarine model tests are presently being performed in the J-basin at
DTMB. Techniques have been developed so that these tests can be conducted in the new
Maneuvering Basin using a model that is internally programmed and has internal recording
equipment.

646551 O—62:——_29

430

8

Time from Ist Execute in seconds
— LSJ
s s

3

$

0

Approach Course

Alex Goodman

Period
shoot I
|
|
!
Reoch
1
| Over-
a shoot kl
1 Turning
1) pa ariee Movement
»

20 0 0 0 20
_ Deporture from Bgse Course ond Rudder Angle in degrees

Fig. 42. Horizontal overshoot maneuver

}~_—_—— Tactical Diameter
(180°)
Transfer
(90°)

Execute (2)
Rudder

Final Diameter Eased

Recovery Course
Regain Speed

Fig. 43. Turning circle maneuvers

Analysis Used in Submerged Body Research 431

For the submerged tests in the J-basin, the model is provided with watertight equipment
(propulsion motors, actuators, and gyros) and is ballasted so as to have only a slight re-
serve buoyancy. The final displacement and trim is achieved by means of the bow and stern
ballast tanks. The power and control cables emerge from the model through a hollow faired
tube and are supported by an overhead follower boom controlled from the carriage. The cable
runs over a pulley and has a counterweight so arranged that the cable feeds out as the model
submerges. A lightweight framework secured to the top of the faired tube supports flashing
neon lights used in photographing (using overhead cameras) the turning path of the model.
Portable control boxes are provided for both the rudder and stern plane so that the most de-
sirable locations on the carriage could be used. The control boxes actuate waterproofed-
rotary actuators, manufactured by Universal Match Corporation, which are mechanically linked
to the rudders and stern planes. A trim angle indicator also is provided to simplify the work
of the stern plane operator as it provides a faster and more sensitive indication then could
be estimated visually. This information is obtained from a Minneapolis-Honeywell vertical
gyro Type GG 33B-2 which is located in the model. The heading and rate of change of head-
ing data are provided by a Humphrey Model F'G01-0203 gyro and a Humphrey Model RGO3-
0117-1 gyro, respectively, and are of value to the operator of the rudder control in obtaining
a straight approach course.

At the beginning of each run the model is floating with only the top of the bridge fair-
water projecting out of the water. After the model has attained the desired test speed, it is
maneuvered down to the test depth by means of the stern diving planes. The definitive ma-
neuvers shown in Figs. 41 to 43, are then performed to evaluate the handling qualities of
the submarine in the horizontal plane.

ACKNOWLEDGMENT

The DTMB Planar-Motion-Mechanism System was conceived and developed jointly by
the author and Mr. Morton Gertler, both members of the staff of the Hydromechanics Labora-
tory of the David Taylor Model Basin. Patent proceedings have been initiated in behalf of
the United States (Navy Department Navy Case Number 28, 121; Serial No. 817, 002) with
the names of Messrs. Goodman and Gertler as originators of the system. The originators
wish to express their gratitude to those members of the Industrial Department of the Model
Basin whose contributions and efforts in the design and construction of components made
the ultimate system possible. Particular thanks are due to Messrs. M. W. Wilson, and R. G.
Hellyer of the Industrial Department for their aid in developing the instrumentation system
and the roll oscillation system, respectively. The. author is also grateful to Mr. Morton
Gertler for his contributions to the development and application of the concepts of the defin-
itive maneuver for evaluating handling qualities.

REFERENCES

[1] Gertler, M., and Gover, S.C., “Handling Quality Criteria for Surface Ships,” paper pre-
sented at the Meeting of the Chesapeake Chapter of the Society of Naval Architects and
Marine Engineers on May 2, 1959

[2] Gertler, M., “The DTMB Planar-Motion-Mechanism System,” paper presented at the Sym-
posium on the Towing Tank Facilities, Instrumentation and Measuring Technique on
September 22-25, 1959

432 Alex Goodman

[3] Orlik-Ruckemann, “Methods of Measurement of Aircraft Dynamic Stability Derivatives,”
National Research Council of Canada Aeronautical Report LR-254

[4] Gover, S.C., “Free-Running Model Techniques for the Evaluation of Ship-Handling
Qualities,” David Taylor Model Basin Report 1350, July 1959

[5] Bird, J.D., Jaquet, B.M., and Cowan, J.W., “Effect of Fuseage and Tail Surfaces and
Low-Speed Yawing Characteristics of a Swept-Wing Model as Determined in the Curves-
Flow Test Section of the Langley Stability Tunnel,” NACA TN 2483, Oct. 1951

[6] MacLachlan, R. and Letko, W., “Correlation of Two Experimental Methods of Deter-
mining the Rolling Characteristics of Unswept Wings,” NACA TN 1309, May 1947

[7] Lamb, Sir Horace, “Hydrodynamics,” Sixth Edition, Dover Publications

[8] Lopes, L.A., “Motion Equations for Torpedoes,” NavOrd Report 2090, NOTS 827, U.S.
Naval Ordnance Station, Inyokern, Feb. 1954

[9] Strumpf, A., “Equations of Motion of a Submerged Body with Varying Mass,” Davidson
Laboratory Report 771, May 1960

[10] Beam, B.A., “A Wind Tunnel Test Technique for Measuring the Dynamic Rotary Sta-
bility Derivatives Including the Cross Derivatives at Subsonic and Supersonic Speeds,”

NACA Report 1258, 1956

[11] Tuckerman, R.G., “A Phase-Component Measurement System,” David Taylor Model
Basin Report 1139, Apr. 1958

[12] Fried, W., “A Rotating Arm for Towing Models of Ships and Other Forms in Circular
Paths,” Experimental Towing Tank Report 299, Feb. 1946

[13] Brownell, W.F., “A Rotating Arm and Maneuvering Basin,” David Taylor Mode! Basin
Report 1053, July 1956

[14] Johnson, C.L., “Analog Computer Techniques,” New York:McGraw-Hill, 1956

[15] Wass, C.A.A., “Introduction to Electronic Analogue Computers,” New York:McGraw-
Hill, 1956

APPENDIX A

Derivation of Phase Angle Required Between Struts for
Pure Pitching Motion

The condition that must be satisfied to obtain a pure pitching motion for a body moving
through a fluid is that the pitch angle varies with time while the angle of attack 0, meas-
ured at the CG, is maintained equal to zero at all times. The motion is one in which the
body CG moves in a sinusoidal path, with the longitudinal body axis tangent to the path,
as shown in Fig. 3c. This motion in some respects is similar to that produced by a rotating
arm.

Analysis Used in Submerged Body Research 433

The system that will be analyzed uses a slider crank mechanism having a large ratio of
length of connecting rod to eccentricity. There is no appreciable error introduced, therefore,
by assuming that the motion is sinusoidal for purposes of analysis. Referring to the notation
and schematic diagram shown in Fig. 5, the motion of the forward and aft struts (taken with
respect to the midposition of the forward strut) can be expressed as follows:

Z, =| ay Sinuat (Al)

Z, = a, sin (at - ,) (A2)

where the crank arms @, and a, are assumed to be variables. The vertical displacement of

the body CG is, therefore,

= Pde ag ie 2
per Naa ee ee (43)
Expanding Eq. (A3) results in
igo ta OY aes ame ae ae
= ee os ets: t
Zz, b x, + a, cos d,} sin wt as sin ?, cos w (A4)

where the strut spacing with respect to the body CG, x, and x, are assumed also to be
variables. The vertical velocity of the body CG can be obtained by differentiating Eq. (4)
and expressed as

: a,X, [/X, a, a,
2 laa ae x, ee GES cos ES es sin wt]. (A5)

For the case of pure pitching the struts are out of phase with each other, and a body
pitch angle results and can be expressed in terms of the motion of each strut as

g = ++ (A6)

or expanding:

eet a2 mes es t
= 5 ( ae @ sin 2 ess cos wt| . (A7)

The vertical velocity of the CG with respect to the inertial axes can be written as
Zz = wcos 0 = a sin ¢ (A8)

oO

and for small angles, where cos @ = 1, sin 0 = 0, and u = U, can be simplified to

434. Alex Goodman

z = w- U6. (A9)

o

Also, since w/U = @ (for small angles), Eq. (A9) can be restated as
z = U(a-@6). (A10)

o

As stated previously, the primary condition that must be satisfied for pure pitching mo-
tion is that © = 0. Equation (A10), therefore, reduces to

Ze at UE) (A11)

Substituting Eqs. (A5) and (A7) into Eq. (A11) results in

CTR eA a's
— + — cos ¢.] cos wt + — sin ¢, sin ot|
U X, a, s ay s

one ay i,
= (1 = ay cos +) ‘Sin wt + a, sin ¢, cos at| : (A12)

Equating sine and cosine terms results in

wx wx a5 a
a cos". = §sin'p)- =)'= (3) 34) 2 (A13)
(=) Ls Dy a
cos $, t\7,]sin ¢, = (3*) . (A14)

Solving Eqs. (A13) and (A14) for sin @, and cos ¢, results in the following relation-
ships:

1
sin ?s = RR oR eae (A15)
=) [a+ FA |
= Dra enn
2
amar ona 4
ag ( % Cr} (A16)
cos ¢, = Srrcreree renew a BP e
Ait =

A boundary condition that must be satisfied is that sin @, cannot be greater than one.
Therefore,

Analysis Used in Submerged Body Research 435

= eeenaenien ees pets ean a
WX

SOR ee re a ea ae Taw i (A17)

The value of w%,/U which satisfies Eq. (A17) is #x,/U=1. This result can be used to
determine the relationship between the crank-length ratio and the strut-spacing ratio by sub-
stituting wx,/U = 1 in Eq. (A15) as follows:

sin ¢, = 1 = ——— (A18)

or, restating Eq. (A18),

a5 1 Xo
Sod GRD .

The present Planar Motion Mechanism was designed having equal crank lengths. There-
fore, in accordance with Eq. (A19), the strut spacing with respect to the body CG must be
equal. For these conditions, the expression for the phase angle between struts for pure
pitching motion reduces to

cos Ps So ee (A20)

and the body pitch angle relation becomes

2 % 2, Ps
G (=s4— (22 sin 3) cos (.¢ = *)| (A21)

where

(2 sts
Flys ee)

is the maximum amplitude of the pitch angle 0.

As can be seen from Fig. 5, there is a given relationship between the point in the cycle
when the pitch angle is zero and the zero point of the resolver (synchronous switch or

436 Alex Goodman

sine-cosine potentiometer). The pitch angle trajectory of the model is a direct function of
the phase angle between struts as shown by Eq. (A21). Therefore, the zero-pitch-angle point
(wt), will vary with strut-phase angle. This requires that the resolver be repositioned and
synchronized with the zero-pitch-angle point for each condition of pure pitching.

The relationship between the zero-pitch-angle point (wt), and the strut-phase angle can
be determined from Eq. (A5). At the point (w£), in the cycle, 6 = 0, z, = zo, and z,isa
maximum. Therefore for x,/x, = @,/a, = 1, Eq. (A5) reduces to

z, = 0 = (1 + cos $¢,) cos wt + sin ¢, sin ot (A22)
and
tan wt = seals Bs (A23)
sin $,
or

(wt) = tan“? |

sin ¢, (A24)
Differentiating with respect to ¢, yields
owt) 1
3d, ra 2 : (A25)
APPENDIX B

Derivation of Reduction Equations

The Planar Motion Mechanism separates the motions of a body moving through a fluid
into the hydrodynamically pure pitching and pure heaving motions as defined in Fig. 3. The
differential equations of motion referred to a moving body axis system are used to establish
a direct and explicit relationship between the various rotary and acceleration derivatives and
the measured quadrature and in-phase components of the forces and moments. The linear
force and moment equations describing the body motions with respect to the initial equilibrium
conditions can be written as:

Transverse Force:

Y= Y.r+ (5 ety, + n) rU + (Y.-m)v

2 gent 4
+ € pr uv, v + (5 et zi pt Y.p. (Bl)

Normal Force:

Rolling Moment:

Pitching Moment:

Yawing Moment:

Analysis Used in Submerged Body Research

: er oe ae
Z-q + (5 0t aie n) qU + (Z.-m)w

+ (5 t’uz,] se (4 et’uz,’ a ae

: Bias ;
(K; — Te )p + (5 et UK,!) p + K, ae

ere ete Silay sree eae
+ (Fot'un, )r + Kv + (5 ot°u,) v.

=

: 4
(MH, - E-)q + (5 ot mj a + H,6 + Me

+ a et'um | w+ (5 et°un ) vty.

(N.-I,)i + (5 et'un.,’) r+ Nyy

1 RE eee eee _
+ (5 ot uv!) v + € pt uN ,') p + Nep.

PURE HEAVING MOTION

437

(B2)

(B3)

(B4)

(B5)

The reduction equations for the pure heaving motion can be derived using Eqs. (A9),
(B2), and (B4). For this motion, the pitch angle 0 is zero at all times. since the phase angle
between the struts is zero.

As shown schematically in Figs. 3b and 4, the vertical displacement of the body CG

can be expressed as

the linear velocity as

and the linear acceleration as

Z, = 4 sin at,
Z_si3= (ae) COS: GE ,
Zi =\=ae* sin at .

(B6)

(B7)

(B8)

438 Alex Goodman

Since 0 = g = q=0, Eq. (A9) reduces to Z y= w. Therefore

W = awcos wt (B9)
and
2

WwW = -aw? sin ot. (B10)

Substituting Eqs. (B9) and (B10) in Eqs. (B2) and (B3), results in

2
Z= -aw*(Z.-m_) sin wt + aw (3 pt uz) cos at + Z, (B11)
and

; 1 3
NS — ae. sin wt + aw (3 pt uw) cos wt + M, (B12)

where m,, is the model mass.

The component of force and moment in phase with the motion can be written as

Z,, = -aw*(Z.-m) (B13)

in

and

Ho = -aw lH. . (B14)

in

The internal balance system, shown in Fig. 6, measures the components of force at two
points spaced equidistant from the body CG. Therefore, Eqs. (B13) and (B14) can be rewrit-

ten as

fs), eas 2 bape
(21), + (25). = -aw*(Z.-m) (B15)
and
es 2
x|(29),, - (21),,] = -aw"M.. (B16)
The heaving acceleration derivatives, written in nondimensional form can be expressed
as
eZ), + 22),
(Z.'-m") = Me NAMEN RS NC eae (B17)
Ww m ow, 7
and

CAC aca et

5 (B18)
ow

2
os |x

Analysis Used in Submerged Body Research 439

where

is the amplitude of the linear acceleration.

Similarly, the reduction equations for the case of pure sideslipping (lateral translatory
motion), as presented in Table 2, can be derived.

PURE PITCHING MOTION

The condition for pure pitching, as studied in Appendix A, is satisfied when the angle
of attack 2, measured at the CG, is equal to zero at all times. For this condition, the re-
sultant linear velocity w and linear acceleration w are zero. The pitch angle, pitching ve-
locity, and pitching acceleration can be expressed as

tei Peds Taedin chu 9a). Pe de
@= ep aT ee sin 2a cos |at - ba ’ (B19)
the angular velocity as
BS q= -0(22 sin = |- sin (oe - $2) ], (B20)
and the angular acceleration as
ahs der é t,
PSM ge var (22 sin £3 lee (wt = fs), (B21)

where
2 =)
B sin >

is the maximum amplitude of the pitch angle 0,.

Substituting Eqs. (B19), (B20), and (B21) in Eqs. (B2) and (B4) yields:

= | é,
- a (22 sin >) (5 pez. + n.,) - sin (.t - = +Z, (B22)

440 Alex Goodman

and

b
- a 2 sin =| (2 ot‘) [ sin (.t - =|

= & sin a (45) - [cos (ut = 3) + M, (B23)

where /,,, and (Mg), refer to the model moment of inertia and metacentric stability, respec-
tively.

As for the heaving case, in-phase and quadrature components of force and moment can
be written as:

P
(Z,) + (2), = ow? 2 sin a Zz. (B24)

in

J BE as Cally wpe
(21) ut v (29) out oS a (2: Boat = (3 pt Zo

A n, (B25)
p
2

‘ are (21), | a wo? (22 sin ) Gln) = Mom {2 sin =) =

x (25), = Cie = -wU (22 sin ) (5 0¢'m,) (B27)

The pitching rotary and acceleration derivatives, written in nondimensional form, can be
expressed as:

nS a2... + (24'), |] —

CBOs pee se k R71 one / (22') oud (B29
qa m = 3q '

(Hy) m
k22"),. . Cay, 5 ee? (B30)

x ly
4 8q_' aw

Analysis Used in Submerged Body Research 441

c 22") ou. I eee

Note SS (B31)
q z ag.
where
Cea (2 sin *) ae (B32)
and
Go ag (22 sin 5) a (B33)

are the maximum amplitudes of the angular velocity and acceleration, respectively.

The relationships for pure yawing and rolling, presented in Table 2, can be derived in a
similar manner.

The model mass m,, and metacentric stability (M4), are evaluated experimentally by
performing inclining tests (standstills). The model moment of inertia], is determined from
oscillation tests performed in air. For these tests, the model-ballast condition is the same
as during the regular tests.

APPENDIX C

Mathematical Operations Performed by Instrumentation System

As indicated in the section entitled “Instrumentation for Static and Dynamic Stability
Tests,” the operation of the present electronic system differs from that described in Ref. 2
in that a true integration of the gage signal is performed.

To illustrate, assume a ‘pure heaving condition which results in a gage signal of the
form such as given in Fig. 4:

Ze => Zy + Ze sin (at = d) (C1)

which can be written as

Zr = 2, + Z, (cos ¢) sin wt - Z, (sin ¢) cos at (G2)

Z, = 2, + Z,, sin wt - Z,,, cos oat. (C3)

ut

The in-phase and quadrature (out-of-phase) components can be obtained by operating on
the gage signal in the following manner:

442 Alex Goodman

27
1
Ba = | Z, sin wt d(wt ) (C4)
0
and
1 277
Zoe = || Zp cos wt d(at). (C5)
‘0

The operations indicated by Eqs. (C4) and (C5) are performed by the force-component
separator, sine-cosine potentiometer, integrating amplifier, and timer diagrammed in Fig. 22.

The result of the integration is provided by the digital readout and is defined as

Zo = Z cose (C6)
and

Zz = -Z, sin ¢. (C7)

Likewise, for the pure pitching condition the resultant gage signal has the form (see

Fig. 5)

Z, = 2, + Z. (cos (at - a - ‘| (C8)

P, ’,
Z, = 4, + (4, cos ¢) cos (wt = 5 + (Z,sin¢) sin (ut - 3) . (C9)

The integration performed is as follows:

(ot) +27

Zi. 7 if Z, sin [wt - (at) |] d(at ) (C10)
(ot),
and
(ot) ,+27
1
Sa aia | Zp cos [ot - (wt) | d(wt ) (C11)
(wt),
where

(wt), = 2 4 ete,

Analysis Used in Submerged Body Research 443

Equations (C10) and (C11) reduce to

Z = Er cos (C12)
and

Be i Ze arg. (C13)

out

Thus, the operation performed by the electronic system is equivalent to determining the
Fourier coefficients of the fundamental of the gage signal.

APPENDIX D

Equations of Motion for Submarines

The differential equations of motion in dimensional form, referred to a moving body axes
system which is coincident with the principal axes of inertia and having the origin at the
center of mass, can be written for the case when the submarine is initially in level flight and
at a steady forward speed as:

Axial Force

° 3 0) = ' 1.
m(u-vr+wq) = oat [xia =) evr it Ziwq]

(uu)
Lao 22 ' 2 ‘ 2 / ae
+ 5¢u Xeses2)3s thee eeye et A seseyee |
ae

Lateral Force
m(v-wp+ur) = 5
Pp 2 rue ' :
+t [¥.'u2 + Y'uv + ¥’.yvivl|

‘

Clrlér)

xe)

U

3 ‘ '
+—4 [viur+¥ ulr|8e+¥0\,,,)lvle+¥eup|

i)

a ae ue Le ah
+ Se [¥:# + Y;p| +e - Y, 5, - (Fy):

444

Normal Force

Alex Goodman

m(w- uq+vp) = EL | Zi + Y;vp|

+

Rolling Moment

Ie Chel ar

Pitching Moment

2 ’ ’ '
ft [Z.u? + Ziuw + ||

‘ ‘

P pi |7'
at [Zia + Bees ay laiPos + Zo swig) |wla|

U

2
(rr)t

en) 2 : P 4
ei 9 Curt sp as Z

Pp 2 2 ‘ ‘ Pp 4 Ue
Stu? 25,8, 4.25453] +54 254+ (F,),.

Ss s

P 9° Bee ' ' .
at [K3e + (N,-M,)qr + Kyr + Ke re lll

p18 5 o ' )
+ at [ Keu + Kyiv + Key vv]

4 [- ',
+ St | sup + Kur + K;7]

pi Qian! :
+ at a Keo + Bz eesing@it Cae

. 5 ’ ' ’
Iq op (OU aIEE yj ol 5t [usa + (K; - Ni yep]

P »3 IG) ' '
+54 [wee + M uw + Heat wi ™ ll

U]

Pp 4 ) ‘
tat [area + Me raissytlalds + Her eigy|la]

‘

VET es 20 reall OM ak
Cow’ emi. Mee yVE

+ Ley

1 2 Pp 4 4 Pp 3 t ‘
9 Pape + at Hoy ah u? [55,4548

+ Bz, sin @ + (oy.

Analysis Used in Submerged Body Research 445

Yawing Moment

. 5 bina r] ‘ i
eee Ch = 1 \pq = ot [wii + (M; - K.)pq + Np

P 3 ' ‘ ‘
+ at [weu? + N uv + eek |

‘

u|r|Sr + Mi i eylvle]

Ppt [a
+ at [Whar eRe ise)

4 ’ 3 ‘ :
+5t [¥ uP + viv] +54 die NaS evects \OU)iae

Kinematic Relations
U? = u2 + v? + w?
Zee sin 6+ v cos 6 sin ¢+ wcos 6 cos ¢
w- ué + vob
Ps pe p sin 0
pt+réa

q- cos 6 sind

cos
ee Tse
‘? ee cig 6 sin eo)
cos 0 cos ¢
Se. g@ i.
Auxiliary Relations
CBO) Pea aria zaons: aU.

or

Cie) au 2 :

646551 O—62,-——30

446 Alex Goodman

DISCUSSION
H. N. Abramson (Southwest Research Institute, San Antonio)

Submerged body technology shows two interesting trends; one is toward increasing fine-
ness and the other is toward higher speed. These two together tend to introduce a new im-
portant factor and this is the elasticity of the structure. In short, hydroelastic considera-
tions may become important for stability and control investigations of submerged bodies. I
would very much like to have the author’s comments on what considerations he and his col-
leagues have given to introducing hydroelastic effects into their theoretical and experimental
studies.

E. C. Tupper (Admiralty Experiment Works)

The author mentioned in his paper the question of certain standards by which the per-
formance of the submarine could be judged. I would like the author if he will, to state what
these standards are in both the vertical and the horizontal plane. Only by studying in a
computer, say, the way in which the performance of the submarine: varies as regards these
standards, can one judge the importance of various hydrodynamic derivatives. For instance,
the author mentioned that we have what are commonly called the static, rotary, and accelera-
tion derivatives. We feel, at the Admiralty Experiment Works, that the acceleration deriva-
tives are not critical. There are basically four for the simple equations of motion in the ver-
tical plane and we feel that two of these can be ignored and the other two can be calculated
with sufficient accuracy and therefore do not need to be measured.

We have not studied the horizontal plane problem completely, but we feel that the most
important point here is the nonlinearity introduced by the very large reductions in speed which
can occur when maneuvering in the horizontal plane. I would like to make a plea, therefore,
for some description of the standards which the author considers to be important.

It would be very interesting if the author could give us any information on comparisons
between results with the oscillator technique which he has described and other methods of
test, in particular the comparison with the rotary derivatives between the oscillator and ro-
tating arm methods of testing.

S. T. Mathews (National Research Council, Ottawa)

I think we should congratulate Mr. Goodman and his colleagues at the David Taylor
Model Basin for developing what I consider the most powerful method of considering this
submarine stability and control problem, experimentally anyway. Surely these experimental
results are most useful or even essential before we further develop the theoretical methods.
I have a few points of detail. I would like to ask Mr. Goodman if the carrying out of static
moment and force tests is not superfluous, because the same results are obtained by the
oscillating heaving tests. I would be glad to have any comments he has on how the results
obtained by the two methods compare. Mr. Tupper took one of my points. I was going to ask
specifically if we could have any comparisons between results obtained for the rotary static
moment and force derivations from the oscillating mechanism as compared with a rotating arm.
It seems to me that no mention has been made of surface models; for linear seaworthiness
considerations one could obtain much useful data using the oscillating mechanism technique,
in pitch and heave, and also we could get results in yaw, I would think, certainly at low

Analysis Used in Submerged Body Research 447

Froude numbers. It would also seem that the technique could be extended to cover non-
linearities if a different sort of recording technique were used.

J. P. Breslin (Davidson Laboratory, Stevens Institute of Technology)

I have a few questions I would like to put to the author and the first one is the question
as to what the David Taylor Model Basin does about strut interference on the planar motion
experiments.

The Davidson Laboratory has been operating a rotating arm now for about 16 years and
has been greatly interested in refining and improving the techniques, particularly for the in-
vestigation of interferences, of coupling effects, and of nonlinear behavior of submarines and
bodies of revolution with a mind to exploring the behavior of such vessels in radical maneu-
vers. In this regard, it is rather interesting that in the model regime the Navy has been able
to get along without very much information in this area of large course deviations for this
long period of time. We look forward with great interest to the developments which will
come from the David Taylor rotating arm, not only for the comparisons for which Mr. Tupper
asked, but also for the predictions of nonlinear coefficients which are necessary for modern-
day treatment of evasive and radical maneuvers.

My second point has to do with the assertion that in regard to accuracy one ought to
look somewhat askance at rotating arm methods for determining static derivatives. The
Davidson Laboratory and the Model Basin have had a number of discussions on this matter
and as a matter of fact the Model Basin has supported us in the very recent past in an ex-
perimental investigation in which we conducted experiments in the straight tank to measure
static derivatives and repeated the experiments on the rotating arm, interpolating or extrap-
olating, depending upon how you interpret the procedure to obtain the derivatives at infinite
radius. I am happy to say that these experiments,* as well as several that have been made
in the past, always give very close agreement. Of course, general statements of a sweeping
nature are all subject to exceptions and perhaps the class of bodies with very small damping
is one in which close scrutiny should be made of the difference between the techniques.

Finally, I would like to call the attention of the participants to the great usefulness of
the small model in such explorations to map out the gross effects of shape and control sur-
face location so that modern concepts of optimizing maneuverability and controllability can
be more readily achieved. The David Taylor Model Basin has certainly done an exceptional
job in developing equipment which is suitable for proof testing of designs and this is a very
important aspect of their part in the naval establishment, but it is urged that the use of the
rotating arm at the Davidson Laboratory and other small facilities be continued for interest-
ing exploratory research.

R. Brard (Bassin d’Essais des Carenes, Paris)

I would like to say how we appreciate the work done by the David Taylor Model Basin
in order to study the maneuverability of the submarines and I would too address warm con-
gratulations to the staff, and particularly Mr. Goodman, for their beautiful results, specially
those concerning the maneuvering of the submarine under the polar ice cap.

*S. Tsakonas, “Effect of Appendage and Hull Form on Hydrodynamic Coefficients of Surface Ships,”
Davidson Laboratory Report 740, July 1959

448 Alex Goodman

I would ask him if he has given reflexion to the following point. If the motion of the
ship is not steady, the hydrodynamic forces and moments on the ship depend, partly, at the
instant ¢, on the motion of the ship before this instant t. Therefore, the equations of the
motion of the ship are not exactly the same as in the case where the forces and moments
would at each instant equal their values in steady motion. One study made in France shows
that the condition of stability is not modified; but the trend to the steady motion is more ac-
centuated. Thus, the transient motions are affected in such a manner that, furthermore,
seems to be favorable.

Alex Goodman

Dr. Todd’s paper raised a question pertaining to the stability and control requirements
of a large submarine tanker. In my opinion, it would seem that such a vehicle would require
a high degree of inherent stability as well as a properly designed autopilot. I noted a touch
of pessimism in Mr. Newton’s comment on Dr. Todd’s paper which I cannot share. I think
that with the techniques and means available today, solutions to the stability and control
problems of such vehicles can be obtained without any difficulty.

I would like to thank the discussers for their comments. I will be brief with my replies
to the various questions that have been asked. Dr. Abramson asked whether the effects of
hydroelasticity on the stability and control of submarines have been considered. We have
not considered such effects. However, prediction of full-scale performance based on rigid-
body dynamics has been very good. Answers to several of Mr. Tupper’s questions are in the
written manuscript as well as in other Model Basin sources. Mr. Tupper questioned the need
for the experimental determination of the acceleration derivatives by stating that theoretical
techniques are available to estimate these derivatives. It is true that theoretical means are
available for estimating the acceleration derivatives; however, in my opinion they are not
accurate enough when it is required to determine the degree of stability. I can only say that
by using the results obtained from tests using the Planar-Motion-Mechanism System we have
been able to achieve excellent correlation with full scale performance. Correlations between
this technique and the rotating arm technique are planned using the same model, essentially
the same support system, and the same force-measuring-instrumentation system. I would
like to thank Mr. Mathews for his compliments. As far as his question pertaining to the use
of the Planar Motion Mechanism for the surface ship model testing, I can only say that some
thought has been given to using this technique for such problems. However, I think we will
first obtain some experience using the rotating-arm technique for the surface ship problem.
Dr. Breslin raised a question regarding the effects of strut interference on the measurement.
An extensive and detailed study of strut interference has been conducted at the Model Basin.
The results of this study led to the strut design which has been incorporated in the Planar-
Motion-Mechanism System, that is, small-chord struts in the vicinity of the model and angles
of attack of the body being taken in the plane of the strut. As I mentioned in my presenta-
tion, this eliminates the mutual interference effects between the strut and the body. There
are small blockage effects. However, by choosing the proper extension of the lower half of
the strut the resistance of bodies can be measured as accurately as is done by other stand-
ard techniques. As to the required accuracy of static coefficients, I don’t recall to which
paper Dr. Breslin was referring, but the one I remember which shows the comparison between
static coefficients, obtained from straight line tests and those obtained by the extrapolation
technique differed by as much as 10 to 20 percent. This is not a small percentage when we
are talking about performing accurate analyses of maneuvers. With regard to Dr. Breslin’s
comment as to the economy regarding small models compared to large models, this is an
argumentative point. DTMB is at an advantage, since we feel that the large model is much
cheaper to construct than a small specialized model. We use standard internal equipment

Analysis Used in Submerged Body Research 449

which can be used from model to model, thereby eliminating the need for special purpose
equipment for each model. There are other advantages associated with size. DTMB is not
only in the business of checking specific designs; extensive systematic studies are also
conducted at DTMB and the Planar-Motion-Mechanism System is very amenable to such
studies and relatively cheap to operate. For example, it takes approximately one week of
testing to completely define all the hydrodynamic characteristics for a specific submarine
design.

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RRGTIUGIE Firs OES OO 14 et

A THEORY OF THE STABILITY OF LAMINAR FLOW
ALONG COMPLIANT PLATES

F. W. Boggs and N. Tokita
Research Center, U.S. Rubber Company,
Wayne, New Jersey

1. INTRODUCTION

A great deal of theoretical work has been done in recent years on viscous flow in the
neighborhood of a rigid wall. This has made it possible to explain the initiation of turbu-
lence through the development of flow instability. Laminar flow has been shown by the
theory to be stable under conditions when it is actually observed, and it has been shown to
be theoretically unstable under conditions when turbulent flow is known to prevail. These
results developed by Tollmien, Schlichting, Lin, and many others [1-7] have been verified by
Schubauer [8] at the National Bureau of Standards. Today the controversy which existed for
many years over the nature of turbulent flow and its relationship to laminar flow seems to be
settled.

The recent work of Kramer [9-11] showing the effect of a flexible boundary wall on fluid
flow has emphasized the need to extend the boundary-layer theory to include this interesting
and important case. This present paper undertakes the development of a theory which will
predict the stability conditions of the boundary layer in contact with a flexible wall under
the same general assumptions regarding the flow that were made by previous authors for the
rigid wall and to the same degree of approximation. The result should, therefore, have the
same kind of validity.

In this treatment the properties of the flexible wall are expressed in terms of its acous-
tical compliances.

One of the significant conclusions which follows from this work is that the conditions
of stabilizing a flow are fairly critical and that all coatings which are flexible do not neces-
sarily have a favorable effect on the flow conditions.

In subsequent papers we will attempt to extend this theory to some of the special cases
of practical importance, but here we will confine ourselves to some very general conclusions.

The theories of boundary-layer stabilization are based on the supposition that the Navier-
Stokes equations hold at all times and that transition from laminar to turbulent flow results
when the laminar flow is unstable for some arbitrarily small perturbation. Stable laminar flow
exists only if the flow is stable for every possible infinitesimal perturbation. It assumes but
does not really prove that when a laminar flow fulfills these conditions, it will be the domi-
nant flow pattern and that the drag coefficient calculated from such a laminar flow is the one
applicable.

451

452 F. W. Boggs and N. Tokita

The question of the lower limit of turbulent flow is not considered by the Schlichting
theory nor do we consider it here. Certainly, before the picture of flexible walls can be
complete, a consideration of their influence on the fully turbulent boundary layer should be
made.

The theory of boundary-layer stability starts by solving the Navier-Stokes equations
and the equation of continuity for a steady state. An arbitrary perturbation developable in a
Fourier series is added to this solution, and the conditions are studied under which this
arbitrary perturbation will either increase or decrease as time proceeds. If no possible per-
turbation can increase, then the flow is assumed stable. Instability will exist if any per-
turbation exists capable of increasing in magnitude without external excitation. Between
these stable and unstable conditions there exists a boundary at which a perturbation neither
increases nor decreases in magnitude. This is known as neutral stability, and it separates
the stable from the unstable conditions. The values of the parameters of the system which
lie on the boundary separating the regions of stable and unstable flow are known as condi-
tions of neutral stability. The plotting of these conditions makes it possible to separate
the regions of stable and unstable flow, and on this basis to predict when instability will
occur. The principal aim of this paper is to express the changes in the curves of neutral
stability which are brought about by the presence of a flexible wall, the properties of this
wall being expressed in terms of its compliances. These compliances may not only be cal-
culated from the nature of the walls, but they may also be measured.

2. PERTURBATION OF TWO DIMENSIONAL PARALLEL FLOW

If we have a flow exclusively in the x direction which, however, depends on y, then we
may examine the effect of a small perturbation on this flow. Suppose that V, satisfies the
Navier-Stokes equation and the equation of continuity and that it vanishes along rectilinear
boundaries extending in the x direction. It has been shown that if the perturbation has the
form

+0 =
0O(0,Y) icax-ft) ,-
Vv =. | at e eee da (OI)

-®

s
i]

+00
-| ia@(d,y) ef °F) ag (2.2)

then the function ®(0,y) satisfies the Orr-Summerfeld equation

v 2 d?v . 4 2
(en -£)(<8- a’s) A et Syipiecnes 2H (ee - 207 Fe, a‘s| (2.3)

a/\ dy? dy? a \dy* dy

It is usual to express the solution of Eq. (2.3) in terms of dimensionless variables appro-
priate to the system. Let us choose a characteristic length which may be the distance be-
tween the edges in Couette and plane Poisseuille flows or the thickness of a boundary layer
in a boundary-layer problem. Let us express the velocity of flow in terms of the maximum

Theory of Stability of Laminar Flow 453

velocity U,,. We will take Y = y/5 as an independent variable and introduce the following
parameters:

8U V
R=—", a= 48, See @=—. (2.4)
Equation (2.3) then becomes
" 2 " i on 2au 4
GS ac) (2) = ar) = B40 + pyr = 200) it Oc@)), (2-5)

The solution of Eq. (2.5) has been extensively discussed in the literature. It has been found
that it may be expressed in terms of solutions of the left-hand side of Eq. (2.5) set equal to
zero (proper attention being paid to the behavior at the branch point) and in terms of solu-
tions of

= 0 (2.6)

where 7 =(a,dR)!/3(Y - Y,), Y, is the value of y for which g/y) = C and a, is the deriva-
tive of g at the point y= Y,. All previous solutions to these equations have assumed that
v, and v, vanished on all boundaries and at infinity if the flow was not confined. In the
case of the boundary layer with which we will primarily be concerned, the conditions are

V, = Vy = 0 for y=0 and y=. This requires four boundary conditions. If we choose an
appropriate solution @ of the left-hand side of Eq. (2.5) which vanishes with its first deriv-
ative for large values of y and a similar solution of & of Eq. (2.6), then it can be shown that
for the familiar solution of the stable equation, we obtain

e o

=| e02 (2.7)

dp! (a,aR)*/3y'
This determinant is a complex function of the three variables 0, R, and C. If for all real
values of @ and R the imaginary part of C is negative, then the flow will be stable. If we
choose C real, we can plot any one of these variables (a, R, and C) as a function of one of
the others for the condition that satisfies Eq. (2.7). The curves so obtained are known as
curves of neutral stability and establish the boundary of stable flow conditions. Diagrams
of this type permit one to predict not only whether the flow will be stable but the wave-
lengths which occur in the unstable regions of the diagrams. Our objective in this paper is
to examine how these curves of neutral stability will be influenced by the compliant sur-
faces. This paper, however, contains only the basic principles which will need to be de-
veloped somewhat more fully.

3. BOUNDARY CONDITIONS FOR COMPLIANT SURFACES

The velocity of flow of a viscous liquid must be identical with the velocity of the wall
at every point of contact. Suppose that 7 is the position of a point on the undeformed wall
and that the deformation caused by forces in the flow is designated by the vector €(r). The

454 F. W. Boggs and N. Tokita

rate of deformation of the wall at the point 7 + E(r) will be given by €(r). This must be
identical with the flow velocity at the position 7+ &. Hence, we will have

Xr) =Urté)= Wr) + We Etres, (3.1):

Since we are considering only small perturbations, we can neglect the effect of € on ¥ and
identify the rate of deformation of the surface with the velocity of flow at every point along
the boundary. This will lead to the approximate relationship

2r) Ca) (3.2)

If the motion of the wall is due exclusively to the forces in the fluid boundary layer,
we must express the surface deformation in terms of these forces. To completely establish
the boundary conditions, we must, therefore, express the surface forces in terms of the flow.
When this has been done Eq. (3.2) will become a differential expression in the stream func-
tion which must be satisfied on the boundary.

If the response of the surface to external forces is linear, we can always express both
the forces and the corresponding response of the surface in terms of progressive waves, just
as we did in the case of perturbation of the flow velocity. If €, and €, are the amplitude of
a wave of wave number © and frequency £ traveling along the surface and if P(a, 8) and
T(d, 8) are the corresponding expressions for the amplitude of the wave of pressure and
tangential force, we can define a set of compliances such that the following equations are
satisfied:

UNL:
x
I

Ma T + Moyea
(3.3)

UN:
<
TT

Ve Spiny. Pe

The constants Y,. are dependent on the parameters @ and Bf and are determined by the
specific nature of the wall. At a later date we hope to present detailed treatments in which
these constants are calculated from the structure of the coating. In this paper we will con-
fine ourselves to their general properties.

To complete the calculation of the boundary conditions, we must express the surface
forces T and P in terms of the stream function. From the perturbation of the steady-state
solution of the Navier-Stokes equation, we obtain

2
='vpVivi: (3.4)

We assume that Vs vanishes on the boundary and that its derivative at this point is given by

a constant depending on the nature of the boundary layer profile. In the case of Blasius

profile this is equal to yU,,/5 where y is a dimensionless constant and the same for all

boundary layers of this type. If we express v, and v, as was done in Kgs. (2.1) and (2.2)

and if we use the dimensionless variable defined in the previous section as the independent

variable and if we use the same dimensionless parameters, it is readily shown that the pres-
sure assumes the form

Theory of Stability of Laminar Flow 455

fap |
RES lis whey Nit (c + iar) + 7009), : (3.5)

The tangential force will be given in turn by

(3.6)

If we substitute Eqs. (3.5) and (3.6) into Eq. (3.3) which expresses the deformation of the
surface in terms of the surface forces, we will obtain a relationship between the velocity on
the boundary and its derivatives along the wall. Every term in this new expression will de-
pend upon the stream function or on its derivatives and on the parameters of the system.
These two equations, therefore, replace as boundary conditions the vanishing of the compo-
nents of velocity on the surface of the body. The conditions at infinity will not be changed
by the flexible wall and will, therefore, remain v, = v, = 0 as in the classical equation.

It was pointed out previously that the stream function ® could be expressed as a linear
sum of two functions ¢ and w. This will still be true since the differential equation is un-
changed by the flexible wall. Consequently, these two functions obey the boundary condi-
tions at infinity both for the flexible and for the inflexible wall and can be used in linear
combination to develop the solution of the equation of the flow in the presence of a flexible
wall. If we substitute into Eq. (3.3). a linear sum of these two functions, multiplying by
arbitrary coefficients, we will obtain a set of two equations in the two arbitrary coefficients
which much be satisfied. The condition of existence of solutions to these equations is the
vanishing of a determinant which is related to the determinant obtained for the rigid wall.
The set of linear equations which must be satisfied are given in

mw U
44 3) ~ Oy0 Vy Eee ae Hates 3" 19"
BY (aaRy'* v'Cy) -UpY,, 2 yl" + (c + BY cary"? y+ 99 |

Up 173 2
a (a,aR) ee = 10)

: ae mt - U_p
A-iag = Up ee iz abe ar is) g' + | = oi v.07}

+ p{-iay - Up io E yn +(c +: A) cau)? y + yw |

(3.7)

Up 1/3
meee) Youd" f = ©.

456 F. W. Boggs and N. Tokita

ae : 1/3 ; ; : :
Retaining only the terms in (a, 0 R) /3 and those independent of this quantity, in other
words neglecting the negative powers of @R, we will obtain the set of equations

A [s' =auee Y,(C¢' + ¥¢)]

+B {cayary? y' =e Yya| -éayy" + C(aR) vy] = 0

(3.8)
A [-ia¢ - U.P Y,,(Co' + y¢)]
. TT . 1/3 f]
tp 18) {- ay YP Woe [-ia, y" + C(a,aR) We sr vy |} = 0.
We make the substitution
e, = U,P Y 52
z (3.9)
en = ULP Yi.
and rearrange terms, and we obtain
A [(1 = ©)" = e,y¢|
1/3 ' : m
+ B [(ajaR) (GI RUE Os ie,a,y"| = 0
(3.10)

A [-e,c¢' - (10 + e,7)9|

3

1/ . UT)
+ B [-e,€¢a,08) Ws (t0 Eee yi ats rear ] = 0.

The condition of existence of solutions in this set of equations is the setting equal to zero
of the determinant of the coefficients of A and B. Again we will note that when e, and e,
are equal to zero this condition will reduce to the one obtained for a rigid plate. This is as
it should be since e, and e, vanishing means zero compliance; the plate cannot be deformed.

It may be readily seen that the matrix of the coefficients of A and B in Eq. (3.10) is
given by

1-e,C -e,7 p' (aaRy¥3 0 we, ay”
+ .. “@eim)
SENG Clann 9) p wy OF Meta wa

To obtain the eigenvalues of the differential equation, we must set the determinant of this
matrix equal to zero.

Since the second matrix in Eq. (3.11) is singular, we cannot divide through by it, but if
the prefactor on the first term is not singular, we may multiply through by its inverse so that
Eq. (3.11) is transformed to

Theory of Stability of Laminar Flow 457

p' (a,aR)¥3 wy! -(ia+e,y) e,y 0 ie,a,y"
1
tz » (3.12)

p wy e,c (Hare, Cy 0 te,a,y"

This step requires examination. If the matrix has no inverse, if the operation cannot be
carried. out, then we have a special problem. In general, the boundary layer will be unstable
if the zeros of the determinant (3.13) have positive real parts:

i e5€ ology

A = SCM ap SINE IC Seay
eC (20 ey)

= -ia + iaCU,p Mag = yU_ p Ee (3.13)

Since we assume the compliances are passive, this will generally not be the case, so that
we will be able actually to divide through by the matrix and obtain (3.12). Our stability
conditions are then given in

dp! (a,aR)1/3 y' ¢' den

i = 0. (3.14)

P Ww OQ NS,

As can be seen the first term is the expression which must be set equal to zero in the case
of the rigid boundary. The second term contains the added conditions imposed by the flex-
ibility. Developing this determinant, we finally obtain the following equation for the condi-
tions of the boundary:

wy db ie, de,¢ ay"
Y sn) a Wee, a 0. (3.15)
Teaeny a | Y_(a,aR)/3 y

We will note that this boundary condition is expressed in terms of three functions depending
upon the hydrodynamic conditions and on the two compliances. Two of the three functions
which are involved are identical with the ones which appeared in the solution for the rigid
body as developed by previous authors. We may rewrite in the form

HCE) = G(a,C) =" — | ze) -foe, V GCa,€) | a H(A

Be Se ae eee (Cont)

458 F. W. Boggs and N. Tokita

y"'( C)

Cw'(f)

1/3
C = -(a,aR) y.

H(C)

(3.16)

FG) =(GCase) Y,, + ia¥,G(a,C) Y,,

22
aan) iyY,, + aCY,, - =—

se!
o

4. SOME GENERAL PROPERTIES OF A SURFACE COMPLIANCE

Rayleigh was the first to show that solid bodies are subject to waves propagating over
their surface. Such waves will exist in the case of a turbulent boundary layer in contact
with a flexible surface and will play an important part in the nature of the compliance.
Basically, there are two factors to be considered; the deformation which occurs at the point
where the pressure is applied and the way this deformation propagates along the surface.

We will consider the effect of both of these reactions of the surface. Let us suppose that
we have a pressure of the type mentioned previously, and for the sake of simplicity, let us
suppose that it varies periodically with the time so that it may be represented by the product
of a complex exponential in the time and a function of x. In this case we may represent the
pressure by an expression similar to

el Pash = =| PGa) eo dae 7 (4.1)

This will cause periodic displacements of the surface velocity in the x and y direction which
may be represented by

4 SHewiGie chu a(n)

-iBt
e

S
|

+0 —
| Y,,(0,8) P(i)e da

(oo)

(4.2)

v.=-ipe ”* € (x)

+0 ts
= = POX YL Ae 2 Bie
| Vos) P(a) e dae

fo)

In Eq. (4.2) the Y’s are the surface compliances for a wave having a wave number a. If we
use the Fourier inversion theorem for Eq. (4.1) and Eq. (4.2), we will obtain

+ 00

PCa) | FGx) en) ax

iH
he
ex)

PAID) MOB) yee Es) eu apex (4.3)

Theory of Stability of Laminar Flow 459

Y,,(@,B) P(a) = - #| E(x) ee dx. (4.4)

Combining these equations we obtain for the surface compliances

Y, (2,8), =——________ (4.5)

Yo % 2) = TP eq aaamOG sous ww vat (4.6)

So far these are purely formal relationships which, for example, would allow us to determine
the surface compliance experimentally from the reaction of the surface to a pressure. This
is useful in that equipment can be designed and has been designed to use this relation for
the determination of the surface compliance of arbitrary surfaces. However, we must put it
in more manageable shape for the determination of experimental quantities, and we will show
how it can be used to relate the surface compliance to the propagation constant of Rayleigh
waves at a given frequency. Let us suppose that the pressure is applied over a very small
length which is allowed then to go to zero. Experimentally, this could be achieved by the
use of a very narrow knife edge as a driving unit. We will have for the pressure

P(x) = = when =e <x < ¢
(4.7)
PCx) = 0 when xX <-€ or X> &.
Using these in Eq. (4.4) to obtain the Fourier integral of the pressure, we have
Bee, 8) 1 p Sin Qe ties
VJ 27 de :
Taking the limit as € > 0 we finally obtain
1 sin de 1
P(a@,8) = ——F lim — = ——F, (4.9)
p Vv 27 630 Qe J 2r

460 F. W. Boggs and N. Tokita

The compliances of the surface will be given respectively by

+00
iB - iQx
Yee Sn = ae SiGe) e dx (4.10)

1B ' - idx
Woe = el eae) 7e dx, (4.11)

The Fourier transforms in this case are carried out from minus infinity to plus infinity in the
complex plane. It is convenient to use a real notation, in which case the integrals will go
from zero to infinity. It is worth noting, however, that the normal displacement will be an
even function of x, whereas the tangential displacement will be an odd function of x. Asa
consequence only the cosine transformation from zero to infinity will be important in the
first instance, and the sine transform will be important in the second instance.

This gives us for the surface compliance

i 2i6 E
7 cee). = ae F E(x) cos ax dx (4.12)
val Gy) ———— 218 | E(x) sin ax dx, (4.13)

V 27 F ,

Let us now suppose that the vertical vibration imparted by the knife edge gives rise to
propagating waves having the propagating constant I. The compliances which apply to the
normal and tangential components will be respectively given by

_ 218 if (8) Ss il'x
Y,2(%,) F

oF e cos ax dx (4.14)
V 477
0

me 2i8 6 (A) = Aine
Y,,(4, 8)

- ——_—__—_ e sin ax dx (4.15)
J 27 F

where f,(8) and g,(8) are functions of the frequency.

Carrying out the integral we obtain

Theory of Stability of Laminar Flow 461

r
teste)
aC E) = = Oy. . (4.16)
a
y 2 1
Oe ode), = 28 & (8) een T\2 5 :
Jam OF (=) ae (4.17)
a

The quantity I"/d is the ratio of the wavelength of the disturbance in the boundary layer to
the wavelength in the coating. One of the points of interest will be to consider the effect
of a large difference between these two wavelengths and also what the behavior will be
when these two wavelengths are close together.

If we calculate the corresponding dimensionless compliances, we obtain

is
we oi! oi? a ire Me) —2 =

Wort Se ae)” si Virmanktbio: Wo Bal CH

q = 1 Vv -1
f be if sy

je, we UB. of &) p ee akey oP aU : for x >> 1

2 Ws ae. © enya m4 aL

ie a f m4

eee 2): AD) Py th Dlaie? R@5)) p E see

eke os 8 (8) 1 oe wiee er SAGE) ie 1
Jar F ye sie ip Es Sn
a Vv
2 2 Ba 6 (5) 7? 4 3 we 8 (Af) 9 r
on 2 F Am. = Jan G F PY mn? for 7 a |

=_- =_ —_. __

646551 O—62——31

462 F. W. Boggs and N. Tokita

It is apparent that e, and e, both pass through the origin when B is equal to zero. It
follows that they will both pass through the origin when C vanishes. Furthermore, when @
is small e, will be negligible and e, will be a function of 8 only. These two characteris-
tics will have some important practical consequences which we will consider in the next
section of this paper. It can also be seen by substituting that similar conditions will be ob-
tained when the propagation constant of waves over the surface is proportional to the veloc-
ity of flow.

5. CURVE OF NEUTRAL STABILITY FOR A FLEXIBLE WALL

If Eq. (3.16) is rewritten in terms of e, and e,, we obtain

e
1
F(t) - G(a,C) eg Mideg YGCa CT ge ee

Sung OR; eae ay Sele Beep wae e al y-c4))
a,H(o) a Corr) ae
2 a

The left-hand side depends on the Reynolds number only through its dependence on ¢,
whereas the right-hand side depends on the Reynolds number through the frequency 8 and
wave number 0, which depend on both the velocity and the Reynolds number. The curve of
neutral stability can be obtained by identifying the real and imaginary parts of both sides of
Eq. (5.1) and eliminating either @ or C. For each value of U,, there will be a different curve
of neutral stability and a corresponding critical Reynolds number which, if it exists, will
depend on U,,. The procedure outlined above represents a formidable amount of work which, |
even with the aid of calculators, would be forbidding. Can we find a procedure for obtaining
bounds for the Reynolds number which will simplify the general discussion?

For a given velocity the right hand-side of Eq. (5.1) will be a function of two variables
only, & and 8. Furthermore, as we saw in Section 4 all the curves will pass through the
origin and be tangential to each other when © and B are small.

If we hold constant we can plot the right-hand side of Eq. (5.1) as a function of C
treating R as a parameter. For a given value of © the right-hand side of Eq. (5.1) will bea
single curve passing through the origin while the left-hand side will give a family of curves,
one for each Reynolds number. The intersection of the curve corresponding to the right-hand
side of (5.1) with each one of the family of curves for the left-hand side will correspond to a
value of @ and of R on the curve of neutral stability. If the family of curves representing the
left has an envelope through the elimination of R, the value of the frequency corresponding
to the intersection of this envelops with the right plotted for % and U,, constant will be an
extreme value for C. The corresponding value of the frequency for each value of & will form
a boundary separating areas of stability and instability. The value of the frequency along
this boundary will be a function of the velocity and of @, and its value will be given by an
equation of the form

B60 koi 22 00, (5.2)

Theory of Stability of Laminar Flow 463

In addition, we must specify that we are on the envelope. This will give a relationship
between @ and C. These two relations together will, therefore, lead to a relationship be-
tween the Reynolds number and the velocity, from which a family of curves of critical
Reynolds numbers can be constructed.

A delineation of the family formed of the envelopes obtained by the elimination of the
Reynolds numbers through the construction of the envelope requires a more detailed anal-
ysis of the way the right-hand side of Eq. (5.1) is formed. If we consider the function given
to the left of Eq. (5.1), it is not difficult formally to construct the envelope.

The analytic expressions for the functions F, G, and H are inconvenient. We have used
an approximate expression for the denominator and have constructed the envelope graphically.
The value of the function in the numerator for a given set of values of © and C is obtained
by simple graphical subtraction. Holding constant and varying C, we obtain a family of
curves with the general appearance given in Fig. 1. This family of curves has an envelope
which is formed by the tangent to the common apices of the curves. There will be one such
envelope for each value of &. This family of envelopes is given in Fig. 2, which was con-
structed from existing data. Each member of this family of envelopes separates the plane
into two domains. Any point in the lower domain will have two members of the family of
which it is an envelope passing through it, a point on the envelope will have only one, and
finally a point above the envelope will have none. The locus of the points of intersection
between the family of envelopes on the one hand and the compliances on the other will form
the boundary between areas of stability and instability. Points corresponding to the com-
pliance which are above the envelope cannot lead to unstable solutions.

In the discussion that follows we will consider the special case when e, vanishes.
For small values of 8 and @, e.g., moderate speeds and large Reynolds numbers, this
will be the most important case.

ted Se a

Fig. 1. Family of
curves from Eq. (5.1)
for constant @ and a
set of values of C

4
0.0
SSS
o2 a=0.4
me)
a=0.5
Fig. 2. Envelopes obtained = 506
as shown in Fig. 1 ;
a=0.7
A
fe

464. F. W. Boggs and N. Tokita

We can solve Eq. (5.1) for e;/id, giving us

1
Fo SooiGiuri bs yhicaniion doth ME sinlandtih GR AG) wort
a,H ey id = (F-G) - iH’
© a5 = i
al
VEE TAG ®
ga EKO for large C (5.3)
F-G 1 1

GL oe eee ties hy vag Gees ten

The family of curves given in Fig. 2 can be transformed graphically by a succession of
projective transformations to give the family of curves given in Fig. 3. This is the appro-
priate form to use for Eq. (5.3). It should be noted that there is one member of this family
corresponding to C equals 0, which is a boundary for all members of the family. The value
of C for which the family of curves passes through the origin corresponds to a minimum
value of C compatible with complete stability for a rigid flat plate. The study of this en-
velope is in fact a convenient way of visualizing the stability problem for a rigid wall as
well as for a nonrigid one and leads to essentially the same results in the former case as
the procedure employed by Schlichting.

Let us now proceed to a consideration of the general effect that e, will have on the
stability of the flow. e, will be represented by a family of curves of the two variables

25 BS) 5 |
ae U
~
~ 4
~
~
~
~
= >
Lig FNS iy
: ? Wat ak A
29 COMPLIANCE OF 0°:
PLATE SURFACE ‘A
CSA
0:
-5 “ee
-15
ZA

Fig. 3. Family of contours in the compliance, obtained by
transformation from Fig. 2

Theory of Stability of Laminar Flow 465

and

a= 5
vJ/R

which we know must be mutually tangent and pass through the origin for small values of
both variables. If now for a fixed value of 4 we describe one of the families of curves, its
intersection or intersections with the member of the envelope for a corresponding value of @
will give points on the boundary line separating stable conditions from unstable conditions.
In general, there will be two such intersections and usually the stable conditions will be
those sections between the points of intersection. If the members of the family of curves of
the compliance do not intersect the corresponding member of the family of envelopes, then
either the system will always be stable or will always be unstable. Since the envelopes
corresponding to unstable conditions for a rigid wall always enclose the origin and since the
compliances must pass through the origin, it follows that there will be at all times at least
one intersection with the compliance unless the compliance is so small as to be completely
enclosed by the envelope. Since, however, in this family of curves the portion closest to
the origin represents unstable conditions, it follows also that in the case where there is no
intersection between the compliance and the envelope that the boundary layer will always

be unstable. Consequently, in the case of interest there will always be at least one inter-
section corresponding to low frequencies between the members of the envelope and the
family of compliances. These curves considered as a function of R, £, and the velocity of
flow will form a family from which the extreme values of the Reynolds number can be deduced.

SUMMARY

We have given a very brief discussion of the principles on which an analysis of the ef-
fect of a flexible surface on fluid flow is based. It shows that through the introduction of
the concept of surface compliance, the conditions of stability of laminar flow may be ana-
lyzed. A more detailed presentation would allow us to give conditions which will be ful-
filled for a stable flow. In general, we show that the flow will be stable when certain con-
tours in the complex plane exclude the origin. The presence of the flexible wall replaces
the origin by a curve and the points of instability are the intersection of this curve with the
contours.

APPENDIX

Reference 4 gives the relation

-0 Pé
v(L) \ | eM? Hy 3[ 3 (iey/"] do dé
_ ¢+0 J+

F eA ee ease Se SE OS ee ee ee

1 A) e
pet Hy 3[4 (ipy?/7| dp

466 F. W. Boggs and N. Tokita

For large values of ¢ it is shown that

F(L) mt ale Dale (A2)

We wish to calculate w"/Cy' in terms of F and its derivatives. Let

wi! 1

=— = =. (A3)
wy CFCC)
In terms of A we obtain
yi" 1 ( 2 =)
—— = ——lh, + 3h’ +—].. A4
igh alk h nee
For large values of ¢ we obtain by combination of (A2) and (A4)
m : -im/4 .
ll ee ange Ta rye d tig = 2 ses (A5)
C C 2t 4l

ACKNOWLEDGMENTS

The authors take pleasure in thanking Dr. Max O. Kramer and Professor P. J.W. Debye
for many helpful suggestions made during the course of this work. .

REFERENCES
[1] Tollmein, W., “Gottinger Monographie uber Grenzchichten,” Pt B 3, 1956

[2] Schlichting, H., “Zur Entstchung des Turbulentz bei der Plallenstromung,” Nachr. Ges.
Wiss. Gottinger, Math. Phys. Klasse 182, 1933

[3] Schlichting, H., “Boundary Layer Theory,” New York:McGraw-Hill, 1955

[4] Lin, C.C., “On the Stability of Two-Dimensional Parallel Flow,” Quarterly of Appl.
Math., Vol. III, pp. 117, 218, and 277

[5] Lin, C.C., “The Theory of Hydrodynamic Stability,” Cambridge:The University Press,
1955

[6] Squire, H.B., “On the Stability of Three Dimensional Disturbances of Viscous Fluid
Between Parallel Walls,” Proc. Roy. Soc. A-142 (1933)

Theory of Stability of Laminar Flow 467

[7] Schlichting, H., and Ulrich, A., “Zur Berechnung des Umschlungs Laminar-Turbulent,”
Jahrbuch d. dt Luftfahrtforshung 18, 1942

[8] Schubauer, G.B., and Shramstad, H.K., “Laminar Boundary Layer Oscillations and
Stability of Laminar Flow,” National Bureau of Standards Research Paper 1772

[9] Max O. Kramer, Communication Jour. Aero-Space Sciences, June 1957
[10] Max O. Kramer, Communication Jour. Aero-Space Sciences, May 1959

[11] Max O. Kramer, “Boundary Layer Stabilization by Distributed Damping,” Jour. Amer.
Soc. of Naval Engineers 72:25 (1960)

DISCUSSION
T. G. Lang (U.S. Naval Ordnance Test Station)

I would like to mention that I have observed Dr. Kramer’s experiments on models utiliz-
ing compliant surfaces in Long Beach Harbor in California and consider his test apparatus °
to be well designed and his reported results to be accurate. When the theory of distributed
damping using compliant surfaces was first proposed by Dr. Kramer, it was mentioned that
sea mammals such asporpoises and whales utilize this phenomena to reduce their drag. Re-
ports by many observers of fish and sea mammals indicate high performance. I would like
to describe a study which was recently conducted by the U.S. Naval Ordnance Test Station
(NOTS) on the performance characteristics of a live porpoise. These results which I shall
present are still preliminary since additional analysis is planned.

The performance tests were conducted by a group of NOTS personnel in the towing tank
at Convair, San Diego. This tank is 315 feet long, 12 feet wide, and 6-1/2 feet deep. It
was filled to a depth of 4-1/2 feet on June 3, 4, and 5 and to 6 feet on June 15 with sea
water. This water was continuously filtered except during tests, and chemicals were added
to prevent growth of plankton and bacteria. The tests were composed of two types. One
type was a peak-effort run down the tank, and the other was a motionless glide through a
series of large underwater hoops. The porpoise was tested both in its natural condition and
with a ring, whose thickness varies from 1/16 to 1 inch, placed around its head section.
The purpose of the thinnest ring was to induce turbulence on the body. The thicker rings
were used to significantly increase the drag of the porpoise by a fixed amount and thereby
aid in determining its horsepower output when the top speed with each ring is known.

Figure D1 shows a porpoise with such a ring placed around its head section. This is a
porpoise similar to the one used in these tests, except it was trained to support itself in the
position shown in the figure for several seconds. Figure D2 shows the dimensions of the
porpoise (Pacific white-sided dolphin) which was tested in this program. It is 6.7 feet long,
has a maximum diameter of 1.2 feet, and weighs 200 pounds. The distance versus time data
were primarily measured by overhead cameras. Two cameras were mounted at the beginning
of the runs behind underwater windows, but much of their data was lost due to camera mal-
function.

Figure D3 shows that portion of the horsepower which was required on the peak effort
runs to produce the recorded acceleration. It is noted that horsepowers up to 1.8 were

F. W. Boggs and N. Tokita

Fig. D1. Placement of ring on the
porpoise head section

QIMENSIONS (WV INCHES

38
0 (PROJECTED)
sae’ %

60.5
(PROJECTED )

WE/EHT OF PORPOISE-200 LB

MEASUREMENTS ARE ALONE BOOY UNLESS
NOTEO OTHERW/SE, C/RCLED NUMBERS AkKE

GIRTH MEASUREMENTS.

Fig. D2. Dimensions of the porpoise tested

ACCELERATION
HORSEPOWER

2

4

Theory of Stability of Laminar Flow 469

RING THICKNESS

eo= NO

a= 1/16"

= 1/2"

v= 3/4"
CONSECUTIVE
DATA POINTS
NON CONSECUTIVE
BUT MAXIMUM
POWER DATA
POINTS

<<

8

Fig. D3. Acceleration horsepower versus velocity

measured, and this does not include the horsepower expended in overcoming drag. This
value of power is in agreement with that which humans can exert for the same time period.
The lack of acceleration data at low speeds is due to the previously mentioned camera mal-

function.

Figure D4 shows the results of the glide runs wherein the drag was calculated from the
deceleration rate as the porpoise glided through underwater hoops. The data are plotted as
drag area versus glide velocity, wherein the effects of virtual mass are included. This drag
area is the drag divided by 1/2 pV”. The scatter of points for any one configuration is be-
lieved due to movement of the porpoise while gliding. For numerous reasons, the maxima
of these points are considered to approach the correct drag value of a motionless gliding
porpoise. Figure D5 is a plot of this same data against ring thickness. The solid lines are
the estimated drag area at various glide speeds. The dotted line is a curve faired through
the maximum drag area data points and is believed to represent the minimum value of the
experimental drag area of the porpoise. The wave drag has not been subtracted from this
data, but it has been calculated to be small at the higher glide speeds.

DRAG AREA, D' (FT®)

TAIL MOVEMENT

SEEN IN MAJORITY@ g2

OF RUNS FOR
D' <0.04 NO RING

75-39

<4

RING THICKNESS

4p 0Cbe
none non
ol
~~
o

THEORETICAL TURBULENT,
NO RING

THEORETICAL 40% LAMINAR,
NO RING

THEORETICAL LAMINAR,
NO RING

4m 18
VELOCITY (FT/SEC)

12) 16920) 24 728:

Fig. D4. Drag area versus velocity

470 F. W. Boggs and N. Tokita

ai: 2m. ( Ato )
Be pXA MENA
XA

THEORETICAL TURBULENT XA lls
(Ato) (A t))

“DRAG AREA FOR
V=10, 20, 30 FT/SEC

0.20 5-39
c
& 0.16
pole FAIRED
< EXPERIMENTAL
i 0.08 ‘ DRAG AREA
ie 5-1 1
2 0.04t |
= 615-19 15-90 H
a fe) Q
(0) 0.2 0.4 06 08 1.0

RING THICKNESS (INCHES)

Fig. D5. Drag area versus ring thickness

Figure D6 shows the maximum recorded speed versus ring thickness. With no ring, the
maximum speed was 25 ft/sec, which is only 15 knots. It is noted that the water depth has
no effect on top speed, since it was 4-1/2 feet for the runs on June 3, 4, and 5, and 6 feet
for the June 15 runs. Figure D7 shows the drag horsepower versus ring thickness. The
drag horsepower was not measured but was calculated using the dotted-line experimental
drag area of Fig. D5 and the top speed from Fig. D6. Of primary interest is the fact that the
2.0-horsepower maximum, recorded for the no-ring condition, agrees well with the recorded
maximum acceleration horsepower. These results tend to indicate that the effective drag
while swimming is essentially the same as that while gliding, and that the boundary layer in
each case is effectively turbulent. It is noted, however, that the horsepower for the runs -
with rings is only 0.8 rather than 2.0. This fact would tend to indicate either that the rings
had a large effect on the boundary layer while swimming, that they caused the porpoise to
exert less effort due to irritation or some other cause, or else that the experimental ring
drag values are low. The time during which the purpoise traveled at top speed was several
seconds longer when a ring was carried than when it was not, so the expended horsepower
with rings would have been expected to be around 1.3 to 1.6 rather than 2.0.

Figure D8 shows the porpoise body movement during a typical cycle while accelerating.
The nose stations are aligned and the body position is sketched from different film frames

MAXIMUM RUN VELOCITY
(FT/SEC)

fo) 0.2 0.4 0.6 0.8 1.0
RING THICKNESS (INCHES)

Fig. D6. Maximum velocity versus ring size

Theory of Stability of Laminar Flow 471

ove
_ Op Vitax
—o— DRAG HP ae

D' = DRAG AREA FROM
FAIRED CURVE OF
EXPERIMENTAL DATA

5-29

~l9-5 95-20

15-9
SCINSUFFICIENT DATA
0 02 0.4 06 08 1.0
RING THICKNESS (INCHES)

DRAG HORSEPOWER

Fig. D7. Maximum drag horsepower versus ring thickness

RUN 15 -2/

(FINS AND Je W. DRAE-RiNE NWOT SHOWN )
(NOT EQUAL INTERVALS )

Fig. D8. Porpoise movements during
acceleration, nose sections aligned

to show body and tail movement. The fins have been removed from the sketches to improve
clarity.

In conclusion, it can be stated that no unusual physical or hydrodynamic phenomena
were apparent in this porpoise study. These tests, however, have not proved that unusual
characteristics do not exist. It is possible that: (a) turbulence in the tank water or the
added chemicals affected the ability of the porpoise to control its boundary layer, (b) the
porpoise did not exert maximum effort, or (c) that unknown factors affected the results due
to the unnatural environment. A report is in process which contains more detailed informa-
tion on the study.

M. Landahl (Massachusetts Institute of Technology)

I want to comment on an error in the boundary condition, but first I want to point out
that the present author here is not the only one who happened to fall in that trap because it
is very easy to do it. The reason for it is as follows: When you have a wall; of course, you
know that the steady boundary layer is such that whatever waves you may have superimposed
on top of this layer, all components of velocity must vanish at the wall. But when you have
a flexible wall you have to consider that you must satisfy the boundary condition on the wall
itself and not on the mean position and this gives an extra term due to the finer slope of the
profile which accounts for this extra term. This problem is really quite intriguing in a way;
when you straighten things like this out it turns out that this problem mathematically is very

A72 F. W. Boggs and N. Tokita

simple and, as a matter of fact, is simpler than the classical problem. The reason for this I
will point out in a minute, I will first mention that other investigations in this area have
been published and are on the way to being published. There is a forthcoming paper in the
Journal of Fluid Mechanics by Dr. Bannerman of Cambridge and I think that his paper is by
far the most comprehensive and has very interesting physical discussion on it. I said that
the mathematical problem is really quite simple, however, the difficulties are in the physical
interpretation of the results which are, I must say, extremely difficult. First of all, the rea-
son why the mathematical problem is so simple is that you can state this problem as a direct
problem instead of as an indirect problem. In this particular case you can, instead of asking,
what is the stability for a particular layer, ask what kind of wall admittance should one
choose to maintain neutrally stable oscillations for waves of a particular wavelength and
wave number. This is quite straightforward and as a matter of fact if you straighten out all
these initial difficulties the results come out to be extremely simple. The results show that
in order to take care of the instability completely you have to supply energy into the bound-
ary layer and consequently a passive wall layer will never do the trick. However, it turns
out that a pure compliance of the surface, a pure spring compliance, that is, a very flexible
wall, will push the minimum Reynolds number up quite a bit. You can push it up, as Dr.
Boggs pointed out, as far as possible, but there is one particular thing here I must say I
don’t quite understand completely yet. It looks as if you say to the boundary layer, I take

5 percent of the energy out of you, and then the boundary layer says, all right, I give you

10 percent. The more energy you take out of it the more energy the boundary layer supplies,
and this is a very peculiar thing, which I think has to be understood more completely before
you can really design these layers with good knowledge.

F. S. Burt (Admiralty Research Laboratory)

At ARL we also discussed this problem of the use of damping in the skin to stabilize
the laminar boundary layer and came to a similar conclusion, that in the normal instability
case one might have to put energy into the layer rather than take it out. The only other
thing I would like to ask Mr. Boggs is if he can give us some figures for the critical Reynolds
number, both maximum and minimum that he was referring to in his figure, preferably in terms
of length to the transition point, which is the more normal Reynolds number used for transition.

F. W. Boggs

To the question of the energy, I would like to point out one thing. We have found, as a
practical matter, that when the mechanical loss in the rubber that we use in the coating be-
comes large, the coating becomes substantially ineffective, maybe even detrimental. As to
how energy might be transferred I wonder whether it wouldn’t be through the propagation; our
feeling is, and my impression is also, from an examination of our data that it is only when
you have propagation that you will have stabilization. If you have propagation of the
Schlichting wave which differs from the propagation of the wave in the coating (let’s sup-
pose, just to make things simple, that there is a factor of 2), then if they are in phase at
one spot, half a wavelength ahead they will be out of phase. So whereas in one case you
would be transferring the energy into the boundary, a wavelength or half a wavelength away
the energy transfer would be taking place in the other direction. I would think that this is
the type of mechanism that might explain it. As to Mr. Lang’s comments, we have discussed
this and we have seen reports on porpoises that travel at very high speeds, but I don’t know,
every time you talk to somebody he gives you something higher. The Scientific American,
where we all get our information now in the United States, had an article on this subject and

Theory of Stability of Laminar Flow 473

they quoted reliably something of the order of 25 knots, if I recall correctly. Two horse-
power is about what the biologists will tell us the porpoise should deliver. I am quite in-
terested that this is the figure that is gotten because this is just about right on the basis of
the body weight and the amount of muscle and:what we know about the muscle efficiency.
The only question is, does the porpoise go 25 knots? The 2-horsepower figure cannot ex-
plain the 25 knots, even though it can explain the 15, so I think this is the question, are
the speeds of the porpoises that we have just heard proverbial fish stories or are they the
truth? I think that Mr. Burt’s question perhaps I have answered already, except for the
Reynolds number. We have rough calculations. The limits, I am sorry, I don’t know in
length; I haven’t made the calculation, but they lie somewhere (and these calculations are
uncorrected by Mr. Landahl’s remarks so they are subject to some doubt) within the area of
maybe between 107 and 10°, about 45 miles an hour (excuse my nonnautical language) for
the coatings which have given us the best results. This is fairly consistent with our ex-
perimental data. The calculations are rough and could easily be off by a factor of two and
they might be off by a lot more, but we do not get 107° or anything like that. These are
modest figures.

4.

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THE FRENCH BATHYSCAPH PROGRAM

Pierre H. Willm
Ingénieur en chef du Génie Maritime, French Navy

INTRODUCTION

The purpose of the third symposium on naval hydrodynamics is to study the latest tech-
nical developments concerning the means of navigation available to all men. Therefore it
seemed reasonable to include, among other topics, the survey of a very new achievement
which enables man to reach the greatest depths actually known, namely, the bathyscaph.
But it was perhaps less logical to ask France to deal with this subject, since the diving
record is now held by the U.S. Navy with the bathyscaph TRIESTE, manned by CDR Don
Walsh and Mr. Jacques Piccard, who reached 11,000 metres in January 1960. I wish to
express my thanks to the organizers of this symposium who asked me to read this paper,
thus emphasizing the part taken by France in the development of this system of underwater
navigation.

In the short historical account with which I shall begin this paper, I intend to give more
details about this French participation, and to sum up the different types of bathyscaphs
actually known.

In the second part, I will recall the principles involved in this new kind of navigation
as well as the operational sequence of bathyscaph diving.

In the third part, I will briefly describe the first French achievement, the FNRS-3, then,
in the fourth part, the new type now under construction, the Bathyscaph 11,000.

I will conclude by mentioning a few scientific results obtained during the 80 dives
made by the FNRS-3 since it was built.

I regret to have no exceptional announcement to disclose to this honorable assembly as
regards hydrodynamics: the only new fact concerning propulsion is the essential part played
by the weight and by the Archimedean principle of buoyancy in vertical movements of the
craft. All movement in other directions is achieved by means of conventional propellers
which are entirely satisfactory in every respect.

HISTORY

The first attempts at very deep diving were made by two Americans, Professors William
Beebe and Otis Barton, who in 1934 succeeded in diving to 908 metres, a remarkable depth
at that time. But the 1.45-metre diameter and 45-mm thick cast steel sphere in which they

475

476 Pierre H. Willm

were enclosed hung at the end of a steel cable. Far from being a safe device, this perma-
nent link with the surface was a source of real dangers; to say nothing of a possible
breaking of the cable, which would have resulted in the total loss of the sphere (which was
heavier than the upward pressure exerted on it), the motion of the surface ship, in a heavy
swell, was amplified by the cable, so that the Bathysphere (as its designers called it) was
very uncomfortable, and it was quite impossible to approach the bottom.

The inventor of the first self-contained deep-diving craft was Professor Auguste Piccard,
a Swiss scientist. He applied to underwater navigation the operating principles of the free
balloon. The observation sphere, which was much heavier than the volumé of water which it
displaced, was attached under a thin sheet-metal tank, filled with petrol lighter than sea
water. This petrol was directly exposed to external pressure. Vertical movement was
effected by jettisoning ballast: steel shot for rising and petrol for sinking. Professor
Piccard, of the University of Brussels, submitted his design to the Belgian “Fonds National
de la Recherche Scientifique” (FNRS) which eventually granted the necessary funds (1939).
The construction, delayed by war, was resumed in 1945 under the joint direction of Profes-
sors Piccard and Cosyns.

The craft had been called FNRS-2 (FNRS-1 had been Piccard’s and Cosyns stratopheric
balloon) and it was shipped in September 1948 to Dakar, where the first trials took place
with the assistance of the French Navy. Unfortunately, many defects in this first construc-
tion were revealed during these trials. However, the FNRS-2 succeeded in reaching a depth
of 1380 metres without passengers, but all hope of performing this feat a second time had to
be given up.

However, this semifailure had established the soundness of the general principles, and
it was decided to undertake the construction of a new craft on the same lines. The job was
entrusted to the French Navy by the Belgian “Recherche Scientifique.” An agreement was
signed in October 1950 between the Belgian FNRS, the French CNRS, and the French Navy
concerning the design and construction of the new craft. Ingénieur Principal du Génie
Maritime Gempp , who was in charge of this work, retained only the sphere of the first bathy-
scaph and succeeded in building a vehicle capable of being towed in a moderate sea and
accessible to a crew when afloat, which was not the case with the FNRS-2, where the pas-
sengers had to shut themselves in the sphere before the bathyscaph was lowered and the
float was filled with petrol, so that the prediving operations were so complicated as to be
unacceptable.

Professor Auguste Piccard remained technical adviser to the French Navy for this work.
He paid several visits to Toulon to join in discussions of the project. But in the beginning
of 1952, he ceased to collaborate, in order to have another bathyscaph built in Italy, the
TRIESTE. This Italian bathyscaph was constructed at Castellamare di Stabia, near Naples,
and in August 1953, it dived to 3000 metres, manned by the famous Professor and his son,
Jacques.

Meanwhile, the FNRS-3, as the French bathyscaph was called in tribute to the Belgian
scientists who financed part of its construction, was planned and laid down in July 1952 by
Toulon Naval Dockyard. It was launched on June 3, 1953 and made its first dives to 2000
metres off Toulon in August 1953. It was then taken to Dakar (French West Africa) and
successfully achieved its first two dives to the maximum depth for which it had been
designed, reaching 4200 metres on January 31, 1954, without passengers, and 4050
metres on February 15,1954 with passengers. I took part in this record dive as the engineer
responsible for the construction, together with Capitaine de Corvette Houot, appointed com-
manding officer of the bathyscaph.

French Bathyscaph Program 477

During this last dive, the bathyscaph descended to the greatest depth ever reached by
man until the beginning of 1960, when the deepest sea floors known to this day (about
11,000 metres off the Philippines) were reached by the TRIESTE bought by the United

States of America.

The U.S. Navy had acquired the TRIESTE with a view to using this powerful underwater
research tool for oceanographic purposes. After the apparatus had undergone important
changes: lengthening of the float, replacing the original sphere by a new one, forged by
Krupp steelworks, in Germany, and enlarging the shot silos, it was an American bathyscaph
which landed for the first time on the bottom of the Pacific Ocean, at the depth of 11,000
metres. )

So, there are now two bathyscaphs in operation: one, the FNRS-3, is French, and the
other, the TRIESTE, is American. Their research programs are quite similar: physical and
biological oceanography, study of the propagation of supersonics and of electromagnetic
radiations in sea water, and the geology of ocean bottoms. A second French bathyscaph is
under construction in the Toulon Naval Dockyard; I shall mention it briefly at the end of
this paper.

As to the existence of other deep-water oceanographic vessels, we can only guess, as
these would be Russian. The USSR has stated several times that it was building bathy-
scaphs, but we have no information on this subject.

PRINCIPLES

To descend to great depths beneath the sea, man must enclose himself in a pressure
hull that is watertight and capable of withstanding the pressure exerted upon it. To with-
stand a uniform external pressure, the best shape is a sphere; that is why the hulls of
bathyscaphs have, up to now, taken the form of a sphere. Allowing for the safety factor
selected and for the inside diameter necessary to meet the accommodation requirements,
this sphere must have a certain thickness, which is determined by the strength calculation
of materials. Owing to this thickness, the sphere is much heavier than the volume of water
which it displaces. It would immediately sink to the bottom if it were not suspended under-
neath a float filled with extra-light petrol, which gives it the necessary buoyancy.

This is the main difference between a submarine and the bathyscaph: in the case of a
submarine, the weight of the pressure hull plus the weight of all the hull fittings and inside
equipment must be equal to the displacement. We shall apply the term bathyscaph to the
machine for which the weight of the pressure hull, including that of internal and external
fittings, is greater than its displacement. Therefore, the float is necessary to ensure the
trim.

Vertical movement is effected by increasing or decreasing the machine’s weight. The
increase in weight is produced by jettisoning petrol (which is replaced by sea water); the
dischargeable petrol is contained in a special tank, which is isolated from the float itself
whose buoyancy must not be impaired; this attitude is absolutely necessary to the lift of
the craft. The decrease of weight is produced by shedding ballast; this consists of small
steel shot contained in vertical silos in the center of the float. This shot is held in at the
base of the silos by the electromagnetic field of an electromagnet called an electrochute,
which lets the shot fall when the current is cut. In addition, there is a safety device,
called an electrorapid-release, which enables the opening of the silo to be freed completely

646551 O—62——32

478 Pierre H. Willm

if the electrochute is blocked. In any case, the ballast is automatically released in the
event of a power failure.

External pressure acts freely on the petrol through a stabilizing hole, in the bottom of
the float. Unfortunately, this petrol has an important drawback: it is very compressible.
When the bathyscaph sinks, the petrol is compressed and sea water enters the float.

The resulting increase in weight is markedly greater than the apparent decrease in
weight due only to the increasing density of sea water. Therefore, once the bathyscaph has
started to sink, it descends quicker and quicker. It is necessary to discharge shot during
the descent, in order to slow down this movement and to avoid sinking into the soft muddy
bottom from which it would perhaps be difficult to break loose. On the other hand, once the
bathyscaph starts going up, its speed increases progressively and it is impossible to stop
it. The maximum speed under such conditions is approximately 1 metre per second.

Sphere and float are joined by a metal frame. The float is surmounted by a narrow deck
running from stem to stern, with a conning tower in the center, in order to protect from heavy
seas, during the towing of the bathyscaph to the diving site, the various external fittings
that have to be placed there and the upper hatch of the entry shaft to the sphere. This
entry shaft runs vertically through the float and enables passengers to enter or leave the
sphere while the bathyscaph is afloat. It also acts as the submersible’s water-ballast; that
is to say, being normally empty on the surface, it fills with water during diving. The
increase in weight when it is flooded with water is enough to make the machine submerge.

Added to this, there are also devices enabling one to tow the bathyscaph on the surface,
to fill the entry shaft (by a flooding valve in the bottom) and to empty it (by means of com-
pressed air), to land on the bottom with the aid of a guide chain, and to drive it horizontally
while under water. There is exterior lighting, for detection of the sea floor and of obstacles
ahead, and means of communication with the outside world while under water or on the sur-
face, various measuring instruments, and so on. I shall give a fuller description of these
accessories in the next section concerning the FNRS-3.

THE BATHYSCAPH FNRS-3

The sphere of the FNRS-3 is made of cast steel containing nickel, chromium, and
molybdenum, with a yield point of 95 kg/mm? after heat treatment. It was built by the
Emile Henricot steel works, at Court St Etienne, in Belgium. Its inside diameter is 2
metres, and its thickness 9 centimetres, reinforced to 15 centimetres around the hatch and
porthole. It is composed of two hemispheres separated by the equatorial plane normal to
the porthole-hatch axis. The joint consists of the two metal faces bearing one upon the
other. The hemispherical parts are held together, with an initial force of 24 tons, by 400
steel clamps gripping two flanges machined on both sides of the joint.

Water tightness at low depth is ensured by a synthetic rubber ring which is fitted on
the sphere before setting up the clamps. The viewing portholes, made of Plexiglas, 150
millimetres thick, are shaped in a frustum of a cone, with two parallel planes and forming
an angle of 90 degrees at the apex. The internal and external planes have a diameter of
100 and 400 millimetres, respectively. At great depths, the watertightness of the joint
between porthole and sphere is ensured by the plasticity of the Plexiglas. The hatch is
made up of a steel frustum of a cone with two parallel planes and an angle of 45 degrees at
the apex. At the center, it has a porthole which is similar to the viewing port. This hatch,
weighing about 140 kilograms, is hinged about an axis and balanced in all positions, like a

French Bathyscaph Program 479

submarine hatch, so that it is comparatively easy to handle. Around the porthole ten transit
holes have been made in the sphere to link the inside with the outside, as follows: electric
cables (Pyrotenax cables with copper sheathing and insulated with pressure injected magne-
sia) and pipes for hydraulic fluid and compressed air.

The float which contains the lifting petrol is made of steel plating 4 to 6 millimetres
thick. It is divided into a certain number of tanks which only communicate at the bottom (to
limit the loss of petrol in case of damage). The shot silos and the entry shaft to the sphere
run vertically through the float. This float has a length of 16 metres and a width of 3.35
metres. The center part is cylindrical and the ends are in the shape of a truncated cone;
two stabilizing keels are fitted on each side of the float to improve the dynamic stability
during vertical movements and in particular in the course of an overrapid ascent. Trials on
models, undertaken at the trial tank of the Service Technique des Constructions et Armes
Navales had revealed the existence of oscillations in the course of ascents in a state of
slightly excessive positive buoyancy (all ballast jettisoned), which might have been dan-
gerous to the equipment and uncomfortable for the crew.

To conclude this description of the FNRS-3, it is necessary to mention the outside
storage batteries housed in two baths filled with oil and exposed to external pres-
sure. These batteries are used to run the propelling motors and the searchlights and supply
about 1000 ampere-hours at 28 volts. The two motors are one-horsepower each. They drive
two horizontal shaft propellers which enable the bathyscaph to be steered and moved hori-
zontally over an approximate range of 50 metres. There are six searchlights of 1000-watt
each. The incandescent bulbs which equip them are enclosed in steel cylinders which are
designed to withstand external pressure.

All the remote controls of this external equipment are housed inside the sphere. Con-
tact is maintained with the surface by an ultrasonic transmitting-receiving set, during the
diving (radio transmission only works while the bathyscaph is on the surface). An echo
sounder gives the distance from the bottom, a log the vertical speed, and a compass indi-
cates the bearing of the craft during the descent. Moreover, an air-regeneration plant
enables two men to live for 48 hours in this confined space; the carbon dioxide is absorbed
by soda lime; the oxygen consumed is replaced by a fresh supply contained in cylinders of
compressed oxygen.

THE BATHYSCAPH 11,000

The Bathyscaph 11,000 is the name we have temporarily given to the new French craft
under construction, to indicate the maximum depth for which it has been designed, that is to
say, eleven-thousand metres.

The sphere of the Bathyscaph 11,000 is made of forged steel, alloyed with nickel,
chrome, and molybdenum, with a yield point of 105 kg/mm? after heat treatment. It was
built by the Compagnie des Ateliers et Forges de la Loire in its works of St. Chamond and
St. Etienne (France).

Its internal diameter is 2.10 metres (2 metres in the case of the FNRS-3) and its
thickness is 15 centimetres (9 centimetres in the case of the FNRS-3).

The hatch, in the shape of a truncated cone, has an opening diameter of 45 centimetres
and is located at the upper part, to facilitate the inner arrangement of the sphere. Three

480 Pierre H. Willm

viewing portholes are provided; one looks forward and the other two are side portholes dis-
posed in vertical planes forming an angle of 50 degrees with the fore and aft symmetry
planes of the bathyscaph. The axes of the three portholes point downward forming an
angle of 20 degrees with the horizontal plane.

To avoid all change in thickness which would impair the aplexism of the sphere, no
stiffening has been provided in the area of the hatch and portholes. Each of these is
equipped with an optical system mounted in a 2]-millimetre-diameter hole, which is plugged
by Plexiglas shaped in a frustrum of a cone, forming an angle of 90 degrees at the apex,
and only 45 millimetres thick.

The watertight lead-throughs into the sphere for cables and pipes have been designed
similarly to those of the FNRS-3.

The float is approximately twice the size of the former. It has been designed to reduce
to a minimum the power required for towing and to increase the towing speeds. That is why
the sphere is no longer suspended below the float but it is now integrated to it, and the only
area which is now visible is that of the three portholes. It will be 21 metres long and 5
metres high and its extreme width will be 4 metres.

As in the case of the FNRS-3, the new bathyscaph will have steel shot in silos running
vertically through the float and held in by electromagnets placed at the base.

The outside batteries will be much more powerful than those of the FNRS-3. They will
be located on the after-part of the float and will be accessible, when the bathyscaph is
afloat, through an after-shaft similar to the forward one leading to the sphere. These
batteries will supply the necessary power to a 30-horsepower propulsion motor driving a
propeller with a horizontal shaft lying in the fore and aft symmetry plane of the bathyscaph.
In addition, a steering propeller with an axis normal to the fore and aft symmetry plane and
a lifting propeller with a vertical axis will each be driven by a 5-horsepower motor.

The 1000-watt pressure-resisting searchlights are still under design. All the measuring,
navigational, and communicating equipment now existing on the FNRS-3 will of course be
fitted on the new bathyscaph. The construction of this craft is financed by the French
“Centre National de la Recherche Scientifique.” It was begun in 1959 and will be completed
during 1961. After trial dives to the maximum designed depths, the Bathyscaph 11,000 will,
like its predecessor, be placed at the disposal of the Recherche Scientifique.

CONCLUSION

Although the main purpose of this Symposium is to investigate the characteristics of
high-performance ships, I think it may be of interest to conclude this paper with the scien-
tific results obtained during the first five years of the FNRS-3’s operational activities.

During this period, the French bathyscaph made more than 80 dives, in the course of
which it was possible for scientists interested in various scientific activities to undertake
their respective researches. Thus, biologists have had the opportunity of studying the
behaviour of abyssal fauna and the distribution of plankton versus depth and physical
characteristics of sea water. Geologists and sedimentologists have been able to observe
the various aspects of the sea bottom, to confirm or refute certain of their hypotheses con-
cerning the formation of underwater canyons in particular. Physicists have installed on
the bathyscaph precision recorders for measuring temperature, pressure, the pH value, the

French Bathyscaph Program 481

speed of ultrasonics and their absorption, and so on. The physicist oceanographers also
intend to measure the speed of deep underwater currents, the existence of which was not
even suspected before the first dives.

I feel that now is the time to state that it is definitely out of the question to carry out
oceanographic research from oceanographic surface ships. Man must penetrate into the
medium he wants to investigate, since he now has the means of doing so. The oceano-
graphic ship of the future will have to be submersible and in order to dive below one-
thousand metres, which is the actual limit for conventional submarines, it will also have to
be a bathyscaph. But, on the other hand, the study of the sea will only prove a poor invest-
ment if each country works on its own, without close cooperation with its neighbor. Ocean-
ography is an international science calling for an international organization. The recent
International Geophysical Year has already pooled the efforts of a great number of nations
towards this specific purpose. This effort must be pursued in the future. I sincerely wish
that the present and future bathyscaphs be designed and operated jointly by all the research
workers who are interested in oceanography. As a matter of fact this symposium is striking
proof that any real progress in the matter should spring from international cooperation. That
is why I wish to thank once more the organizers of this symposium for inviting France to
present her program concerning bathyscaph.

DISCUSSION

M. St. Denis (Institute for Defense Analysis, Washington)

Man will always try to excel himself and, having gone high, he will try to go higher, or,
having gone deep, he will try to go deeper; but fortunately in this case there is a bottom to
things. This idea of breaking records may be sufficient justification for building a bathy-
scaph, but if one tries to give a scientific reason for going deep, one must ask himself
what he is to gain thereby. When man goes deep, all he can do at great depths is to look
and to collect, and it would seem to me that he can see and collect with a lot less complica-
tion than a bathyscaph by using instruments remotely controlled from the surface. But be
that as it may, and accepting the author’s viewpoint that it is advantageous to have a bathy-
scaph to go deep to do certain things, it becomes evident that a manned vehicle is going to
grow in size because one will want to carry more and more instruments or more and more
people. And with a growth in size of the bathyscaph there follows a growth in size of the
float and eventually the vehicle is going to become bigger and bigger and costlier and
costlier. So one must ask oneself if this trend in size and in cost cannot be reversed or at
least retarded. It seems that this can, to a large extent, be done, and it can be done by the
proper choice of materials. Steel, even high-tensile steel, has not the same strength-to-
weight ratio of, say, aluminum which is slightly superior to it; and titanium is superior to
aluminum in this respect and fiberglass is superior to both. I have just a simple question
and that is this: In constructing the bathyscaph, was any thought given to the use of these
alternate materials, and if so, why were they discarded?

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a cindou pe copdedh. fev svi ry aah ta iia | serosa

A METHOD FOR A MORE PRECISE COMPUTATION OF
HEAVING AND PITCHING MOTIONS BOTH IN SMOOTH

WATER AND IN WAVES

O. Grim
Hamburg Model Basin (HSVA)

BACKGROUND

In the Symposium “On the Behaviour of Ships in a Seaway,” in Wageningen, 1957, a
paper “Durch Wellen an einem Schiffskorper erregte Krafte” was read by the author, which
will now be continued. In part II of that paper ideas were deduced which may be considered
an improvement of the well-known strip method. At that time some simple numerical results
were given on the basis of this improved strip method although the method was not entirely
completed.

In the meantime, some progress.has been made and, in addition, numerical work could
be done to a much greater extent since an electronic computor is now available. This paper
gives the results obtained starting from a complete summary of the method which is deemed
useful since the representation given in 1957 is no longer considered satisfactory.

The methods which have been applied to the theoretical treatment of the problems of the
ship waves may be crudely subdivided into two groups, viz., (a) methods of singularities and
(b) strip methods. The methods mentioned under (a) have as a basis a representation of the
ship body by periodical singularities [3,4,5]. They enable us in an elegant manner to con-
sider certain important parameters within an expression for the velocity potential and then to
examine the influence of these parameters. The weak point of these methods consists in the
loose connection between the motion and the shape of the ship on the one hand and the dis-
tribution of the singularities on the other hand. As a consequence the condition on the sur-
face of the body is hardly sufficiently satisfied. Therefore, these methods can only be ex-
pected to give qualitative information as to the influence of the main parameters and not
quantitative results as to the influence of the shape of the ship.

The methods mentioned under (b), viz., strip methods [6], assume solutions to be known
for corresponding problems on two-dimensional bodies of which the sections coincide with
the sections of the ship body. The results for the three-dimensional body will then be ob-
tained by adding those obtained from the two-dimensional bodies. It must be considered an
advantage of these methods that one can start from relatively accurate results and that the
influence of the shape of the sections can be allowed for. The well-known strip method,
however, has several disadvantages, e.g., that the influence of the three-dimensional flow

can scarcely be taken into account and that the influence of the speed of the ship cannot be
established exactly.

AQQ

484. O. Grim

In the method proposed by the author the advantages of the two methods shall be com-
bined and their disadvantages shall, if possible, be avoided. A strip method will be applied
for the distribution of the singularities; i.e., for each section of the ship the singularity will
first be chosen to satisfy the condition on the surface of a two-dimensional body of the sec-
tion in question. The distribution of singularities so obtained is not yet accurate, however,
even if an accurate representation of the two-dimensional cases is taken as a basis, because
the representation of the three-dimensional ship body requires a somewhat different distribu-
tion of singularities. An improved distribution will be obtained with the help of an integral
equation.

The method will be described here for the speed V = 0 only. For V £0 it is necessary
to complete the formulae for the flow potential and, in addition, to allow for the speed in the
boundary condition. These extensions have been worked out; since, however, numerical re-
sults are not yet available, this extension of the method will not be discussed here.

The following problems will be treated: the heaving motion, the pitching motion, and
the forces generated in a vertical direction by waves. An ideal fluid free from friction will
be assumed and, further, the problem will be linearized. It should be mentioned that the
boundary condition at the body cannot be satisfied exactly. These approximations as well
as additional simplications will be discussed in the course of the paper.

I. TWO-DIMENSIONAL PROBLEMS
Introduction

Since the treatment of the complete three-dimensional problem requires solutions of two-
dimensional problems to be known, it is necessary to briefly discuss the latter. The method
used is the same, in principle, as deduced by the author in 1953 [10] on the basis of which
later published results were computed [12]. As now an electronic computor is available it
was possible to carry out the computations both more precisely and for a sufficient large
number of transverse section contours.

Computations have been made for three problems, viz.,
(a) for the periodical heaving motion of the body in smooth water,

(b) for the vertical force generated on the restrained body by a transverse surface wave,
and

(c) for the hypothetical case of a two-dimensional body in a “longitudinal wave” which
cannot be realized physically. For this case which is important relative to the
method described under II results have been given by Abels [9].

Description of the Method

The system of coordinates is fixed in space. Its origin lies in the plane of the waterline
when at rest and in the midst of the contour, the y-axis being horizontal, the z-axis being
vertical. The computations have been carried out for transverse profiles which can be rep-
resented by the transformation formula of Lewis. On the basis of this formula the coordinates
of the profile can be expressed by means of the parameters a and b and of the coordinate 0
running from 0 to —7.

Computing Hearing and Pitching Motions 485

The formula for the boundary of the contour is expressed in nondimensional form as
follows:

(riz =) 6 ales 2° oF tbteg 2", (1)

The following formulae for the potential and for the stream function caused by a heaving
motion are applied:

foo)
Kz
=U [A, Lim | £8082 BY) wag
10 Koy te Stl

ie) fo)
n-1
+ Pa A. | Ke J (K+v) e*? cos (Ky) ck

n=1 0
(2)
oo) .
enue a im or Shisarie (Ks)
¥ | ep ” K-~v+ ip
© (0)
2(n-1) em ee
+ A, K (K+v) e*? sin (Ky) cK
n=1 0

The time factor e’®* has been omitted. U describes the amplitude of the oscillatory ve-
locity of the body. Both the condition of continuity and the condition on the free surface
are satisfied. The problem is now to define the coefficients a such that the condition on
the boundary of the contour is also satisfied. This requires a sufficient number of terms in
the rows to be considered.

The condition on the boundary of the contour for the heaving motion is

we Uy: (3)

In case (b) — the restrained body in transverse waves — a surface wave with the orbital
velocity 1 is assumed so that

486 O. Grim

evz
Oy =, Cone
(4)
iahue
WY, = > «Sin (vy).-

The whole potential or the stream function, respectively, consists of the potential of
the wave (4) and the potential (2) by which the deformation of the wave caused by the body
is described. In Eq. (2) the factor U will then be omitted. For instance:

©
ene eens ’ eX sin (Ky)
a sin (on) + A, in | eee dK

2 ©
+ peed. | Kon CK + y)e® £ ceding (Rag) ae
n=1 0

Of course, the coefficients A have different values in this case as for the heaving
motion.

In case (b) the condition at the boundary of the contour is
W= 0, (6)

In the hypothetical case (c) the basis is a representation of a transverse section of the
three-dimensional ship in a longitudinal wave. The potential of the nondeformed wave with °
the orbital velocity 1 is

Vz

Qi = = cos (vx). (7)

From this it follows that the velocity of the water particles in a vertical direction
amounts to

e”’? cos (vx) (8)

and the hydrodynamic pressure to

Vz

- ipw cos (Vx). (9)

The value of cos (vx) may be understood as the phase shift for the following and may
be omitted in Eqs. (8) and (9).

The following question may be asked: Which two-dimensional potential D(y,z) describes
such a velocity on the contour of the section of a two-dimensional body that the boundary

Computing Hearing and Pitching Motions 487

condition will be satisfied by this velocity together with Eq. (8)? This potential, for which
Eq. (2) will be applied putting U = 1, can be defined as a transverse deformation of the
longitudinal wave. At the boundary of the contour the following condition will be used:

dy. =. \e" 7 dy. (10)

The boundary conditions (3), (6), and (10) cannot be satisfied exactly (except for w = 0
or @ = 0), It has been proved that the following procedure is well converging.

Into the boundary conditions (3), (6), or (10), Eqs. (2), (5), or (2), respectively, will be
introduced. Within these equations the coordinates of the boundary of the contour can be
replaced by Eq. (1). Then only one coordinate appears, viz.,9. The equations are written
as follows:

[oo]
B. (F+ 6) + ) B, sin (2n 8) = 0, for -7 <6'S0. (1)
n=1

Of course, the coefficients A are linearly included in the coefficients B.

The boundary condition is satisfied if all coefficients B vanish in Eq. (11). In Eqs.
(2) or (5) the series are cut off after N terms so that N unknown coefficients A are included.
Then Eg. (11) is also cut off after N terms and N linearized equations are: obtained:

B = 1) B. =80: (12)

By Eq. (12) 2N equations are represented since the coefficients A or B, respectively,
are complex numbers. Solving these equations the potential is found. The boundary condi-
tion is nearly satisfied. An error remains which changes sign on the contour several times
and this more frequently the larger the N that is chosen.

Having computed the unknown coefficients A the problem is now to determine the hydro-
dynamic force. To obtain the force in a vertical direction the following integration around
the contour is required:

f@® dy. (13)

In case (a) this integration yields only the hydrodynamic force. It is possible to add
both the hydrostatic and the inertial force of the body and then to deal with the total force
which is responsible for the heaving motion.

In case (b) only a hydrodynamic force exists. Therefore the integration yields immedi-
ately the force in a vertical direction which is caused from the wave on the restrained body.

In case (c) the force which follows from the pressure in the undisturbed wave is added
to the force from (13). This is necessary since (13) yields only the force which arises from
the deformation of the wave.

488 O. Grim

Representation of the Results: Figs. 1 to 24

More convenient than the parameters a and 6b of the transformation formula are the param-
eters H and B for the designation of the section contours:

H = B/2T denotes the ratio of the half breadth of the profile in the waterline to the depth
of the profile

B = (area of the section)/BT denotes the fullness of the section profile.

Together with the transformation formula (1) the two parameters suffice to exactly define the
sections.

The following results of the computations are represented in Figs. 1-24:

For the heaving motion: R, A ,n/B, C, and A:

R=R,+ iR; represents the total force required to produce the heaving motion (real and
imaginary part). R is made nondimensional by the amplitude of motion and by 5B.

SOM

@O

Fig. 1. Heaving motion; plot of force R for B= 0.5

Computing Hearing and Pitching Motions 489

1.0
B=06

2.2
1.8
1.4
12
=< 10
SS 08

SVQ

6

j

an
YON

| ONTOS SS PSS I

2.2
18
1.4
1.2
1.0
-05
08
0.6
-1.0
04
H= 02
OM ON O4Ie = OCR SOM wanes

HEEB:
2

Fig. 2. Heaving motion; plot of force R for B= 0.6

Therefore, in the statical case, i.e., a =0, R equals 1. For a periodical heaving
motion, viz.,

z=Z eit (14)

fo) fo)

the force which is required to generate the heaving motion is

RigiBeZ: e7 2, (15)
This force is identical with
Cnt) Czar Ni) 2 + Bz. (16)

This form is known to be the one side of the equation of motion. It contains: the hy-
drodynamic force m"z,+ NZ,, the hydrostatic force 5Bz,, and the inertial force mz,.

A,7/B is the coefficient of the first term in (2) in nondimensional representation

490 O. Grin

0.6

0.4

yv:2
2

Fig. 3. Heaving motion; plot of force R for B= 0.7
C represents the hydrodynamic mass in nondimensional form, i.e.,
m
Tp? (17)
8 B

A represents the ratio of the amplitudes, i.e., the ratio of the amplitude of the surface
waves drifting away from the body to the amplitude of the heaving motion. Between A
and R; the relation

R, = pee aie
w*B

(18)

holds because the work done by the force in unit time equals the energy dissipated by
the waves.

For cases (b) and (c), viz, the restrained body in transverse or longitudinal waves, respec-
tively, the following results are represented:

Computing Hearing and Pitching Motions 491

B-08
WN 2.2
1.8
OS OSS 4
SSS OSS :
—a ON :

ZB ANSE 1.0
wy 7 08
ASS

4
A Tt SSS

2.2
1.8
-0'5
1.4
1.2
1.0
0.8
0.6
=1.0 0.4
H =0.2
(e) 0.2 0.4 0.6 0.8 1.0 1.2 1.4
V-B
2

Fig. 4. Heaving motion; plot of force R for B= 0.8

E = E, + E; which is the force in the vertical direction generated on the body by a sur-
face wave of amplitude 1. This force is made nondimensional by 5B. In the case of a
very long wave (w > 0) E necessarily equals 1. For this case the force can be looked
at as the hydrostatic buoyancy which corresponds to the additionally displaced volume.
This force has first of all a physical meaning only for the case (b).

For case (c) there is also plotted

A ,7/B which is the coefficient of the first term in the formula for the potential in a non-
dimensional representation. (The diagrams for case (c) are not given here since they

are published in Ref. 9.)

All results are plotted against a frequency parameter which includes the beam B, viz.,

B 2B
3 os . (19)

In cases (b) and (c) this frequency parameter can also be expressed by the wavelength

492 O. Grim

1.0
\
B=09
05
fo)
22
-05
1B
H=\0.2 04 \o6 \os \to \L2\14
22 Fi
7 18
Os H=1.4
-03 lon
08
-0.2 g 06
P
()
ps | ar ies ee SS
0 C204 OG mOS (ome 1.4
V-B
2

— = ae (20)

With the help of these results the heaving motion of the free body generated by trans-
verse surface waves may be determined. If the radius of the orbital circle of the surface
wave is denoted h and the heaving motion produced by the surface wave z,, the following
equation must be satisfied:

z OR S/n Fh (21)

From this the heaving motion of the free body can be computed both in magnitude and phase.

Convergence of the Method

To prove the convergence, first four terms in the series for the potential and then five
terms were taken into account. The difference of the results is not significant. It, therefore,
can be expected that both the convergence and the accuracy of the method are sufficient.

493

Computing Hearing and Pitching Motions

S°0 =9 Joy g/u°p wornqInstp
Ayren3uts yo yord ‘uorjow Suravay *L “31

80 O1 2! 1

O'L =¢ 10} Y 040] Jo yOTd ‘uotjow Buravayy *9 “31y

81

c2

ie)

10)

33

646551 O—62

L'0 = 10} g/u°y voLNqinstp 9°0 =9 10} g/u°y UorInqIstp
Ayizepn3uts jo 307d fuopjouw Juraeoyy *g “31y

Aqtzenguts jo jojd fuoyjow Sutavey “6 “SIy

v | ra Ol 80 90 vo CHO) (0)

Ol Ziv 178 I" 22=H

O. Grim

494

495

60 =9 10} g/u°p uorngiasip 8'°0 =9 10} g/u°p uormgqiusip
Aqrzepn8uts jo yord fuomow Buraeay *T][ “Sty Aqtre[n3uts jo yo,d ‘uomow Buraeay “OT “9Ty
eons 4

aa aa

Computing Hearing and Pitching Motions

O. Grim

496

G°0 =9 10}
V Jo pue + jo sioyd ‘uorour SuLAva “ET “S1q

re)
20
140)

OT =_ 10} g/v°p uorinqrustp
AyizepnSuts jo j0;d ‘uotow sutaeoyy “ZT “Sty

v | 2} Ol 80 90

a ee ed
SSS Se

sees
aa

————

SS Sa

ee ee

(0) 20 oe
L) 0
rate)
£0
vO
iox(0)

497

Computing Hearing and Pitching Motions

Ae)

90

80
Ol
Zl
vl

8'|
ee

L°0 =9 403
¥ Jo pue 2 jo sjo[d ‘uorjoul Suraeoyy “oT “31y
BS
aa
ral ral ol 80 90 v0 re) %
=H
rx)
vO
g 90
80
v Ol
Ci
aa
tl rl Ol 80 90 v0 zo O%
Ae)
PK NYO
eee) (oR)
= —— 30
———
=
01
sf 21
20 =
bl
gi» |
Z2:H

a

OF Oa no
FV JO pue + jo siojd fuonjow Suraveoy ‘py *3ty

Ge cre
EV Jo pue 2 jo sjod fuonou Suraeey “ZT “Sty

2
‘ : : ay ell ;
bl ZI Ol go * ? 2 <2),
a=
20 i
26
== vo
|
= ; “Ee

90

a
5 ae

O. Grim
T
N
je)
@
fe)
(do)
o
Tt
to}
N
S

°

so

|
|
“3 S

498

BO menace}
¥ Jo pue + jo siojd ‘uonow Suravay “oT “Sty
ces
aa
vl i) O1 80 90 vO rile) 05
Se ae
Aye)
Ae)
vO
vO
90
90
80
80
(ol
Vv
oy 7 Ol
v|
8'| 4
222H— i
Al Cal Ol 80 90 Ae) Cae) Os
£0 —c0
I —~ 7~¥0
ee ae,
——= = 390
ee BO
go-¢g a re
cilne
bis |
81
2?=H

499

Computing Hearing and Pitching Motions

S°0 =9 103 O'I =F 103

gq e010} jo yo]d ‘eaem asioAsUeI], “6] “B1yq Y Jo pue + jo siojd ‘uonouw Buravoy “gt “Sly
eee
dle ek Hime ete) O90 +
a | 80 9 = vO a, 95
cO
pe Ke)
aac
vO cO
30 £0
gO ze
OT Ao
All 90
al v ie)
81
80
Z2=H 2o=
aa 60
v1! al oO} 80 g0 vO rae) 0

sO

ait WN 01

vO S|

20O=H

L'0=9 403 9°0 =¢ 10}

gq 9010} Jo Jo7d ‘aaem assJoAsSuBIy, “TZ “ST q 20103 Jo JoOTd ‘aaem asiaASUeIT, “OZ “ST
ae, nem
[0/4 aa
val All Ol 80 90 vO Xe) % al Z| Ol 80 90 vO 20 O%
=a Saas
iif) 0)
cO=H Z20=H
cO co
120) Z a6
= A
=—ZeA vO
—=——- SO
i |
zz sl bl zl Ol 80 2) lle

O. Prim

500

501

Computing Hearing and Pitching Motions

all

ral}

Ge 3G!
q e010} Jo jo[d feaem asioasuely, “EZ “3TY

ace

a2

2 Ol 80 90 vO 20
St

Ae) Loo

90

8'0

pl=H S81 22

8°0 = 10}
q 2010} yo Jo[d ‘aAem oesioASuUBI], °Z7Z ‘S1q

502 O.. Grim

Fig. 24. Transverse wave; plot of force E
for B= 1.0

Summarizing Remarks to Part I

It is certain that sufficiently exact solutions are obtained in the way here described.
Recently, a Japanese study by Tasai [7] has been published in which numerous numerical
results are presented which agree closely with those given in this paper.

It should be mentioned that the method may be applied in the whole range of frequency
from w = 0 tow =co. It appears necessary, however, that the sections are amenable to con-
formal mapping. In this respect difficulties may arise for sections of which the contour is
not perpendicular to the waterline at their intersection.

The first term in Eq. (2) for the potential represents the potential of a periodical source
iny=0, z=0. The coefficient A,, therefore, represents a measure for the strength of the
source. It is now extremely remarkable to learn from Figs. 7-12 that the coefficient A, is
complex and depends very strongly on both the shape of the section and the frequency. It is
therefore a very crude and inaccurate approximation if only the shape of the waterline and

the amplitude of the motion are considered for a method of singularities when choosing the
singularities.

Computing Hearing and Pitching Motions 503

Il. THREE-DIMENSIONAL PROBLEMS
Heaving and Pitching Motion in Smooth Water

The motion of the body is determined by the velocity U,(x) in a vertical direction. This
velocity is constant and independent of x relative to the heaving motion. Relative to the
pitching motion this velocity grows linearly when x increases. The time factor e!?is

omitted.

In Ref. 11 the formula
®(x,y,z) = 3 ae i UCLA Ge) 0. (x- €).y.z] dé (22)

is chosen for the flow potential. The functions 9,[(~—€),y,2] within the integrals are de-
noted partial potentials and are expressed as follows:

0, [(x- Jz

co (oa)
Vm? +K* z
= A) cos |[m(x - €)| in | SUMO EL ACOs) ak for 0
0 0 fm? + K%* - vy + ip
(23)
Q, [(x- &.y.2]
2(n-1
ape | : } 322 ~ toyz
ea [(x- €)? + y2 + z2| ae [(x- €)? + y* + z2| dis

This formula satisfies both the condition of continuity and the condition on the free
water surface outside the ship body. When applied to the limit case of an infinitely long
two-dimensional body this formula is identical with Eq. (2). The transition to the limit is
performed as follows:

1. The distribution functions A,(x) become constants A, which are placed before the
integrals.

2. These coefficients A, are chosen such that they are identical with those computed
in part I.

3. The integrations are carried out from —~ to +.

The formula furnishes a nearly exact solution in this limit case. Relative to a ship of
finite length the distribution functions A (x) are chosen identical at each section x with
those distributions 4 (x) which hold for the corresponding motion of the two-dimensional
body. In Eq. (22) there are, therefore, no unknowns. It may also be expected that the

504 O. Grim

condition on the surface of the body is satisfied to a sufficient degree as long as the ship
body is sufficiently slender, i.e., as long as L/B and L/T are great and the angle between
the tangent on the waterlines and the plane of symmetry is small.

Relative to the methods of singularities an improvement is obtained because the rela-
tion between the singularities, the frequency and the contour parameters is satisfied to a
higher degree.

To better recognize the improvement, Eq. (22) is transformed into the identical formula

+0

Wx. z= 4h) Ux) 4,00 { 0, [(x-).y.2] dé
n=0

- ©

+| {ce AE) - U(x) A(x) 9, [(x- 8.2] dé} . (24)

The first term represents the nearly two-dimensional flow around the section x because
the integrals are independent of x. Only in consequence of the functions A (x) a velocity
component in x-direction arises. By this velocity component, however, the boundary condi-
tion is not considerably disturbed and it is nearly satisfied if the distribution functions are
correspondingly chosen. Only the first term is taken into account for the common strip
method. If the hydrodynamic pressure and the force in a vertical direction generated on a
section are computed only from this term the same force as in part I is obtained. Applying
the representation R used in part I one obtains

5
< i) U, BR dx (05)
L
for the total force to generate the motion with velocity U, or for the total moment
° ( xU, BR dx
w fo) y (26)
L

Here it is assumed for the computation of the moment that the weight of the body is
distributed over the length in the same manner as the displacement. Of course, a different
distribution of the weight can be taken into account.

The common strip method will be improved if the second term in Eq. (24) is also ap-
plied. This improvement is already included in Ref. 11. An additional improvement is dis-
cussed in the following considerations.

The second member in Eq. (24) may be explained physically as a flow generated at x by
a distribution [A (€) — A ,(/| situated outside x. Since this distribution lies in y = 0, z = 0
and since the sia body is slender (i.e., the distribution does not change too rapidly with €,
this additional flow at x will depend only a little on y inside the ship body, i.e., for

Computing Hearing and Pitching Motions 505

|y| $ B/2. Only a small error arises if in this second member of (24) 9 [(x-€),0,z] is writ-
ten instead of 9 ,L(x—),¥; z]. Since it is well-known that circular waves are generated by a
periodical singularity of which the dependence on the coordinate z is described by the fac-
tor e”? (provided that the distance from the origin of the circular waves is not too small) it
is suggested that the same dependence on the coordinate z holds for the second member of
Eq. (24). For all singularities within this member have the same frequency and there are no
singularities at x itself. It, therefore, may be permitted to further simplify the partial poten-
tials in this second term, viz.:

0, ((x- €).y,2] + e”* 9, [(x- €),0,0 (27)

In this way the coordinates y and z are removed from the integrals. This term can then
be described as a product of a function of x times the exponential function e””. This holds
within the region of the ship body, i.e., for ordinates y which do not go far beyond B/2.

Since the condition on the surface of the body is well satisfied by the first member of
Eq. (24) the second member causes a disturbance of this boundary condition. To reduce the
error an additional velocity U as function of x will be introduced. U may be explained as the
velocity of an additional deformation of the water surface (depending on x and z for |y| $ B/2)
so that a certain section of the ship body has the velocity (U, + U) relative to the water sur-
face. This deformation is called “additional” because a deformation (depending on y and z)
takes place already in the two-dimensional case. The additional deformation diminishes when
the depth increases following the function e”* because the influence of singularities with
circular frequency @ lying far outside of x will be eliminated.

Therefore, the following formula instead of Eq. (22) will be chosen for the total poten-
tial:

wWx,y.2) = 4) if U4, (2) + UA, CE) 0, (x- 2.9.7] ac (28)
n=0 L

Two different distribution functions 4, , and A, . are introduced. The second indices
of these functions denote the functions belonging to cases (a) and (c) treated in part I.
Since the additional deformation U decreases when the depth increases, the distributions
corresponding to the two-dimensional case (c) of the body “in longitudinal waves” will be
chosen for this part since the motion of the water particles decreases with the same expo-
nential function. This formula has been transformed in the same manner discussed previously
into the identical formula:

1 2 , +0
O(x,y,2) = 5 a” {uc 454609 | On [(x- €),¥.2] dé

n=0

+00 +a

On ((x-€),y.2] dé | U.ce4, (ES)

- @

U(x) A(x) |

-@

+ UE) A, CE) - Ux) ACD - Ux) A, CO] On (X-O).Y2 ae}. (29)

506 O. Grim

Again, as mentioned before, the common strip method results if only the first row is
taken into account. The second row, again, represents essentially a plane flow around the
sections.

To carry out the computations a further simplification, besides the one already men-
tioned, is applied for the third row. In this row only the member for n = 0 of the series over
n members will be considered. The reason is that this member causes the greatest long-
distance effect since the functions 9, fade away at a greater distance from the origin of the
disturbance more rapidly the greater n is. Besides, the potential of a periodical source is

described by 9, and only by this term circular waves, which transport energy, are described.
Therefore, instead of the third row in Eq. (29),

Vz

+00
wiEB | [U(E) A, (CE) + UE) ALE) - Cx) ASC

- U(x) A,(x)| 9, [(x- 4).0,0] d& (30)
is written and, the integral being a function of the coordinate x only, this row can be written
e’?2 F(x). (31)

The next problem is to determine U such that the condition on the surface of the body
is satisfied to a sufficient extent. This means that the influence of the second member in
(24) on the boundary condition is approximately eliminated.

A simplification is introduced relative to this boundary condition; viz., the velocity in
longitudinal direction is neglected so that

ae ae - By = U, dy
S

Oy OZ (32)

where S = the surface of the ship body.

The potential (29) is introduced into the left-hand side of this simplified boundary con-
dition. This can be done without difficulty since solutions are known for the corresponding
two-dimensional cases.

; o® dz + 2 ay] =U dy + Ue dy. + yer EC x)idye
Ss

“oy (33)

The sum of these members equals U, dy from Eq. (32). The simplified boundary condi-
tion leads, therefore, to the following equation for the unknown function U:

U(x) + vF(x) = 0. (34)

Computing Hearing and Pitching Motions 507

The application of the hypothetical two-dimensional case (c) is again confirmed by the
appearance of the function e”? in the second and third row of Eq. (33).

In the equation for U, only the coordinate x appears. This equation is an integral
equation.

Having determined U the task of determining the hydrodynamic forces remains.

Having satisfied Eq. (34), one may write for the potential instead of (29)

+0

ce cr
0 = 5U, 2 AOD | on x= e.¥.2] de
n=0 -@

© +0

gc) ay ie cal P(e Seyi y nz Orde wil (35)

This expression does not satisfy the continuity condition any longer and is, therefore,
not exact. However, the expression may be used for coordinates which correspond to the
surface of the ship. For these coordinates this equation certainly represents a sufficient
approximation. The computation of the hydrodynamic pressure and the integration of this
pressure over the contour of the section in transverse direction to determine the force act-
ing in a vertical direction has already been carried out in part I.

From the first member in Eq. (35) one obtains as in the common strip method:

5— UR. (36)

@

From the second and third member in (35),

B
6— UE. (37)

@

for, the second member in (35) describes a “longitudinal wave” at x of orbital velocity U
and the third member the additional two-dimensional potential computed for (c) in part I.
E , represents the nondimensional parameter of the force which is related to (c) of part I
and which is generated by the “longitudinal wave.”

The total force in a vertical direction on a section of a ship body amounts then to

B
On [UR =F UE | (38)

The second member represents the correction to the common strip method. U, R, and E are
complex functions.

508 O. Grim

The force and the moment for the whole ship body are obtained by corresponding inte-
grations of Eq. (38) over x.

The Force Generated by Waves on the Ship Body

To compute the force it is assumed first that the ship body is restrained. The generat-
ing surface wave is supposed to run in longitudinal direction from fore to aft. The method
can, however, also be applied for all running directions.

As compared to the Froude-Kryloff method an essential progress would be obtained if,
instead of the quasi hydrostatic force computed from the nondeformed wave, the force E .
computed for case (c), viz., the ship in a “longitudinal wave” is introduced.

A different method is proposed in Ref. 11. Also this method could be improved.

The formula for the potential of the nondeformed surface wave of orbital velocity 1 is
as follows, omitting the time factor e’®*

® = EY, (39)

The following formula will be used for the total potential of the wave and deformation
caused by the body:

Bem fA OW oy ee

This potential still exactly satisfies both the continuity equation and the condition on
the free water surface. G(x) represents a presently unknown complex function in a similar
manner as U(x).

Using the same simplifications as before, which are deemed permissible relative to
points on the surface of the ship, Eq. (40) is simplified to

: evz 1 = bas
© ~ eivx { eta G(x) A (Cs) i om (x- ).y,2] dé
n=0 - 0

Ls | EV es COrZa©®

= G(s) AUR @s) | one 1))0.0) ts| (41)
The last member in this equation can be written as follows:

Coe sHiGx) (42)

Computing Hearing and Pitching Motions 509

since the integral in this row is a function of x only.

The following simplified condition on the surface of the ship body S will be used:

ae +o w| =a.
Ss

oy Oz (43)

The right-hand side of this boundary condition is zero since the body is assumed to be re-
strained and since the whole potential will be introduced. The required computations have
already been carried out or can easily be carried out. The boundary condition furnishes the
relation

e”? dy + G(x) e”* dy + vH(x) e’* dy = 0 (44)
or the definition equation for the unknown function G(x), viz.,
1 + GCx) + vHCx) = ©. é; (45)

This is an integral equation for the unknown function G(x).

After having determined this function G the next problem is to determine the exciting
force.

Into the formula for the potential H(x) will now be introduced

: ev? 1 - ous
® ~ ei’* G(x) ‘ 5 i 5). Ax) | Pn [(x- E).y,z| c} : (46)
n=0 21

Although this formula does not yield an exact value for the potential, it gives a very
close approximation for coordinates of points at the surface of the ship.

The hydrodynamic force per unit length in vertical direction has, for this expression,
already been determined in part I under (c). This force amounts to

ee (47)

The result Eq. (46) may be conceived a two-fold deformation of the longitudinal wave
caused by the ship body. The first member, viz.,

age: er ztix)
7

(48)

may be considered the potential of a longitudinal wave with variable effective wave ampli-
tude of which the reduction factor relative to the oncoming wave amounts to —G(x). This
646551 O—62 34

510 O. Grim

member represents, therefore, a deformed wave which, in the region of the ship body, is in-
dependent of the coordinate y. The second member in (46), however, mainly represents a
deformation of the wave which depends on the coordinate y. It may be convenient, although
not quite exact, to denote G(x) an effective wave amplitude.

Free Moving Ship Body in Longitudinal Waves

All hydrodynamic problems relative to the free moving ship body are already solved in
the foregoing. considerations since a linearized treatment is assumed.

Some hints will now be given how the motions are determined. All forces or moments,
respectively, may be found by an integration of the forces per unit length over the length of
the ship body. The exciting forces are found by an integration of Eq. (47), the restoring
forces by an integration of Eq. (38). The results for the nondimensional heaving and pitch-
ing motions (made nondimensional by the wave amplitude or by the amplitude of the wave
angle, respectively) are as follows:

A, || BCR + UE) dx + | xB(xR + ULE.) dx = | BG eee dx (49)
E L L
and

z, | xB(R + U,E,) dx + vf xB(xR + UE.) dx =| x BG e*”* E dx. (59)
L L L ’

where E, G, R, and U are complex functions of x. The values of U are nondimensional
values of the function U(x) which follow from Eq. (34) for the case U,,, = 1 or U,p = *, re-
spectively. The indices H and P indicate that the equation for U is satisfied either for
heaving or for pitching.

Using these values for the amplitudes of the motions two interesting functions* can be
determined, viz., the resulting wave deformed in a longitudinal direction:

Ge*”* - z.U,- WU, (51)
referred to a space fixed zero-line or
Ge 7s iz .U = ie iz ex (52)

referred to a zero-line moving with the ship.
The function (51) appears for the case of the free moving ship body instead of the func-

tion Ge'¥* for the case of the restrained body. Taking the amplitude of the function (52)
this function may be conceived as envelope of the wetted ship surface.

*Of physical significance.

Computing Hearing and Pitching Motions 511

Resistance

Since the deformation of the longitudinal wave has been determined one could try to
compute the mean force generated by the wave in direction of the negative x-axis, i.e., the
resistance WV. However, it is necessary to simplify the distribution of the pressure for this
purpose.

(52) G6" 2, Uy - pUp- 20-92

For the free moving ship body (see sketch) the contour of the surface of the wave is
given by (51) relative to a line fixed in space and by (52) relative to a zero-line moving
with the ship. An element of the ship body of length dx, bounded by two sections, displaces
the volume:

V = [BT6 + B(Ge*”* - 2,0, - yU, - z, - Yx)] dx. (53)

The simplifications now are introduced, when computing the resistance, that the magni-
tude of the buoyancy force equals SV and that its direction is perpendicular to the contour
of the surface of the wave. These assumptions lead to the following formula for the force
in the direction of the negative x-axis:

pe 5 J [BT 6+ B(Ge*”*- 2,U,,- WU, ~ z,-x)] d(Ge*”*- z,U,-yU,). (54)

Only the time mean value of this force is of interest, viz.,

y= — P dt: (55)

The first member in (54), viz., BBT does not contribute to this mean value. The re-
mainder may be written as the following sum:

W= HtoW,

We ee sface, = We) pAGGers cos z Uy - rm)
M

512 O. Grim

The index M indicates that in both cases the mean value over the time is to be taken.

The first part can be looked at as caused by the reflexion of the wave on the ship body
and the second part by the phase shift between the motions of the ship and the wave. This
representation of the resistance is known [1,8] and it now appears possible to compute both
parts of the resistance.

Example

The results for an example which have been computed from the method discussed be-
fore will now be given in detail. The ship body chosen is as follows:

L/B = 6.4; B/T = 2.8; T = constant for all sections; B = 0.9, constant for all sections;

section
weer (for 10 sections) = 0.33, 0.70, 0.90, 0.99, 1.0, 1.0, 0.99, 0.90, 0.70, 0.33.

max

This ship body is symmetrical about x = 0. The 10 given sections lie in the midst of
10 equally long intervals.

Each integral equation (34) or (45), respectively, has been transformed into a system
of linear equations and has then been solved. The 10 given sections were chosen as sup-
porting points for this system of equations. Each system of equations contains, therefore,
the 10 complex values U_, or Up or G, respectively, of these sections as unknowns and,
therefore, three times a system of 20 equations for 20 real unknowns had to be solved.

Figures 25-31 show the results in a nondimensional representation.

Figure 25 shows the deformation of the smooth water surface caused by the heaving and
pitching motions. This deformation would not exist for an infinitely long body. The veloc-
ity of the deformation represented on the diagram is made nondimensional by the velocity U |
of the motion and the velocity of the bow is chosen for the pitching motion, i.e., U, at
x =L/2. Further X/L is chosen a parameter. The ratio \/L is defined by the relation

since the wavelength A has in this case no physical meaning.

Figure 26 gives the deformation at the point x = 0 which arises from the heaving motion,
plotted against the frequency. This diagram is presented to show the course of U,,/U, for
«0. From this course it follows that the virtual mass does not go to infinity for w > 0,
as it does with the common strip method. For the virtual mass, the virtual moment, the
damping force, and the damping moment about the same values are obtained for w > 0 as
given in Ref. 11.

Figure 27 shows for the section x = 0 the coefficient C of the virtual mass and the hy-
drodynamic damping force both for the two-dimensional and for the three-dimensional case.
The latter is computed on the basis of the former by means of the additional velocity.

Computing Hearing and Pitching Motions 513

HEAVING MOTION
REAL- PART IMAGINARY - PART

PARAMETER X/L

g 2
WL De
Mins
Pe XIE ya a ae = a Ae
-05 -025 ) 025 05 -O5 -025 ) O25), 105
PITCHING
MOTION
Up
Upx=L/2 fe}

Fig. 25. Heaving motion and pitching motion

HEAVING MOTION

0.2

Ol IMAGINARY - PART ~,.

REAL - PART

A/L

-0.2

Ke) 0.75 0.6 OS

3.0 |

(Ce eae => B .[-tn-(1.783- +1)-in] =

o cles = e ot aoa
313 7 [ Ine (Hl.2 Xr?) in|

I
(eo)

Fig. 26. Heaving motion

In Fig. 28 the amplitude of the motion of the sections of the ship body in vertical direc-
tion, viz., z, + w, is plotted, made nondimensional by the wave amplitude A.

In Fig. 29 the amplitude of the deformed wave is presented, made nondimensional by
the wave amplitude. The upper diagram holds for the restrained body and the lower diagram
for the free moving body. The amplitudes of the wave deformed in a longitudinal direction
are, as is expected, less reduced relative to the undisturbed wave in the case of the free
moving ship than of the restrained ship body. These differences are, again as is expected,
smaller in the case of small wavelengths than of large ones. The enlargement of the wave
amplitude for A/L = 1.0 to 1.5 in the force part of the ship body in the case of the free mov-
ing ship is striking.

514

AMPLITUDE OF VERTICAL MOTION MADE NONDIMENSIONAL

O. Grim

HEAVING MOTION

VL

12 UB

2
Fig. 27. Heaving motion; coefficient C and force
R,; for the section x=0, two- and three-dimensional

Z, +¥X Zo+¥x

1.50

1.00

0.50

(e)
-0O5

[m/sec?]

/)
\ [1.50

0.50

a ;

oO +0.5
X/L

Fig. 28. Vertical motion of the sections of the ship

ACCELERATION FOR

515

Computing Hearing and Pitching Motions

O/T =V/ 4% JO Seaem soy drys ay3 Jo yISus]
ey Aq [BUOCISUSUIIpuOU Spell sdeByMs aAeM JY} pues Apoq

ay} UseMjeq UOT}OW BVATIB[aI ay Jo apnriitduy ‘og ‘3Iq

so -
0

10'0

£00

epnitjdue eaem pouloyapun ay} Aq [euoIsuoutp

-uoU opel dABM poulloyep ay1 Jo apnitjduy ‘67 “314

SOt+t

1/XK
(e)

00°2 = 1/X
AGOs SNIAOW -33y4

AGOs G3aNivyilssay

sO -
lo}

Ae)

140)

90

80

Ol

Zl

516 O. Grim

W/A

0.002

0.001 }—

My

075 1.00 1.25 1.50 (75 2.00

Fig. 31. Resistance arising from reflexion made
nondimensional by the weight of the ship for waves of

2h/A = 1/40

Figure 30 deals with the amplitude of the relative motion between the deformed wave
and the ship body. This amplitude is not made nondimensional with the amplitude of the
wave but with the length of the ship. The ratio of the wave height to the wavelength is
taken 1/40. This representation has been selected because the envelope of the plotted
curves shows to which maximal degree the water surface can move relative to the ship body
for all waves of the ratio chosen.

Figure 31 shows the ratio of the resistance to the weight of the ship for a constant
ratio of wave height to wavelength of 1/40. The first part of the resistance only arising
from reflexion is represented.

Calculations for this example were previously carried out for a subdivision of the ship
body in 8 sections and applying a quite different program for the computor. The results of
both the computations closely coincide for both U and G, i.e., for the deformation of the free
water surface in consequence of the heaving and pitching motion and for the deformation of
the wave in the case of a restrained ship body. In the case of the heaving and particularly
of the pitching motion somewhat greater values have been obtained with a subdivision into
10 sections. These deviations, as checking computations have shown, are mainly caused by
the different subdivision of the ship body. It can be stated on the basis of this BOERS OL
that the accuracy of the calculation is sufficient.

ACKNOWLEDGMENT

This paper has been spensored both by the Deutsche Forschungsgemeinschaft and the
Office of Naval Research.

Computing Hearing and Pitching Motions

NOMENCLATURE

X,Y,z
Bi

@

r

=!

EBT

coordinates

same coordinate as x (used only within integrals)
circular frequency

wavelength

wave amplitude

2a 27
MRO? ON:

density of water
specific weight of the water
length, beam and depth of the ship

beam of a section (if used within integrals)
fullness of section

heaving displacement

angle of pitch

velocity potential

partial potential

517

coefficients of singularity distributions in the two-dimensional case and dis-

tribution functions in the three-dimensional case
amplitude of the velocity

velocity of a section in the three-dimensional case

velocity of the water surface in the three-dimensional case which arises from

the finite length of the body

amplitude of the wave deformed by the restrained body

mass per unit length
virtual mass per unit length

coefficient of the hydrodynamic damping force

518 O. Grim

R the total nondimensional force which is necessary to produce the heaving mo-
tion in the two-dimensional case

E nondimensional force excited by the wave in the two-dimensional case
C coefficient of the virtual mass

A ratio of amplitudes

REFERENCES

[1] Havelock, T.H., Trans. Inst. Naval Architects 86 (1945)
[2] Havelock, T.H., Trans. Inst. Naval Architects 97 (1956)
[3] Havelock, T.H., Trans. Inst. Naval Architects 99 (1958)

[4] Hanaoku, T.S., “Proceedings of the Symposium en the Behaviour of Ships in a Seaway,”
Wageningen, 1957

[5] Eggers, K., Ingenieur Archiv, 1960
[6] Korvin-Kroukovsky, B.V., and Jacobs, W.R., Soc. Nav. Arch. Marine Engrs., 1957
[7] Tasai, F., J. of Zosen Kiokai, July 1959
[8] Kreitner, H., Trans. Inst. Naval Architects 80 (1939)
[9] Abels, F., Jahrbuch der Schiffbautechnischen Gesellschaft 1959
[10] Grim, O., Jahrbuch der Schiffbautechnischen Gesellschaft 1953

[11] Grim, 0., “Proceedings of the Symposium on the Behaviour of Ships in a Seaway,”
Wageningen, 1957

[12] Grim, O0., Forschungshefte fur Schiffstechnik 1957, S. 99

DISCUSSION
John V. Wehausen (University of California)

The following remarks are supplementary to that part of Dr. Grim’s paper which deals
with the heaving motion of two-dimensional bodies. However, they are also supplementary -
in an important sense to the whole paper, for they deal with the applicability of theory to
experiment. The work described below was carried out by W. R. Porter as part of a doctoral
dissertation at the University of California, Berkeley.

Computing Hearing and Pitching Motions 519

The aim of this work coincided in some respects with that of Dr. Grim in that we
wished to obtain information for forms of shiplike section which could be used later in com-
putations for three-dimensional bodies by a strip method or some modification. However,
there was a further aim. Experimental evidence confirming the applicability of perfect-fluid
theory with a linearized free-surface condition to oscillatory motion of a body in a real fluid
with a free surface seemed to be very scarce, and we felt that more was needed. In order to
make a comparison of theory and experiment, configurations were needed for which the the-
oretical calculations could be made with controllable accuracy and for which the correspond-
ing experimental measurements could be carried out with available equipment. Two-
dimensional shapes had several advantages for the experimental work. Furthermore, it was
known how to generate by conformal mapping of a circle families of shiplike sections, the so-
called Lewis forms, Landweber forms, Prohaska forms, etc. In addition, Ursell* had al-
ready carried through computations of added mass and damping coefficients for a circular
cylinder by a method which seemed likely to lend itself to further generalization to include
the forms mentioned above. This turned out to be the case. Further modification allowed
the inclusion of a horizontal bottom.

In order to have a more sensitive test of the perfect fluid theory, it was decided to com-
pute and measure the pressure distribution around the cylinders as well as the added mass
and damping coefficients. The computations were programmed for and carried out on an
IBM 704. We believe them to be accurate to within 0.5 percent.

At the time this work was carried out we did not know that Dr. Grim was also engaged
in similar computations for infinite depth by the method which he had proposed much earlier
in 1953. However, Tasai’s paper in J. Zosen Kyokai 105:47 had come to our attention, after
this work was well under way, and we recognized that the two approaches were identical.t
However, since Tasai had not included the pressure distributions, in which we were espe-
cially interested, or finite depth, we decided to continue with the computations. Although
Porter’s computations show some small discrepancies with Tasai’s, they confirm them gen-
erally, as well as those of Dr. Grim.

The experimental measurements have at this time been carried through only for a circu-
lar cylinder (10-inch radius). However, other shapes, especially U-shaped and bulbous sec-
tions, will be tested later. In carrying out the force and pressure measurements a retreat
was necessary in one respect. Although, a phase resolver had been designed, constructed,
and tested, it was not feasible during the first set of experiments to make use of it because
of excessive noise in the signal, a difficulty we are confident of being able to overcome
later. As a result, the experimental points shown in Figs. D1 and D2 are for the amplitudes
of fluctuation of the total pressure and force. For these the agreement between theory and
experiment seems to be very satisfactory. Although, extremely square U-sections may show
greater deviations, it seems likely that perfect-fluid theory can be used to give adequately
reliable predictions of the motion of stationary oscillating bodies.

A detailed description of this work may be found in the report of W. H. Porter, “Pres-
sure Distributions, Added-Mass, and Damping Coefficients for Cylinders Oscillating in a
Free Surface,” Inst. of Engrg. Res., Univ. of Calif., Berkeley, Series No. 82, Issue No. 16
(July 1960).

*Quart. J. Mech. Appl. Math. 2:218 (1949).

t An English version of Tasai’s paper (Rep. Res. Inst. Appl. Mech., Kyushu Univ., Fukuoka, Japan,
8(No. 26):131 (1959)) was later brought to our attention, and presumably renders our translation of
the Japanese version (Inst. of Engrg. Res., Univ. of Calif., Berkeley, Series No. 82, Issue No. 15
(July 1960)) superfluous.

520 O. Grim

(0) 0.5 1.0 1.5
OSCILLATOR FREQUENCY (CPS)

Fig. Dl. The calculated and measured amplitude of
total pressure fluctuation

NORMALIZED VERTICAL FORCE

0 0.5 1.0 1.5
OSCILLATOR FREQUENCY (CPS)

Fig. D2. The calculated and measured
total vertical force

O. Grim

Prof. Wehausen’s report about the work carried out at the University of California by

W. R. Porter must be welcomed. I have seen this work only recently and did not know about
it before the symposium. Indeed this work represents a helpful contribution to the problems
dealt with in my paper. It is important that the mentioned experiments will be continued be-
cause only a few experimental results with respect to the forces on oscillating two-dimen-
sional bodies are known. Such experimental results, however, are needed. Tasai, too, has
carried out such experimental work. Since no severe differences seem to exist between the
known theoretical and experimental results for the two-dimensional cases, we can be

Computing Hearing and Pitching Motions 521

encouraged to continue the theoretical work for the three-dimensional or any other more com-
plicated cases. [ am very pleased with this contribution.

P. Kaplan (Technical Research Group, Syosset, N.Y.)

We have received in this rendition by Dr. Grim, quite an outstanding piece of work in
the sense that it covers many problems that have been considered in the past 10 years from
the point of view of ship motion analysis. Dr. Grim, in the past, has provided the largest
amount of available data for use in the predictions of motions of ships in waves. This
latest work is an improvement in the sense that three dimensionality is rather important,
since there are not many three-dimensional solutions for determining the coefficients and
the resulting motions of ships in waves. One of the main points which I see here, from my
point of view, leads me to a question which I would like to place before Dr. Grim. The very
fine thing that you have found is a creation of a distortion in the wave pattern, even in the
case of a ship which is restrained in waves. This means you have taken account of the in-
fluence of the free surface in computing the excitation. Now, this has never been done in
all the latest work used for computing the motions of ships and also it has not been taken
into account in determining the bending moments which act upon ship forms. Therefore, I
would like to know if you can tell me if you have made a comparison of the magnitudes of
the exciting forces on a restrained ship with your theory and compared it to the simple slen-
der body theory which just replaces the effect of the interaction by a dipole and does not
take into account the influence of the free surface. Secondly, I would also like to make a
point here concerning Fig. 27 in the paper. I gather here that what has been found is an ex-
pression for the magnitude of the virtual mass coefficient and also the damping coefficient
at the particular section. Two dimensionally, one finds the value at a section all the time,
but now you have a three-dimensional effect which shows the inclusion of interactions. The
point I would like to make is that the appearance of close values for the damping terms in
both the two-dimensional and three-dimensional cases is not necessarily to be taken as say-
ing that there appears to be only a small difference between two-dimensional and three-
dimensional values when you look at a section. It happens to be so for this particular case.
I can report, and will some time in the future, the fact that there are large differences in the
local values of damping; that is, the distribution along the hull is quite different, yet the
total values are about the same. So [ think it important for this one case to see this close-
ness yet realize local values must be looked at with care. That, of course, is most true for
bending analysis, which is of vital importance for the structure of ships.

O. Grim

Dr. Kaplan has put two questions: At first he asks how large the differences are between
the normal strip method of computing the forces and the bending moments without any defor-
mation of the wave and the other computations with the deformation of the wave. I have not
made such a comparison, but I think the last figure in which the reduced wave amplitude is
represented will enable us to make a judgment about this comparison. It may be that the
forces and the bending moments will be reduced in about the same magnitude as the wave
amplitudes. Dr. Kaplan’s second question is whether the added mass and the damping force,
mentioned in Fig. 27, are typical. I think that this is so and that the differences between
the two-dimensional and the three-dimensional results for heaving and pitching motion are
only important for the range where the frequency is near to zero, while in the other range
these differences are not so important. More important are the differences for the ship in
waves, and the reason for this is the following: For the ship which undergoes a heaving

522 O. Grim

motion with a large frequency, the forces exerted by the ship upon the water are in the same
phase along the whole length of the ship. At each section of the ship an oscillatory force
acts upon the water. The force at each section excites an elementary wave. If the ship
makes a heaving motion with a large frequency, the resulting wave is small due to the phase
relation between the elementary waves. However, for the ship in head seas the sum of all
the elementary waves, which gives the deformation of the original head wave, is largely due
to the phase relation between the elementary waves, which is different from the phase rela-
tion for heaving motion. This is the reason why the deformation is large for the ship in
head seas and why the deformation of the surface of the water in the longitudinal direction
is small for heaving motion with large frequencies.

J. B. Keller (New York University)

My comment is based on the following consideration. Dr. Grim’s work begins with the
solution of a certain two-dimensional problem and then he shows how to utilize a solution
of the two-dimensional problem to solve the three-dimensional problem. I would like to
point out another method of using two-dimensional problems to solve a three-dimensional
problem. The method I have in mind is especially useful for short waves, i.e., waves that
are short compared to the dimensions of a ship. The method is that of geometrical optics.

I would like to suppose that Fig. D3 is the waterline of a ship which we are looking at from
the top and I would like to consider the forces exerted on this ship by a wave coming in at
some oblique direction. Then, according to the principals of geometrical optics, the wave
can be thought to travel along rays and each ray will hit the waterline and be reflected ac-
cordin g to the law of reflection and give rise to a reflected ray. The calculation of the re-
flection coefficient, that is the amplitude and phase of the reflected wave, is a local affair
and depends only upon the geometry of the ship in the neighborhood of the point of reflec-
tion. Furthermore, since waves of short wavelength penetrate a very short distance into the
water, it is only the geometry of the ship very near the surface at the point of reflection that
determines the reflection coefficient. Since the geometry of a ship at a single point near
the waterline can be described in terms of the radius of curvature of the vertical section and
the slope of the ship at the waterline, and also, of course, the curvature of the waterline it-
self, those three quantities — the two radii of curvature and the slope — will determine com-
pletely the reflection properties for this particular ray. The radius of curvature of the water-
line will be taken into account because neighboring
rays get reflected in slightly different directions.
The lateral divergence of the reflected rays will ac-
count for the curvature of the waterline. In order to
compute the reflection coefficient for a given ray, it
suffices to solve a two-dimensional problem, namely,
reflection from a circular cylinder with the same
curvature as the ship has in a vertical section at the
waterline. The circular cylinder should cut the water
with the same slope the ship has at that place. That
is a two-dimensional problem that hasn’t been solved,
but never mind, [ just promised to tell how to relate
a two-dimensional problem to a three-dimensional
WATERLINE one. If we could solve that two-dimensional problem
for each position along the ship, then by such a geo-
eee metrical construction we could calculate the re-
flected wave in the three-dimensional problem. The
Fig. D3. Geometric optics analogy result for the two-dimensional problem might also be

REFLECTED
RAYS

Computing Hearing and Pitching Motions 523

determined by measurement or by someone solving it. Thus, it is possible, by solving a
relatively simple problem, to get results that should be useful for a relatively complicated
one. I would like to conclude by saying that the same ideas can be applied to calculating
the waves produced by the motion of the ship in pitching or heaving or other periodic mo-
tion. Again, in each case it is necessary to find the corresponding waves produced by the
motion of a circular cylinder with the same radius of curvature and slope as the ship has at
the point of interest and that problem hasn’t been completely solved.

O. Grim

I can imagine that this is a good idea as a way to compute the wave propagation for
any large frequency. However, for a prediction of the motions of a ship a more complete
solution than only for the wave propagation is needed. It could be helpful to obtain by
Prof. Keller’s proposed simple method results which confirm those obtained from a more
complete solution.

Owen H. Oakley (U.S. Bureau of Ships)

I am sure that if my colleagues, that is, my practical naval architectural-type colleagues,
could see me standing here with the temerity to discuss Dr. Grim’s paper they would be
amazed, and I assure you I am too. I have two points to make, one trivial and one more on
the serious side. First of all, in 1957 I was a shipmate of Dr. Grim on an icebreaker in the
Sea of Bothnia. We were concerned with vertical movements of the ship, because, installed
in the bow of this ship was an eccentric weight device known as the Stampflanage which,
when properly tuned at the remote control station in the pilot house, created a pitching mo-
tion in the bow of the ship and thus helped to break ice. Dr. Grim was at the controls and
occasionally he would turn the knobs the wrong way and a fore and aft surging motion would
result which was a bit disturbing to say the least. I wonder whether Dr. Grim has consid-
ered the matter of surge in this present analysis; I should think it would be something he
would be very much aware of. Now a more serious comment. As a general rule practical
naval architects do not have occasion to give serious thought to this sort of theoretical
work. However, occasionally one encounters problem where such mathematical assistance
would be most welcome. Such an experience was the design of the escort research ship.
We would have liked very much to have been able to calculate the response of this ship to
waves, but since we could not, we had to build models and go to the trouble of testing them,
finding where we were wrong, correcting it, and trying again. The theoretical and empirical
development of this subject has been going on for a number of years; many excellent minds
have worked on it and produced a prodigious quantity of papers on the various aspects of
the problem. I hope that some day all of this work to which Dr. Grim has contributed so
greatly will bear fruit and we will be able to sit down with pencil and paper or with computer
or what have you and predict quantitatively the amplitudes of response of arbitrary ship-
forms. I am sure that the contribution which Dr. Grim has told us about today is another
significant step in that direction.

O. Grim

I am glad to hear that Mr. Oakley likes to remember the time we spent together on the
icebreaker in the Sea of Bothnia and I hope he forgives me for the wrong way in which I

524 O. Grim

twiddled the knobs of the pitching plant of the icebreaker. To the other point of Mr. Oakley’s
comments I have to say that I also am a naval architect and I too hope that the results of
such complicated investigations can, in the future, be used by naval architects.

SEMISUBMERGED SHIPS FOR HIGH-SPEED
OPERATION IN ROUGH SEAS

Edward V. Lewis and John P. Breslin
Davidson Laboratory
Stevens Institute of Technology
Hoboken, New Jersey

To evaluate methods to obtain high speed at sea without large in-
creases in ship size, several hull configurations were surveyed; in par-
ticular semisubmerged craft designed to operate at or just below the sur-
face. By increasing judiciously the amount of submerged hull-volume,
both high-speed resistance and pitching motions in rough seas can be
reduced.

The hull configurations that were surveyed included a craft that resem-
bles a surfaced submarine, a slender ship with large bow and stern
bulbs, and a craft (semisubmerged) that is similar to a shallow-running
submarine with a permanent surface-piercing strut for air supply and ex-
haust. The semisubmarine, potentially, can exceed submarine speeds
because of its low resistance and good sea performance. In addition, the
air-breathing power plant of the semisubmarine weighs less and requires
less space than a nuclear power plant.

The study of the calm-water stability characteristics of the semisub-
marine at intermediate and extreme speeds shows that depth and direc-
‘tional stabilization can be achieved with tail fins of reasonable size,
except at low and intermediate speeds, where stabilization by fin area
alone is not attractive; however, active controls readily can provide op-
eration at constant depth in calm water. Analog computer studies show
a peculiar dependence upon the distribution of stabilizing surfaces and
other parameters. Therefore, precise and extensive model tests are re-
quired to explore the near-surface dynamics of this craft to fully develop
its advantages.

INTRODUCTION

The successful development of the true submarine has resulted in a craft that can re-
main submerged almost indefinitely and operate at high speeds in a three-dimensional realm
of steadily increasing depth. When a submarine runs far below the surface, surface wave-
making resistance is virtually eliminated; the remaining resistance is caused almost entirely
by skin friction. Although a submarine can be made squat in comparison with a surface ship
to minimize the ratio of wetted surface to volume, its surface area is greater than that of a
comparable surface ship. This is a disadvantage at moderate speeds; however, at high
speeds, this disadvantage is more than compensated for by the elimination of surface wave-
making. Power is directly proportional to speed cubed, even at high speed; therefore, speed
is limited only by the weight and size of the power plant.

525
646551 O—62—_35

526 Edward V. Lewis and John P. Breslin

The development of the submarine puts a new light on the problem of surface ship de-
sign. During World War II, the submarine and the destroyer were evenly matched. This is
no longer true. Submarines have distinct advantages in speed and maneuverability, partic-
ularly when seas are rough. In addition, submarines have added advantages in evasiveness
because they operate in a three-dimensional realm. To meet the challenge of the submarine,
naval architects must develop high-performance surface ships. This problem has frequently
been viewed with pessimism. It has been said that the only way to cope with the submarine
is to go above the surface, as with a hydrofoil craft, or to go below it with another subma-
rine. In this view, ships on the surface can only be fast if they are large, which increases
cost.

There is another viewpoint, however, that suggests the need to give serious attention
to the possibilities of more radical types of small surface craft that might be able to attain
high speed even in rough seas.

POWER REQUIREMENTS

Figure 1 shows three directions for seeking higher ship speed than can be obtained with
a conventional surface ship — for example, a destroyer. One direction is to go well below
the surface to eliminate surface-wavemaking resistance. Another is to stay on the surface
but to reduce surface-wavemaking resistance drastically with a longer and more slender hull.
The third is to raise the hull above the surface with either a planing-type hull or hydrofoils.

In addition, Fig. 1 shows that there are a multitude of different possible types of craft,
with successively larger submerged volumes, that can be designed to operate at the air-
water interface. Other possibilities have been suggested by Boericke [1].

It is of interest to consider the comparative powering problems of these diverse water-
borne craft. Because it is difficult to determine reliable figures for the propulsive coeffi-
cients, the comparison of the power required for these craft (Fig. 2) is based on effective
horsepower (EHP). Displacement is a reasonably good index of size; therefore, EHP’s have
been estimated for the craft shown in Fig. 1. Each craft is assumed to have a displacement
of 2844 tons, which corresponds to the displacement of a DD-692 class destroyer. This dis-
placement was selected as the desirable maximum displacement for high-speed craft.

Figure 2 is a rough comparison of the calm-water resistances of these waterborne craft
at speeds up to 70 knots. The craft that has the lowest resistance is an ideal, deep-running
submarine with a minimum of appendages, optimum hull form, and an assumed length-to-
diameter ratio of 7.0. Not only is a deeply submerged submarine the lowest in resistance of
any type of craft, but even at 60 or 70 knots the EHP is far from astronomical.

For an ideal submarine form (H) running with its center 1-1/4 diameters below the sur-
face (that is, the upper surface of the submarine is 3/4 diameter below the surface) there is
a pronounced hump in resistance that corresponds to the maximum wavemaking speed. The
submarine is close enough to the surface to produce a surface wave that causes this rise in
EHP. As the speed of the submarine increases further, the wavemaking resistance becomes
a decreasingly important part of the total resistance.

For an ideal submarine (about 210 feet long) with a strut (about 30 feet long) piercing
the surface (G) there is a sizable increase of EHP, which rapidly goes up with speed. Infor-
mation is still rather sketchy on the resistance of struts piercing the surface, but the plotted
values are believed to be roughly correct. The disturbance of flow by the strut is not

Semisubmerged Ships 527

meee =
CO AN 8 gece Fo

DESTROYER LENGTHENED DESTROYER

SEMI-SUBMERGED SHIP

SEMI-SUBMERGED HULL WITH LARGE STRUT

SEMI- SUBMARINE

SHALLOW - RUNNING SUBMARINE ©)

DEEP-RUNNING SUBMARINE

Fig. 1. Possible directions for seeking higher speeds at sea

wavemaking resistance in the usual sense (because the speed is very high for the length of
the strut), but there is separation of flow, ventilation, and even cavitation that cause a very
large increase in resistance, usually designated spray drag, which shows very emphatically
the reason for trying to make the strut as small as possible.

For a typical destroyer (C), the EHP is considerably higher than for a submarine with
a small strut, especially as speed goes up. In addition, this ship is slowed more than a
submarine in rough water.

Figure 2 also shows that a surface ship could achieve very high speeds if it were de-
signed to be much more slender than a destroyer (K). This would reduce the surface

528

EHP

20

Edward V. Lewis and John P. Breslin

SUBMARINE
LARGE STRUT

C37 IDEAL SUBMARINE -
SHALLOW

30 40 50 60
SPEED IN KNOTS

Fig. 2. Effective horsepower required versus speed

for a group of bodies with A = 2844 tons

70

Semisubmerged Ships 529

wavemaking resistance and thus compensate for increased frictional resistance. As the
speed of this ship goes up, the EHP curve almost coincides with the curve for the shallow-
running submarine with a small strut. This suggests the latter type of craft has no resist-
ance advantage over a slender destroyer designed specifically to go at very high speed.
However, the very slender destroyer would run into trouble at high speed in rough water,
and the near-surface ship with strut may have advantages.

A submarine with a very large strut (F), as suggested by Boericke [1] and investigated
by Mandel [2], shows comparatively high resistance, especially at high speeds. An all-strut
ship — that is, a ship of the same displacement and about the same draft as the others but
with its entire hull extending through the surface — has a very high EHP. Figure 2 also in-
dicates that a large planing boat (B) of optimum-breadth would be better only at very high
speeds (near the limit of this figure).

The semisubmerged hull form (E), reported by Lewis and Odenbrett [3], and the slender
hull with large bow and stern bulbs (D) are capable of high-speed supercritical operation in
rough seas with comparatively small pitching motions. The first of these ships, (E), shows
rather poor calm-water EHP performance at deep rough-water draft, but the latter, (D), is
remarkably good up to the limit of the speed range so far tested.

Therefore, a number of types of craft intermediate between conventional ships and sub-
marines are feasible that promise remarkably low power requirements. However, the larger
the proportion of the hull that is submerged, the more difficult the ship design problem be-
comes; available hull volume is curtailed and problems of transverse stability (static) be-
come increasingly difficult with greater submergence. The cost of the craft can also be ex-
pected to increase correspondingly. Hence, it is desirable to consider and compare all types
of craft thoroughly to determine the most satisfactory compromise for any particular need.

ROUGH-WATER PERFORMANCE

Of course, it is not enough for a surface or near-surface craft to show favorable calm-
water resistance at high speed. If it is to compete with a deeply submerged submarine, the
ship must be able to maintain good speed in rough-storm seas. This may be, in part, a pow-
ering problem. However, in high-speed craft, it is usually mainly a matter of avoiding or
minimizing ship motions and indirect effects — such as wet decks, slamming, propeller rac-
ing, local high accelerations, etc. The ideal craft, therefore, is one that provides the best
compromise between low-resistance characteristics and outstandingly good performance in
rough water — capable of high speed in any sea condition.

(As background for the discussion of the problems of rough water performance, the bal-
ance of this section has been adapted from Ref. 4.)

Ship motions at sea are usually resolved into six components for study: the angular mo-
tions of roll, yaw, and pitch and the translatory motions of heave, surge, and sway. Some
are more troublesome than others, but when one or two are reduced the others are apt to be-
come more noticeable. Rolling has received particular attention ever since steam replaced
sails and the steadying effect of canvas was lost. Because the forces involved in rolling
are small, it has proved feasible to reduce these amplitudes drastically by various antiroll-
ing devices. The simplest device is the bilge keel, which has been generally accepted in
shipbuilding for many years. Its effectiveness can be explained on the basis of the theory
that irregular storm seas contain many regular component waves of a wide range of periods
superimposed on one another. A ship will respond much more violently to one particular

530 Edward V. Lewis and John P. Breslin

band of wave periods (or frequencies) than to any others. This period is the natural rolling
period of the ship. Consequently, the most serious rolling occurs in the natural period of
the ship and, therefore, may be classed as synchronous rolling. In any oscillating system,
simple damping is always effective in reducing synchronous oscillation. Therefore, by in-
creasing the damping of roll, the bilge keel markedly reduces serious synchronous rolling.

Another damping device is the passive antirolling tank system, in which there has been
a notable resurgence of interest recently. Such tanks installed in naval auxiliaries have
proven to be remarkably effective. Important features are their low installation cost and
equal effectiveness hove to and at forward speed. However, reduction rather than elimina-
tion of rolling is the best that can be expected.

For moderate and high-speed vessels, a much more effective method of reducing rolling
appears to be controllable fins. These devices are so effective that the rolling problem ap-
pears to have been solved in principle, except for low-speed vessels for which an activated
tank or gyroscope system is applicable. This does not mean that further research is not
needed. There are problems of the interaction or coupling of other motions, particularly
yaw, of obtaining increased effectiveness with reduced weight and cost, of avoiding struc-
tural failures of fin shafts, etc. Furthermore, the system for the control of roll must be co-
ordinated with the system for the control of yaw, that is, the steering gear and gyro-pilot.

Heaving and pitching motions are the most difficult to overcome and present the most
serious problems because they involve large vertical accelerations and cause shipping of
water, slamming, and propeller racing. Furthermore, when other motions are controlled or
reduced, they become even more noticeable. Consequently, attention will be focussed on
these symmetrical modes of motion, assuming that methods are available for solving the
other motion problems.

Model research in regular-head seas has brought three important facts into prominence
[5]: (a) the amplitudes of motion are greatest in the vicinity of synchronism between the
period of encounter and a ship’s natural period of oscillation, (b) phase relationships lead-
ing to wet decks and slamming also are characteristic of synchronism, and (c) waves appre-
ciably shorter than the length of a ship do not cause serious motions even at synchronism.
On the basis of these general facts, there are two possible methods by which significant
reduction of pitching and heaving amplitudes can be sought: avoiding synchronism with
waves of ship length or longer, and reducing the magnification effect, which causes in-
creased amplitudes near synchronism.

Although damping devices such as bow fins can reduce the magnification of motions
somewhat, the most effective method of reducing motions is by avoiding synchronism. Ina
regular swell, this undesirable condition can be avoided by changing either course or speed.
If the speed is changed, synchronism can be avoided by an increase or a decrease. If the
speed for pitch synchronism is the critical speed, slowing brings the ship into the subcriti-
cal range; increasing speed brings the ship into the supercritical range. Figure 3 (from
Ref. 5) shows the relationship between pitching period and speed for synchronism in head
seas. For other headings, a simple cosine correction must be introduced. The curves that
define the conditions for synchronous pitching show that (a) the longer the wavelength, the
higher the critical period, and (b) in any particular wavelength, the lower the ratio of T,/VL,
the higher the critical speed.

When a ship encounters irregular storm seas, the situation does not remain so simple.
Oceanographers have shown that storm seas can be considered as composed of a great many
regular-wave trains of varying length and direction of travel, all superimposed on one

SPEED-LENGTH RATIO (V/V )

Semisubmerged Ships

PAAR A AT

ZONE OF SEVERE
MOTIONS AND WET -
DECKS IN IRREGULAR
STORM SEAS

26 Saas
eR RAY

ZONE OF MODERATE
MOTIONS AND DRY
DECKS IN IRREGULAR
STORM SEAS

2.0

0.8

0.6

0.4

WAVE LENGTH
SHIP LENGTH

0.2

PERIOD-LENGTH RATIO (T/ //— )

Fig. 3. Theoretical speed-length ratios for synchronous oscillation
in regular-head seas of different lengths (from Ref. 5)

0.4

0.3

FROUDE NUMBER (V/ / )

531

532 Edward V. Lewis and John P. Breslin

another [6]. They also have confirmed the observations of seamen that shorter waves are
formed first in a storm. As the wind continues to blow, longer components are formed with-
out seriously affecting the smaller components. For a particular wind velocity, the sea
reaches a limit when it attains its fully developed state. If the wind increases in strength,
not only are the component waves believed to be higher, but — when the fully developed
state is reached — longer components also will be present.

If a ship is able to attain a speed sufficiently high that its period of encounter with the
longest important wave component is shorter than the natural pitching period, the ship will
be in the supercritical condition for that particular storm. Most ships can attain this condi-
tion only in moderately heavy seas — that is, in light winds or in stronger winds of short
duration. In general, whether or not a ship can attain the supercritical condition depends
both on the sea state, indicated by wind velocity and duration, and on the ship’s natural
pitching period. Speeds for supercritical operation are shown for the case of head seas in
Fig. 4, which is taken from Ref. 5. At other headings, the ship must reach even higher
speeds to attain the supercritical condition.

Because short waves are present in both severe and moderate storm seas, it is impos-
sible to avoid synchronism with all component waves by reducing speed. However, model
tests have shown that waves appreciably shorter than the ship do not cause serious motions
even at synchronism. Therefore, speeds that are low enough to avoid synchronism with
waves of ship length and longer cause moderate motions. This condition may be termed the
subcritical range. For head seas, it depends mainly on the ship length and the natural pitch-
ing period, that is, on the ratio T /VL, as shown in Fig. 3. At times, all ships must reduce
speed in storm seas to attain the subcritical condition of moderate pitching.

In following seas, most ships steaming at ordinary speeds are always in the subcritical
range. This explains the advantage of heaving to with wind and sea astern, provided that
the ship can be kept under control. However, Mandel [7] has pointed out that unusual ship
forms intended for supercritical operation in head seas may encounter critical conditions in
following seas.

SUPERCRITICAL OPERATION IN ROUGH SEAS

It has been shown in Ref. 5 that for most ships the most promising method of reducing
pitching and heaving motions is to use hull proportions that raise the critical-speed limit
and permit higher subcritical speeds. This means increasing the length in relation to dis-
placement, which results in a reduced period-length ratio, T,/VL. A ship can then go at
higher speed before synchronous response to waves of near ship length is experienced

(Fig. 3).

Thus, in Ref. 8 a model 25 percent longer than a conventional DD-692 destroyer, but
with the same displacement, had lea ta = 0.20 instead of 0.25. It was concluded that in a
storm in which the conventional destroyer could attain a speed of 20 knots (V//L = 1.0),
the longer vessel could attain a speed of 30 knots or V/\/L = 1.4 (Fig. 3). The longer
destroyer was an abnormally slender ship, with A/(L/100)3 = 30. Tremendous problems of
stability, structure, and hull arrangements would be encountered in the design of such a
ship. It hardly seems feasible, therefore, to think of large increases of speed to 50 or 60
knots by going further in the direction of longer and thinner ships.

The most promising direction then to obtain really high speeds in head seas, where mo-
tions are most severe, is to aim at supercritical operation. At the same time, care must be

533

Semisubmerged Ships

n”
2
o
e
[2
2
oO
oO
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q
oO
=
ec
oO
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a
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SLONY NI G33dS

PITCHING PERIOD (Tp) IN SECONDS

Fig. 4. Ship speeds for critical and supercritical operation,
based on a Neumann ideal spectra (from Ref. 5)

534 Edward V. Lewis and John P. Breslin

exercised to remain in the subcritical speed range in following seas. The basic idea is
simply to attain high ship speed and a long, natural pitching period. The problem of high
speed involves the question of reducing wavemaking resistance, which has already been
discussed.

To consider all the possible means of increasing the natural pitching period GG ) it is
well to examine the equation given in Ref. 5:

(1)

in which V is displacement volume, g is acceleration of gravity, B is breadth, and L is
length. The coefficient C, is the inertia coefficient so that the longitudinal mass moment -
of inertia J, =-Ve(C,L)?, or C, = 1I/L J, /Ve; (C 7k, is an analogous coefficient of virtual
inertia by which VpL?2 must be multiplied to give the longitudinal mass moment of inertia of
the entrained water. C,, is the coefficient of longitudinal waterplane inertia defined by the
relationship
3
EY BOB

From Eq. (1) it can be seen that the following trends increase the natural pitching

period:

1. Increase of displacement

2. Reduction of length or beam

3. Increase in virtual inertia (C,7k,)

4. Reduction in waterplane inertia (C,, )

5. Increase in mass moment of inertia (C,).

Because a ship in supercritical operation tends to plunge through the waves rather than
to ride over them, it is difficult to attempt to keep water off the foredeck. Accordingly, low
freeboard and a heavily built deck is one way to cope with shipping of water, which at the
same time permits a very narrow waterline. The body plan in Fig. 5 illustrates a possible
design for a supercritical ship of this type, (Z) of Fig. 1, a ship like a surfaced submarine,
made longer and more slender to attain high surface speed. The normal waterline would be
used for good weather operation; in bad weather, large peak ballast tanks would be filled to
lengthen the pitching period by increasing the displacement, by increasing the radius of
gyration, and by reducing the waterplane area (items 1, 4, and 5 above). Characteristics are
compared with a typical destroyer in Table 1.

This semisubmerged model had T p/VE = -0.38 in the deep-draft condition. Results of
tests in regular- and irregular-head seas [3] confirmed theoretical expectations that a super-
critical condition of moderate motions could be attained at speeds of 35 to 40 knots in mod-
erately rough irregular-head seas (Fig. 6). Heaving motions were more pronounced than

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536 Edward V. Lewis and John P. Breslin

Table 1

Characteristics of Unusual Ship Forms

DD-692 Semisubmerged | Large-Bulb
Destroyer Ship Ship

Length overall (ft)

Length on WL (ft)

Length of each bulb (ft)

Beam (ft)

Draft (ft): hull, normal displ.
bottom of bulb, normal displ.
hull, deep displ.

Displacement (long tons):
hull, normal

bulbs

total

deep
A/(LWL/100)°
Waterplane coefficient
Longitudinal gyradius
Natural pitching period, T, (sec)
Period-length ratio, T,//LWL

Natural heave period (sec)

*At deep draft.

pitching, but heaving accelerations were not unusually high in spite of the high average fre-
quency of wave encounter. In irregular waves, the highest single value of heave accelera-
tion was 0.45 g at 25 knots and 0.26 g at 40 knots. The reason for these moderate values
was that the motion occurred primarily in the natural heaving period.

No following-sea tests were run, but the natural pitching period is such that subcritical
operation is anticipated at speeds up to 45 knots. However, as previously noted, the power
requirement at deep draft is exceedingly high, even in calm water, for practical considera-
tions. Furthermore, wave impact on the superstructure would be a serious structural prob-
lem, and it might be necessary to raise the deckhouse up on columns to permit the waves to
pass under it. Hence, though this type of craft looks interesting as a supercritical ship,
there are serious problems to be solved to make it practicable.

Another type of supercritical craft is the slender hull with large bulbs mentioned earlier,
(D) of Fig. 1. Available results of a model investigation of such a ship now under way at the
Davidson Laboratory will be summarized. The objective here was to increase the natural
pitching period by increasing the virtual mass moment of inertia of the hull, without incurring
a resistance penalty in the process. That is, k,, cf Eq. (1) was greatly increased. The
bulbs appear to be too deeply immersed to provide appreciable benefit in the form of damping
of motions.

The hull form selected was the well-known ideal form developed by Wigley [9] for which
the theoretical wavemaking resistance has been calculated. In this form, the underwater
sections are parabolic, the waterlines are sinusoids, and the above water form is wall-sided.
For the shape of the bulbs, a body of revolution was selected with length/depth =.5. The-
oretical wave resistance of this type of body has also been calculated. The size of each

537

Semisubmerged Ships

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538 Edward V. Lewis and John P. Breslin

bulb was fixed at 20 percent of the volume of the hull alone. Thus, both bulbs represent
28.6 percent of the total displacement. The bulbs were attached to the hull in a manner that
permitted their fore and aft location to be varied.

Figure 7 shows the model with 40 percent of the length of the forward bulb extending
forward of the FP and 60 percent of the aft bulb extending aft of the AP. Full-scale char-
acteristics of the large-bulb ship are given in Table 1, and compared with those of the typi-
cal destroyer and semisubmerged hull.

Fig. 7. Slender ship with large bow and stern bulbs

At present, neither the testing nor the analysis in this project has been completed. Re-
sults presented here are preliminary and subject to further evaluation when the testing pro-
gram is completed. Because the forward position of the bow bulb showed the least resistance
of the positions tested, it was selected for the regular 2.0L wave tests. Figure 8 com-
pares the pitching motions of three types of ships. The worst pitching motions for the
large-bulb model occur at about the speed for synchronous motions, 2.4 fps or 9.8 knots. At
intermediate speeds, the motion of the large-bulb model was somewhat greater than the semi-
submerged model, but at high speeds it was somewhat less.

Figure 9 shows one real shortcoming of the model. Worst heaving motions occur in the
vicinity of 34 knots ship speed, where the amplitude approaches 2-1/2 times the amplitude
of the wave. Synchronism for heave occurs at 6.8 fps or 28 knots ship speed. While the
heaving was severe, maximum acceleration amidships accompanying the motion was 0.45 g
in an LWL/41 wave, no worse than the semisubmerged model. However, the large heave am-
plitudes may actually cause the middle body of the ship to emerge from the water at speeds
where maximum heaving motions occur. Nevertheless, this type of craft appears to offer
real potentialities of high speed in rough water, and research is continuing on possible
means of improving performance further. It has the advantage over the semisubmerged de-
sign in having less hull-volume limitations, and sufficient freeboard and flare can be pro-
vided forward to keep water off the foredeck.

SEMISUBMARINE
Characteristics

Finally, we may consider the possibility of larger proportions of submerged hull. Mandel
has considered a hull with a surface-piercing fin almost as long as the hull ((F) of Fig. 1)
with some success [2].. However, it is believed that the power requirements are excessive
and that the only hope for success with this type of craft is to reduce the fin to the absolute
minimum. Mr. E. Frankel stated at the Seminar on Ship Behavior at Sea, Stevens Institute of
Technology, June 1960, that a model is now being investigated at M.I.T. in which the fin has
been reduced to about 1/6 the length. Resistance and control problems for such a craft ((G)
of Fig. 1) are now being investigated at Stevens.

539

Semisubmerged Ships

SeABM 7(0'Z Je[Ndal ur suoyjow Buryoyrg “g “Bry

(SLONN) G33dS dIHs

dIHS Q3983NENS— INE

3d071S 3AVM WNWIXVN
SONLINAGWY HLid

540 Edward V. Lewis and John P. Breslin

3
‘3! VW, i g
»
©
on ai \!
"alk ai j . ge ,
& ij
sh EN 3
3
r=]
ro) ne 8
@
< |
oo. ieee, 0
= A —
ow
i
b 3
2 °
e
°o
° 7) ° n 9 id °
Ca) rt] a = = oO

SGALIIDNY JAVA
Santi

SHIP SPEED (KNOTS)

Fig. 9. Heaving motions in regular 2.0L waves

Semisubmerged Ships 541

Before discussing control problems, it is important to consider in what way this craft
would be better than a submarine. The question is, why should we take the trouble of hav-
ing a strut piercing the surface, paying the penalty of additional resistance? In short, why
not use a submarine? The answer is, for the moment at least, that there seems to be no
doubt that the weight and size of nuclear power plants are greatly in excess of air-breathing
plants of comparable power. In fact, the weight and space requirements of the present type
of nuclear power plant seem to be two or three times as great as air-breathing plants. Fur-
thermore, we have seen definite possibilities that the weight of air-breathing power plants
can be further reduced. One of the most helpful developments here is the hydrofoil craft
which, regardless of its own merits, is certainly stimulating the development of lightweight
power plants that would make the air-breathing submarine even more advantageous.

Because the high-speed near-surface craft with surface-piercing strut (semisubmarine)
has not been previously discussed in the literature and because of the many interesting
technical problems associated with this type of ship, the balance of this paper will deal
with this particular craft.

The appearance of the semisubmarine might be roughly as shown in Fig. 10. It would
have a submarine-like hull, but not designed for high hydrostatic pressures. The strut
would have to go high enough above the surface so that the crests of the waves would not
cause difficulty. Its main function would be to take air down into the hull and carry exhaust
gases out. It would be necessary to accept from the very outset the need for control devices,
for we cannot expect it to run at a constant depth below the surface, even in calm water.
The reason for this is, of course, that there are surface interaction effects. Any body mov-
ing at high speed, or even at moderate speed, below the surface experiences forces (which
can be resolved into a moment and force) related to the wavemaking on the surface. These
forces probably reach their maximum at the condition of maximum wavemaking resistance,
but they are still present at very high speed. Fortunately, the effects at very high speeds
appear tentatively to be generally favorable. To explore this problem at the Davidson Labo-
ratory, simplified linear equations of motion have been set up and solved on an analog
computer.

Fig. 10. High speed semisubmarine

646551 O—62——36

542 Edward V. Lewis and John P. Breslin

Design Problems

It should be noted that the control of a semisubmarine involves more than a problem of
stability in the usual sense. Suppose the heading of a stable ship is disturbed downward;
if the craft is stable, it will steady down on a new course that will take it deeper and deeper.
Similarly, if it is displaced upward, it will steady down on a new course upward so that it
will eventually break surface. The ship should have a tendency to come back of its own
accord to the original course and depth. This quality is here referred to as depth stability.
It will be shown later that this can best be achieved by means of control surfaces with auto-
matic control, which can keep the ship running level. This problem appears to be roughly
the same difficulty as the control of completely submerged hydrofoils on a hydrofoil boat. It
will be discussed in more detail in the next section.

If there is a large surface-piercing fin, or a smaller fin near each end of the craft, it may
be expected that the increase or decrease of buoyancy associated with changes in immersion
would assist in the depth control of the body. However, with a single minimum fin, the buoy-
ancy effect appears to be small and reliance must be placed on dynamic control. A small
amount of positive overall buoyancy would be desirable in the submerged condition for safety
in case of power or control failure.

The next problem, after the consideration of maintaining a level course under a calm
sea, is the behavior of the ship in a rough sea. Here we find that the head-sea case is no
problem. The runs on the analog computer indicated that this particular design had a pitch-
ing period of about 20 seconds, or at least when it was disturbed it oscillated in a period of
about 20 seconds, which is a sort of dynamic, natural pitching period. It signifies a very
low response to outside disturbances, so that the high frequencies encountered in head seas
will not cause motions of any consequence in the craft. However, in following seas we may
expect that it will run into the critical zones of Fig. 1 of Ref. 7. The problem of providing
vertical control so that the ship will run level or approximately so will be discussed further
in the next section. In long waves it may be desirable for the ship to contour the surface,
moving up and down, just as a hydrofoil craft must do.

Another problem to be investigated is that of lateral control and turning. A fore and aft
location for the strut must be determined which, in combination with other fins, will provide
satisfactory directional stability in a horizontal plane and at the same time will not lead to
serious heeling in a turn. This problem can easily be investigated by the use of conven-
tional model tests on a rotating arm. Recent work by Strumpf suggests that an arrangement
of two fins as shown in Fig. 10 might be the most satisfactory solution in combination with
the surface-piercing strut.

In addition to the purely hydrodynamic problems discussed, there are many other de-
sign problems to be solved, including the following:

1. Compact, lightweight air-breathing power plant,
2. Air supply and exhaust,

3. Structure of surface-piercing strut,

4. Static transverse stability,

5. Minimization of resistance by selection of optimum hull proportions and obtaining the
best balance between depth and strut drag,

6. Automation of operation to minimize crew requirements.

Semisubmerged Ships 543

Because of the high ship speeds and relatively low thrust loading considered, it does
not appear difficult to obtain a conventional propeller of good efficiency. A single shaft
should be used to obtain a high propulsive coefficient. There are virtually no limitations
on propeller diameter (other than construction) and rpm may be selected to give a good effi-
ciency along with low machinery weight. Submergence depth enters into the cavitation cri-
terion to some degree, but there appears to be no difficulty in avoiding cavitation up to 45
or 50 knots. Cavitation is a problem for control surfaces, however.

The installation of a high-power propulsion plant dictates a minimum weight and space
allowance. Gas-turbine main engines seem to satisfy these requirements. For naval pur-
poses the ship must be able to achieve a high speed for short time intervals and must also
be able to cruise at a considerably reduced speed for long time intervals. A single gas-
turbine engine does not have the ability to operate efficiently over this considerable power
range. Hence, to achieve high engine efficiency, multiple engines should be considered. At
top speed all engines would be running. For this operation, engine life is a prime considera-
tion. At low speed, fewer engines would be running and the idle ones could be disconnected
from the reduction gear with a fluid coupling to facilitate maintenance and to eliminate wind-
age losses.

The reduction gears to reduce the turbine shaft speed to a practical propeller speed in-
volve very strict weight and, more important, space limitations. These considerations sug-
gest an epicyclic gear train with high surface hardness and high K factor. This type of gear
should have the ability to transmit power efficiently over a large range of powers. Even so,
the gearing will undoubtedly be the largest single item of machinery weight.

Another possibility for reduced weight is to design a ducted type of propeller operating
at very much higher rpm, perhaps in the supercavitating range. This would permit a drastic
reduction of gear weight.

Air supply and turbine exhaust involve serious engineering problems to minimize pres-
sure losses and yet keep the surface-piercing strut to minimum size. In any case, the duct-
ing will certainly be a big item in machinery weight.

It is proposed that a compact control station be located at the top of the surface-piercing
strut for good visibility. Permanent ballast of about 10 percent of displacement might be
necessary to provide satisfactory static transverse stability.

Basic trends of semisubmarine characteristics have been studied in a preliminary way
in connection with work of the Panel on Naval Vehicle Systems, Undersea Warfare Committee,
National Academy of Sciences-National Research Council. Tentative design features of a
series of craft of varying size were worked out on the basis of certain assumptions regarding
technical potentialities [10].. These assumptions will be summarized before presenting the
results.

Resistance

Resistance was estimated from model tests of simple bodies of revolution, with axes
1-1/4 diameters below the surface. An Ogive strut was selected, the size was established
by the estimated engine air requirements. A length of 14 percent of ship length was used
for the series. Strut resistance was calculated from Ref. 11, neglecting the elimination of
tip losses by the attachment of the strut to a hull. The characteristics that were selected
were not necessarily optimum.

544. Edward VY. Lewis and John P. Breslin

The complication of a controllable pitch propeller was not considered necessary; how-
ever, for backing and for emergency propulsion, small auxiliary screws were assumed driven
by medium-speed diesels. The speed that can be achieved with this propulsion will be
around 5 or 6 knots.

For these tests the following characteristics were selected.

Hull-Form Coefficients — Length/beam ratio = L/B =:7.0; submergence beam ratio =
h/B = 1.25 (to centerline); prismatic coefficient = Cy = V/(7/4) B2L = 0.650; wetted surface
coefficient = K, = W.S./7BL = 0.774.

Ogive-Strut Coefficients — Thickness/chord ratio = t/c = 0.167; submergence/chord
ratio = h/c = 1.30; c/L = 0.139.

Machinery — Main engines: GE converted aircraft engines, type MS240B, MS240A, 6000
rpm; reduction gears: double reduction, epicyclic type, small with low weight (estimated on
the basis of Timmerman second reduction gears); overall fuel consumption: 0.55 /SHP-hr at

V = 45 knots and 0.58 /SHP-hr at V = 20 knots.

Propulsion — For this study the top speed was chosen at 45 knots and the cruising
speed at 20 knots. The propeller and propulsion characteristics chosen (not necessarily
optimum) were as follows:

A = 6000 A = 3000 A = 1500 A = 750
Propeller diameter Dalhelsy ate 16.8 ft 13.4 ft 10.0 ft
Shaft rpm 200 250 300 400
Propeller efficiency
at 45 knots 74% 75% 76% 76%
Assumed hull
efficiency 140% 120% 120% 120%

The appendage resistance allowance chosen was 10 percent in all four cases. All propellers
were checked for avoidance of cavitation.

Weights
Assumed hull, machinery weights, and other data are shown in Table 2.

For any vehicle, whether for military or commercial service, the basic characteristics
of interest are speed, range, and payload. Various types of plots can be prepared to show
the relationship among these variables for different craft. However, for ASW purposes the
particular need seems to be for a top speed between 45 and 50 knots and a cruising speed
between 15 and 20 knots. Hence, in the present study fixed speeds have been assumed and
trends have been determined of range versus payload for a family of vehicles of different
sizes. The results are shown in Fig. 11.

The plot shows that for fixed speed and size, the payload available for weapons, detec-
tion gear, etc., varies inversely as range. And for any given range, the larger the ship the
greater the payload must be. To be more specific, it appears that a 1500-ton craft for

Semisubmerged Ships 545

Table 3 .
Weight Data for Semisubmarine Series [10]
Displacement (A) in tons 3000 1500
Length (ZL) in feet 272 216 W715 136
Beam (B) in feet 38.8 30.8 24.4 19.4
SHP 130,000 70,000 40,000 25,000

Weights in Tons

Steel weight (0.35 A)

Outfit weight (0.10 A)
Salt-water ballast (0.05 A)
Lead ballast + margin (0.10 A)
Main propeller machinery

Crew and all stores

Deadweight = payload + fuel

More Optimistic Design*

Payload + fuel

* All weights reduced by 20 percent.

example, could perhaps carry a payload of about 100 tons over a range of 2500 to 4500 miles
(including 10 hours at 45 knots). The larger ships could carry much more.

Even better results and higher speeds might be attained with high-rpm and supercavitat-
ing propellers. Although the design data given are only tentative, it is hoped that the sug-
gested trends show the potentialities of unusual naval craft if the same level of technical
effort is applied as in current hydrofoil boat designs.

CONTROL OF NEAR-SURFACE BODIES

The depth control of a body moving at high speed below, but close to, the water surface
is an interesting problem. Therefore, it may be of value to summarize the theoretical work
carried out so far for both the calm-water and following-wave cases. It is hoped that this
will serve to clarify the problem and to specify areas that require research.

Dynamic Stability in Calm Water

The following analysis considers only small disturbances of equilibrium conditions,
which the craft is assumed to have when proceeding at a constant speed (U) along a path

xen
Kot
RCE

wont

Semisubmerged Ships 547

(z, = A) parallel to the undisturbed free surface. For the heave and pitch analysis assume
that there are no lateral excursions.

Stated briefly, the equations of motion for heave and pitch are

m'(w' - q') = Z'(w',q',w',q',z,,0) (2)
Lig! SE wiya’ omg", 2150) (3)

where m' is the nondimensional mass of the craft (2m/e°), w’ is the vertical velocity of
the center of gravity in fraction of speed (U, positive downward), q' is the nondimensional
angular velocity about the transverse (y) axis, zi is the instantaneous distance from the
free surface to the center of gravity in fraction of the length of the craft (2, positive down-
ward), @ is the pitch angle (positive nose-up), Ti is the nondimensional mass moment of
inertia about the y axis (I, =] y/(e/2)45), Z' is the nondimensional total vertical force
(force 1/2e¢7U?), M' is the noddimenaiaaal total pitching moment (about the y axis), and
the dots above a symbol indicate differentiation with respect to nondimensional time

(s = Ut/4).

Motion near the free surface induces forces and moments that are dependent upon the
distance (x,) and possibly on the pitch angle (0). These dependencies are, of course, not
present when the ship is deeply submerged.

When the ship is near the free surface, the force and moment become dependent on z,.
There is also a possible dependence of the force and moment upon @, which does not arise
in the case of deeply submerged bodies. In view of the dependence on depth below the sur-
face, it is advantageous to write the equations with respect to an axis system (x 595925)
that is fixed in the free surface, rather than with respect to a system (x,y,z) that is fixed in
the body as is usually done. Expanding the right sides of Eqs. (2) and (3) in Taylor expan-
sion and retaining only the linear terms yields:

(Zi-m')%, + Zit) + Zt zi 4+ 216+ (214+ 2/04 (21+ 238 = 0
MZ) + HZ, + My Zo + Ma-1,)0 + (My, + Me + CM + Mg)2 = 0 65)

where all the coefficients are derivatives with respect to the indicated subscripts (except m
and I,). To obtain Eqs. (4) and (5), the kinematic relation between the motion referred to
axes fixed in the craft and those fixed in the water surface has been used; therefore,

WS oe) O and G=1G.

In principle, these equations can be solved analytically, but the results are somewhat un-
wieldy. They can also be solved on an analog computer. In our study we have attempted to
glean some understanding from the analytical approach and have also used an analog for
more rapid exploration of the influence of fin area and control on the response of the body
when disturbed as it cruises at 30, 40, and 60 knots at a depth of 1.25 diameters (to the cen-
terline). The following discussion will first reveal what has been learned from the hand-
turned mathematics and will be followed by a recounting of the results of the analog studies.

548 Edward V. Lewis and John P. Breslin

Equations (4) and (5) represent a pair of linear coupled equations in z, and 0 of the
form

LZ, + L,6 -

it
i=)

(6)

zene = 0

where L, 4 are the linear second-order operators displayed in Eqs. (4) and (5). The
solution for either z, or 0 involves integration of single, fourth-order, linear differential
equations of the following form:

Z
(oe tp} ps 0 (7)
! IC

where (upon omission of the primes and the subscript o on 2)

(Zeer) DZD) raz
Eo = 2,02 (2, + 2.) +.(Z& 2)
3 (8)
Lz, = M.D“ +M,D+ Mm,
EL, =. (4; ~1,)D* + (He + H,yD +. + Mg)
and D = d/ds.
Thus, the equations for z and @ are in the form
Zz
(aD* + bD3 + cD* + dD + e) \- 0 (9)
c
and the solutions are of the form
4 o.s $ o,;,s
mh YT ne oa nes (10)
i=1 1=1
where h is the original depth and the o,’s are the roots of the quartic equation
aot + bo3 + co* + do +e = 0. (11)

For stability, or no exponential divergence, a, b, c, d, and e must each be greater than zero.
In addition, for no oscillatory divergence

bed - (ad? + b7e) > 0.

Semisubmerged Ships 549

The last inequality is Routh’s discriminant, which requires the real parts of all complex
roots to be positive.

At this point we must note that stability as used here refers to directional and depth
stability, that is, depth stability means that after an arbitrary excursion in either z or 9,

zy as a or S > @

and directional stability means

ek essa(0) as t or S25 Coe

This is to be distinguished from the case of the deep-operating submarine, where the
body is considered stable as long as it does not go into a diverging trajectory but settles
on a new path that is straight but not in the same direction as that prior to the disturbance.

If we note that M ., Zy M gare small and that Zg = 0 (as may be proved for the doubly
symmetric body under consideration) the coefficients of the characteristic equation take on
the relatively simple form

Bye |
Wh M
oO ew, toe,
ee mM. ° ON
2 y
ele zh [M.Z,- M, + (m+ Z,)] - =
Dimay, ne, (12)
ite eee Mewhiy 7 ough
- le (tt. - 1,2)
2imenyz
e = gar (42. - ¥,2,)
omy,

where M, =m — Z,;, the virtual mass in direction z, and N, =/
ment of inertia.

y ~ Mg, the virtual mass mo-

It can be noted that a, b, and c are the same as will be found in the equation of motion
for the deeply operating body, with the exception of the term —Z,/M,. The quantity d is
seen to be an effective damping coefficient that is arrived at by coupling the damping force
and moment derivatives with the free-surface force and moment derivatives, and e is an effec-
tive spring constant arrived at by a similar coupling of the static force and moment deriva-
tives with Z, and M,. It is now necessary to say something about the characteristic of Z,
and M, as functions of Froude number [12]. The force rates (Z, and M,) arise from the asym-
metry of flow about the body when it is moving parallel to the water surface. At very low

550 Edward V. Lewis and John P. Breslin

Froude numbers, the surface acts as a rigid ceiling, which causes a suction or attractive
force and virtually zero moment. This attractive force is considered negative because the
positive direction is downward. Because this attractive force becomes less negative as Z
increases, Z, is positive at low Froude numbers. At high Froude numbers, the reverse is
true, that is, the force is one of repulsion and Z, is negative. The force rate changes sign
at a Froude number of F = U\/gl = 0.55. Again, with increasing Froude number the moment
changes from a small bow-up moment (M, < 0) at low Froude numbers, to a bow-down moment
(M, > 0) for all Froude numbers greater than about 0.33.

Physically, the vertical attractive force at low Froude numbers tends to be destabilizing
because any excursions toward the free surface will cause this force to increase. At high
Froude numbers, the repelling effect is stabilizing. If we consider the effect of M, in the
absence of the force, then the moment is destabilizing at low Froude numbers and stabilizing
at high Froude numbers in regard to simple exponential divergence. However, large values
of M, can lead to oscillatory divergence or instability.

The magnitudes of Z, and M, at 30 knots (U//gl = 0.62) are significantly different from
their values at 40 and 60 knots so that we may expect, and do find, a considerable difference
in the stability or the fin area required for stability over this speed range. Although, other
derivatives can be expected to be Froude-number dependent, such as M,,, Mz, M;, Z,» and
Z.;, they have very weak variations with Froude number at the submergence ratio and over
the Froude numbers considered.

Exponential Instability

To establish stability criteria, consideration must be given to the requirement that each
of the coefficients a, b, c, d, and e of the quartic equation in o be positive. Equation (12)
shows that a and b are positive; Z,, and M, are both negative. Figure 12 is a graph of b as
a function of tail area; note that b is large.

The coefficient c is composed of three terms; the last term is found to be small and, be-
cause the two remaining terms are composed of those derivatives which are considered to be
independent of Froude number, the relation for stability arising from c > 0, that is,

MZ, - M,(m+Z,) > 0, (13)

is exactly that required for the body when operating deeply submerged. Figure 13 is a graph
of c as a function of fin area; a fin area of 220 ft? is the minimum required to prevent ex-
ponential divergence. In computing the changes in the various coefficients as a function of
fin or tail area, a conservative lift rate of 2.0 was chosen to take into account the attrition
expected from partial cavitation.

An inspection of the equation

i 1
d = i, (1,2. ) H,2, (14)
y

shows it to be positive for high Froude numbers but negative for those ranges of low Froude
numbers where M, is negative. Thus, this term can cause exponential divergence at low

Semisubmerged Ships 551

FOR ALL SPEEDS

ee “ga
See ir
i ea
Vi Fal A
i fares

STABLE FOR ALL AREAS

° 200 400 600 800 1000
TAIL AREA IN FT°

Fig. 12. Coefficient b versus tail fin area

Froude numbers but is not troublesome in regard to this type of instability at high Froude
numbers. However, for Froude numbers where M, is maximum, the attending increase in d
can contribute to oscillatory instability. Figure 14 is a graph of das a function of tail area.

The coefficient

(15)

is negative for zero tail area. Figure 15 shows that the minimum tail area to prevent ex-
ponential divergence at 60 knots is 255 ft. Thus, the free-surface effects at high speed
require a bit more tail area than the condition imposed by c for deep operation.

Oscillatory-Exponential Instability

If a pair of the roots of the quartic characteristic equation are complex, either damped
or undamped sinusoidal oscillations will occur subsequent to an initial disturbance, depending

552

Edward V. Lewis and John P. Breslin

FOR
( 40 knots
eae es aa a

ao eee,

re) 200 400 600 == Yeo) 1000
TAIL AREA INFT@

Fig. 13. Coefficient c versus tail fin area

—————

a

Semisubmerged Ships 553

FOR 60 knots

NOTE: d= 4.8 AT |OOOFT
AT 40 knote

STABLE FOR ALL AREAS

ry 200 400 600 800 100(1
TAIL AREA IN FT@

Fig. 14. Coefficient d versus tail fin area *

554 Edward V. Lewis and John P. Breslin

FOR 40 knots :

FOR 60 knots

STABLE

0 200 400 600 800 1000
TAIL AREA IN FT@

Fig. 15. Coefficient e versus tail fin area

Semisubmerged Ships 555

Ry bed —(ad~ + b’e)

> O FOR STABILITY
< O FOR INSTABILITY

acieun
cae

—————_.....

FOR 40 knots
(UNSTABLE)

0 200 400 600 800 1000
TAIL AREA IN FT

Fig. 16. Variation of the condition for oscillatory stability, Ry

upon whether the sign of the real part of this complex root is negative or positive. Routh’s

discriminant (Rj) provides the following relationship, which the coefficients must satisfy
for stability.

Riosn beda= (ad? .b2eyachw 0 (16)

It can be seen that the quantities d and e, which represent coupling of the free surface ef-
fect with the body damping and static force rates must not become overly positive or the
product bed will be overcome and oscillatory instability will result. Inspection of Fig. 16
shows that Routh’s discriminant is positive at 60 knots for tail areas greater than 200 ft”,
but that even a tail area of 1000 ft? is insufficient at 40 knots. The loss of stability with
decreasing speed is due mostly to the large change in M, which, with some assistance from
Z,, produces a 12-fold change in the coefficient e, thus valine it much more positive (even
for a tail area of 300 ft”) and resulting in a negative value for Routh’s discriminant. It is
also to be noted that increasing b, which is really the deep-water effective damping coef-
ficient (depending on Z,, and uM), will cause oscillatory divergence. This is also a

556 Edward, V. Lewis and John P: Breslin

surprising result, which stems from the coupling of the body coefficients with the free sur-

face spring terms (M, and Z,). Thus, making the body overly stable for deep operation pro-
vides a very stiff effective spring coefficient (e), which appears to be most responsible for

divergent oscillatory oscillations.

Results of Analog Studies

Exploration of the influence of fin area, distribution of fin area, the sensitivity to cer-
tain coefficients, and the effectiveness of control can quickly be assessed by an analog com-
puter. The results of such studies are discussed in the following subsections.

Responses with Fixed Fin Areas — Results of the analog studies of responses with fixed
fin areas can be friefly summarized as in Table 3.

Table 3
Analog Computer Results for Fixed Fin Areas

Speed All Fin 3/4-Area Aft, 1/2-Area Aft,
(knots) Area Aft 1/4-Area Forward 1/2-Area Forward

Fin Area = 300 ft?
30 Divergent
oscillations
40 Divergent
oscillations
60 Damped
oscillations
Fin Area = 1000 ft?
40 Slightly divergent Damped Divergent
oscillations oscillations oscillations
60 Lightly damped Heavily damped
oscillations oscillations

Since the case of 300 ft? of fin area has been discussed, the following remarks will be
directed toward giving some rationalization of the curious result of very slight stability at
60 knots with 1000 ft? of fin area aft as compared with very marked stability when this area
is split into half forward and half aft. To begin with, this large area was selected as an
intuitive attempt to stabilize against the oscillatory instability at 30 knots without regard to
its implications on the stability of the deeply operating submarine. An inspection of the
criterion of stability for deep operation,

M,Z, - M,(m+Z,) > 0,

which is essentially the quantity c, shows that by adding too much tail fin, m + Z, can be
made negative (by making Z, so negative that it overwhelms m) and, at the same time,

Semisubmerged Ships 557

making M_,,, change from positive to negative so that the second product in the above ine-
quality contests with the magnitude of the positive product M,Z,,. Thus, it is possible to
alter the stability in deep operation little beyond a certain maximum by a piling-on of fin
area. At the same time, the quantity b increases enormously, and though b plays a minor
role in determining stability in the deep operating case, in this case of coupled motion in
the neighborhood of the free surface, it is very disadvantageous to make b large. This can
be seen by again referring to Routh’s discriminant

bed - (ad? + be) > 0

which shows that the quadratic behavior of b can be overriding, provided the value of eis
not too small.

As we take area from the stern and put it on the bow, c decreases only about 20 percent,
b is large and does not change, and e decreases by a factor of 8 (d changes but is not deci-
sive). Thus, the role of M,, (which has, with equal areas forward and aft, a destabilizing
tendency with regard to deep operation) is to provide a much softer effective spring value e
to this fourth-order system and thus permit quick decay of oscillations.

Control Effectiveness — The foregoing has shown that exponential instability will exist
at Froude numbers below 0.3, that excessive tail area is not the answer at F = 0.6, and that
reasonable tail area will only stabilize at F ~ 1.2 (which corresponds to an unlikely speed
of 60 knots). This leads to the necessity of applying controls that obviously would be
needed anyway for rough-water operation. A systematic study of the effectiveness of con-
trols proportional to z, z, 0, and 6 could and should be made to arrive at the best combina-
tion. In view of the shortness of time available for this study, computer runs were made only
for a control function proportional to 6. An all movable control area of 300 ft? is considered
to be actuated without time lag according to the equation

Tas vAngtie=— 0. —wekGs

When the gain factor k was varied to determine a suitable value, it was found that k = 1.0 is
just enough at 30 knots to prevent divergent oscillations and that a value of k = 2.0 gives
good control, requiring a tail angle of about twice the maximum pitch angle. Results of com-
puter runs for this gain factor (& = 2) at different speeds are as follows, the values listed
being the ratios of amplitudes of any cycle to that of the previous cycle:

0.92 0.35

0.30 critical damping

critical damping critical damping

It therefore appears possible to control the craft in calm water over the speed range inves-
tigated by means that are quite reasonable. A more thorough analysis that involves the use
of z, z, and 0 may well reveal a still more effective means to achieve controllability.

Sensitivity of the Analysis, and Recommendations Regarding Stability and Control —
Runs were made on an analog computer to investigate the sensitivity of the response of the

646551 O—62——37

558 Edward V. Lewis and John P. Breslin

stable system to variations ‘in the virtual (or effective) mass and virtual mass moment of
inertia. It was found that the rate of decay of stable oscillations is not sensitive to virtual
mass nor to virtual moment of inertia, which is in contrast to studies on deeply submerged
bodies. Also for the case of 60 knots with 300 ft? of tail area, the sensitivity of the re-
sponse to variations of M' and Z/ of +20 percent was determined and found to be quite
small. Because several of the stability characteristics of this craft are so profoundly dif-
ferent from those for which experience has been built up, it would seem worthwhile to con-
duct model tests to check the predictions made here and at the same time to widen the scope
of the study. This might be done through the use of an ultrasensitive motion-following ap-
paratus or with the use of freely operating models. Certainly, the stability in the horizontal
plane will be very important, particularly in view of the excitation in roll and yaw that will
be provided by the long strut. These programs should be backed up by rotating-arm experi-
ments to determine static and rotary derivatives near the surface and also by oscillator tests
to determine the influence of heave and pitch frequencies. Seaworthiness investigations
must also be made, particularly for the case of following seas, where critical operation may
be encountered.

CONCLUSIONS
The possibilities of small, fast ships operating on or near the surface appear excellent.

To permit the design and development of such craft, energetic research should be under-
taken in the following hydrodynamic problems:

1. Propulsion by means of high-rpm machinery,

2. Drag of surface-piercing struts,

3. Forces and moments on high-speed bodies near the surface,

4. Depth control of near-surface bodies in following waves,

5. Optimization of control surface arrangements on submerged bodies,
6. Lateral control of a near-surface body with surface-piercing strut.

Parallel research and development should be carried on in other areas that affect the de-
sign of high-speed craft, particularly in the area of lightweight power plants.

The Navy should be encouraged to design and build experimental limited-purpose ships,
not intended to undertake normal operational duties in the fleet. These ships would be of
inestimable value in guiding future research, development, and design.

ACKNOWLEDGMENTS

The authors wish to acknowledge the assistance of many members of the staff of David-
son Laboratory, in particular Messrs. Edward Numata, Clayton Odenbrett, Paul Van Mater,
and Robert Zubaly, for experimental work on different types of models; Dr. Pung Nien Hu and
Mr. Paul Spens for theoretical and analog computer work; Mr. Charles Garland (now associ-
ated with the J. J. Henry Company, Inc., Naval Architects, N.Y.) for design studies of semi-
submarines, and Mr. Albert Strumpf for his advice concerning directional stability.

Semisubmerged Ships 559

REFERENCES

[1] Boericke, H., Jr., “Unusual Displacement Hull Forms for Higher Speeds,” International
Shipbuilding Progress, Volume 6, 1959

[2] Mandel, P., “The Potential of Semi-Submerged Ships in Rough Water Operation,” New
England Section Paper, Soc. Nav. Architects Marine Engrs., Mar. 1960

[3] Lewis, E.V., and Odenbrett, C., “Preliminary Evaluation of a Semi-Submerged Ship for
High-Speed Operation in Rough Seas,” Davidson Laboratory Report No. 736, Mar. 1959

[4] Lewis, E.V., “Possibilities for Reducing Ship Motions at Sea,” J. Amer.:Soc. Naval
Engrs., Nov. 1958

[5] Lewis, E.V., “Ship Speeds in Irregular Seas,” Trans. Soc. Nav. Architects Marine Engrs.,
1955

[6] St. Denis, M., and Pierson, W.J., “On the Motions of Ships in Confused Seas,” Trans.
Soc. Nav. Architects Marine Engrs., 1953

[7] Mandel, P., “Subcritical and Supercritical Operation of Ships in Waves and the Coinci-
dence of Maximum Damping,” J. Ship Research, June 1950

[8] Numata, E., and Lewis, E. V., “An Experimental Study of the Effect of Extreme Varia-

tions in Proportions and Form on Ship Model Behavior in Waves,” ETT Report 643, Dec.
1957

[9] Wigley, W.C.S., “Ship Wave Resistance, A Comparison of Mathematical Theory with Ex-
perimental Results,” Trans. INA, 1926

[10] Lewis, E.V., and Garland, C., “A Preliminary Parametric Study of High-Speed Schnorkel
Submarines,” Davidson Laboratory Note 582

[11] Breslin, J.P., and Velleur, J.W., “The Hydrodynamic Characteristics of Several Surface
Piercing Struts, Part I, Analysis of Drag at Zero Yaw,” ETT Report 596, Jan. 1956

[12] Wigley, W.C.S., “Water Forces on Submerged Bodies in Motion,” Trans. INA, 1953

DISCUSSION

H. N. Abramson (Southwest Research Institute, San Antonio)

The subject of this Symposium is, of course, devoted to problems of hydrodynamics of
high performance ships. Thus far during the Symposium, however, I have heard a great num-
ber of discussions concerning economics and propulsion and even some structural topics, as
well, and therefore I hope the organizers of the Symposium will not chastise me too greatly
if I too digress, if only briefly. I hope that the authors will also forgive me if I give you a
little story that is incidental, more or less, to the subject of the paper, but which, never-
theless, has some bearing. One final remark by way of introduction: The paper by Professor
Mandel was on the hydrodynamics of a deep-diving submarine, which design was carried

560 Edward V. Lewis and John P. Breslin

through at my Institute. I hope that by my discussion, which is going to be very brief, you
will not get the impression that our institute is concerned only with such very unusual types
of vessels.

What I want to get to is the following consideration: The common yellow mineral known
as sulphur exists in great abundance throughout the world, particularly along the gulf coast
of the United States and Mexico, and in France and Italy. Sulphur is mined by the so-called
Frasch process in which hot water is pumped down into the ground and the sulphur is forced
out in molten form. This sulphur is then usually piled up in very large heaps in which it
solidifies. Then one places a dynamite charge in the sulphur and blows off big chunks which
are then loaded either on barges or on railroad cars and transported to other parts of the world.
We began to think of a different means of transporting sulphur and for this we needed to know
something about its structural properties, but we found out that even the largest sulphur
producers in the world had little or no knowledge concerning the mechanical properties of
sulphur. To make a long story short, sulphur is as strong as fairly high-grade Portland
cement concrete: it has a very high compressive strength and a moderately good tensile
strength. Our idea was, and we have investigated only certain of the economics involved
here, to bring the sulphur up in molten form and to cast it in the form of a ship or boat and
“sail” it to some other port. or destination.

Because of its high compressive strength sulphur has some desirability for a semisub-
merged boat. Unfortunately, its low tensile strength would require some reinforcing material,
but since we had concrete barges during the war, why not a sulphur barge? To reduce wave
impact forces and to improve operation in rough seas, why not a semisubmerged boat made
of sulphur? It might be found rather than self-propelled, perhaps in train, and thus result i in
very large savings in transportation costs.

So, while I am not talking about particularly high performance ships of the semisub-
merged type, and while [ am talking about commercial aspects rather than military, I thought
you might be amused by these speculations and considerations, bearing somewhat on the
subject of semisubmerged ships, that we have been talking about. The stability studies and
other considerations of the authors will certainly be of extreme interest to us if we pursue
this subject further.

DESIGN DATA FOR HIGH SPEED DISPLACEMENT-TYPE HULLS AND A
COMPARISON WITH HYDROFOIL CRAFT

W. J. Marwood and A. Silverleaf
National Physical Laboratory
Teddington

This paper is intended as a preliminary design guide and as a prelude
to a more comprehensive research programme on high speed displace-
ment-type hulls.

_ It includes a short survey of available NPL data on the resistance,
propulsion, and running performance of this type of hull and a brief
discussion on the shallow water effect.

The research programme planned at NPL is outlined and a descrip-
tion given of the first of a series of torsionmeters designed at NPL
specially for use on trials of small high speed craft.

Finally some estimates are given of the power requirements of a
suggested form of mixed craft in which the total weight is supported
partly by the buoyancy of the hull and partly by fully submerged
hydrofoils.

INTRODUCTION

Many small boats, such as patrol launches, work boats, and pleasure craft, operate at
speed/length ratios of 1.2 to 3.5 (or Froude number F, about 0.4 to 1.2). These boats tradi-
tionally have round-bilge displacement hulls, but there is very little published information
about them to help their designers. While systematic series of model experiments have been
made for most types of larger vessels, the relatively high cost of model tests has made it
difficult to persuade owners and builders of small high speed boats to commission model
experiments for new designs. Consequently there has long been a tendency to base hull
designs for small boats on previous forms presumed to have been successful, without any
proper basis for assessment of performance.

This paper is intended as a preliminary design guide and as a prelude to a more com-
prehensive research programme on this type of hull form which is being carried out at the
Ship Hydrodynamics Laboratory of Ship Division, NPL; the large high-speed towing tank
there will enable experiments to be made in better conditions than previously, including
systematic measurements in head and following seas. The present paper is a short survey
of available NPL data on the following aspects of the design of high speed displacement-
type hulls:

1. Resistance of Round-Bilge Forms

(a) Average results for models

(b) Effect of beam/draft ratio and LCB position
561

562 W. J. Marwood and A. Silverleaf

(c) Appendage resistance
(d) Effect of changes in scale.

2. Propulsion

(a) Components of propulsive efficiency
(b) Ship-model comparison.

3. Running Performance

(a) Effect of spray strips
(b) Design of a round-bilge form for good seakeeping.

4. Shallow Water Effects.

This account of the information available at present is followed by an outline of the
programme of research now commencing at NPL, and by a description (in the Appendix) of
the first torsionmeter specially designed at NPL for power measurements on small boats.

High speed hydrofoil boats are not uncommon, and are being actively developed at
present. However, the use of hydrofoils to give partial lift to heavy displacement-type high
speed boats operating at speeds less than the planing speed is not well-established. Some
preliminary estimates of the value of such devices are given in this paper.

RESISTANCE OF ROUND-BILGE FORMS

Normal round-bilge hull forms designed to operate at speed/length ratios between 1.2
and 3.5 (F, 0.4 to 1.2 approximately) have the following distinctive characteristics:

1. The afterbody has a flat or convex bottom having a dihedral angle (or rise of bottom)
up to 45 degrees which runs smoothly into almost vertical sides with a rounded-bilge shape,
not a sharp-angled chine.

2. The buttock lines are usually straight and almost parallel to the centre buttock line
which usually runs aft at 10 to 15 degrees to the waterline.

3. The stern has a flat or rounded transom, except for low speed/length ratio forms
which may have a cruiser-type stern.

4. The bow above water is designed for minimum interference with the bow wave. Any
chine in the forebody is high in profile so as to reduce the breadth at the deck line, and
will not extend as far aft as midships.

High speed round-bilge forms incorporate an unusually large number of design param-
eters. Besides such basic parameters as beam/draft and length/beam ratios, block and
prismatic coefficients, features such as depth of transom and angle of rise of afterbuttocks
may have considerable effects on resistance and propulsion characteristics.

High Speed Displacement-Type Hulls 563

Average Results for Models

In a first attempt to examine the effects of these design parameters, the measured model
resistances for a varied group of about 30 round-bilge forms were plotted as in Figs. 1-11.
The resistance coefficient © for a form of length L = 100 feet is throughout given in
terms of the displacement/length ratio A/(0.01L)* or, what is the same thing for L = 100
feet, the displacement A in tons. Figures 1-6 give the resistance coefficients for a series
of speed/length ratios V/\/[, from 1.7 to 4.0, with the specific value of the beam/draft ratio
B/d shown for each form. These diagrams show no clear evidence of any systematic varia-
tion in resistance with 3/d at any point in.the speed range. In Figs. 8-11 the same results
at V/V/L = 1.7 are replotted, with the specific values of block coefficient, angle of
entrance on waterline, position of LCB and length,/beam ratio shown in turn. Again there is
no clear evidence of any predominant form parameter.

If resistance data were available for a larger number of hull forms, say about 150, it
should be possible to determine the influence of each of the principal form parameters by
carrying out a multilinear regression analysis, using a high speed computer. With the
limited data presently available this is not possible, and as a preliminary step mean lines
have been drawn in each of Figs. 1-6. In addition, lines representing 5 and 10 percent
deviation from these mean lines have been drawn. These show that most of the results lie
within 5 percent of the mean lines. These mean lines are replotted in Fig. 7 to give, for
ships of length 100 feet, average resistance coefficients for speed/length ratios from 1.4 to
3.5 for a range of displacements from 50 to 200 tons. It is suggested that for preliminary
estimates these average values can be used with discretion to give the resistance generally
within 5 percent for good standard hull forms.

Effect of Beam/Draft Ratio and LCB Position

The data in Figs. 1-11 indicate that it is probably unwise to assume the predominance
of one parameter, say B/d, as has been done previously [1], and to ignore other form param-
eters. However, an attempt has been made to assess the effect of beam/draft ratio by com-
paring resistance coefficients for two models, each run at two displacements, which had
different B/d values but were otherwise similar. The comparison is shown in Fig. 12 and
Table 1, and suggests that at V/\/L = 2.5 resistance varies approximately as the cube root
of the beam/draft ratio but at V//Z, = 3.0 the difference is much smaller. A similar com-
parison for two forms having a chine instead of a round bilge, and a fairly high chine line
forward, given in Fig. 13, showed a much smaller variation with beam/draft ratio. Although
these data are inadequate, they have been used to provide preliminary guidance on the
effect of beam/draft ratio, as shown in Fig. 14,

Table 1
Comparison of Resistance Coefficients for Two 100-Foot Round-Bilge Models
with Different B/d Values

Se i ee ©) 100’ (B-A)/A

B-A
(B-A)/A

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Avos ,O0} IN3SWN25vVIeSI0

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565

High Speed Displacement-Type Hulls

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W. J. Marwood and A. Silverleaf

566

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567

High Speed Displacement-Type Hulls

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W. J. Marwood and A. Silverleaf

568

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High Speed Displacement-Type Hulls

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W. J. Marwood and A. Silverleaf

570

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646551 O—62——38

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W. J. Marwood and A. Silverleaf

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High Speed Displacement-Type Hulls 577

20%

ROUND BILGE BOAT
CHINE BOAT

7. INCREASE
IN. RES.

le) {1-0 20
CHANGE OF 9%,
Fig. 14. Comparison of the effect of B/d on
round-bilge and hard chine boats

The available information does not permit any reliable estimate to be made of the effect
of changing the position of the longitudinal centre of buoyancy (LCB), although this is gen-
erally considered to be an important parameter. As shown in Fig. 15 two forms, otherwise
similar, with LCB positions 6.25 percent and 8.81 percent of the length aft of midships had
almost identical resistance coefficients, possibly because the maximum advantage to be
gained by moving the LCB aft had been achieved in the first of them. In both forms the for-
ward waterlines were straight, and by moving the LCB further aft the half angle of entrance
at the waterline, initially 11 degrees, was reduced to 9-1/2 degrees, probably the practical
minimum. It is suggested that so long as the LCB is far enough aft to permit straight
waterlines, there is little to be gained by further movement aft.

Appendage Resistance

Appendages for high speed displacement-type hulls generally comprise: shaft brackets,
either A or / type, propeller shafts, usually angled to the flow, stub bossings at the hull,
skegs, rudders, either single or twin, bilge keels, and bar keels.

The separate resistances of these appendages were measured in one case at NPL by
removing each one in turn from a fully fitted model. The measured model resistances,
expressed as percentages of the naked hull resistance, were: twin “A” brackets, 7.5%;
shafts and stub bossings, not measurable; twin rudders (large), 10.5%; bilge keels (large)
4.0%; bar keels, 5.0%; for a total of 27.0%.

In accordance with current NPL practice for such appendages, this total was halved in
estimating the additional full-scale resistance. At the design speed/length ratio 1.35, this
gave an additional resistance coefficient 5 © of 0.27. For two other models the full-
scale addition 5 © was 0.21; one of these had a single rudder and the other twin rudders,
though neither had bilge keels, which are not usually fitted to launches and similar vessels.
In all three cases the appendage resistance coefficient showed little variation with change

W. J. Marwood and A. Silverleaf

578

Ss

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‘©

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“31

High Speed Displacement-Type Hulls 579

of speed. These measurements suggest that the total resistance for typical appendage
arrangements is more closely estimated by a constant 6 © rather than by a constant pro-
portion of the naked hull resistance. A tentative value for the resistance coefficient of
normal stern arrangements, excluding bilge keels, is thus 6 © = 0.20.

Effect of Changes in Scale

The average resistance coefficients © in Fig. 7 apply directly only to vessels of
length L =100 feet; for vessels of other lengths a skin friction correction must be applied.
The conventional relation for this is

©y00' - Ox = (M91 - %) O Ov%- 779

where the subscripts 100’ and L refer to vessels of length 100 feet and L feet respectively,
“0” is Froude’s friction coefficient, and ©) and © are Froude’s circular wetted-surface
and speed/length constants respectively. This does not hold for high speed displacement-
type forms, but a modified relation does give friction corrections with practical accuracy.
This modified relation is

© 409" - Ox = MOgq - %) © O17

where the factor & may be taken as a function of displacement/length ratios A/(0.01L)?
alone with sufficient accuracy for preliminary design purposes. Values of é and of Froude’s
“0” are shown in Fig. 16. It should be noted that the wetted surface area of a round-bilge
displacement-type form does not vary sufficiently with speed to cause appreciable errors in
using a fixed value of wetted area. For preliminary design purposes it is generally adequate
to estimate the wetted area S from a simple formula such as the Denny-Munford relation

S = L(1.7d + BC)

where Cg is the block coefficient and the other symbols are as previously defined.

PROPULSION
Components of Propulsive Efficiency

Components of propulsive efficiency have been derived from the results of propulsion
experiments made with eight different models. In these experiments the thrust was measured
in the direction of the propeller shafting, and corrected to eliminate the effect of the weight
of the propeller. The resistance was measured as the horizontal force in the direction of
motion, but no attempt was made to compute the fore and aft component of the thrust. Open
water experiments with propellers alone were made with the shaft horizontal.

Values of the thrust deduction fraction t, the wake fraction w based on thrust identity,
the relative rotative efficiency 7p, and the hull efficiency 7, are given in Figs. 17-20 in
terms of the speed/length ratio V/\/L. These show that the thrust deduction fraction ¢
tends to vary much as the resistance coefficient © , with a maximum near V/V, = 1.5 and

580 W. J. Marwood and A. Silverleaf

- 0-02

-0-01

oo SHIP (TONS)

+0-01

+0-02

{00
LENGTH (FT)

Fig. 16. Values of & and of Froude’s “0” for a round-bilge boat

decreasing at higher speeds, finally approaching open water conditions. The wake fraction
w is small, as expected, and slightly negative at high speeds.

From these factors and the propeller open water efficiency 7)» Which depends princi-
pally on the permissible propeller diameter or the engine-propeller gear ratio, the quasi-
propulsive coefficient 7 can be estimated as

71 = No IR:

Ship-Model C omparison

Good data from full-scale trials under ideal conditions are required to determine the
correlation factors linking predicted performance from model experiments with that measured
on shipboard. Little reliable data for high speed displacement-type forms are available, and

581

High Speed Displacement-Type Hulls

yeog a3]tq-punos w roy T/A/A YIM 7 UONOBIy UONONpap isNsyI oY Jo UOTIEIIEA “ZT “SIq

W. J. Marwood and A. Silverleaf

582

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583

High Speed Displacement-Type Boats

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W. J. Marwood and A. Silverleaf

584

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High Speed Displacement-Type Hulls 585

an attempt to obtain further results is being made by the Ship Division, NPL. For this
purpose two portable torque meters, of strain gauge type, have been designed and construc-
ted at NPL to fit propeller shafts having diameters from 1-1/4 inches to 6 inches. A
description of the first of these is given in Appendix A.

Using this torque meter, good trial results have recently been obtained for two twin-
screw launches, one 7] feet in length and the other 51 feet. Resistance experiments were
made at NPL with both forms, and propulsion experiments with the model of the smaller
vessel.

The results of the trials with the 71-foot launch are given in Fig. 21; this shows the
measured power for one shaft (only one torque meter was available) and also calculated

DIMENSIONS OF VESSEL
71'BP X 19-75 BMX5-42 DRAFT X 70 TONS

MEASURED TRIAL POWER

SHAFT HORSE POWER

KNOTS

Fig. 21. Results of trials with a 71-foot round-bilge launch in a water
depth of 50 feet

values of the power absorbed by the propeller for assumed wake fractions wy of 0, 0.05, and
0.10. Closest agreement between measured and calculated power occurs for wy = 0.05; the
estimated value was 0.03. The measured power at the highest speed is greater than the
absorbed power calculated for ws = 0.05; this may have been caused by a 2-1./2 percent
difference in the shaft revolutions. Had both shafts run at the higher speed of that with the
torque meter, the total power, and consequently the speed, would have been slightly higher.
In this case ws = 0.05 would have given close agreement at the highest speed.

586 W. J. Marwood and A. Silverleaf

The full-scale power was estimated directly from the model results, taking 7p = 0.99,
7, = 0.65, 7, = 0.83, and appendage resistance of 0.20 © corresponding to a factor of
0.10. This gave a delivered horsepower (dhp) equal to 118 for a speed of 10.77 knots; wind
resistance estimated from measured velocities increased this by 3 percent to dhp = 121.5.
The measured power was 138; the correlation factor (ratio of measured to estimated power)
is thus 1.135, slightly higher than the current NPL value 1.10 for this type of vessel. Part
of the discrepancy may be due to a small difference in trim between model and full scale.
These trials were run in water of depth about 50 feet, considered sufficient not to introduce
any depth effect.

The trials with the 51-foot launch were run in water of depth 20 feet or less while the
model experiments were made in water of 45-foot equivalent depth. A possible shallow
water effect thus confuses the ship-model comparison; Fig. 22 shows that the greatest
discrepancy between measured and predicted powers occurs at speeds close to the critical
speed for water of depth 17 feet. In this trial also only one torque meter was fitted, but it
was here assumed that both propellers absorbed equal powers. A power estimate based on

250

POWER CALCULATED FROM
GAWN CHARTS USING “SeN

ipa MODEL
200 EXPERIMENTS.

fore sir (10 CORRELATION FACTOR.

{50

SHAFT HORSE POWER

{OO

DIMENSIONS OF VESSEL
5\‘BP x (3-3 BM X 3-42’ DRAFT X 21-5 TONS

10 if {2 3 18
KNOTS

Fig. 22. Results of trials with a 51-foot round-bilge launch in a water depth
of 20 feet or less

High Speed Displacement-Type Hulls 587

a correlation factor of 1.10 and a measured model wake fraction of 0.05 gave good agreement
with the measured power, particularly at higher speeds.

The results of these two trials suggest that a ship-model power correlation factor of
1.10 is reasonable; further trial] data are needed to confirm this estimate.

RUNNING PERFORMANCE
Effect of Spray Strips

At speed/length ratios of 3.5 and above (F, 2 1.2), a round-bilge displacement-ty pe
hull develops a troublesome bow wave. This occurs at about 25 to 40 percent of the length
aft of the bow, and for a 100-foot vessel can be 4 to 5 feet above the still water level.
Unfortunately, this wave often is covered by a fine spray film which starts at the bow
profile and in still water clings to the hull above the main wave profile. In bad or windy
weather this film breaks away from the hull as spray which can cause considerable wetness
of the afterstructure.

Model experiments were made with spray strips in an effort to combat this effect. On
the form shown in Fig. 23 a strip 1 inch x 1 inch (for a 100-foot vessel) was fitted at about
3 feet above the still water line from the bow to midships. This prevented the film rising to
the top side of the model, and enabled the speed to be raised from 30 to 42 knots without
serious trouble.

Fig. 23. Model 2084; 100 feet x 16 feet x 5.48 maximum
draft x 100 tons

It has been found that spray strips either reduce the resistance of a model or have no
measurable effect. Model trim is increased by about 0.5 degrees at the highest speeds.

588 W. J. Marwood and A. Silverleaf

Design of a Round-Bilge Form for Good Seakeeping

For a small vessel, less than 100 feet in length, designed to operate at sea, the per-
formance in waves can be more important than in smooth water. Near-synchronous pitching
conditions may well be met, and waves no more than 3 feet in height can cause slamming
decelerations up to 9 g, injurious to both structure and personnel, as recorded at sea [3] and
observed in model experiments.

The requirements for a small vessel with good seakeeping qualities include: (a) a dry
deck, especially forward of the wheelhouse so as not to interfere with navigation, and (b)
avoidance of slamming, and (c) minimum resistance so that available power is efficiently
used in maintaining speed in a seaway. Other qualities are also required, but these are not
discussed here.

An attempt to design a hull form with these qualities was made some time ago at NPL.
Initial experiments were carried out with models of hard chine forms similar to those used
on wartime air-sea rescue launches. However, for a design maximum speed/length ratio of
3.4, it was soon found that a round-bilge form with a fine bow and a transom stern had a
superior smooth water resistance. Such a form, designated Model 2084, was designed as
shown in Fig. 23, and for comparison a hard chine form, designated Model 2117, shown in
Fig. 24, was also evolved; boats with this hard chine form were known to have had

Fig. 24. Model 2117, 100 feet x 18.8 feet x 4.80 maximum
: draft x 100 tons

to reduce speed considerably in moderate seas. These two forms had the following main
dimensions:

High Speed Displacement-Type Hulls 589

LWL B on WL Max. d A

Round bilge 2084 100 16.0 5.48 100
Hard chine 2117 100 18.8 4.8 100

Smooth water resistance data for the two forms are given in Fig. 25; the round-bilge
form behaved well up to 30 knots (V//L = 3.0), and a spray strip fitted as described pre-
viously helped to keep the hull clear at higher speeds.

Experiments with 1/15 scale models were also made in head seas. In these each
model was towed by a bridle held by hand on the towing tank carriage. Hand towing was
preferred to a rigid attachment to allow slight retardation of the model when its resistance
increased in passing through a wave.

Figures 26-28 show extracts from continuous film records taken of both hulls in the
wave conditions indicated in the figures. These film extracts have been chosen to show
the vessels in their worst position, low in the wave with spray being thrown upwards or to
the side.

The chine form is “stiff” in waves and tends to slam violently. The low chine forward
throws water forward and up, obscuring wheelhouse vision and producing a wet ship. The
round bilge form pitches more but this reduces slamming. The flare forward, which was
designed with particular care, is very effective in throwing water away from the hull. The
film records clearly show the superior wave-performance of the chineless, round-bilge form.
Although the behaviour of the hard chine form could be improved by raising the chine line
forward, it was not possible to reduce its resistance to that of the round-bilge form. It is
hoped that a vessel having this round-bilge form will shortly be built.

SHALLOW WATER EFFECTS

Many builders of small boats do not appear to be aware of the marked effect of depth of
water on wavemaking resistance. Trials are frequently run in river estuaries over accurately
measured distances in water between 10 feet and 15 feet deep, which is considered to be
ample for vessels with drafts of 2 feet to 3 feet. Not surprisingly, often the designed maxi-
mum speed is not achieved.

The 51-foot twin screw launch mentioned earlier demonstrates this shallow water effect.
Its draft was 3.42 feet, and trials were run in about 20 feet of water, for which the critical
speed (v = VgD ) is about 15 knots. Figure 22 shows that the measured and predicted
power curves differ most at about 13-1.,/2 knots. Saunders [2] quotes a method for assessing
the depth of water effect both below and above the critical speed; this is based on data from
trials of a German destroyer, the only available information for speeds above the critical
value. Using this method, and taking the depth of water as 20 feet for the 51-foot launch,
the shallow water effect given in Table 2 has been calculated.

The considerable differences between the power ratios directly measured for the 51-foot
launch and those deduced from Saunders’ data are not surprising, since Saunders states that
other model and full-scale data do not agree with the German results. It is clear that further
information is necessary, and a series of NPL trials in different depths of water with a
27-foot launch are being completed.

646551 O—62——39

W. J. Marwood and A. Silverleaf

590

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High Speed Displacement-Type Hulls 591

MOOEL 2117 MODEL 2084

MODELS OUT OF WATER

SMOOTH WATER
SPEED {5 KNOTS

WAVES - LENGTH {30FT HEIGHT 5 FT
SPEED (5 KNOTS

Fig. 26. Behaviour of the two models of Figs. 23 and 24 in waves

592 W. J. Marwood and A. Silverleaf

SPEED 34 KNOTS
MODEL eli? MODEL 72084

SMOOTH WATER

WAVES — LENGTH 45 FT HEIGHT 4:5 FT

WAVES — LENGTH SOS FT HEIGHT 4:SFT

Fig. 27. Behaviour of the two models of Figs. 23 and 24 in waves

High Speed Displacement-Type Hulls 593

SPEED 34 KNOTS
MODEL 72117 MODEL 72084

WAVES — LENGTH {30 FT HEIGHT 4-SFT

WAVES — LENGTH {90 FT HEIGHT 45 FT

WAVES -LENGTH 315 FT HEIGHT 4-5 FT

Fig. 28. Behaviour of the two models of Figs. 23 and 24 in waves

594 W. J. Marwood and A. Silverleaf

Table 2
Shallow Water Effect for a 51-Foot Launch in 20 Feet of Water

Power in Shallow Water

Speed/Depth Ratio Power in Deep Water

1.285
1.364
1.330
1.313
1.270
1.120
1.035
Probably
becoming ©
<ol

FUTURE WORK ON ROUND-BILGE FORMS

While it is hoped that the preceding summary of data will be of use to the designer, it
has also emphasized the scanty nature of the information presently available. Consequently,
systematic investigations into high speed round-bilge forms are to be carried out as part of
the research programme of the Ship Division, NPL.

The round-bilge form Model 2084 has been taken as the parent form. Systematic changes
in form will cover variations in displacement, beam/draft ratio, and position of LCB, and
features such as the depth and width of transom and the slope of afterbuttocks will also be
varied. The displacement range will cover that shown in Fig. 7, and speed/length ratios
from 1.2 to 3.5, and to 4.0 in certain cases, will be investigated.

Most of the model experiments will be made in the new large No. 3 tank at the Ship
Hydrodynamics Laboratory of NPL. Its greater depth will allow higher speed/length ratios
to be reached without encountering serious interference at the critical speed v/V gD , its
greater size will enable larger models to be run so that reasonable size propellers can be
fitted for propulsion experiments, and the wavemaker control system will facilitate experi-
ments in irregular head and following sea conditions. The propulsion experiments will be
made with at least three propellers of different pitch and diameter for each form so that data
appropriate to different engine-propeller gear ratios will be obtained. Some detailed meas-
urements of hull pressures are also planned.

DISPLACEMENT HULLS WITH PARTIAL HYDROFOIL SUPPORT

Recent studies of hydrofoil supported craft have included some interesting comparisons
with high speed displacement-type hulls (e.g., Ref. 4). These naturally show that the effec-
tive lift/drag ratio, or displacement/resistance ratio, is higher for displacement hulls at low
speed/length ratios, and then significantly higher for hydrofoil craft at high speed/length

High Speed Displacement-Type Hulls 595

ratios (say above V//L = 4.0). The hydrofoil craft considered in these comparisons have
been those in which the complete hull is above the water at the design operating speed,
only a minimum of foil and support structure remaining immersed.

It seems worthwhile to consider whether a mixed-type craft has any advantages. In this
the total weight of the craft and any foil support structure would be balanced partly by
buoyancy, as in a conventional displacement vessel, and partly by dynamic lift generated
by hydrofoils below the surface. Such a mixed type craft would thus move through the water
rather as though it were a displacement vessel operating at a lighter draft than in its deep
load condition. It seems possible that such a craft might have operating advantages com-
pared with a fully supported hydrofoil craft, and consequently some preliminary estimates of
the resistance of such a craft are given here.

Among the available NPL data for high speed displacement hulls are some results for
one form at different displacements. These indicate that, at a fixed speed/length ratio the
displacement/resistance ratio is approximately constant between full and half displacement.
This enables an estimate to be made of the variation of resistance with displacement over
this range. For any assumed reasonable value of the lift/drag ratio for a hydrofoil support
system (which could closely resemble the simple arrangement familiar in ship roll stabil-
isers), it is then possible to estimate the total resistance of the partly supported mixed-
type craft. Making a reasonable allowance for the additional weight of the foil support
structure, it is then possible to compare the total resistances of a series of partly supported
craft based on a single parent normal displacement-type hull. This simple analysis may be
expressed thus:

where r — Resistance of hull remaining in water
A — Displacement of hull remaining in water
L, D — Lift, drag respectively of hydrofoil system
€ — Lift/drag ratio of hydrofoil system
R — Total resistance of “mixed” craft
W — Weight of vessel excluding foil and support structure
S — Weight of hydrofoil and support structure
k — Weight ratio of hydrofoil and support structure to lift of foil system.

Figure 29 shows a group of resistance curves derived on this basis from the data obtained
from a model of a displacement-type craft having a full displacement of 80 tons and length
100 feet. At each speed/length ratio selected, estimates have been made for L/D = 10, 15,
and 20 for the hydrofoil unit, and for k = 0.15. Variations in L/D significantly affect the
result, but reasonable variations in & have little effect.

The shaded areas in Fig. 29 represent regions in which resistance, and thus power
reductions are achieved. It is obvious that as L/D increases, so the possible reductions
also increase. It is also clear that if displacement-type hulls can be designed so that the
displacement/resistance ratio decreases with the displacement, then significant gains may
also be achieved; this is now being investigated.

596

W. J. Marwood and A. Silverleaf

4/1 1417/777/7 SAXDED PORTION REPRESENTS GAIN

IMMERSEO HULL

W - WEIGHT

4 - DISPLACEMENT
T - RESISTANCE

FOIL SYSTEM
$ - WEIGHT
L - WET

o- enue
Es

he NW

COMBINED SYSTEM
W+s 2 AtL
R = RESISTANCE

=T+D

TONS A FOR W 80 TONS

Fig. 29. Possible performance of a partially immersed 100-foot boat and fully
immersed hydrofoils

Preliminary calculations suggest that hydrofoil support systems to carry half the weight

of a conventional displacement type vessel should present no serious hydrodynamic or
structural design problems, even if the foils are restricted to noncavitating types.

ACKNOWLEDGMENTS

The work described in this paper forms part of the research programme of the National

Physical Laboratory, and the paper is published by permission of the Director, NPL. The
assistance of Messrs. James and Stone, Ltd., Brightlingsea, Essex, in providing facilities
for trial measurements on two launches, is gratefully acknowledged.

NOMENCLATURE
General
A = Displacement
L = Length (waterline), feet

SS
Il

Speed, knots

Speed, ft/sec

= Draft, feet

High Speed Displacement-Type Hulls

B = Beam (waterline), feet
g = Gravitational constant, ft/sec”
D = Depth of water, feet
Cg = Block coefficient
Resistance

V/VL = Speed/length ratio

B = Froude number V/ VeL

0100 = Froude skin friction coefficient for 100-foot ship
Or = Froude skin friction coefficient for L-foot ship
© = Wetted surface x 0.09346 / A248

@© = =Froude speed/length ratio = 1.055 V/V.
2
ee 0 (+) L

1.319

VL] Ae
r = Resistance, pounds
© = 2epx 427.1

V3A 28
Propulsion

t = Thrust deduction
Nh = Hull efficiency
1, = Relative rotative efficiency
We = Open water propeller efficiency
n = Quasi-propulsive coefficient
wy = Froude wake fraction
We = Tayler wake fraction
dhp = Delivered horsepower at propeller

ehp = Effective horsepower = © V2A2/ 427.1

Note: The constants used above are for ships in salt water.

REFERENCES

[1] Kafali, K., “The Powering of Round Bottom Motor Boats,” International Shipbuilding
Progress, Vol. 6, No. 54, Feb. 1959

597

598 W. J. Marwood and A. Silverleaf

[2] Saunders, H.E., “Hydrodynamics in Ship Design,” Vol. 2, p. 408

[3] DuCane, P., “The Planing Performance, Pressures and Stresses in a High-Speed
Launch,” Trans. I.N.A. 98:469 (1956)

[4] Crewe, P.R., “The Hydrofoil Boat; Its History and Future Prospects,” Trans. I.N.A.
100:329, Oct. 1958

APPENDIX A
NPL Small Portable Torsionmeter No. 1

This torque meter (Fig. Al) was designed to be readily portable, and to fit shafts
having diameters from 1 to 6 inches without any need for removing any part of the propeller
shaft. To meet these requirements a strain-guage-type meter was designed, with the gauges
mounted directly on the shaft and their output leads taken through a readily dismountable
slip ring assembly.

Four small resistance strain gauges are bonded to the shaft by an appropriate strain
gauge cement at a position close to, and inboard of the stern gland. The gauges lie on 45-
degree helices on the shaft surface, so placed that there is symmetry relative to the shaft
axis and also to a plane normal to the shaft. Hence gauges respond only to torsion in the
shaft and the gauge output is not affected by bending or thrust in the shaft. The gauge

Fig. Al. Torsionmeter No. 1

High Speed Displacement-Type Hulls 599

leads are taken to a terminal block fixed to the shaft which is, in turn, connected to the
inner unit of the slip ring assembly.

This slip ring assembly has two main units: an inner unit fixed to the shaft which
carries the slip rings themselves, and which rotates with the shaft, and a fixed outer ring
which carries the pickup brushes. Both units are made in two halves, so that they may each
be placed around the shaft and bolted together. The inner unit of the slip ring assembly is
large enough to fit around a 6-inch-diameter shaft; it is fitted to shafts of smaller diameter
by inserting a distance piece in one half, and clamping the other half in place by adjusta-
ble, screwed feet. The outer unit carrying the pickup brushes is kept in position by rubber-
tired rollers both radially and axially.

The five slip rings are of silver-plated brass, accurately scarphed at the joints of the
two half units. Four rings are used for gauge leads, and the fifth for counting revolutions of
the shaft. The brushes are of silver graphite. The gauge outputs are measured either on a
Baldwin and Southwark SR 4 or a Siemens strain indicator. The complete meter is calibrated
statically by applying known torques to the shaft close to the propeller when the vessel is
out of the water.

The system has proved very reliable. The slip rings have behaved well, the strain
readings are steady, and there is no appreciable zero drift.

DISCUSSION

Peter du Cane (Vosper Limited, Portsmouth)

We at Vosper have been interested in this paper as it discusses the range of high
speed craft in which we have specialised over the last 25 years.

Within reason we agree that for V/\/Z, = 3.4 a round-form hull as suggested by the
authors would be our selection though only because if 3.4 is assumed as a maximum the
probability is that for continuous cruising V/V, = 2.5 might be expected. At V//I = 3.4
we could certainly produce a hard chine planing form with better resistance qualities than
the round form and in our opinion with at least comparable seakeeping qualities.

It may be conceded that the subject of the relative merit of hard chine versus round
form in waves is very relevant to this Symposium so that it is proposed to discuss it at
some length:

Reference 4, which is a paper I submitted to the R..I.N.A. in 1956 and which was
mainly concerned in recording extreme values, is quoted in justification of the following
judgement on the qualities of the hard chine planing form: “waves no more than 3 feet in
height can cause slamming decelerations up to 9 g, injurious to both structure and person-
nel, as recorded at sea.”

It is felt there must be some misunderstanding of the data recorded in Ref. 4. Cer-
tainly there was no question of 9 g being recorded in 3-foot waves. If this is inferred

\

600 W. J. Marwood and A. Silverleaf

anywhere it is incorrect, though it is conceivable that such an acceleration could be
recorded at say the stem in exceptional conditions of speed and irregular wave formation.

In the particular case of this reported research in Ref. 4, however, the objective was
to find extremely adverse conditions which did not prove easy and eventually resulted in a
slightly hazardous attempt to run across the bow wave train of the “Nieuw Amsterdam” at
35 knots.

As this bow wave train was approaching us at a speed estimated to be 14 knots the
conditions of encounter were equivalent to the launch meeting the waves at (35 + 14) = 49
knots.

Under these circumstances it is not surprising that 6 g was experienced in the wheel-
house. The maximum acceleration at the stem was recorded as 8.28 g but this was in
exceptionally adverse conditions.

The relative merits of the round form and the planing form (so called) are discussed at
some length in Ref. 4 and in the discussion thereto.

In this connection it is interesting to mention N.P.L. Report S.H.MV. 5 of Feb. 28,
1955, where a hard chine and round form were tested in waves. It was reported that there
was little to choose as regards vertical accelerations between the hard chine and the round
form. If anything the accelerations appeared less in the case of the planing form and cer-
tainly the spray on decks was reported as less in the case of the planing form.

This, of course, does not accord with the comment in the paper we are now discussing
which reads as follows:

“The chine form is ‘stiff’? in waves and tends to slam violently. The low chine forward
throws water forward and up, obscuring wheelhouse vision and producing a wet ship. The
round bilge form pitches more but this reduces slamming. The flare forward, which was
designed with particular care, is very effective in throwing water away from the hull. The,
film records clearly show the superior wave performance of the chineless, round-bilge form.
Although the behaviour of the hard chine form could be improved by raising the chine line
forward, it was not possible to reduce its resistance to that of the round-bilge form. It is
hoped that a vessel having this round-bilge form will shortly be built.”

Among other comparisons between planing and hard chine craft which are available in
Ref. 4 is a closely reasoned and cautious judgement made in the course of the discussion
by Dr. Gawn based on his great experience at the Haslar Tank and his collated data from
sea reports. His contribution is offered verbatim:

“The author makes no secret of his preference for a hard chine hull and it does appear
this type can be drier and generally no less satisfactory at sea than the round bilge. There
is a need for firm fact to replace some of the contentious opinions often expressed on sea
behaviour, and the author’s tests go some way in this direction. However, the issue is not
clear cut in a general sense but depends on the size and speed of the craft. If, for example,
the speed-length coefficient is less than about 3.7 there is much to be said for the round-
bilge form because of its resistance advantage in calm water. If, on the other hand, as in
the launch dealt with in the paper the speed is much greater, then the hard chine hull can
have a clear advantage as regards resistance in calm water and in my judgement is to be
recommended, provided there is no overriding emphasis on economic cruising at low speed.”

High Speed Displacement-Type Hulls 601

Since the date of that paper, quite a lot more evidence on this matter has reached me,
both from actual seagoing experience and model tests in waves. In particular and resulting
from model tests carried out in irregular waves at the Davidson Laboratory of the Stevens
Institute in 1959, Figs. D1 and D2 were reproduced in a report of the “Development and
Running of the ‘Brave’ Class Fast Patrol Boat” read at Goteborg in February 1960.

SO fee as ee a
— |

MODEL TYPE
56 VOSPER HARD CHINE
57 PLANING FORM,
WIDE & ROUND FORWARD
ROUND FORM L+20%

eit eal

AVERAGE BOW ACCELERATION (G's)

0 10 20 30 40
AVERAGE SHIP SPEED (KNOTS)

Fig. Dl. Average bow acceleration versus average ship speed
in an irregular head sea; Beaufort scale sea state 5; average
height, 5.16 feet; signal height, 8.10 feet

MODEL TYPE
56 VOSPER HARD CHINE
57 PLANING, WIDE, :
ROUNDED FORWARD
58 ROUND FORM

MAXIMUM BOW ACCELERATION (G's)

0 10 20 30 40
AVERAGE SHIP SPEED (KNOTS)

Fig. D2, Average bow acceleration versus average ship speed
in an irregular head sea; Beaufort scale sea state 5; average
height, 5.16 feet; signal height, 8.10 feet

This shows remarkably little difference between the round form 58 and the Vosper hard

chine 56, especially when it is realised the round form was 20 percent longer for the same
displacement.

To summarise, there is little to choose between the forms for speed/length ratios

between say V/V7, = 2.5 to 3.8. After this the hard chine planing form is a “must” for
reasons of resistance.

602 W. J. Marwood and A. Silverleaf

The planing form probably can be made to slam more unless trim is adjustable by
transom flap or similar device. Equally the round form can undoubtedly be made to bury her
forecastle in the waves and wash down her bridge. Directional stability is much better in
the planing form and there is less liability to yaw and broach in a following sea. In head
seas the hard chine craft will lose more speed due to the energy lost in generating spray
which undergoes an acceleration relatively forward and outward from rest. This feature
tends to keep the decks and bridge dry.

So far we have only discussed the relative merits of the round form and hard chine
planing craft and it may be appropriate to point out here that in the light of development
over the years the difference between the two classes of craft which used to be clearly
defined are becoming very much less.

For instance, both types are in fact planing craft in the generally accepted sense. In
both types it is fully appreciated that the waves must be met from ahead by relatively soft
V-shaped or rounded sections. Equally, to obtain good resistance qualities the aft sections
in both types are relatively flat and wide and run to a “chine” at the sides. To control the
amount of spray on deck both types work a substantial flare in forward sections leading to a
“knuckle” or chine.

In fact it is probably fair to say that in theory the optimum would be represented by a
rather wide form incorporating the forward sections of a round-form craft with the aft sec-
tions of a planing form. The draw-back here is that the round form leads to wetness on deck
and width leads to rather excessive slamming.

Such a compromise form is represented in Figs. D1 and D2 by model 57.

As the subject of the Symposium covers high speed craft it may not be out of place
here to state the admittedly personal opinion that no hydrofoil craft of the surface piercing
type has given any real evidence of being able to deal with substantially rough conditions
in the open sea more successfully than the round form or hard chine.

It seems possible the submerged foil arrangement as demonstrated by Dr. Hoerner
during this Symposium may be superior in this respect but at the moment this awaits prac-
tical proof at sea, at least in the larger sizes.

The ground effect machine may seem to be limited at the present state of the art by the
very severe effect upon lifting efficiency likely to be experienced if cushion heights suit-
able for passage over the ocean waves are to be generated.

Finally as a constructive suggestion we should like to suggest to the authors that we
could submit for trial a hard chine planing form which would prove a much more serious com-

petitor to the round form proposed than is the form exemplified in model 2117.

It would also be most valuable to run a hydrofoil configuration of comparable dimen-
sions and load-carrying capacity in comparison with the above forms.

D. Savitsky (Davidson Laboratory, Stevens Institute of Technology)

I would like to comment on the interpretation given by the authors to the relative sea-
keeping abilities of the so-called hard chine and round-bilge boats. It is the authors’

High Speed Displacement-Type Hulls 603

contention that the round-bilge boat exhibits superior performance (less pitching motions
and considerably less bow spray) than the hard chine boat. The point that I would like to
clearly emphasize is that the existence of a hard chine or round bilge per se has no rele-
vance to its behavior in waves. Heavy bow spray and large pitch motions are primarily de-
pendent upon the fullness of the bow. For very sharp bow forms (high deadrise forward) the
rough water behavior will be relatively mild and the development of bow spray very small.
The reverse is true for a full bow (low deadrise forward). It is to be noted from the authors’
paper that their round-bilge boat, which exhibited relatively mild rough water motions was
indeed designed with large deadrise forward while their poorly performing hard chine boat
was designed with low deadrise forward. With such design features it is expected in advance
that the hard chine boat will perform poorly in waves. If the full bow is, for reasons of con-
struction, necessarily inherent in a hard chine boat then the authors’ general comments on
the relative merits of each boat are correct. However, if this is not the case, that is, if high
deadrise forward can be designed into each boat, then general conclusions on rough water
behavior of round-bilge versus hard chine boats cannot yet be made.

In reply to the authors’ plea for basic force data on planing boats, I would like to sug-
gest the many reports prepared by the Davidson Laboratory of the Stevens Institute of
Technology on this very subject. These basic experimental and analytical studies were
sponsored at our laboratory by the Office of Naval Research.

E. V. Telfer (Technical University of Norway)

All I want to say is once more in connexion with presentation. In this paper the
authors make the very positive statement that “again there is no clear evidence of any pre-
dominant form parameter.” Against this I would like to suggest that what the authors really
mean to say, but clearly do not, is that because they have adopted a basic presentation in
terms of the displacement of the 100-foot ship, which is of course the Froude @ = L/V¥8
value in another form, this is clearly in itself the dominant form parameter and any other
usual form parameter is likely to be of minor importance. Had the authors’ diagrams such as
Fig. 8 been presented in terms of @ the importance of relative length on constant dis-
placement would have been more evident and the distribution of the individual spots would
have more uniformly covered the @ base.

The issue is, however, not quite as simple as this. It should be noted that while at
any constant displacement the © values correctly grade the corresponding powers, a com-
parison of the © value at one displacement does not give a direct comparison of the cor-
responding power ratio at any other displacement. The authors have made their © com-
parisons under the condition of constant V/\/Z,, or since L = 100 feet, under the condition
of constant speed. It follows therefore that to get the relative power ratios P and P2 for
two different displacements A and Ag, the respective © values must first be multiplied
by the two-thirds power of their displacements. Alternatively to express the true relative
power per ton displacement over the displacement range covered the © values firstrequire
multiplication by @ . When values of © @ are plotted to the displacement base, then
the comparison is both correct and compatible. The authors’ presentation is neither. For
example, in Fig. 8 for 200 tons displacement a © value of 4.9 may be indicated, while at
50 tons the © value is probably only 2.45 at most for the same B/d ratio. The correspond-
ing © @ comparison is thus 4.9 x 17.1 = 84 against 2.45 x 27.2 = 66.6. The 50-ton boat
thus requires 79.4 percent of the power per ton displacement needed by the 200-ton boat.

604 W. J. Marwood and A. Silverleaf

The authors’ presentation might have tempted the unwary to believe that the 50-ton boat
was twice as efficient as the 200-ton boat. It follows therefore, since the authors’ presen-
tation has very seriously exaggerated the apparent importance of @ value, that the influ-
ence of other factors will not necessarily have quite the insubordinate influence which the
authors are led to believe. I am not, of course, suggesting that the authors themselves in
their specialised tank practice would be at all unwary; but I do feel very strongly that all
data presentation for general professional consumption should be visually correct, so that
he who runs may read, and read quickly and accurately.

\

R. N. Newton (Admiralty Experiment Works)

Any paper which provides data which can be usefully employed by a designer is very
welcome at any time or place. This paper by Messrs. Marwood and Silverleaf is just such a
paper and any opportunity to add to the information they present should not be missed, for
the same reason.

Experiments on high speed planing forms having been carried out at A.W. for many
years, with forms of different lengths and coefficients, in calm water and in waves, and with
various devices for improving performance, I will attempt to take this opportunity to qualify
some of the statements in the paper and add to them, in a general way, in terms of the ratio
of length of wave to length of craft and in terms of Froude number.

Here let me hasten to add that while I incline very much to Prof. Telfer’s opinion that
we should use more rational performance coefficients and parameters to show the trends
arising from changes in form and displacement for the present purpose I must adhere to the
commonly used ones, such as Froude number and length of wave to length of ship ratio.

First, then, speaking generally, on the effectiveness of steps in the bottom, a large
number of resistance experiments conducted at A.E.W. between 1935 and 1944 led to the
general conclusion that although stepped hulls had advantages over unstepped hulls in some
ways, they also had inherent disadvantages and particularly increase of resistance at low
speeds.

Experiments conducted with two, three, and five steps in the bottom, at different
spacings and positions along the bottom indicated that these gave generally inferior perform-
ance to a single step on the same form, particularly as regards an increased tendency to por-
poise and, judged by tests of one form in waves, inferior behaviour in waves.

There is an optimum position for the step which effects a comparison between reduction
in resistance above planing speed, increase in resistance below planing speed, and the
liability to porpoise. The risk of porpoising increases as the step is moved forward. As
the step is moved aft the speed at which planing starts increases until when the step is
well aft the craft can hardly be said to be planing at all in the usual sense of the word.

Secondly as regards resistance in calm water, the position of the center of gravity is
critical and a small movement has a marked effect. Generally speaking a moderate stern
trim in still water is an advantage at planing speeds but adds to resistance at lower speeds.

Too large a dihedral angle amidships adds to resistance due to increased running in-
cidence and reduced rise. A low dihedral angle is good for calm water performance but
detracts from behaviour in waves.

High Speed Displacement-Type Hulls 605

A narrow transom gives increased running trim but the effect on resistance is compli-
cated. There is an optimum transom width depending on speed, trim, loading, and other form
parameters. For high speed a wide transom is desirable, but if the width is excessive stern
trim is impeded. If the width exceeds 0.8 of the maximum beam the sides of the hull may
not be “clean” at top speed.

Running attitude at a given speed is mainly governed by loading and static trim,
although chine dihedral, curvature, and width of transom can have appreciable effects as
noted previously. Excessive trim causes high hump resistance which can lead to “locking
up” of the engines. The effect of propellers is to increase the running trim by as much as
1 degree.

Next as regards resistance and motion in waves, as the authors have shown, a form
which has a low calm water resistance by virtue of high running trim can be a poor boat in a
seaway, particularly as regards pitch and slamming. On the other hand too small a trim
leads to wetness and pounding even though slamming and motion may be reduced.

In L/30 waves maximum motion in ahead seas usually occurs at A/L between 2.5 and
3.0 when the pitch is about 1.] to 1.2 x maximum wave slope, out to out, and the heave
about 1.1 x wave height.

In L/20 waves the maximum motion occurs at A/L about 2.0 or less, pitch being about
1.35 x maximum wave slope and heave 0.9 x wave height.

The motion is characterised, at planing speeds, by notably small movement at the stern.

Mention is made in the paper of accelerations as high as 9 g and this has been referred
to by previous speakers. Such accelerations can be produced of course but conditions on
board would hardly be tenable and there would be grave risk of structural damage. Local
accelerations of this order occur of course when the craft slams.

For a given maximum acceptable acceleration, which may be decided for strength or
other reason, there is a limiting speed and wave slope; e.g., in a 70-foot boat if the maxi-
mum acceptable acceleration is 2 g the boat should not be driven into seas steeper than
L/20 at speeds greater than F, = 0.7. Steeper waves can only be encountered, to keep
below 2 g, if they are shorter than ship length.

Now as regards the advantages of round bilge and hard chine forms there is much that
can be added to the information given in the paper and among some of the more important
facts are these.

In calm water the round bilge form is less resistful than the hard chine form up to at
least V//L = 3.2 (F, = 0.96). Thereafter the hard chine form shows to advantage. This
agrees with Fig. 25 of the paper being discussed.

In waves the resistance of the round bilge form is still less than the hard chine form up
to V/VL = 3.7 (Fn = 1.1), but only in waves up to 3.0L when the difference is of small

magnitude only.

As regards motion in waves it is necessary, I think, to qualify the statement in the sub-
section of the paper titled “Design of a Round-Bilge Form for Good Seakeeping” in terms of
Fe and E/Ls:

646551 O—62——40

606 W. J. Marwood and A. Silverleaf

Up to a certain speed, usually about F, = 0.7, both pitch and heave in ahead seas are
greater for the round-bilge boat, and less than the hard chine boat beyond this speed. This
applies in waves up to 1.5 or 2.0L. In longer waves the round-bilge form continues to pitch
more but there is little difference in heave between the two forms.

In the following seas the pitch of the round-bilge form is slightly greater in waves up
to 2.0L and slightly less in longer waves. The reverse is true of heave.

In shallow seas the accelerations at the fore end of the round-bilge form are slightly
higher than for the hard chine design, and this corresponds with the slightly greater pitch
and heave. Accelerations at the after end are much the same for both designs. In steeper
seas the accelerations of the hard chine form are greater both at the fore end and at the
after end.

Slamming is generally more pronounced in the hard chine form for the same still-water
loading condition as regards displacement and running trim.

With regard to wetness, as would be expected the water is thrown low and clear by the
hard chine forms and higher and nearer the hull in the round bilge form, although both are
generally dry in seas dead ahead. The effect of wind on the bow is to increase the degree
of wetness, and the round-bilge form is wetter than the hard chine in this condition.

Finally there is one feature of high speed craft, of which the paper makes no mention,
which I should like to remark on briefly, viz., the use of a transom flap in the form of a
plate hinged at the lower edge of the transom so as to be adjustable and generally about
1,/20th the length of the craft in width.

Model experiments show that in calm water running, stern trim and rise are reduced by
the use of a transom flap at all speeds, from which it might be expected that the resistance
would also be reduced at all speeds. In fact the measured model resistance is reduced by
the use of a flap only up to a moderate speed, beyond which it increases, compared with no

flap.

Nevertheless sea trials have shown that the flap improves the speed up to full power
due to the change in interaction between hull and propellers having a favourable effect upon
the hull efficiency elements. For a model of a 70-foot-long hull the augment of resistance
was reduced and the wake increased to give an increase in hull efficiency of 8% at top

speed, for a flap angle of 5 degrees.

Both model experiments and sea trials have demonstrated, however, that as the maxi-
mum speed increases, or as displacement decreases, the flap becomes less effective and
smaller angles of incidence of the flap are required to obtain maximum effect. Ultimately a
condition will be reached when the flap is of no propulsive advantage. One explanation, or
part explanation, for this may be that in designs with high top speed, not fitted with a flap,
the running trim ceases to increase above a certain speed and then begins to reduce, although
rise may still continue to increase. Full consideration must be given to this fact when con-
templating the incorporation of a transom flap in a new design.

As regards behaviour in a seaway in waves less than ship length and height, in the
region of L/30 the flap has an appreciable effect upon pitch or heave at any speed. In
waves of critical length about 2.5L when motion is usually severe the flap reduces the
pitch and heave by about 10 percent.

High Speed Displacement-Type Hulls 607

In longer and steeper waves around 2 ship lengths and height L/20 and at F;, greater
than 1.] there is a more significant decrease in both pitch and heave of the order of 30
percent and 45 percent respectively.

At these high speeds, however, there is a grave danger of the craft plunging. In fact
during one experiment with a model at F, = 1.4 the model buried so deeply into the slope
of the oncoming wave as to cause it to sink. Obviously in such conditions the ship would
suffer grave damage or loss.

In following seas the use of a flap or wedge introduces a distinct tendency of the
model to bury into the slope of the wave —much more so than in ahead seas. For instance,
in the case mentioned previously the same model, in waves 1.5L using 5-degree flap inci-
dence, plunged and sank at F, = 1.4, The same occurred also at the same speed in waves
2.5L. In longer wavelengths the model could only be controlled with difficulty and then
only up to F, = 1.06, when there was still a tendency to broach-to.

As a result of close study of films of models in waves and experimental data a general
conclusion can be drawn. This is that whereas a transom flap can prove of advantage pro-
pulsively in calm weather, the degree of this advantage being determined by consideration
of the top speed and displacement, before the flap can be used to improve the motion in
waves, i.e., to make conditions on board more comfortable, it is essential for the operators
to be able to recognise the sea conditions at the time, or to be provided with some scien-
tific indication that the craft might be in danger. There is no doubt that with long experi-
ence of such craft, judgment can be placed upon the “average” length and height of sea
which may be running, but this is very different from being able to predict the length and
height of oncoming waves. For the time being, therefore, until much more research has
been done into the statistics of wave spectra in a given ocean area, and until such time as
the spectrum can be indicated with some degree of reliability, the use of transom flaps in
rough weather cannot be recommended —at least not for craft designed for really high speed,
say beyond 45 knots.

In conclusion it is perhaps pertinent to remark, in the light of these general observa-
tions that any change of form, or device, introduced into a new design, although it may
improve performance from one aspect, may well detract from it in another aspect, and a
compromise is very often necessary, and can only be found by careful model experiments.

K. C. Barnaby (John I. Thornycroft Company, Southampton)

The authors of this interesting paper referred to the difficulty of finding a suitable
parameter for expressing their data. It is, of course, obvious that a conventional constant
such as © is quite unsuitable. Once dynamic lift has taken charge, the wetted surface is
no longer constant, but must diminish rapidly since the product C,%pSV* must always be
less than the original displacement weight, by at least the amount of submerged hydrostatic
buoyancy. Also, the resistance to motion increases with approximately V and very defi-
nitely not with V?. Thus both the terms A2/3 and V3 become incorrect at high speeds.

The one factor that cannot change as the speed increases is the actual weight and this
forms the basis of a suitable parameter. If one plots the ratio R/AV, that is, the total
resistance expressed in pounds per ton divided by the speed in knots, for a common parent
form at various displacement ratios and speeds, a family of curves is obtained that, at suf-
ficiently high speed/length ratios will merge into one line that is nearly independent of

608 W. J. Marwood and A. Silverleaf

displacement ratio and varies only slightly with speed. In general, there will be a slight
increase with speed, except in the case of round-form hulls, which usually show a slight

falling off.

This parameter R/AV is conveniently expressed in the modified form of K = V/ohp/A >
where K is a constant depending on the type of craft and the static waterline length (or per-
haps more correctly the beam as being an unaltered dimension). I have examined several
hundred cases. Those giving R/AV directly have mainly come from tests at Mr. Thornycroft’s
Fort Steyne Model Basin. The more numerous K values have come from trial results and from
published data. In no case have I found any wide discrepancy from Table Dl. These latter
are of course only approximate, but in naval architecture practice it is often more useful to
have an approximation that can be made in a few minutes than a more elaborate method which
takes a considerable time. The approximation can at least show whether a proposal is fea-
sible or quite impossible.

Table D1
Values of the Constant K in the Formula* K = V//bhp/A
for Planing and Semiplaning Types

Value of K for Various Types of Hull and Limits of Speed/Length Ratio

Round bottom,
transom stern,

V-Chine,

Stepped Hydrofoil

very flat aft sigplese
V/VL = 2.5-3.5| V/VL =2.75-4.5|V/VL = 3.5-6.5 | Up to 60 knots
3.6 5.3
3.96 5.5
4.3 5.7
4.6 5.9
4.8 6.1

*/\ in tons and V in knots.

Surprise may be expressed at the inclusion of round-bottom craft in the table. It is
however necessary for such a type to have a transom stern and a very flat stern in order to
reach speed length ratios of over V/\/[, = 2.5. Under these circumstances, the stern lines
are not very different from those of a normal V-chine type and a nearly comparable amount
of dynamic lift can be obtained at high speeds. The increased drag due to the round bilge
is however reflected in lower K values. It is assumed in all cases that the form is suitable
for the speed and runs cleanly with a suppressed bow wave that does not lap high up the
side but is deflected outwards. It is also assumed that the propulsive efficiency is normal,
say, about 0.6.

Powering figures stated by Mr Rader, during the meeting, give a K of 4.76 at 40 knots,
increasing to 4.90 at 55 knots. These were for a 100-ton V-chine hull of about 50 percent
increased length over the maximum table length. They are thus in line with the table
values.

High Speed Displacement-Type Hulls 609

J. B. Hadler and E. P. Clement (David Taylor Model Basin)

The authors have suggested that a high speed displacement-type hull combined with
partial hydrofoil support may show performance which is superior to that of a conventional
boat. A number of years ago the Model Basin gave consideration to the same suggestion
and established a program of investigation under our Bureau of Ships Fundamental Hydro-
mechanics Research Program. A partial support hydrofoil system, composed of two sub-
merged foils, each on a strut attached to the side of the hull were installed on a model of
the U.S. Navy 52-foot aircraft rescue boat. These foils were arranged such that their fore
and aft location could be varied as well as their angle of attack. The optimum location for
these foils was 32.7 percent of the length forward of the center of gravity, and at a minus
3-1/2 degree angle to the keel of the boat. At this condition the resistance was reduced
throughout most of the higher speed range. For speeds below 15 knots, the resistance of
the hybrid craft was somewhat greater. Between 15 knots and 30 knots the resistance
decreased to about 75 percent of that of the conventional craft. Between 30 knots and 40
knots, the average reduction in resistance was about 25 percent. Although these results
indicated favorable performance for the hybrid craft, no further work was undertaken at the
Model Basin because of the urgency of the hydrofoil program itself. Further research should
be continued in this area, particularly in investigating the performance of the hybrid craft in
seaways and with foils of the surface-piercing type. The full details of this work are con-
tained in a Model Basin report entitled, “Tests of a Planing Boat Model with Partial Hydro-
foil Support,” TMB Report 1254 dated August 1958.

The resistance data for high speed displacement-type hulls which are made available
in the paper are very welcome. A similar presentation was made by H. F. Nordstrom in his
1951 publication, “Some Tests with Models of Small Vessels,” but in that paper the maxi-
mum speeds are considerably below the maximum speeds for which data are given in the
present paper. The data on the components of propulsive efficiency are particularly inter-
esting and valuable, since little information of this kind has been previously made avail-
able. It is noted that there is considerable scatter in the plots of these data, however,
presumably because of significant difference in the hull forms and appendages of the boats
represented. The usefulness of the data would be enhanced considerably if the authors
would provide some additional information on those designs which were self-propelled. A
profile drawing of the stern, including the appendages, would be particularly helpful. This
would make it possible for designers to select values of wake and thrust deduction for the
design most like the one with which they were concerned at the moment, and thereby obtain
assistance in making accurate predictions of performance.

W. J. Marwood and A. Silverleaf

The authors first wish to thank all those who contributed to the discussion of this
paper; their comments have greatly enhanced its value.

Several contributors have discussed the relative merits of round-bilge and hard chine
hull forms; however, the aim of the paper was solely to provide data on high speed
displacement-type hulls, and there was no desire to make comparisons of a general nature
between different basic forms. Commander du Cane’s views on this important practical
point naturally deserve close attention, but it is necessary to use carefully defined terms
when comparing hull forms. The round-bilge form as defined and described in the paper is
without any chine or knuckle line which could affect the flow, and is in extreme contrast to
the hard chine form used in some of the experiments in waves. Forms of these two types

610 W. J. Marwood and A. Silverleaf

can be designed to differ much less, and the differences between them in performance will
then also be much less. Indeed, the round-bilge form as quoted by Cdr. du Cane retains a
chine line forward and aft intended to influence the flow pattern.

There seems to be some doubt as to the correct interpretation of some of the results
given by Cdr. du Cane in Ref. 4. Table IV of that paper gives for Run 35 the maximum
acceleration in the fore peak (position Al) as 8.36 g in waves 2 to 3 feet high. On the
other hand, Table III gives a value of 2.86 g for the same run; this may be an average value
of the measured acceleration, and agrees well with model values measured at N.P.L.

The experiments quoted by Dr. Graff, carried out in Duisburg with high power round-
bilge forms in deep and shallow water, are of considerable interest. A full account of these
experiments has been given,* and the results show the same characteristics as those given
here for similar experiments. However, detailed comparison shows some differences in the
absolute values of resistance coefficients, and these discrepancies emphasise the need for
further investigations of shallow water effects.

Mr. Newton’s contribution is a most valuable addition to the paper, for which we are
very grateful. In drawing on the extensive store of information available at Haslar he has
raised many points which we are not qualified to discuss, and we shall restrict our com-
ments to four of the topics he mentions. First, we agree that a narrow transom does give
increased running trim, but only for speed/length ratios V/\/L greater than 1.8, and it then
also increases the resistance. However, at lower speeds, particularly below the resistance
hump around V/VZ, = 1.5, when there is little or no hydrodynamic lift, a narrow transom is
found to decrease resistance. Indeed, at lower speeds still it becomes advantageous to fit
a cruiser stern. Second, we are slightly surprised at Mr. Newton’s statement that propellers
increase running trim significantly. Provided the model is towed along the shaft axis when
the resistance and trim are measured, we should not expect the trim to alter appreciably
when it is self-propelled, unless the propellers have a marked influence on the average sur-
face pressures over the after body. Third, we agree that at planing speeds the pitching
centre is well aft; indeed, it can be aft of the transom. Fourth, experiments made at N.P.L.
support Mr. Newton’s views about transom flaps. Attempts to turn or “hook” the extreme
afterbuttock lines have also succeeded in reducing calm water resistance. However, we
believe that, in a vessel designed to operate mostly at one speed, transom flaps are only
needed when the basic hull design is faulty. For a vessel intended to operate at several
different speeds, the fitting of flaps is more easily justified. Some interesting effects of
flaps on heave and pitch have been observed in experiments at N.P.L., and these agree well
with the comments made by Mr. Newton.

Mr. Savitsky points out that the terms round bilge and hard chine are not satisfactory
descriptions for comparing ship performance in waves. We agree with this criticism, and
agree also that the deadrise angle forward is a more realistic criterion to adopt. The
principal justification for the descriptive terms used in this paper is that they are commonly
employed by naval architects, and it was hoped that they would be sufficiently precise for
the purposes of this preliminary report. We must also apologise for not referring explicitly
to the Davidson Laboratory reports on planing craft.

*W. Sturtzel and W. Graff, “Systematische Untersuchungen von Kleinschiffsformen auf Flachem

Wasser im Unter- und Uberkritischen Geschwindigkeitsbereich,” Report 617, Versuchsanstalt
fir Binnenschiffbau, Duisburg, 1958.

High Speed Displacement-Type Hulls 6l1l

Professor Telfer rightly draws attention to our statement that “again there is no clear
evidence of any predominant form parameter.” We agree that, for clarity, this would have
been better if qualified by the important reservation “apart from displacement/length ratio
or its equivalent @ .” We also agree that a presentation based on @ would be better
than one with displacement alone as basis. Professor Telfer’s advocacy of © @ as the
best criterion for power comparisons at different displacements also has much to commend
it. However, we were primarily concerned with presenting the basic information as simply
as possible, and presumed that those who used it would know how to do so correctly.

Mr. Barnaby also advocates an alternative basis of comparison. However, his factor
K = V\V/A/P also has disadvantages. It has the dimensions of a velocity, and it seems
that a nondimensional factor K/\/V would serve Mr. Barnaby’s purpose better. This is
closely related to what is sometimes called transport efficiency, and when the values
quoted by Mr. Barnaby are converted into this form the very valid practical advantages
which he claims for this type of comparison factor are demonstrated more clearly. It is also
worth pointing out that much of the data given in the paper are for speeds at which hydro-
dynamic lift is relatively unimportant, and to that extent © is still a reasonable drag
criterion.

The comments of Mr. Hadler and Mr. Clement also form a valuable addition to the paper.
We hope that further work will be undertaken at DTMB and other establishments to examine
the potentialities of hybrid craft with partial hydrofoil support. The additional information
requested on details of propulsion arrangements will be included in a subsequent paper
intended to give results of systematic propulsion experiments with models of high speed
displacement-type hulls.

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FR, Neodieal ved Bf 1, “Syotanee ar ante mechs i hgberre wegen BL Lp sh a hia, sah = a
Waewane ivy Lhe Bd, CP hemticnhig buy tempera Nepean imam abn betty
Vea Mes wae ES So nan hating SOR f
‘ 7

AUTHOR INDEX

Boggs, F. We. wecevecrescesescee 451 Sonim, So of ooaonogonoupnooas ty;
ingly, del Ro obo oooopoOoood oO B74) Schwanecke, H: ........ io oucmionnee: | WMI
Stlvemlleats Ace \tcneieheis sietelebenc se cieliciis DOL

Chaplin, Harvey R. ---+.-+--s pop co UL Sparenberey J. Ae cers eenas cree ces 45

(Gina, Os 60 616-016 Gta Gong coo oaGrolo A ano eG Rs3s)

Goon Nan ALEX! isl leilciicl ss) 's) 6) clei esis 09 Timea, Ros ssw le velietel oe, e040 Bhetemeeveie, Om)
Me6Gl ITI soaednoocadugog0ge . 341
AOSV ING GooGG0c Go0mdCoCoGGo dou Sei
UROOS Is goaocdnnbbdoccaDoe SG Beli
Wubi; MERIC MIT IDS Gooodoonooodan wll

Hadjidakis, A. ..........-. Goooo 4 Alyl

Karp, Samuel ....2-ccccccscscces 75
LGuliliare, AVG (UG GlGlo iON Ol nOIOnG. tl CACNONOsONCECE me. sia 34
lexi. EES 6 GNaeG oo olonond OO OnCLOLONO GG 1/5)
van Manen, J.D. ....... Siienenteneisiat ls 23
ReNas, edward Vi. Pres eaters ele) ewe SLO von Karman, Theodore ......2.++s+-+- Vii
HERE) Te) Soh RI Cae yee Ce 5 Von Schertel, H. ....... eeodoo me Mss

Reet Gi ok wee ceed hues 307 VOBSIBOTIS Gin cte sais) ere 6.crobele wcimeemw. (OF

IMarwOOd Wie dic «<< 6 « «<6 Ca Sono ad “Soil
Wennagel, Glen Jo ssesceesesese = 205

OakleyspOweny He slew «6c oo oe ew 1 Willm;, Pierre H. os see sence sases 405

613

U.S. GOVERNMENT PRINTING OFFICE: 1962 O—646551

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