Non-linear ship wave theory

15 SEP.1912 .C)

v;

:

ARCHIE

F

To be presented

at

the Ninth Symposium of Naval Hydrodyanmics Paris, August 1972. jotheek van

Onderad1t'

;sche Hogesch00 DOCUMETATIE DATUM:

NON-LINEAR SHIP WAVE THEORY by

C. Dagan

Technion-Israel Inst. of Technology and Hydronautics-Israel Ltd.

Lab.

v

ScheepsbouwbtT

Technische Hogeschool

Deift

4I

ABSTRACT

Systematic attempts to extend ship wave theory into the non-linear range are described. The basic derivations are carried out for two-dimensional flows

and the bearing of the results on corresponding ship wavetnaking problems is dis-cussed. The main topics are: Ci) the derivation of the second order wave resis-tance for a body generated by an arbitrary distribution of sources, _{(ii) the}

wave resistace at low Froude numbers. _{A uniform solution, valid at low speeds,}

is for the first time presented and (iii) a few preliminary experimental results on the bow breaking wave.

CONTENT

INTRODUCTION

ThIN BODY EXPANSION

SMALL FROUDE NUMBER PARADOXES

WAVE RESISTANCE AT LOW SPEEDS

PRELIMINARY EXPERIMENTS ON ThE NO-DIMENSIONAL BREAKING WAVE. GENERAL CONCLUSIONS

ACNOWLEDGEMENT

LIST OF SYMBOLS REFERENCES

INTRODUCTION

The linearization of the problem of free surface gravity flow past a Ship body has (unlike the equivalent aerodynamical -problem) a two-fold effect: not

only the body boundary condition is simplified, but also the free surface

boun-dary condition is linearized. Although these two simplifications are associated with the same first order term in a perturbation expansion in which the uniform

flow. is th _{zero order leading term, the mathematical difficulties associated with} the nonlinear..ty of each one are quite different. _{It is relatively easy with the}

present large computers to derive a solutIon which satisfies the boundary

cóndi-tion of zero normal velocity on the full body. It is extremely difficult, if not impossible, to êatisfy the nonlinear free surface condition, even by numerical approaches, _{It is no wonder, therefore, that effort has been spent in the last}

years for solving the flow past ship like bodies, while keeping the free surface

condition in its linearized version (the so called Neumann-Kelvin problem). The

aim of such studies was to determine the range of validity of the usual

linea-rized solutions and to improve them, when necessary. This -way it, was hoped that a better agreement between theory and experiment could be achieved,.

The present work is dedicated mainly to the influence of the nonlinearity of the free-surface conditions on the wave resistance. Since at this stage we are -interested in elucidating problems of principle and basic concepts, we have

carried out the derivations for two-dimensional flows. _{Two dimensional solutions}

are obtained much easier than the three-dimensional ones due to the use of the

powerful tool of analytical functions. _{They permit to find in a simple way quicq}

answers for problems which in three dimensions need a tedious numerical treatment.

It is realized, however, that the final conclusions about the applicability _of the results derived here to flow past ships could be drawn only after _their

ex-tension to three dimensions. _{We consider, nevertheless, at each stage of the} pre-sent study, the implications of the results to associated ship problexns

ThIN BODY EXPANSION

1. General

We consider n _{inviscid two dimensional flow past a submerged body KFig.}

5a).. Let z'x'+iy' be a complex variable, f'='+iq' the complex

potential-and w'='u'-iv'=df'/dz' the complex velocity. We limit our considerations to a synetrical body parallel to the unperturbed free surface: _{h. is its}

sübmer-gence depth,

2W

its length and 2T' the maximum -thickness. With U' the velocity of uniform flow far upstream, we make var-iables dimensionaless as

id-lows:

y"y'/L',

Z=Z'/L',

wu-iv='w'/U', n=n 'IL'

f'+i4,=f'/U'L', h=14/L',. "T'/L', FU'/(gL')1'2,

FU'/(2gL'

ri'(x.') (Fig. 5a) being the free surface elevation above y'=O.

The exact boundary conditions satisfied by v(z), which is analytical in

the flow domain yn(x), given here for convenience of reference, are as fol-. lows ImEiF2()2 -w] 0 liii wdz = 0 J (y=ri)

u+1

mi wdz = 0

_(IxI<i,

y=±ct(x) )

where t(x)=t'/T' is the dimensionless thickness distribution

_an4

wu+iv,

For a given body shape w depends on z,c,h and F. we consider now a "thin body expansion",. i.e. an expansion of w for c'o(l) and F,h=0(l). This is the basic linearization procedure used in ship wave resistance theory.

Hence, with

.w(z,c,F,h) 1 + cv1(z,F,h) +c2w2(z,F,h) +

f(z.,c,F,h) = z + cf1(z,F,h) +c2f2(z,F,h) + ... (6)

n(x,c,F,h) = n1(x,FJ).

+

c2n2

(x,F,h)

+

the free surface (2), (3), radiation (4) and body boundary cond.tion (5) become

at first and second order

df Irn(1F2

;r;1

= p1(x) Ti1 = f1-'-O ql = ±t df Ini(iF2

=0J

(r'O) (x+oo)

(IxI<1,

y=-h±0)

(3(ui?+(vi)uivi]l

(11)

J(YO)

₁₂₎ (13) (IxI1,y=-hO) (14) (1) 1/2

3

Eqs. (7)-(I4'have been written, after a simple integration, in terms of

f1, f2 rather than w1,w2 (the derivation may be found, for instance, in Wehausen & Laitone, 1965).

The main purpose of this section is to derive the. second order solution and the associated wave resistance, with applicaUon:.to a' few particular shapes.

Such computations have been carried out previously by Tuck (1964) for a

submerged cylinder and Salvesen(1969) for a submerged hydrofoil. The subsequent developments in this section are. a continuation of their work. The cylinder is

an extremely blunt shape whose three dimensional counterpart 'is'e sphere. In the

case of the hydrofoil it was found that a major nonlinear contribution comes from

the vorticity associated with the Kutta-Joukovaky condition We have been

inte-rested to extend the previous computationS to the case of elongated

two-dimen-siona]. bodies ressembling Bhips and. we have also considered only 'the co.ntribution

of thickness, since the trailing edge coudiion has no direct counterpart in three

dimensions. In contrast with the previous works, we have' been able to deriv

in a closed analytical fOrm and we are inclined to believe that the method

enip-loyed here maybe efficiently used in three dimensional cases.

To obtain simple results we replace the body by a distribution of an

arbit-rary number 'n of discrete Sources Of strength q, located at

z'x-ih'

(j=1,. . . ,n) (Fig. 1) Under this scheme the two boundary conditions (10) and

(14) become '

f ='z + IZ_ZjI13ccj (15)

where cj_q3/UTI_2At/T'_2atj is the

ch*nge

of' the relative thickness at

Zj.

Obviously, by (15) the actual body is replaced by one undergoing abrupt thickness

changes at each source, but by taking the spacing Xj+lxj' sufficiently small in

the portions of steep variation of t, we can achieve any desired accuracy.

Again, in view of àur interest in three dimensional applications, we do not take

into account the trailing edge Kutta-Joukovsky condition, Our problem reduces,

therefore, in determining andf, 'subject to (7)(9),(1I),,(l3) and (15).

The profile of the far free waves as obtained, subsequently from (8) and (12) with

2.

' '

We derive now and the potentials related to the flow past two

sources of strength CYCK at z.x_ih zKxKh

respectively. The advantage of the body discretization is the easy exteaion of the 'solution for n2 to any

other n. The first order solution is given (Wehausen and Laitone, 1965) by

C _Ek'

j- ln(z_z) + ln(z-z). - ln(z-z-2ih) - 'ln(z_zk_2ih)

-z-z -2ih _Ek

ZZk2ih

-

_______ -

_P2

The function w() is defined as

- + iB

eEt(i)

e1j Ce

/A)dA (17).

-where the A plane is cut' along the real' negative axis and, the integration is carried out below' the cut,'auch that (9) is satisfied.

The equatiOn of the tar downstream wave is obtained from the w terms of

(16), by the residue in (17), as follows

1,jk

1,jk'

_""

.

2eh/P2L(ccos 4+CkCOB

)cos

+(csin 4+CkSifl

)sin V2] (,cc) (18).

The second order potential is now split, into two

_{parts f2jkt4jk+fjk}

where is the body correction with the free surface condition kept in its homogeneous form (7) and is the free surface correction related to the

linearized pressure of (11) 'in the absence of the bo4y.

We begin with the bo4y correction. To satisfy' (15) we have to cancel

ar-round the èource on z_zj I-cc the velocities induced at by the first

order terms of (16), excepting the first term representing the source 'itself,. In other 'words we require w=dfljkIdz_Cj1n(Z_Zj)/2t14O for

Iz-zjt".

To sa-tisfy this requirement at second order we have to superimpose a source of strength

c2cu(xrh)

to the original source of strength ce and also a vertical doublet of strength

C3Cji(xj ,h).

Since we consider only second order terms, we

dis-regard the vertical doublets

which

'contribute only at" third order. Adding the appropriate sources at and z1 and carrying out the computations (for

de-tails' see Dágan, 19'72a) 'we obtain finally, for the second order streamfunction far downstream,' the following expression

2jk(mO) '- I_a' e_ix/F2('(Ceixj/P2+eexkiP2)iBb +

'+ cicki (ei

XlrjP)j

Ek+eixkfl i L]}

(x-) (19)

where

2e31

h/F2

Elk(F2+4h2) + 281jk

(20)

b -3h/F2

I. = 4e cosl.

In deriving (19) and (20) we have assumed that k>j, _xk>xj and

ljk=(Xk_xj)/F>0.

We consider now, the second order free surface correction which results

from the linearized pressure _{p2(x) in (11), acting on the free surface. In the} case of a pair of sources p2 can be written as follows

£2 2 _{E C}

k "

2,jk = 4rr jj + kk + 4ir2 jk (21)

the different tertnsresülting from the substitution of the real and imaginary

parts of df1 _{kI1Z Into the last term of (11). The complex potential of the}

flow generated by

jk' for instance, is (Wehausen and Laitone, 1965)

5

I

=

-and ', corresponding to

and of (20), respectively, are

ob-tamed from by letting

xfxk.

Carrying out the detailed computations (see Dagan, 1972a) yields for the

streatnf unction. far downstream

= Im

e''2{(c ei/F2+c ek'F2) (AS+iBS)

_{.+ cjck[(ei-rek)Ck}

j k

IXk

+ (exi _eixk/F)1E;k + e

_{i(Ik+iKk)]}+(c+c)G8}

1-+ CjCkGcosljk (x-co) (23) where z-s' Pjk(s;xi ,x,,h,F2)w (-.j-ids. s 2 -h/F2 2h. F2

A =-e

iT [ln(2h/F2)+0.8272-2cz(0,-.)+ r] s -h/F2 B 2e ( s 2 _h/F2[l 4h2 1 0)-2ct(1 -C = - e -

n(l-f-

.y)+c(-jk jk iT jk 2h F2)+ h l +4h21 (22)

with

-6

Ek =

e_h[_tan_ - (4jk,O)+28(_ljk,_ )+ ljk(9k+4h2) (24) ijk

4 e_h(1+2e_2h)cosljk

Kjk = 4

e_2(_i+2e_h)sjnhjk

= _2-2h/F2

3. _{The wave resistance of n sources}

In the case of n sources (Fig. 1) the streamfünction _{far downstream is} ob-tained from the solution for two sources as a finite sum. _{If we write for n} sources A ₊ sin sinx

(x)

(25) ' cos _conat *2(X,01 _*2 COSX+ ainx + *2 (x-)

where 4'c0'8t _{results from the last two terms of (23), then the wave resistance}

at second order (Salvesen, 1969) is given by

D = C2D1 +

1 COS 2

D1=[(*1

) +

1 cos cog si-n sin

(*i *2 _{1 '2}

where D=D'/pgL'2 and

D'

is the

wave drag.

By using the results for two sources, _{and D2 for n sources are found} after some manipulations (Dagan, 1972a) as

= e 2h/F2

fl

j=1 k-i Lick COSlik

(28)

D2 = D + D

_e_h'/F2{

_C

c[A8sinl +(o

] +

I

j=ik=l

jk

un-i

n

-+ c c c [C8 (sin1 +sinl)+(Ek+Ek)(cosl _{-cosl ) +}

m1 i-i k=j+l in j k jk mj mj - ink

7

and are obtained in (29) by the selection of the coefficients with

the appropriate upper index. _{All the coefficients are given in an analytical} closed form in (20) and (24). _{(29) permit the computation of the nonlinear Wave}

resistance of a body of arbitrary thickness distribution _{at any desired accuracy.} The function a and 8 _{(18) may be taken from the tables in Abramowitz and} Stegun (1964), taking into account that w(c)=e(2wi-E1(-ic)],or may be easily calculated by using the appropriate power series of _Ei(i).

4. _{E21ication to bodies of different sha2es}

We consider first the simplest conceivable case, i.e. the wave resistance

of an isolated source. _{We immediately obtain from. (28) and (29) with n1, c=2}

-2h/F2

(30)

The wave resiStance (30), of a blunt semi-infinite body in _our

approxima-D1 - 4e

=

_16e_4k'/F2

=

_l6e_2h/F2(l2e_21)/p2

tion (Fig. 2a), is represented in Fig. 2c. We

drag coefficient CD=D/puT=D/CF. With

C

CD=Dl/F2, CD2=D2/F2. We have also taken in t

F2U'2/gh' and c=T'/h'.

have used there the more common

CD1+e2 (DD2+CD2) we have

his case L'=h', i.e. h=l,

On the same Fig. 2c we have represented the coefficient of wave resistance for a semi-infinite body having a fine leading edge of _{a wedge shape (Fig 2b),}

created by distributing ten. sources of equal strength at constant spacing. This

way we could estimate the influence of the fineness of the bow on the nonlinear wave resistance. In Fig. 2c we have represented _{CD1 as well as the ratios}

C2/C2 and

_D2/CD1. The first ratio is a measure of the rel.ative importance.

of the free surface correction versus the body correction. .The second ratio rep-resents the relative magnitude of the second order correction.

In these examples there are no interference effects 1ecause the bodies are

of semi-infinite length. _{The next case considered was of a closed body generated}

by a source and a sink of equal strength (Fig. 3a and 4a). With n=2 and

in (28) and (29) we obtain in this case

-h/F2

8e _{(1-coa(2/F2)] (31)}

D2 _(e_F /F2){(2A5..2C8 -I-K5 )sin(2/F2)+( 2Eb+Ib..2E8+15

) (1-cos (2/F2) ] }

-8

Again, we have represented the wave resistance in Figs. 3 and 4 in terms

of the more common coefficient _{CD=D'/2pU'2L'=D/2F2.} Hence. with CDImC2CD1+C3CD2

we have this time _CD1=Dl/2F2 and. C,2=D2/2F2. In Fig. 3b C1]P _CD2 and

are represented as functions of the Froude number for a body of length

submer-gence ratio 2L'/h'=20. In Fig. 4b the same curves are represented for the case

2L'/h'='lO. It is emphasized that the scales of the varIous quantities are

dif-ferent In Figs. 3b and 4b.

5. Discussion of results and conclusions

Fig. 2 permits 'to draw a few conclusions on the effect of the bow shape on the nonlinear wave resistance. First, it is seen that the free surface

correc-a

_.b

tion _CD2 is larger than the body correction _CD2 by a factor of three at suf-ficiently large. F=U'/(gh'.)2. When decreases this ratio begines to increase

In a very steep manner. Hence, any conclusion retarding nonlinear effects which

Is based on the body, correction solely is completely misleading, particularly at small Froude numbers.

_irhe

total nonlinear correction CD2 is a

small

part of CD1 at large F. Again, the nonlinear correction becomes unboundedly large as

FO

tn fact, from (30) we have _{CD2/cDi1I_16/F} as F-' and _{CD2/CDl'aC2/CDf}

-4/F2 as F-O. The influence on the nonlinear wave resistance of making the bow

fine Is manifest in tlie medium raüge of F values, when the bow length and the wave length are of. the same order of magnitude. In tha.t range, for a fine bow _CD2/CJ1

Is almost constant over a large stretch of Froude numbers and is smaller than

CD2/CD1 of a blunt bow. At small and very large F the behavior is similar to. that of an Isolated sources. Finally the second order effect is always negative, i.e. it diminishes the wave resistance.. Moreover,, if c is not sufficiently

small

C1CCDl+t2CD2

(Figs. 3 and 4) may

become negative, which is obviously an

aburdIty. . . .

Figs. 3 and

4 display clearly the interference effects. The

nonlinear

ef-fect is very large for the large length submergence ratio of Fig. 3 (2L'/h'w20)

and becOmes significantly smaller for 2L'/h'i:lO (Fig. 4). Obviously, these large ratios have been selected in order to emphasize the nonlinear effect. To

render it relatively small, the body has to be execeedingly thin or not so blunt.

Again the body correction

C2

is generally smaller.than C2, especially at

small F. The nonlinear term _CD2 tends to sharpen the peaks of the resistance

curve, and to widen its hollows. The nonlinear effect becomes very large in

corn-parisonto the first order wave resistance for amal. F.

Again, we may arrive at

negative wave resistance near the zeros of

9

One of the striking results of our computations, which has been observed

previously by Salvesen (1968), is the singular behavior of the waves amplitude

and wave resistance at small Froude numbers. Ife is kept constant, and no mat-ter how small, the second order wave resistance becomes unboundedly large in

com-parison with the first order wave resistance as F-'O. Hence the linear theory,

as well as the second order correction, become inadequate at small Froude number,

although both _{CD1 and CD2} tend to zero

as

F-'O.

This

effect is called subse-quently "the second small Froude number paradox".

Finally, we believe that our method of computing the wave resistance of n

sources by starting with the solution for two sources offers a possible efficient

way of attacking three-dimensional problems.

III. SMALL FROUDE NUMBERS PARADOXES

1. Introduction

We have seen before that the computation of the yave resistance by the thin

body expansion, which is the method universally used a(present as far as the

free surface condition is concerned, becomes doubtful at small Froude numbers.

This could be observed only after evaluating the second order terms. Experiments

also support the conclusion that the linearized theory fails to predict correctly

the wave resistance at low speeds. The aim of Chaps. III and IV is to elucidate

this problem. The same subject has been considered previously by Ogilvie (1968).

Some of his ideas are validate4 by the present study, but his solution is Shown

to be incomplete.

2 Solutions

As long as we seek solutions of two-dimensional flows it is more convenient

to operate in. the potential plane _f=++i*, as the plane of t1e independent vari-able, rather than the physical plane zx+iy, in order to derive results of

prin-ciple. The advantage stems from the fact that the free surface is kept at the fixed and knon location Tjs-O. Hence, we consider now the solution of v(f) (Fig. 5b) analytical in the half plane O cut along $<i, p--h±O satisfying the following condition, equivalent to (2), (3) and (4)

Im(iF2()2w

f

-w] - 0 (4,-O) (32)

10

-Here, the variables are made dimensionlesa With _{respect to U'}

_and

_{V (Fig.5b)}

and h is defined as h'/L'.

The physical plane is mapped on f. with the aid of

I

which leads to an unwieldy integral _{equation replacing the boundary condition (5).} We shall see, however, that in different approximations the body boundary

cOndi-t-ioubecopes quite simple.

We. consider now two basic types of. perturbation expansions of w(f ;c Ph) aimed to linearize (32):

the thin body expansion, considered in Chap. II,

w(f;c,F,h) 1 .cw1(f;P,h) + c2w2(f;P,h) + ... (35)

for o(1), F0(1) and

the naive

small Froude expansion

w(f;c,P.,h) _w0(f;c,h)

₊

_{F2w1(f;c,h) +....}

(36)

for F'=o(l), cO(l).

Our aim is to study the solutions obtained

_{for different limits c0, P-'O.}

2.

Thethjnbodj

.

The thin body expansion in the potential plane

_{yields results similar, to}

tho8e obtained in the physical. plane.

The

mapping

_{(34) becOmes}

z

f +

+

+

I..

(37)

where

.

-Jw1df

Z2 U

_J[v2_(w1)2/2]df,

Uaing these

relationships

_{. we obtain the following èet}

_{of equations for}

V1

_and

similar to. (7)-(l4)

dv

. Im(iP2

_1-w1)

0 _{(aO) (38)} -.0 .

(,)

. (39) Im

_±()

. (14

kl,=-h±o)

(40) dw Im(iF2

1a_2

0

1mw2

p2 .4[3(u)2+(v)2] . (i=O) (41) (44-co) . (42) (I 41cl,4=-h±O) (43)

where

TdtIdzI,

is the slope of the body contour and under the linearization, process

L'L'+o(L,

h"h'+o(h) such that. h, c and F _{retain their meaning of}

(1). Equations similar to (38)-(43) may be written for higher order _terms.

Due to the similarity between (38)-(40) and (7)-(lO) the solution for may be written at once as

Jt(s)

1 (1 r(s) ds +

i.J

_t(s)w()ds.

f+ih-s ds -J f-ih-s ,(44) -

4

4

where w

is defined by(17) (for details see Dagan, 1972b).

w2, obeying (41)-(43) may be again found like in Chap. Ii by a discrete

source. -distribution. It will display the same siügular behavior for small F as w2(z).

- Let us consider now a small Froude number limit of

w1(44), i.e. an expan-sion of the, type

V1 = w + F2w1+

(45)

for F2"o(l). _{To carry out the expansion of. (44) we have to find the asymptotic} expansion of w for large arguments.

It can be shown' that the function w(C)c1'EI(ir) _{has the following}

aayinp-totic expansion for 'large c

Ic! k=O

(j)k+l

w(C) = ki 2irie k=O (ic)

+

(-1rargc-6) (46) (-6<argir)

+ fths)ds

F2 -if/F2 (45<arg(f-ih+l)ir) (48)

where is an arbitrarily small angle (see Erdely, 1956).

Hence, by using (46,) we find that w1 has the f011owing expressions

1

V1 = _t(s)(f+ih

+ f-ih-s1 (-r.arg(f-ih+l)-5) (47)

- 12

Hence (47) is valid in the f lover half plane, excepting a "wake" attached

to the image of the body across the free-surface (Fig.6). In this wake we have to add the last "wavy!' term of (48) to (47). In particular. (47) is not uniform

along any line parallel to 1'=O and below *-h. In other words, no matter how

Small i F, it is always possible to find, for *h, a sufficiently large $ such that the wavy term of (48). becomes arbitrarily larger in comparison wit the first term. In fact (48) is a uniform asymptotic expansion in the region

-n+arg(f-ih+l)Ew.

The far free waves associated with w1(44) or w(48) are given by _ZL(37)

as

-

_(4ie''

T(s)e52ds]e'

₍₄₉₎

and the coefficient of wave resistance iS

CD1 - 1A112/8F2 (50)

Again, these well known expressions continue to exist if F0 , in virtue

of (48).

4. The naive small Froude number exEansion

We turn now to (36), valid for a body of finite thickness moving at low

speeds. Substituting (36) into (32) and (33) we obtain

Imw°0

(4p.) (51)

w0l

(+_co) (52)

liz v1 - <u0)

vi + 0

v0 is therefore the solution of uniform flow at infinity past the actual body beneath a rigid wall at gi-0, briefly "the rigid wall solution". w1

des-cribed a flow generated by a source distribution along =0, (53) being a

stan-dard Neumann condition. It can be shown that the higher order approximations sa-tisfy the same type of free-surface conditions (53). Moreover, it has been shown.

(Dagän and TuMn, 1972) that the total flux of the sources in (53) and higher

or-der approximations is zero such that w0,v1,...

are 0(l/f2

_I)

as _IfI-' for a closed body (in the absence of circulat-ion). In particular no free-wave are present and the wave resistance is identically zero.

type

We can now consider a thin body limit of

It. is easy to show that

13

-i.e. an expansion of the

1 + cw: + .. (55)

(56) represents the rigid wall solution for the flow past the linearized body.

o,u '

-It is easy to ascertain that w1

(f)w

(f).

We have arrived to what we call "the fir8t small Froude number paradox": the thin body limit (c-O) of the naive small Froude number solution w (56) is not 'equal to the small FrOude number limit of the thin

body

expansion w

(47,48). The two solutions dif fete in the "wavy wake" of Fig. 6.

5.' Discussion of results

(56)

In thepreceeding sections we have defined the two small Froude number

pa-radoxes occurring as F-sO, in the solution of the. problem which .depeu4s on the two small parameters c and, P. The nonuniform behavior of the solution may be

related to' the fact that in 'carrying out the naive small Froude number expansion

(36) we have lowered the order of the boundary condition (32), the derivative

dis-appearing in the 1.h.s. of'(5l) and (53), similarly to well known boundary layer,

problems. This observation is strengthened by the inspection of the wavy term -if/F2

(49): the function e changes its order by differentiation and the two

terms

of (38) become of the same order of magnitude .no,matter how Small is F (this observation has underlain OgilvLe's(1968) study). The nonuniformity present in our problem is, however, different' and more subtle than. that of other singular expansions (Van Dyke, 1964'; Cole, 1968) with a few respects: (i) ye cannot detect the nonuniformity of the solution from the naive, small Froude number expansion which is well behaved in the entire flow domain. We cannot, therefore, ru1e out

this solution at the present stage; (ii) for a submerged body the "wavy wake" is attached to the fictitious image of the body (Fig. 6) rather than the. body itself.

It intersects the flow domain *cO only far behind, the body and has an 'exponen--if/F2 4/F2 -i/F2 tially small effect upon the body itself; (iii) the wavy term e "se e

has, for F-sO, the character of an exponential boundary layerdecay for 4<0 and

a rapidly oscillating behavior in the. $ direction. It displays, therefore, a complex pathological behavior as 1-'O.

o,u

ol

1 11

r(s)

1 1 1

r(s)

ds V1 W]

+ V1

-

_f-ih-s ds + -f+ih-s

14

-It is also worthWhile t&. point out that only for hO (submerged body)

is the amplitude of the free waves decaying exponentially for FO. Otherwise,

for h=O the decay may be algebraic at best.

We are going now to determine the origin of the small Froude number

para-doxes and to derive uniform solutions for the wave resistance as F=o(l) and

E=O(l) or c-o(l).

IV. WAVE RESISTANCE AT LOW SPEEDS

1. The model 2roblern

Like in other nonlinear problems of hydrodynamics we seek a "model"

prob-lem which has the same features as the basic nonlinear probprob-lem, but can be solved

exactly. We define our auxiliary problem as follows: determine the complex func-tion (r;c1,c21h) of the complex variable' C.+ix, analytical in the whole plane cut along x=h, <l (Fig.7a), satisfying everywhere the di:fferential equa-tion

ic2[1+a(c;c1c2,h)]ft - w = .p(ç;c1,c2,h) (57)

subject to the condition

w-, 0 (F+_oD,X<h) - (58)

o and p are given functions, holomorphic in the entire plane excepting the

slit _IIcl, 'and

0(1/Ic!2)

as

k

Moreover, a and are bounded

álong.the slit, and l+a does not vanish there. To simplify matters we assume

that a and p

admit

expansions in power series in and of the type

a(c;c1,2,h) = k=1 j-0 p(c;c1,c2h) = '

(2)ipi(;1,1)

= (c j_O j=O k-1 l/(l+a) 1 +

(e2)j(;c1,h)

'l + j=O j=O k=l

uniform in the entire _c plane cut along nh, IIi and convergent for 'finite

and c2, not necessarily small,.'

The "model" problem is similar to our original flow problem,(57) being

sitni-lar to (32), c2,c1, w and c corresponding to F2,e,

w and

f, respectively, and p representing the body effect. Two major simplifications are present,

15

-however, in (57): the coefficient

of dw/d

is given, the equation being thus linearized, and equation (57), 'iith analytical coefficients, is valid in the

en-tire plane and not merely at p=O. Due to these simplfications (57) admits an

exact closed solution.

Our purpose is to establish, by using the exact solution, how can uniform

expansions of w for c2-o(l) be obtained from the expansion of (57).

If we expand (57) for c1o(l), c20(1) (corresponding to the thin body expansion) with

e1w1 + (c1)2w2 + .... (60)

we obtain for '... equations and solutions similar to (38),(39),(41),(42),

(44). In particular, if we let subsequently

_{2° into}

(60), i.e.

we arrive at different asymptotic expansions in the two regions

of the plane (Fig.7a) exactly like in (47) and (48), "wavy" terms being pre-sent in the shaded region of Fig. 7 If we Start with an expansion of (57) for

c2=o(1),'c1=0(1) (corresponding to finite thickness, naive small F expansion)

with we obtain equations similar to (51)-(54) and solutions with

no waves. Furthermore, if we let afterwards

e1O,

i.e.

we obtain limits which are different from those of the preceeding expansion, in

the shaded zone of Fig. .7. Hence, our model problem leads to the same "paradoxes"

as the prototype, nonlinear problem.

Now, let us consider the exact solution for w, satisfying (57) and (58),

which can be written at once as

-

I.

''l"2'

A d

1A'h

exp[_'f

l(V;Ch)]}dA

(61)

where the integrals are carried out belOw ReAl, ImA=h

_{and Revl, Imvh}

in the A and v planes, repsectively. We can use now the series (59) to rewrite

(61) as follows

- _{Lexp(....i/c2)}

_J

_{(p+c1p+e2p+a.}

_.}exp(iA/c

exp[-1

J

(+cip+c2pI+ci2+...)dJ]dA

(62)

valid for finite c, _C2 We'are now in a position to expand (62) for small

and/or c. The detailed analysis may be found in Dagan (1972b), Herewith, the

16

-(1) the limit _i=o(l)

cfO(l)

of (62) yields the same results at first order, c1w1., as the solution obtained by expanding (57) if, and only

, c1/c.2o(1). This last condition stems from the existence of the

ratio in the last exponential of (62);

the limit cfo(l),c1=O(l) . of. (62), w oes not coincide with that obtained by the naive expansion of (57). The uniform solution differs from the naive solution in the "Wavy wake" and does comprise "waves".

Moreover, to obtain a first order complete solution we have to retain

in the last exponential of (62) all the written terms, up to (c2)2, in

particular c2p, The "far waves" are obtained by contour integration

(Fig.7b) as follows

= - f-

exp(_iC/E2)4(p0(A)+p0(A)).l0(A)] exp(IA/c2)

exp{f- _I (i°(v)i-c2i1(v)]dv}dA

(Reç)

(63)

Again, it is emphasized that in the last exponential E2M contributes at 0(1) because of the division with C2 Going in reverse, the

dif-ferential equation which yields. the uniform first order solution, obtained

by the appropriate expansion of.(57),is .

dw0/dC + (i/c2)(l+ia0+2p1)w0 - (L/c2)(p°+p%0) (64)

where terms up to (c2)2 have been retained in l/(l+a), the coeffc1ent

of(59).

the

limit c1o(l),e2=o(1) of

(62) is not defined unless we specify the or der of c/c2. _{For !li!2n0(1)} We obtain again at first order a solution with a "wavy" term1 the latter having the expression

Wuniform

Cll,unifOrm +;;

A

e*P(_ic/c2)JP(A).

exp (iA/c2) .exp[-j2

J

(v)dv]dA +

0(4tQ

(Rè-I.o) (65)

This solution differs from that àbtained by taking. the limits

first and c2-O afterwards in (57). Moreover (65) is obtained from the

solution of the differential equation, derived from (57),

- 17

o(1), c2'=o(l), £1/2ó(1) yields by the expansion of

i

.X

j çÀ

expl-1

J C (v)dv.] 1 + __L udv + ... in (65) the usual

£2 £2 'ç

linearized approximation

4

0

-

c2

exP(_/c2)JP(A)exp(iA/c2)4A

(67)

satisfying the differential equation

d4/dc + (.1/c2)w1 = (j/c2)p (68)

2. AEElicationo.f results to the noniinearroblem (2otential Elane)

Due to the similarity between .(32) and (57) the results obtained in the

mo-del problem can be extended to the hydrodynamical problem at once (for details see

Dagan, l972b). With w=1+W the uniform solution, the key to obtaining first

or-der uniform approximations W0(for F2=o(l),c=O(1) ), CV(for c=o(l),F2=O(l) ) and cW(e=o(1).,F2o(l),c/F2=O(1) ) is to expand the coefficient

()2w of

dw/df in (32) in a naive, small Froude number expansion and to ret.ain the appropriate number of terms. By doing that we obtain the follOwing unifotm asymptotic approximations

for W:

c.'o(1),F=O(l), i.e. c/F=o.(l) (thin body, finite length Ftoude number,

large thickness Froude number U'2/gT'), WcW1+.., coincides with (44), and the usual thin body approximation is, therefOre, uniform.

F2=O(l), e-O(l) (small Froude number,finite thickneSs), W=W0+... sa-tIsfies the free-surface boundary conditiOn (similar to 64)

Im{iF2((u0)3+F2(uO)2(3u1+iv1)+...] - w°} = 0 (4=O) (69)

and along the.body l+W°"w°, where w0(51,52) is the.najve small F solU-tion. v° is not an uniform solution in the "wavy wake". The solution of W0 subject to (69) is very difficult.

F2"o(l), e"o(l), c/F2=0(l) (small Froude number, thin body, thickness Froude numberS U'2/gT' of order one), WmeW°1+... '.

satisfies the free-surface condition (similar to 66)

Im[iF2(1+3eu

_{W]= 0}

. (4s=0) (70)

where u=Re w1°

is the naive linearized Small F solution (56). Also,

18

-integration by parts W _{has been found in a close form as follows}

w'1

- + _{exp(_if/F2)J}

w'l(A).eXp(iA/F2),exp[_

-- _J (71)

For F2-'O _{W+%4+o(F2) excepting the "wavy wake". There, we obtain}

Siini-larly to (67), by contour integration like in Fig. 7b,

W = cW + ... = _{exp(_i(f_ih)/F2]Jw'(sfih).exp(is/F2).}

s+ih -1

exp(-

!-J

v(v)dv]ds + ... (Ref-oo) (72)

where w°'' and w10,l _{are given by (56)..}

is obviously not a unifprm

solution.

Only, and if only, c/F2=o(l) (72) _{degenerates into (47,48). The} implica-tions of the different limits are discussed in the following secimplica-tions.

3. UnifOrm solutions in the pZsical plane

it was advantageous to carry out the basic derivations in the potential

plane. _{In applications it is convenient (and in thràe dimensions it is essential)} to operate in the physical plane. It is easy to transfer the previous results to

the physical plane. With w(z;c,F2,h)=l+W(z;c,F2,h) we have the following limits:

(1) c=o(1), F2=O(l) (thin body, finite Froude number), WECW1+E2W2+...,W1,W2, satisfy equations similar to (7)-(14). W1=df1/dz is the usual thin body

solution, W2df2/dz is the second order solution (see Chap. II).

(ii) F=o(l), e=O(1) (Small Froude number, full body), _{W'=W0+F2W'4... . The}

complete first order term W0U0-iV0 _{satisfies the free-surface boundary}

condition

p2((uo)2+2F2uou11i!L+ 2Fu0v1 V0 = 0 (y=O) (73) on the unperturbed free-surface, similar to (64). w0au0_iv0 ts the rigid

wall solution for flow past the actual body and w1 is the next term of the

naive small Eroude number solution.

(73) has a simple physical interpretation: it _{represents the equation}

satis-fled by waves generated on a stream of variable speed _{v0+F2w', beneath}

y=O. _{Ogilvie (1968) has retained only the first term in (73), i.e. has} rep-laced (73) by

Im[iF(uO)2 _{-WI = 0 (y0) (74)}

dz

He has based the derivation of (74) on the intuitive reasoning that.at small

F the wave length of the free waves becomes small compared to the body length

scale (which governs the rigid wall solution) and, therefore, the waves are

traveliling on a basic stream of varying velocity. Although the argument is valid in principle, (74) is not an uniform asymptotic approximation, as shown

in the preceeding section. To determine W0, satisfying (73), is a diff i-cult task which is not pursued here.

(iii) £.=o(l), F=o(l). /F2=D(l) (thin body, small length Froude number, finite thickness Froude number),

WwcW+... .

By analogy with (70)

W(z)

satis-fies the boundary condition.

19

-dW0

Im[iF2(l+2c4)ri_W]

0 (yO) (75)

the radiation condition

along the skeleton of the body. w, the linearized rigid wall solution has

the expression

. (78)

may be found by analytical continuaion across y=0. After some

manipula-tions (Dagan, 1972b) the solution is found to be

W(z;h) o,l -

+

e'

J.wU(A)eiA/Fe_2if.1(A)IFdA

(79)

-which is analogous to (71). The profile of the free waves NacN+.... is

derived from (79) as

= Im

Ae'

= Im(2f

Uei

Fe_2i

dA]e"

(x)

(80)

A

In. (79.) and (80),

f=J

v(v)dv and

S is the cut Ixici, y=h±0 in the A plane. Along S

+00

_o,uo,u+i

and

f1=f' +°"±it

Hence,

we can write

for A

the following expression

A 4Ji+iheiA/F2e_2ic(f cósh (81)

-l+ih

(I

xJ

(x+.co) ₍₇₆₎

and the body condition

20

-The wave resistance coefficient _CD is given by

CD

=c2IAj2/8F2

(82)

(iv) c=o(l), F2ao(l), c/F2=o(1) (thin body, small length Froude number, large thickness Froude number). This limit can be obtained in two ways: by

expan-ding W1 of (i) for F20 or via (79) for c-0. Hence, the usual linea-rized approximation usd in ship resistance theory may be extended in'the

range of small Froude numbers only if c2/F =gT'/U'2 is small. 0To keep C

fixed and to let F2+0 is tantamount to wrIte in (80)

e_212l_2iC4/F12

which is obviously illegitimate if c/F2 is not small.. For this reason the

second order approximation '2ic 4/F2 may become large compared to unity and

the expansion of (79) may diverge. For

c/F2o(l)

(81) leads to

A,1j4ie_hJ

)eiSds

i.e. the usual complex amplitude of linearized waves.

4. Illustration of results: wave resistance of a biconvarabolical bod1

To illustrate the main features of the thin body small Froude number

solu-tion (79,80,81), presented

here\for

the first time, we have computed the wave

re-sistance for a parabolical body (Fig. 8) with the thickness dIstribution t=1-s2

(Iski),

r=-2s. The linearized rigid wall solution in. this case is expressed by

+ f"

₌ {2z -

((z_ih)2_l)ln

::

_[(z+ih)2_l]in}

(84)

and w=44/dz.

By using (81)'we have computed CD .(82) by a simple Simpson

in-tegration, for c=0 .05 (Dagan, l972b). CD/cZ 'as function of FU '/(2gL') is represented in Fig. 8. On the same figure we have represented _CD

_Uz'

based on

the usual linearized approximation (83). In addition we have represented _CD1C2=

CD j.fl+CCD2 based on an expansion up to c2 of (82), i.e. on an illegitimate ex-pansion of the exponential in (80).

Our small Froude number thin body solution (81) differs from the usual one

(83)

(83) with two res'ects: while

waves

e"''

generated.

elementary waves

ei)2

(83) may be regarded as a summation of elementary by each element of thickness dttds, in (81) the

have phase and amplitudes depending on A in a corn-plex manner. in particular waves are generated by the parallel part of the body

(t=0,t=const) and the amplitu4e changes if the direction of motion of an asymet-rical body is reversed. The- 'rave resistance is always positive, while, in an

ille-2icf

IF2

CD is close to _Dlin (Fig.8) at relatively high Fr (i.e. shifted towards small F The peaks of the resistance curve

Tt

ti

excepting the highest peak,much smaller than those

The disastrous effect of using the second order approximation of the thin

body expansion in the region of small Froude numbers is illustrated clearly in

Fig. 8.

It is worthwhile to point out that the wave resistance (81) is integrable

only if the leading (or trailing ) edge singularity is like v1'.(z-ih+l) where a<l. Hence, a parabolical nose (=0.5) is acceptable, while a box like shape

(ci=l) is not integrable.

5. ExtensIon Of results to three-dimensional flows

It is easy to proceed along the same lines and to derive the free-surface con4itions satisfied by the various unifOrm approximations as 0 in three di-mansions. With u,v,w the velocity components and z _{a.vertical coordinate the} exact free-surface condition, cOunterpart of (2), may be written as follows

F2(u2u,x+Uv(v,x+U,)+V2v,+UWW,+VWW,1

+ v 0 (z=n) (85)

The naive small Frou4e number expansion (u,v,w)=(u0,v0,v°)+F2(u1 ,v' ,w1')+.. .

t.F20+,.. leads tq u°,v0,w0 as a rigid wall solution for flow past the actual bo-dy. With c _{a fineness or slenderness parameter, a. further expansion of} . (u,v,w)

yields u°=1+c4+...,(v°,w°)=c(i4,4)+... . The usual linearized approximations

is obtained from (85), by expanding the velocity near the uniform flow u=1+eu1+...,

(v,w,)=c(v11w1,ri1)+. as folloWs

F2u

+w 'O

l,x 1,

21

-small c2/F ), but of the uniform .solu-of the usual theory.

(z=0) (86)

Let now l+V be a uniform small Froüde number approximation. By analogy

with (73), in the limit F+O, c=0(l) satisfies the following free-surface

con-dition

F2.{[(u0)2+2F2u0u1]9 + 2[u°v°+F2(u°v1+u'v°)D9 +

+ (.(v0)2+2F2v0v1] + F2u°w'9

4 F2v°w'9} +

= 0 (z=0) (87)

may be represented along the actual body surface by' a distribution of singüla-rities derived from the rigid wall solution. The rigid wall sOlution is asymptotic

to as F-'O excepting a "wavy wake" which this time, is generated by rays

ema-nating from the body image across z0 towards x- at an angle-s (arbitrarily

small) with the horizontal plane. To determine representing.waves over a

The simpler approximation of thin (slender) body c=o(1), small Froude

num-ber Fo(l),

and finite beam (and draft) Froude number c/F2'O(l), _{18. obtained} from (87) like (75)

l-2cu0

+ 1 =

may be represented by the source distribution of the rigid wall solution on the body skeleton (central plane, or axis). _{(88) is the extension of the usual}

linearized free-surface equation (86) (which is the basis of computation of ship,

wave resistance via Michell integral) into the range of small Froude numbers,

where (86) becOmes invalid. The solution of is the object of future studies.

6. _{Discussion of results and conclusions} 22

-(z=O) (88)

The two small. Froude number paradoxes have been explained with the. aid of our model problem.

It was 8hown that the naive small Froude number expansion does not yield

a uniform solution, the region of nonuniformity being the "wavy wake".

The elucidation of the second paradox has led to the important conclusion that the usual thin body first and higher order approximations _{are valid only for} large thickness Froude numbers. Fo moderate values a new first order

approxima-tion. has been derived: it results from taking in the free-surface condition _a

ba-sic variable speed, rather than a uniform flow, as an "inpurturbed" state. This

new approximation is the natural extension of the thin body theory into the _range

of' small Froude numbers.

The basic equations of flow past' a body of finite thickness moving _{at low} Froude number have been also derived. Again, ,to obtain a uniform solution one

has to satisfy simultaneously the boundary condition _{on the full 'body and a} free-surface condition in which a flow of varying velocity is taken _{as the basic state.} This basic flow is derived by solving fOr the two terms of the naive small Froude number expansion (rigid wall and first order Neumann type problem).

To solve for flow past the actual body, while keeping the free-surface

con-dition in its usual linearized form, may lead to _{erroneous results in the range of}

23

-V. PRELIMINARY EXPERIMENTS ON ThE TWO DIMENSIONAL BREAKING WAVE

An important free-surface nonlinear effect, present in the case of blunt bow

ships, is related to the breaking wave. At the 8th Naval Hydrodynamics Symposium we have presented (Dagan & Tulin, 1972) theoretical models of the breaking wave

inception and of the bow jet. Recently, experiments have been conducted at

Hydro-nautics Inc. under the supervision of Mr. M. Altman in order to visualise the

two-dimensional breaking wave. The detailed results of these experiments will be re-ported elsewhere. Herewith a few preliminary observations.

A rectangular body has been towed at constant speed in the small Hydronautics

towing tank. The water depth was 38cm and the model has been submerged at (1)

2.5cm and (ii) 1.25cm beneath the unperturbed level, such that the effect of the

bottom was negligible. The model has been towed at six different speeds in the range 0.61-1.46 rn/sec. The model motion has been recorded through the channel glass wall on a 16mm color film at 64 frames per second. Taking the pictures has

started after 3.5m of run (tit tank total length is 24m). In Fig. 9 we have

rep-rOduced two pictures for the 2.5cm model: at 0.61 m/sec and at 1.46 m/sec, the

cor-responding draft Froude numbers being 1.22 and 2.93, respectively,

The free-surface in front of the body had vertical pulsations which became

more violent as the speed increased. This made quite difficult the definition of

the average free-surface profile. It seems that the oscillations are related to

gravity effects since the periods for the two submergence depths were roughly in

the same ratio as the square root of the drafts. Satisfactory Froude number

Si-militude has been obtained for the free-surface'elevation near the body for the

two drafts.

The breaking wave inception apparently occurs at a Froude number somewhere

between 1.20 and 1.50, which corelates quite well with our theoretical prediction

of 1.50. Separation at the corner of the body profile, visible at high speeds, makes difficult the definition, of the body shape for an inviscid flow calculation. We did not reach a "spray regime" in the range of considered speeds. The study of

the complex flow pattern of a developed breaking wave is the object of future

24

-VI.

GENERAL CONCLUSIONS

We have dicussed the pertinent conclusions at the end of each of the pre-ceeding chapters. Here, we will try to discuss their bearing on ship wave

resis-tance.

The usual thin (or slender) body first order linearized approximation,

lead-ing to the Michell integral, is valid for sufficiently. large beam (and draft) Froude numbers. In its range of. validity this approximation may be improved by taking into account the second order term. It seems that the contribution of the

free-surface correction is of the same order of magnitude as that of the body cor-rection in this second order term. As the shape becomes finer, the Froude number

limiting from below the range of validity of the linearized approximation, as well as the second order correction, become smaller.

For nioderatè beam (and draft) Froude numbers and, henceforth, small length

Froude numbers, the linearized solution is no more valid and the second order cOr-. rection worsens the results, rather than improving them. To obtain a first order

Uniformly valid solution for a thin (or slender) body in this case, one has to

take a variable velocity distribution, rather than a uniform, as the basic unper--turbed distribution in the free-surface condition. This basic flow, as well as the singularity distribution along the center pláe (or axis), may be computed by

solving for a rigid wall flow past the linearized body.

Linear free-surface conditions with variable coefficients have been derived

also for the case of small length Froude number flow past the actual body (finite

beam, or draft, length ratio). To obtain a uniform solution, the basic nonuniform flow (on which the variable coefficienta of the free-surface condition are based)

has to include the rigid wall,

_as

well. as the next term, of a naive small Froude solution of flow past the actual body. The singularity distribution on the body

surface may be taken from the rigid wall solution solely. This results suggests that solving for the acutal body shape, but with a linearized free-surface condi-

-tion with constant coefficients (the Neumann-Kelvin problem) does not yield a uni-formly valid solution at small Froude numbers.

The above conclusions are based on the assumption that the results obtained

in the two dimensional case may be extrapolated to the..ship wave resistance prob-lem, at least in principle. Only solving for actual three-dimensional flows wili

make the conclusions valid in both qualitative and quantitative terms. _Such hree-dimensional solutions pose, however, difficult mathematical problems which have

25

-The picture of the nonlinear ship wave resistance theory is not complete

unless we refer to two components which are somehow related to viscous effects: the bow breaking wave and the wake. Only the first component has been considered

in our studies.

ACKNOWLEDGMENT

The present work has been supported by ONR under contraàt No.

N00014-71-C-0080 NR 062-266 with Hydronautics Inc. Most of the material is based on Hydro-nautics.Rep. 7203-2 and 7203-3 (Dagan l972aand l972b in References). I.wish to

express my gratitude to M.P. Tulin for the stimulating discussions we had on the

subject and for his collaboration in the different stages of the work.

LIST OF SYMBOLS.

Dotted variables have dimensions; undotted variables are dimensionless.

A amplitude of free Waves for downstream.

CD - coef. of wave resistance.

D' - wave drag..

- complex pqtential.

F - Froude number based on half length. F - Froude number based on length.

n

h,h'-submergence of body axis beneath unperturbed level.

- submergence of stagnation streamline far upstream

q - strength of source located at z.

- dimensionleBs distance between two sources. L' - body length.

- length related to the body image in the potential plane.

N - free-surface elevation in a uniform small F expansion. t'(x') - thickness distribution.

- body maximum half thickness. velocity components.

U' - unperturbed velocity.

- complex velocity in two dimensions; vertical velocity component in 3d.

W - perturbation complex velocity in a uniform small F expansion. - horizontal coordinate positive in the direction of flow.

-

vertical upwards coord. in 2d, horizontal in 3d.

26

-- real and imaginary parts of ui.. - angle arbitrarily small

c - slenderness parameter.

c1,c2- perturbation parameters.

velocity potential in two dimensions.

velocity potential of a uniform small F expansion in 3d.

- streamfunction in 2d.

()'e1Ei(iC)

.

EI()

is the exponential integral.

- free-surface elevation above the unperturbed level. - auxil-iary variables.

a,p auxiliary functions. - slope of body contour.

REFERENCES

Abramowitz,M. & Stegun,I.A., Handbook of Mathematical Functions, Dover,

1964.

Cole,J.D., Perturbation methods in applied mathematics, Blaisdell Pubi. Comp., 1968..

Dagan,G., NOnlinear effects for two-dimensional flow past submerged bodies

moving at low Froude numbers, Hydronautics Inc., Tech. Rep.

7103-1, 15p. 1971.

DaganC. & Tul-in,M.P. Two-dimensional gravity free-surface flow past blunt

bodies, J.F.M., Vol. 51, p.3, pp. 529-543, 1972.

DaganG. A study of the nonlinear wave resistance of a two-dimensional source generated body, Hydronautics Inc., Tech. Rep. 7103-2, 1972a.

Dagan,G., Small Froude number paradoxes and wave resintance at low speeds, Hydronautics Inc., Tech. Rep. 7103-3, L972b.

Erdely,A., Asymptotic expansions, Dover, N.Y., 1956.

Ogilvie,T.F., Wave resistance: the low speed limit, Dept. of Naval Arch; & Mar. Eng., Univ. of Michigan, Rep. No. 002, 1968.

Salvesen,N., On higher order wave theory for submerged two-dimensional

-27-1O. TuckE.O., The effect of non-linearity at the free-surface on flow past a submerged cylinder, J.F.M., Vol. 23, pt.2, pp. 401-414, 1965.

Van DykeM. , Perturbation methods in fluid mechanics, Academic, 1964.

Wehausen,J .V., & Laitone,E.V., Surface waves, in Encyclopedia of Physics, Vol. IX, pp. 446-779, Springer, 1960.

LIST OF FIGURES

Fig. 1 A distribution of sources.

Fig. 2 Wave resistance of a body of semi-infinite length: (a) a source

gene-rated body, (b) wedge shape leading edge, and (c) wave resistance curves.

Fig. 3 Wave resistance of a source-sink body: (a) the body shape and (b) wave resistance curves for 2L'/h'20.

Fig. 4 Wave resistance of a source-sink body: (a) the body shape and (b) wave resistance curves for 2L'/h'lO.

Fig. 5 Two-dimensional flow past a submerged body: (a) the physical plane and (b) the potential plane.

Fig. 6 Regions of uniformity of the small Froude number solutions.

Fig. 7 (a) Regioné of uniformity of small solutions of the model problem

and (b) integration path for large Ret.

Fig. 8 Wave resistance of a biconvex parabolical body by different approximations.

Fig. 9 Flow in front of a rectangular body: (a)

Ut/(gT)L'2l.22

and

Figure

1

A distribution of sources

0 05 13 2 2.5 3

Cc)

Figure 2 - Wave resistance of a body of semi-infinite length

(a) a source generated body, (b) we4ge shape leading

edge,, and (c) wave resistance curves.

S S

xli

£ 200 l0_ tOO -_---. Th 2T,...

-.

(1 II I. (a)

r

Il

I \

Ii

I CocUI9u*.&COIE3COO

ii

I EvTSL ! 219h.2Q A 1.0 1.5 I- I -d3 2.0 2.5 -I - I (b)

-Figure 3 - Wave resistance of a source-pink body :

(a) the body

shape and (b) wave resi8tance curves for 2L'/h' = 20.

I I

i'/1

.\

_/

\. I,, t I

/

/ \ I

/

t

/

- _--

_r

-ø

(a) d.6

uivr

Tf1212I -2CDI*E3CD2 11L arnI. to 0

-

'S.- -'nv d7

uW

Figure 4

Wave resistance of a source-sink body :

(a the body

shape and (b) wave resistance curves: for 2L'/h.' = 10.

25 2i0 10 -2 Q2 Ib)

______ 2L ______

2U

(a) (b)

Figure 5

-

Two-dimensional flow past a submerged body:

(a) the physical plane and (b) the potential plane.

SI

1-ih

Figure 6

- Regions of uniformity of the small Froude number

(a)

Figure 8 - Wave

resis-tance of a biconvex

parabólical body by

different

approxi-mations.

Figure 7 - (a) Regions of uniformity of small

₂

solutions of

the model problem and (b) integration path for large

Re

as 04 0.3 0.2 0.1 0 £C&E2 03

,

1 $ r S

(b)

t.raL Q.flt h.KIt Pn.ui r

c,,u0129U'L E.0.05

h.0.20

pt o.ds.

d .we3)

Iid

-.'.

'

-

.4-I

1/2

(a) U'/(gT')

= 1.22

(b)

Ut/(gT)h1'2 = 2.93