R-933

DAVIDSONLABORATORY

Report 933

WAVE-RESISTANCE REDUCTIONOF NEAR-SURFACE BODIES

byKing Eng and Pung N. Hu

March 1963

u _

ii

DAVIDSON LABORATORYII REPORT 933

March 1963

'IWAVE -RESI STANCE REDUJCTION

OF' NEAR-SURFACE BODIES

byKing Eng and Pung N. Hu

Prepared forOffice of Naval ResearchContract Nonr 263(10)

(DL Project FX2531)

Reproduction in whole or in part is permittedfor any purpose of the United States Government

Approved,

D -1JohBrsiDirector

ABSTRACT

An analytical study of the wave-resistance charac-teristics of near-surface bodies was conducted to determine

1) for a given length and displacement, what changes in

body-surface geometry are necessary to cause wave-resistance

I reduction, and 2) how geometrical change affects the wave-resistance behavior with Froude number and submergence depth.

I The general wave-resistance expression for a perturbed ellips-

oid with the constraints of constant displacement and length

is formulated. A digital computer solution of this variational

problem is obtained for the case of the spheroid due to avail-

able computer-size limitations.

The effects of fineness ratio and submergence-to-

length ratio on the Froude number behavior of the wave re-

sistance for a range of perturbations is demonstrated. Sub-

stantial reduction in wave resistance is possible for all

I Froude numbers above and slightly below the optimum Froude

number for a particular perturbation distribution. For

Froude numbers lower than approximately 10% below the optimum

Froude number, a large increase in wave-resistance coefficient

V! may be obtained depending upon the perturbation used. Since

this generally occurs at low Froude numbers, the actual in-

crease in total resistance experienced for perturbations

yielding acceptable geometrical changes should be quite ac-

ceptable. Depending upon the optimum Froude number, the geo-

metrical changes required for wave-resistance reduction fall

into two classes: 1) midsection bulge with finer bow and stern

for Froude numbers below 0.32; and 2) above 0.32 Froude number

a midsection pinch with bulging bow and stern.

R-933I -ili-

NOMENCLATURE

a,, a2 , a. semi-axes of an ellipsoid

A, Am perturbation parameter

D diameter of spheroid

e, e eccentricity of ellipse

U Froude number

g acceleration of gravityko -I-- - i

ki longitudinal added mass coefficient

r., r, r, radius

R , R wave resistance

Ro', R', R' wave-resistance coefficient

-U constant uniform stream velocity

x1 , x2 , x3 rectangular coordinates1 ' 2e rectangular coordinates

0,0,01,4,$ velocity potential

APoP1 strength of doublets with axes in the positivex -direction 1.

Snormalized strength of doublets with axes in thepositive x -direction

Smass density of fluid

Subscript m denotes optimum value i

ii

R-933-iv-

TABLE OF CONTENTS

Abstract iii

Nomenclature iv

Introduction 1

Theory 2

Geometry of Perturbed Spheroid 12

Results and Analysis 13

Acknowledgments 15

References 16

o Appendix A-1

k Figures

II

R-933-- V--

INTRODUCTION

As an integral part of the research program on

high-speed ship forms at Davidson Laboratory, an analytical

investigation into the reduction of the wave resistance of a

submerged body moving close to the water surface was con-ducted. The major problems of interest in the investigation

were:

(1) For a given volume and length, how should thesurface geometry be changed in order to cause a reduction in

wave resistance?

(2) How does the geometrical change affect the

wave-resistance behavior with Froude number and depth of im-

mersion?

The linearized theory of wave resistance for bodiesmoving near the surface has been well established by Michell,Havelock, Lunde, etc. These theories impose a linearized

free-surface boundary condition on the velocity potential.The wave-resistance expression is an integral with a quad-ratic integrand consisting either of functions that definethe shape of the hull, or functions that define some type ofhydrodynamic singularities by which the hull is generated.

The latter type of integrand, being mathematically moretractable, is used in this study. The purpose here is to

find a hull geometry of minimum wave resistance. Therefore,the investigation becomes a variational problem, and it is

apparent that the variation should be in the hydrodynamic

singularities.

Weinblum' treated such a minimum problem by con-sidering a family of hull curves whose doublet distributionwas expressed by polynomials having several arbitrary pa-rameters. His result shows that, for a given Froude number

and immersion depth, the doublet distribution and its

R-933-1-

1

corresponding wave resistance can be evaluated in terms of

a table of functions. However, no comparison can be madewith his results because the hull displacement is not con-strained.

The general case of a perturbed ellipsoid is con-sidered in this analysis. It is approached as a variationalproblem with constant displacement as a subsidiary condition.The ellipsoid is represented by doublets distributed over theconfocal ellipse. This doublet distribution is then perturbedsuch that the perturbation will have no influence on thevolume of the ellipsoid, and will produce a new hull with lesswave resistance.

THEORY

Throughout the discussion, the axes x,, x2 and x3

of a right-hand Cartesian coordinate system are fixed on themoving body. The origin 0 has been taken at the geometriccenter of the body with Ox, parallel to the direction ofmotion and Ox. vertically upward. The fluid is assumed to beincompressible and inviscid. The motion is irrotational andcharacterized by a velocity potential 0 which defines thefluid velocity 4 by 4 - -VO. The wave height on the freesurface is taken to be small in comparison to the wave length.

Conridor an ellipsoid with semi-axes a1 >a2 >a3 movingin an infinite fluid at a constant speed U along thex.-direction. The velocity potential which describes the

absolute motion of the fluid is given by2

where

R-933-2-

UD1 11) 12)2] /2(2

Sale a~e

e2 =_ (a)2 (3)

a,j2 . (a. )2 (4)a2

D• a., (5)1 2-al) e•

0o ( dXCL, = ala 2 a3 (6)

1( al+X) 4 a~+X) (a2-i-) (a3+ 6

r 2 = (x 1 -• 1 ) 2 + (x 2 -e )2 + (X3-ý 3 )2 (7)

The surface integral in eq. 1 is taken over the confocal

ellipse

)2 +1 2 (8)

tL on the plane 0 =0. Let the ellipsoid be perturbed such thatthe perturbation can be represented by a doublet distribution

i�(�,� I lA2) in addition to the original doublet distribution

o1,0(O2)• •((,• 1 2 ) is bounded by the same confocal ellipse

given by eq. 8. The perturbation potential is then

The resultant potential for the perturbed ellipsoid becomesS• (i) d~z ~ (10a)

4t(xlx 2 pX3 ) = e r, Ie

e3=0

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-3-

i

where O(xx 2 ',x3 ) 0 (x 1 ,X 2 ,XP3 ) + 0 1 (xlx 2 ,x 3 ) (lOb) 1and 14( 1 i 2 ) = Ao(,l'e2) + Al(I", 2 ) (lOc)

is the resultant doublet distribution. It is required that

the volume remains constant upon this perturbation; therefore,*

ý3 =0"-

By letting ,e = Mo( 1 ,• 2 )Q., where Q =

is an arbitrary function to be chosen later, eq. 10 becomes

7rxx2L8)- a2 e r e2dd

ý3=o (12)

According to Taylor's added mass theorem,

UpV( l+k 1 ) =41p #( M +A1 )d•• 1 d• 2 =UpVo(l+k°) +4-7rp 1de lde 2

where kj, V and kP,Vo are longitudinal added mass coef-

ficient and volume of the perturbed and unperturbedellipsoid, respectively.

Then V(l-k.) - V (1+k°) = 4, =0 by eq. 11 ji

o A1 o112q.1

or V = 0oI-k+)"

For elongated bodies of approximately same length and L/D,ki and k? are small in comparison to unity; also they areof the same order of magnitude, i.e, kl-k° is very small.Therefore,

1+ko'- " 1 orV'~Vo

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When the body is moving below a free surface, the

velocity potential must satisfy the linearized boundary con-

di ti on

+ ko 6 0 (13)

where k. =

on the free surface (x, = f).

The Green's function, satisfying this condition (eq. 13), is

Ii given by 51s

G(xl,x 2 ,x,; , 1 r 1

0- 2 k[(x 3 + %s-2f)+i(x1 -ej)cosi]C knL0 Re seO cos [k( x2-e2)sinedkde

7T f, k- k 0 see2 e

0 0- (14)where ri = (xi-•i) 2 + (x 2 -) 2 + (x3+s-2

I Physically, the Green's function represents the velocity

potential of a source moving at a depth f below the free sur-

I face. Therefore, to a first-order approximation, the velocity

potential of the body, represented by the doublet distribution

I (,) moving below a free surface may be expressed as

S1G (15)

•s= 0

Substituting eq. 14 into eq. 15, the potential 0 for (x1 -e 1 )>O

and (x 1 -• 1 ) <0 becomes, respectively,

R-933-5-

I

I-l

4 0 JIJ ksecL cos [k(x2 -1 2 ) sinO]- F f k-ko seee 2(

00

ek[(xs-2f) + i(xj-ýj1)cosO]dejde2dkdO

and (D = I( e (x)"L) x-- =) djdýd2

4 k+kosec2O cos [k(x2 -e 2 ) sine]

0 0

e-k[(x3-2f) xix- I cos3e] d~ 1 de dkde7r

+ o0 # 1'(ýV2) cos ekk( (x- -1) secO]. cos k0(x2-

seceO sine] sec 3 O eC d~jde2de

From ref. 6, the wave-resistance expression derived

from a consideration of the energy expended in the production

of waves Is given as f (-

P 2 ]- (16)01X iJ2 ý2 6X ~2

From ref. 5, the velocity potential of (D at x1-4-- can be ap-

proximated to the form

-R-933-6-

VIi

IiII 2(n x1 ' x2 'xr-0 0

eX e -2f)see2 e (17)secS • e dld•2 d(

where q, = koa e sece ; q2 = k0a2 e sec2 e sine

Substituting eq. 17 into eq. 16, one gets:

-2kofsec 2 6R 167pk4 +p2 )sec'Oe de (18)

0owhere

~P a( = Af( ')cos q (Le )cos q2( )dý d2 (19a)

P2 )sin ql("e)cos q2( d 1d ý2 (19b)~12 ale a2 ae

e2IP3 = (ill,2) sin q1 ( e)sin q (;.4de dj 2 (19c)

ale1 1(FE

P f ,2) cos q(e)sin q (a.-.L.)dx dd

co 11d l 2 a~ 2 el d)

The Pi terms in eq. 18 are all positive definite quantities.Therefore, each P2, term will contribute to wave resistance.

However, if Ait, e) is an even function with respect to el andI P 2" ~ i.e., M(1I, 2 ) = I-•i,-•2), then P2, P3 and P4 vanish

identically which to some extent reduces the resistance R. As

a result, doubly-symmetric bodies are better forms as far as

wave resistance is concerned.

Consequently, the expression of the doublet distribu-

tion takes the form:

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-7-

Tr• (7-1e)• (a• e+M({i,{•) S-• [1 - ae -(ia~ )

Q will be chosen as an even function with respect to J, and

t2" Obviously, the choice of Q is not unique under these con-straints. For the sake of mathematical simplicity, the choice

of Q for the present study is

I = - A cos X(a1-.J) cos (20)2ale a 2 e

where A, X and V are arbitrary parameters to be determined.The doublet distribution expression now takes the form

L( '2) -- cos2 Xt2-)212

(21)

Substituting eq. 20 into the constraint equation (eq. 11), oneobtains

-(-U--'.)A _( )2A( 2 -cos 2( e)COS v(-.-,-)dtd2 = 07ra e a2e a eae 1 2

which can be satisfied if (see Appendix)

-3/2

A (ý X•-V) J 3 / 2 (4+ý72) = 0

For nontrivial solution of A, X and v, one has

4X2 + v2 _ tan .4X2 + V2 (22)

Substituting eq. 21 into eqs. 19a through 19d, the results are

(see Appendix)

P = UD (o - A*) (23)

P2 = 0, P3 = 0, and P4 = 0 (24)

where

R-933-8-

3/- j 3 2 (4q 2) a, aa 2a.

i 7r IJs, 2 (4(X~q 1 )2+(v-q 2 )2) J' 1 2(4(X-ql)2+(v+q 2 )2)It~0 =q% /2 + q2)____ 2/ -+%

-((Xq)+ -•q2 (-4" + )2+(v.q)2) • 1

Js / 2 ( (+q,)2+( v-q 2 ) 2 ) J 3 / 2 (q(X+qj)2+(v+q2)2)/2 +/2

(S/WX+q1) +(vv-q 2 2) 2) J31 ( q(X+ql)2 +(v+q) 2 )

and q, = koale sec e; q2 = k -a2e sec2 sin2 0

Substituting eq. 23 into eq. 18, R becomes7T

"R = 167rpk(U)I) f (*- A%)2 g(e) d e (25)0

where g(e) = sec 5 ee -kofseC2 0

2 Vt1FU2 1 RLet L = 2a, F - 2koaI and R 111 ~ 2 0a 1 pu 2idS! - pU2 L2

Where F is the Froude number, and R'is the wave resistance

coefficient. Then7r

R (*7 - Af )2 g(e) dOe (26)

where K= 7 l e 12 T1) (1_2) F8

Denoting

Ii

R-93,3-9-

7r

Ro = K J 02 g(e) de (27a)

0

R I= K "f*0g(e) d 0 (27b)

and R2 = K j 2 g(e) de (27c)

R'can be written in terms of a perturbation parameter A* as

follows:

R'(A) = Ro - 2ARI + A2 R 2 (28)

The first and second variaticns of R'(A) are given, respectively,

as

6R'(A)= 2(- Ri + AR2 ) 6A (29)

and

6 2R(A)= 2R 2 (6A) 2 (30)

since R2 is positive definite as seen from eq. 27c, eq. 30

shows that if R'(A) has an extremal, it is a minimum. The mini-

mum of R' occurs atR**

A = Am = - (31)R2 .

and has a valueR 2

R'(Am) = Rm -R (32)

*R is also a function of X and v, but the major parameter whichreduces wave resistance is the parameter A. , ,X,v)

"**Since R1 =R1 (X,v) and R2 =RH(X,v), therefore Am=Am(Xv) = - __ P1 ~R2(X>- I Vthen for various values of (X,v) that satisfy eq. 31 and theconstraint condition eq. 22, a family of hull forms of similarwave-resistance characteristics will result.

R-933(10-

Since it is not possible to obtain a closed form

ii solution of Ro, RI and R2 , the problem soluticn requires the

use of a large capacity digital computer. Due to the limited

S capacity and speed of Stevens Institute's 1620 IBM computer,

numerical results are presented only for the general case of

i. a perturbed spheroid.

The equations of a perturbed spheroid can be obtained

by taking limits of the perturbed ellipsoid equations, i.e.,

let n = 0, t 2 0 and e--0. The spheroid equations then be-

come for doublet strength:

U(a) I e) (1 - t) (i - A cos Xa) (33)

1 --

1- where t =- and -1i- ý < 1,

a e

and for the potential in an infinite fluid:

I

where r 2 = (x a eE)2 + y2; y2 = X2 +1 2 3

jThe characteristic equation resulting from the constant volume

L constraint reduces to:

X = tan . (35)The expressions Ro, R , R2 , R',A and R'I remain unchanged, but0 2 m mK, 4' and 4P are different.

7t

Ro=K *4,o 2 g(e) de (36a)

1 0

R-933-11-

i

7T

R2 =K * 2 g(O) d0 (36c)

R' R - 2AR1 + A2 R2 (37)0, 2

A R= (38)m R

R' Ro -R (39)

I

sin(X-q 1 ) - (X-q 1 )cos(X-q 1 ) +sin(X+q 1 ) - (X~iq 1 )cos(X.1q 1 ) "

(Kq 1 )3 ( e+q) 2

whee l esec 6.•2 2F e

GEOMETRY OF PERTURBED SPHEROID*

The doublet distribution between foci

=o• Ua~e 2 (1 - a2 ) , - x (40)Fre 7 _ tn(4lýe] aI e

L_ e 2

is the image of the uniform stream -U within the spheroid:

r20 = a2(i - e2) q1 - e q•2)

* This method is suggested by Pr•ofcssor- L. Landweber, fromthe State University of Iowa.

R-933-12-

jj An increment in the doublet distribution AM will produce a

change in the ordinate of the spheroid which, it will be assumed,

|i is given by modified Munk's formula:1! l k+ k2

AM= U(r) (42)

where k is the added mass coefficient of the spheroid given

tI by 12

2 . 1-er2 2e _ n(1+)e (43)1i +k e -1k 1 e3 Lle

Taking AM = jit = -AAo(O)cos Xe, together with eqs. 40, 42 and43:

.(r 2 ) Aa•(l-e 2 )e

combine eqs. 41 and 44 and get:

r 2 (e) = r2 + 6(r 2 ) = a•(l -e2 ) [-e '2- A(l-e 2 )cos X145)

r(x) = (D) 'i- ()- Ae e- 2(X)2] cos a (46)

where r(x) is the radius of the perturbed spheroid along the

x-axis(T) is the slenderness ration of the undisturbed

[ spheroid.

p! RESULTS AND ANALYSIS

The optimum perturbation parameter Am is plotted

V iversus Froude number in Fig. 1. At Froude numbers below .30,Am is almost independent of both slenderness and immersion

ratio. For Froude numbers .28< F< .30, Am shows little changeand has a value of approximately Am = -. 20. For F >.30, Am

R-933-13-

varies significantly with depth, f/L, but varies very little

with fineness ratio, D/L. This result is expected because at

infinite depth, there is no wave resistance.

By examining the doublet distribution, one can obtain a

good indication of what the approximate hull form will be like.

Therefore, from eq. 33, one may conclude that:

1) for A <0, the hull forms bulge out at the midsection

and are narrow at the ends,

2) for 0 <A <1, the hull forms neck in at the midsection

and bulge out at the ends,

3) for A >l, the hull form may be imaginary.

There is no clear-cut dividing line as to what type of body

geometry a hull may have and still be considered reasonable.

However, the perturbed bodies with A <0.5 generated from a

spheroid could easily be considered reasonable forms.

The fact that the body geometry for minimum wave

resistance varies with Froude number and immersion ratio makes

it apparent that there is no single hull that can have mini-

mum wave resistance over a range of Froude numbers and sub-

mergence depths. However, from Fig. 1 and eq. 33, for arbit-

rary values of A ranging from - .20< A< l, there is associated

a hull form which will have a minimum at some F and f/L. For

example: for A = - .20, the minimum will occur between F = .28

and .30. With this in mind, A = -. 20, .25, .50 and .75 were

chosen to illustrate the results of this analysis. The Froude

number and submergence ratio corresponding to the minimum for

the above perturbation parameters are shown in Fig. 2. The

associated normalized doublet distributions and the approxi-

mate hull form are shown respectively in Figs. 3 and 4.

Figures 5 and 6, respectively, show the wave-resis-

tance variation with Froude number and immersion ratio. As

R-933

-14-

1)

would be expected, the wave resistance over the entire Froude-number range is affected by the geometrical change resultingfrom the perturbation. For A >0 the wave-resistance coeffi-

11 cient, in general, has a reduction at high Froude-numbervalues, but shows considerable increase at low values, es-pecially when the body moves toward the free surface. Also,

for A <0, the wave-resistance coefficient reduces at low but

I increases at high Froude numbers.

I ACKNOWLEDGMENTS

The authors wish to acknowledge their gratitude to

L Professor L. Landweber, consultant for Davidson Laboratory

and Mr. Earl Uram, staff scientist at the laboratory, for

f their suggestions and review of this report.

R-933

-15-

1

REFERENCES

1. Weinblum, G.P.: "The Wave Resistance of Bodies of Revolu-tion," DTMB Report 758, May 1951.

2. Newman, J.N.: "The Damping of an Oscillating EllipsoidNear a Free Surface," Jour. of Ship Research, Vol. 5, No. 3,December 1961.

3. Havelock, T.H.: "The Wave Resistance of an Ellipsoid," Pro-ceedings of the Royal Society, A, Volume 132, 1931,pp. 480-486.

4. Havelock, T.H.: "The Wave Resistance of a Spheroid," Pro.of the Royal Society, A, Vol. 131, 1931, pp. 275-285.

5. Havelock, T.H.: "Wave Resistance Theory and its Applicationto Ship Problems," Trans. SNAME, Vol. 59, 1951, pp. 13-24.

6. Lunde, J.K.: "On the Theory of Wave Resistance and WaveProfile," Skipsmodelltankens Meddelelse NR. 10, April 1952.

7. Lunde, J.K.: "On the Linearized Theory of Wave Resistancefor Displacement Ships in Steady and Accelerated Motion," _!Trans. SNAME, Vol. 59, 1952, pp. 25-76.

8. Peters, A.S. and Stoker, J.J.:"The Motion of a Ship, as aFloating Rigid Body, in a Seaway," Communications on Pureand Applied Mathematics, Volume X, No. 3, August 1957.

9. Karp, S., Kotik, J. and Lurye, J.:"On the Problem of WaveResistance for Struts and Strut-Like Dipole Distributions,"Proceedings of the Third Symposium on Naval Hydrodynamics,October 19-22, 1962, Scheveningen, Holland.

10. Pond, H.L.: "The Pitching Moment Acting on a Body of Re-volution Moving Under a Free Surface," DTMB Report 819,May 1952.

11. Pond, H.L.: "The Moment Acting on a Rankine Ovoid MovingUnder a Free Surface," DTMB Report 429, September 1951.

12. Lamb, H.: Hydrodynamics, 6th Edition, Cambridge Univ.

Press, England, 1940.

13. Watson, G.N.: "Theory of Bessel Functions," 2nd Edition,1952.

R-933

-16-

APPENDIX

1 iEvaluate the surface integral of the form

I - c)o-00QXx cos~ydxdy

over the surface of an ellipse

(12+(J)21

where a and A are arbitrary constants.

Let x = asin~cos O and y = bcosk, then the element surface

becomes ds = absin2 sinnedde.

Upon change of variables, the integral I can be written as

I = ab)) sin3 0sin20 cos[ aasinO)cos cos(bjcos0)d~d0

I Denoting = aasinO, and integrating with respect to e:

I P(y) = sin•2e cos(ycose) d o

[ o - Tsingcos(-ycosO) d(7008(9)

7[ Integrating P(y) by parts, we obtain

P(Y) = S c0s8 81n(ycose)de = J )

Substitute P('Y) back into I; then I becomes

I- f wb sin2 J 1 (aasinO) cos(bccosO) dO

V but cosO = J-, (6)

Ii' therefore

R-933A-1

7rI - - Jl(aasino) J (bccoso) Cos2 sin 2 do •

0

From ref. 13, under Sonine's second finite integral, the

expression I is of the formj /(4(aL) 2+(bp)2)

I (7rN'ab) 3/2

(4(aa) 2+(bg )2) "

R-933A-2

00

Ii

U)

-Sr

00

o Uo w

-~ 0I I2

iacc

00

R-3

I In

14~II 4e.

00

Hw W

0 C0

IwZ x z

9 LL

0

2 'og~~~vw OS3FE

d wR->9U3

Ii PL(()-(I-e)(I-A COS XC)

1.2II.O- AA.00 (SPMR0.5)

I F" A, O.7 5

0.4-

00.2 0.4 0.6 0.11 1.0

S~~FIGURE 3. NORMALIZED DOUBLET DISTRIBUTION, 11 •)

R-933

I iiSPHEROID SLENDERNESS RATIO L/b'S

'iiAsO (SPHEROID)

ii+2

FIGURE 4. CONFIGURATION OF PERTURBED SPHEROID

R-933

diti

CLi

ILI

0 00

0 .3£.L l~

0i .N-jjAo

R-3

SPHEROID SLENDERNESS RATIO, L/D: 5

A: -0.20

|.0-

lIIIO

II A:O.O f__O.15__

A 0.L 20.150.9 - L

U :0.30 -- - -

o 0.6-

0.7 AA:O.25

•" t).25

w f AO.SO

z

0.5 Av1.75

0.4

0.3-

S~A-'O.0

0.2-

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0FROUDE NUMBER,F

FIGURE 6. WAVE RESISTANCE COEFFICIENT OF PERTURBED SPHEROIDI VS FROUDE NUMBER.

R-933

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