0. E.

L

Symposium on

Naval Hydrodynamics

Sponsored by the

OFFICE OF

NAVAL RESEARCHandthe

NATIONAL ACADEMY OF SCIENCESNATIONAL RESEARCH COUNCIL

^III

iiiiir-

'MARINE BIOLOGICAL

September 24-28, 1956Washington, D. C.

LABORATORY

LIBRARYWOODSifa

HOLE, MASS.I.

W. H. 0.I-

T'-^

Publication

515RESEARCH COUNCIL

NATIONAL ACADEMY OF SCIENCES

NATIONAL

Washington, D. C.

1957

Library of Congress Catalog

Card Number: 57-60045

PROCEEDINGS

SYMPOSIUM

ON NAVAL HYDRODYNAMICSSeptember 24-28, 1956

FOREWORDThere are few disciplines having the far-reaching and intimate consequences whole spectrum of Navy operations, vehicles, and implements as does Hydrodynamics. Moreover, it is a contemporary characteristic of the field that the basic physical and mathematical problems now recognized as the central ones are also among the very ones faced by the designer of the modern Fleet and its myriad accouterments. In recent years, we have seen a rapid exploitation of scientific and technological breakthroughs in the development of Fleet components and Naval strategic and tactical concepts. It may be fairly stated that advances in Hydrodynamics have played a significant part in the evolution of the new Navy; on the other hand, the need for intensification of research in this discipline has been emphasized by the hydrodynamical problems which must be solved if the most effective Naval system compatible with the unprecedented developments in other fields is to be provided.in the

Ations

result of the increasing

importance of those aspects that have been character-

to highlight their significance in Naval applicabeen a continually growing need for meetings devoted exclusively to these aspects. To meet this need, and as one of the ways of fulfilling its mission of insuring maximum contribution of basic science to Naval effectiveness, the Office of Naval Research has developed a series of symposia of which these Proceedings are the initial outcome. This first "Symposium on Naval Hydrodynamics" differed in an important way from the usual scientific meetings in that the papers were specifically intended to contain

ized here as "Naval

has

Hydrodynamics"

surveys of various aspects of Hydrodynamics rather than the results of isolated and basic nature of the meeting, the National Academy of Sciences National Research Council joined with the Office of Naval Research in co-sponsoring the Symposium.critical

pieces of research. In recognition of the widespread scientific interest

In addition to the motivations mentioned above, there were several other obSymposium. In order to derive the greatest possible benefits for those both in research and in design, we sought the following goals in the several contributions: firstly, critical reviews and analyses of the status of each area including the results of important recent researches; secondly, an interpretation, where appropriate, of the significance of these results for future design applications; and, finally, guidance for future research based on the authors' conclusions and opinions regarding the most fruitful directions toward which future research might be oriented. The intimate role of the foundational scientific aspects of the field in Naval and marine applications is naturally revealed in the choice of topics finally selected for presentation. Thus, invitations were issued for papers on the basic physical and mathematical aspects (turbulence, cavitation, geophysical hydrodynamics) as well as more obviously applicatory topics (ship motions, water-based aircraft research, underwater ballistics problems,jectives for theetc.).iii

most important, motivation and purpose of this meeting was an awareness among those in related disciplines of the many scientifically rewarding and technologically important problems still extant and the new ones that seem to arise faster than the classical ones are solved. In this way, it is hoped that additional activity in classical Hydrodynamics will be stimulated in this country and, as a consequence, increase the scientific potential ultimately needed in support of Navalfurther,to develop

A

problems.

During the early planning stages, as the scope and objectives began emerging sprung up a spontaneous desire on the part of his colleagues, past and present, to choose the occasion of this first Symposium on Naval Hydrodynamics to honor Captain Harold E. Saunders, USN (Ret.) for his contributions in creating a stimulating atmosphere for research in Hydrodynamics in the U. S. Navy. Capt. Saunders, through his foresight and inexhaustible energy, can be credited perhaps more than any other man with the existence in the U. S. Navy at the start of World War II of the facilities and environment so urgently needed for dealing with the hydrodynamic problems suddenly faced by the Fleet and which had been neglected in prior years. In spite of, or perhaps because of, his uncompromising demand for the ultimate detailed solution of the most diflficult problem, he holds the respect and affection of all those who know and worked with him, whether or not they agreed with him in all matters. While Capt. Saunders has received well-deserved honors from the U. S. Navy and various institutions, it was a privilege to make the Symposium banquet the occasion of a more personal appreciation by his friends and scientific colleagues. The testimonial, delivered by RAdm. A. G. Mumma, USN, Chief of the Bureau of Ships, is reproducedclearly, there

as

an important part ofIt is

this

volume.

our privilege to express the appreciation of the Office of Naval Research and the National Academy of Sciences to the authors, each an authority in his field, for their graciousness in accepting our invitations. Special thanks are due Prof. L. M. Milne-Thomson for his willingness to accept the difficult assignment of presenting a critique of the theoretical foundations of the field.

acknowledge the contributions of the members of Symposium. The technical program and physical arrangements for the formal sessions were the responsibility of the ONR Mechanics Branch. The many administrative details, the banquet, and arrangements for personal comfort were kindly undertaken by the National Research Council. Special recognition is deserved by Mr. Marshall P. Tulin for his contributions toward developing and formulating the technical program. Dr. Frederick S. Sherman for undertaking the task of editing the proceedings, Cdr. Waldo S. Simons, USN, and Mr. John M. Crowley and Dr. Howard M. Berger for the physical environment and aids and the Mrs. Lucille Rittue, Yetta Hassin and Audrey Steinman for their good humored secretarial assistance throughout the trying year leading up to the meeting. All are members of the ONR Mechanics Branch. Special thanks are also extended to Mrs. Alice Mclntyre and Mrs.Finally,it

is

a pleasure to

the

team

directly responsible for the

Catherine M. Parrish of the NAS-NRC for their assistance to the Executive Secretary, NRC Physical Sciences Division.

Phillip Eisenberg Head, Mechanics Branch Office of Naval Research General Chairman

John

S.

Coleman

Executive Secretary Physical Sciences Division National Academy of SciencesNational Research Council

Co-Chairman

PREFACEThankstributors, these

to the deft pens and rather widespread editorial experience of our conProceedings have required very Uttle editing on my part.

Discussions of the papers appear nearly as they were given orally, although most contributors availed themselves of the opportunity to polish the actual stenographic record. It is hoped that the resulting gain in clarity and brevity has offset any loss in color and informality.

My

gratitude and admiration go out to Miss

for her herculean efforts in recording the oral proceedings,

Ruth Benning, our steno-typist, and to Mrs. Jane Cochrane

and Mrs. Audrey Steinman forFinally, an apologyis

their aid in preparing the clean typescript.

to expectthis

may have been led by past performance an earlier publication date for these Proceedings. It is sincerely hoped that volume will have a lasting value which will compensate for its tardy appearance.due those whoF. S.

Sherman

PROGRAMMonday, September 24, 1956Welcoming Address,

RADM

R.

BENNETT, USN,Session 1

Chief of Naval Research.

Chairman:

J.

ences,

J. STOKER, Professor New York University.

of Mathematics, Institute of Mathematical Sci-

M.

J.

LIGHTHILL; W. H. MUNK, M.

J.

TUCKER

and

F. E.

SNODGRASS.

Session 2

Chairman: K.stitute

S.

M. DAVIDSON, Director Experimental Towing Tank, StevensJ.

In-

of Technology.

G. P.

WEINBLUM;

V.

WEHAUSEN.

Tuesday, September 25, 1956Session 3

Chairman: R. W.C.

L.

GAWN,

C.B.E., Superintendent, Admiralty Experiment Works,

Haslar, England.J.

NIEDERMAIR;

H. W. LERBS.Session 4

Chairman: F. W. S. LOCKE, Hydrodynamics Consultant, Research Division. U. Navy Bureau of Aeronautics.J.

S.

B.

PARKINSON;

R. N.

COX and J.

W. MACCOLL.

SYMPOSIUM BANQUETToastmaster:

RADML.

A. G.

MUMMA.

USN,

Chief, U. S.

Navy Bureau

of Ships.

Speaker:

PROF.

M.

MILNE-THOMSON,

C.B.E.

Wednesday, SeptemberSession 5

26,

1956

Chairman: W. R. SEARS, Director, Graduate School of Aeronautical Engineering,Cornell University.

H. M.

FITZPATRICK

and M.

STRASBERG;

D.

GILBARG.

Session 6

Chairman: E. H.

KENNARD,

Chief

Scientist, Structural

Mechanics Laboratory, David

Taylor Model Basin.

M.

S.

PLESSET; H. G. SNAY.

Thursday, September 27, 1956Session 7

Chairman: G. B.Standards. C. C. LIN;S.

SCHUBAUER,G. K.

Chief, Fluid Mechanics Section, National Bureau of

CORRSIN;J.

BATCHELOR.Naval Research.

Closing remarks: F.

WEYL,

Office of

Friday,

September 28, 1956Carderock, Maryland.Officer

Tour of theGreetings:

DAVID TAYLOR MODEL BASIN,E. A.S.

CAPT.

WRIGHT, USN, Commanding

and Director.

Tour of the U.Greetings:

NAVAL ORDNANCE LABORATORY,

White Oak, Maryland.

CAPT. W. W. WILBOURNE, USN, Commander.

vni

PLATE ONE

IX

To Captain H.

E.

Saunders, USN,

Ret.

byRear Admiral A. G.

Mumma,

Chief

Bureau of Ships

WeI

honor tonight a very

special

memberI

of the family of Hydrodynamicists,

because of many,

manyget a

reasons.

wish

thisall

gentleman to stand while

make

a

few remarks concerning him,

so that

you can

good look

at

him.will

Captain Harold E. Saunders,

you please stand?

It has been a personal pleasure of mine to have been associated with this gentleman for over fifteen years, and I would like to tell you a little of his history.

When he graduated from the Naval Academy in 1912, he had amassed an academic scholastic record there that had never been equaled, and has never been equaled since. The record he broke, scholastically, was that of David Watson Taylor. This resulted in prizes in seamanship, international law, ordnance, gunnery, and many,manyother prizes, as a result of this scholastic achievement.

After graduation he specialized in Naval Architecture and Naval construction. at the Mare Island Naval Shipyard in the Hull Division, 1916 to 1920, during the period in which we were engaged in World War I. During this time this Yard built fourteen destroyers and the USS CALIFORNIA.

He worked

Later he was assigned to the Bureau of Construction and Repair, in charge of submarine design and construction of submarines. He executed the preliminary design for the then revolutionary USS V-4, which was later renamed the USS ARGONAUT, and which was the largest submarine ever built at that time.

From 1933 to 1935 he was Force Constructor on the staff of the Battle Force, of the U. S. Fleet, and there effected revolutionary changes in our procedure for damage control of ships in warfare.1940 he was Liaison Officer for the Bureau of Construction and new model basin at Carderock, Maryland, so that here we have the father of the David Taylor Model Basin. He served there as Technical Director for a period of six years, and then later, as Commanding Officer and Director and Technical Director, until 1947.to

From 1938

Repair, in connection with the design and construction of a

I

would

like to

read you ais

little bit

of what was said

when he

left that estab-

lishment, in 1947. This

a quote.field

model

basin' existed only

"Ten years ago Carderock was a barren, weedgrown on paper.its

and

'the

new

"Today wide expanses of well-kept lawns surround manyings that house the greatest research establishment of

stately build-

kind in the world, the

David W. Taylor Model Basin. "In the recently ended World War II the fruits of our labors contributed largely to the victory and we have become well and widely known forour high standards in research.xi

"You, more than any other person, are responsible for these achievements; your knowledge and vision in planning, and your unremitting labor and infinite capacity for taking pains have made them possible. We, who under your firm but kindly guidance have shared in these achievements, are grateful for theprivilege.

are about to relinquish your command, we who remain shall uphold the high standards you leave us as our heritage. May that assurance and the consciousness of a job well done add to your happiness and contentment in the future."strive to

"Though you

Xll

CONTENTSPage

ForewordPreface

iii

vvii

Symposium ProgramTribute to Captain Harold E. Saunders, RAdm A. G. Mumma, USNI

USN, Ret

xi

SOME PROBLEMS AND METHODS IN HYDRODYNAMICSL.

1

M. Milne-Thomson, C.B.E.17

II

RIVER WAVESM.J. J. J.

Lighthill

DISCUSSIONStoker, G. K.

40

Morikawa4556

III

REMARKS ON THE OCEAN WAVE SPECTRUMW. H. Munk, M.J.

Tucker, and F. E. Snodgrass

DISCUSSIONM. W. IVLonguet-Higgins, R. Pierson, W. VetterS.

W.

L.

Gawn,

J.

Pierson,

CONTRIBUTION OF SHIP THEORY TO THE SEAWORTHINESS PROBLEM ;.G. P. Weinblum

61

DISCUSSIONP. Kaplan, R.

98

W.

Szebehely,

J.

F.

Gawn, P. Golovato, E. V. Lewis, V. G. Allan, M. C. Eames, B. V. Korvin-KroukovskyL.

V WAVE RESISTANCE OF THINJ.

SHIPS

109133

V. WehausenJ. J.

DISCUSSIONT. B. Benjamin,Stoker, R. F. Chisnell,

M.

C.

Eames139144S.

VI

HYDRODYNAMIC BARRIERSJ.

IN SHIP DESIGN

C. Niedermair

DISCUSSION,

H. E. Saunders, J. F. Allan, F. H. Todd, H. de Luce, K. Davidson, R. W. L. Gawn, M. C. Eames, E. Moberg

M.

VII

ON THE DEVELOPMENT OF THE THEORY OF MARINEPROPULSIONH. W. Lerbs155 165

DISCUSSIONA. J. Tachmindji, A. Silverleaf, T. Y. Wu, S. F. Hoerner, R. Hunziker, J. Martinek and G. C. K. Yeh, J. S. Florio,J. J.

Eisenhuth, R.

W.

L.

Gawnxiii

PageVIII

HYDRODYNAMICS OF HIGH-SPEED WATER-BASED AIRCRAFTJ.

181

B. Parkinson

DISCUSSIONW. A.J.

208Schoech, C. L. Fenn, R.

D. Pierson, D. Savitsky,

M. Hopkins, M. E. Eames

R.

W.

L.

Gawn,215233

IX

RECENT CONTRIBUTIONS TO BASIC HYDROBALLISTICSR. N.

Cox and

J.

W. MaccoUBurt,J.

DISCUSSIONG. Birkhoff, F. H. M. BergerS.

D. Nicholaides, A. G. Fabula,

X HYDRODYNAMIC SOURCES OF SOUNDH. M. Fitzpatrick and M. Strasberg

241

DISCUSSIONByard, G. K. Batchelor, M. J. Lighthill, C. A. Gongwer, T. B. Benjamin, F. R. GilmoreS.

276

XI

FREE-STREAMLINE THEORY AND STEADY-STATE CAVITATIOND. Gilbarg

281

DISCUSSIOND. Nicolaides, C. J. Cohen and L. D. Gates, van Dyke, T. Y. Wu, M. P. Tulin, I. ImaiJ.Jr.,

292

M. D.297318

XII

PHYSICAL EFFECTS IN CAVITATION AND BOILINGM.R.S. Plesset

DISCUSSIONH. S. Preiser, P. Eisenberg, G. K. Batchelor, M. Strasberg, E. H. KennardL. F. Herzfeld, F. R. Gilmore,

W.

Gawn, K.

XIII

HYDRODYNAMICS OF UNDERWATER EXPLOSIONSHans G. Snay

325 347

DISCUSSIONG. Chertock,S.

G. Reed,

Jr.,

G. E. Hudson.

XIV

ON THE STABILITY OF THE LAMINAR BOUNDARY LAYERC. C. Lin

.

353 363

DISCUSSIONW. Wasow,S.

T. B. Benjamin, P.

S.

Klebanoff..

XV SOME CURRENT PROBLEMSCorrsin

IN

TURBULENT SHEAR FLOWSW.

373401

DISCUSSIONK. Wieghardt, L. Landweber, F. R. Hama, R. Silberman, G. K. Batchelor, M. P. TulinStewart, E.

XVI

WAVE SCATTERING DUE TO TURBULENCEG. K. Batchelor

409423

DISCUSSIOND. Mintzer, R. H. Kraichnan, M.List of ParticipantsJ.

Lighthill

XIV

SOME PROBLEMS AND METHODSL.

IN

HYDRODYNAMICS

M. Milne-Thomson, C.B.E.Brown University

When I received Admiral Bennett's invitation to address you tonight I felt almost as uncomfortable as General Burgoyne must have felt at Saratoga with the thought of the North behind me and hydrodynamicists massing in the South. Indeed I began to reflect on the distance from London to York which as you know, or ought to know, is a long way, and even non-stop trains take several hours to cover it. Once upon a time in one such non-stop train sat an American and an Englishman the only occupants of a compartment. For about an hour each looked out of the window.Thenthereplied

American said "Do you mind if I talk to you" to which the Englishman "What about?" You see my difficulty. An after dinner speech which conveyed

any information whatever is outside the range of my experience. An after dinner speech concerning Hydrodynamics must be unique in the history of dining and I therefore feel honored to be chosen as the first to make such history.an exact science started with Archimedes. It is true that he work remains today a correct piece of applied mathematics and indeed a giant achievement for his time.as

Hydrodynamics

treated the particular case of zero velocity, but his

Scoffers have said that Archimedes' chief claim to

fame

is

that he took a bath

and then forgotIt is

to dress in spite of his principles.

not

my

purpose nor indeed

am

I

the point of view of the historian, who, asinto eternity sternfirst.

you know,

competent to discuss hydrodynamics from is an amiable gentleman advancing

Rather I should like to say a few words concerning some specific problems and methods available for tackling them. Omission of a name or a topic does not imply undervaluation by me. It merely proves that a choice has to be made. Many other choices would have been possible.Itis

becoming increasingly evidenttensors.at least

that the

most

insight giving statement ofis

the equations of motion of continuous media in general and of fluids in particular

by means of

There areparably superior.

two ways of regarding

tensors,

namely

as quantities attached

to a coordinate system, or intrinsically, the latter being to

my way

of thinking incom-

The

intrinsic definition of a tensor T'"> of

rank n

is

recursive.

AX,

tensor of rank

n

is

a linear operator which operating

on an arbitrary vectoris

by

scalar multiplication, gives rise to a tensor of rank n-1.

This definition, together with the statement that a tensor of rank zero completely characterizes tensors of all ranks.

a scalar,

Thus for example a vector atensor of rank zero.

is

a tensor of rank

1

since a

x

is

a scalar, or

of rank

Similarly the dyadic product pq;q combines with x to give pq (q x) a tensor 1 and so pq;q is a tensor of rank 2, a 2-tensor, indeed the very importantI

:

momentum

transfer tensor.

As an

application the equation of steady motion under no

body forces can be written

[1]

Vwhere

[*

-

p{q;q)]

=

0,

(1)

$ =is

-

7?/

- %m(V

q)I

+

M(V;q

+

q;V)

(2)

the stress tensor,

I

being the idemfactor or unit 2-tensor.

form of the equation of motion, by integration over a sphere of large radius, an expression for the force on a moving solid is readily obtained. By applying Oseen's approximation at a distance we next analyse this force into a lift and a resistance or drag. The resistance is of particular interest since its expression is VF where V is the velocity of the body and F is an inflow of liquid into the sphere, predominantly an inflow into the wake behind the body. Moreover the result is of an asymptotic character improving in accuracy as the radius of the sphere is increased. The twodimensional form of this result was obtained by Filon [2] 30 years ago.this

From

Tensor expression motion[3]

also

pinpoints

the

Lagrangian form of the equation of

d;r

/ dh-

__.dro

dp __F +- =\ It

1

0,

(3)

\dt^

p dror,,.

where r

is

the position vector at timeis

of the particle originally at

The equation

of continuity

P

(

-

)

=

Po,

(4)

\dro / IIIwhere the notation indicates the third scalar invariant of the tensor Integration from to r leads directly to Weber's transformationderivative.

d;r dro

q

go

=

dx,

(5)

dro

whereT'i

rdv

)

F= -

Vfi.

(6)

The Lagrangian form of the equation of motion has been applied to the onedimensional motion of a gas and more recently to free surface problems which I shallmentionlater.

The pointseemto indicate.

that I

repulsive as the three equationsfication.

make here is that the Lagrangian form is not quite so which result from its expression in coordinates would In the hydrodynamic case we have p zz p,^ and a consequent simpliwantto

The equation, in my opinion, should repay further study. The two great weapons of general fluid mechanics are the theorems of Gauss[4]

and Stokes

and

jdSwhere, in the

Sto

their vector

forms are suggestiveo

X

= JV

Xdr,

jdCC

X

= JidS^V)S

o A',

(7)

first the closed surface S encloses the region r, and in the second the diaphragm S spans the closed curve C. Here the small circle indicates scalar, vector or dyadic multiplication and ySf is a general function of position, scalar, vector or tensor.

.

For thesame

rate of

change of circulation

in a circuit

which always

consists of the

fluid particles

we have

circ Cdt

d

^ = - f dC -Vp ^ f dS P

(8)

Jcso that

P

Js

where

P VPaV(

P

),

P

is

a vector along the intersection of surfaces of con-

stant pressure

and constant

density.

Also

v

*

P =

defines tubes of constant intensity.

Thus we have

the

of Bjerknes that the rate of change of circulation in P tubes which C embraces. For plane flow the vector notation leads directly to the complex variable [5]. The use of the complex variable in two-dimensional problems has a long history, but it is only in recent years that full advantage has been taken of the methods of

and so by Gauss's theorem P famous meteorological theorem C is measured by the number of

function theory as opposed to resolution into equations in x and y. What is beginning now to be more fully realized is that the variables most generally useful are not x, y but the conjugate pair z. Y. For example if a> (x, y) is a plane harmonic function, it is the real part of a holomorphic function /(z). The identitycp(x,y)

^ jiim + M]jiiz)

(9)

leads tof(z)

=

2^(}k,

-

"y/

(gh),

no gravity wave withis

c=Vlowered.rises

exists,

and so the wave-

making

resistance of a two-dimensional obstacleis

Similarly, the drag of a

barge on a narrow canalRussell [28] describes

reduced when

its

speed

above \/(gh), and Scottproprietor's

howits

this

was

firstit,

discovered due to a horse taking fright, and

galloping off pulling

barge behind

when

it

was observed "to the

astonishment, that the foaming stern surge which used to devastate the banks hadceased,resistance

and the vessel was carried on through water comparatively smooth, with a very greatly diminished." The proprietor went on to achieve "a large

increase in revenue" as a result of this scientific observation.7.

Stationary waves on non-uniform streams

I might remark that Mr. J. C. Burns and I investigated [4] what relation corresponding to c = V a stationary wave must satisfy in a stream whose velocity varies with depth according to a one-seventh power distribution typical of turbulent flow over

smooth bottom. For small X/h the waves are on the surface only, so that for them to we must have c ~ surface velocity 1.143 F, (F, = mean velocity). As X/h increases, c drops (Fig. 2) till it equals F, for X/h = 10 and is 0.966Fot when X/h -^ 00 this lower value is because the oscillatory motion for very long wavesa

be stationary

;

tends to be slightly stronger in the low-speed region near the bottom.

20

VE LOCITOF

Y

STREAM

OF CRESTS RELATIVE

TO MEDIUM

OF WAVE ENERGY RELATIVE TO MEDIUMOF WAVE ENERGY RELATIVE TO OBSTACLE(a)

< U-c

C-U

Pattern of

waves generated.

0.9

(b)

Graph which

plots

against X/h, for stationary waves of length X on a stream of depth h,ratio to

with one-seventh power distribution of velocity and

waveFigure 2.

velocity c

mean speed Vm, the which waves of that length would have on still water.

Vm

of the

Two-dimensional wave making.

8.

Three-dimensional wave pattern due to an obstacle

I will now discuss briefly the three-dimensional pattern of stationary waves due an obstacle at a point in a running stream (Fig. 3). Oblique waves are then present, and for them to be stationary those of velocity c must set themselves at an angle

to

21

Figure 3.

Three-dimensional gravity-wave making due to an obstacle O.

Waves generated

t

seconds

ago are shown.so locus ofis

OPt'

=: Vi,

PQ =

cf,

since