Low speed wave making theory by slender body theory

ARCHEF

PAPERS

OF

SHIP RESEARCH INSTITUTE

Low Speed Vave Making Theory by Slender Body Theory

By

Iliroyuki ADACHI

February 1981 Ship Research Institute

Tokyo, Japan

Lab. y. Scheepsbouwkun.L

Techuische Hog9schooI

LOW SPEED WAVE MAKING THEORY BY SLENDER

BODY THEORY* By

Hiroyuki ADACHI* *

ABSTRACT

The low speed theory among the wave making resistance theories is treated

in this paper. This theory is now considered to be the most successful one that actually gives reasonable numerical estimations for ship wave resistance

at relatively low speed. In order to calculate wave resistance more precisely and to investigate flow field near ship hull, it is necessary to develop a higher order low speed theory. One of the important phenomena near ship hull is the so-called local nonlinear effect which arises from the fact that waves generated at bow near field must propagate over flow field around ship.

In order to handle the exact boundary value problem for the low speed problem, the iteration method will be used which leads to the 3-D

Neumann-Kelvin problem. However the 3-D Neumann-Kelvin problem seems not to be easily calculated numerically at present. In this paper, to avoid the 3-D Neumann-Kelvin problem, further assumption that a ship should be slender is introduced to utilize the slender ship theory. And by making use of the method of matched asymptotic expansions we can obtain desired formulas in the far field with which both diffraction and local nonlinearity effects can be

treated.

1. INTRODUCTION

The problem of the wave making resistance for a ship steadily running

through calm sea surface may be formulated as a complicated nonlinear

boundary value problem. Even if the viscosity of fluid were neglected the

problem still remains to be unsolvable because of the nonlinearity

ap-peared in it. Therefore the perturbation scheme should have been applied

to the problem so as for the boundary conditions to be linearlized. Only

the perturbation scheme based on small parameters pertaining to ship geometry has been examined until the low speed wave making theory

came out.

The feature of the new theory is marked by the fact the perturbation around a basic flow is considered which is equivalent to flow field around the double body of a ship in the infinite fluid.

In that theory, it is

re-quired that the ship speed should be small for the problem to be tractable.

To be more specific, it must be assumed that the waves generated by a

ship can propagate over the basic flow field.

Usually the wave resistance formula by the low speed theory is given

by the far field potential which represents wave field in the region far

from a ship. Then it is said that the detailed near field potential is not needed as far as the wave resistance concerned. And some calculated examples of wave resistance based on the low speed theory seem to be

very promising. However, the importance of the local nonlinearity con-cerning wave propagation around ship body has become recognized and it is now considered as one of clues to the full understanding to the wave making phenomena of a ship. From this point of view, the near field

potential should be studied more throughly.

The exact formulation of the low speed theory has been done by

Maruo (1977) [1] who gave the expression of potential valid in either the near field or the far field. However, he did not solve the problem in the

full formulation, nor in the approximated form in the near field. Such

attempt to calculate near field flow field under the low speed theory has been done by Shimomura & Kitazawa (1979) [2]. Although their basic

equation is sligthly different from the exact formulation, they can show

the effect of the diffraction potential which explains the reflection of waves due to the pressure potential on the ship's hull surface. So far

only the pressure potential has been considered for the wave resistance calculation in the low speed theory, so that the attempt by Shimomura

& Kitazawa including the reflection of waves on the hull surface in order

to satisfy the exact hull boundary condition may certainly improve the

theory.

Their basic idea is similar to Maruo's, in that the nonlinear free

sur-face condition is deformed to the same form as the classical linearlized

one defined on the still water surface by putting together the remaining

nonlinear quantities into the right hand side term of the condition and

the Green's function which is the potential due to a point Havelock wave

source is introduced to make use of the Green's theorem in the region

below the still water surface. This procedure will lead to an integral

equation for some unknown quantities such as the potential value on hull

and the source density functions both on hull and free surfaces. In

practice, it seems to be impossible to solve the integral equation all at once for all the unknown quantities, so that the approximation such as the iterative procedure must be devised.

In formulating the iterative procedure for the low speed theory, even-tually the so-called 3-D Neumann-Kelvin problem3 that is the most

labourious chorues in the stage of numerical calculation will appear. As

far as we are concerned to the first step of iteration, only the pressure

potential is our interest. At the first step there is no interaction between the waves due to the pressure potential and ship hull. The 3-D

Neumann-Kelvin problem comes out at the second step of iteration. This fact

3 speed theory can be applied to any arbitrary ship shapes without such

additional trouble as solving the 3-D Neumann-Kelvin problem.

However, if we want to study the diffraction effect as well as the local nonlinearity that is considered to affect the propagation of waves

generated by the pressure potential over the basic flow field, the second

approximation must be treated. That is, the 3-D Neumann-Kelvin problem

for an arbitrary ship hull form must be solved. At present it may not so easy task to calculate the problem by computer. The only trouble seems to lie in the solving method for the 3-D Neumann-Kelvin problem. Then at the second step of the approximation, it may become advantageous

to introduce the perturbation scheme pertaining to ship hull geometry to make the 3-D Neumann-Kelvin problem tractable.

The introduction of the perturbation by the small parameters related to ship hull form in the low speed theory has been considered by Newman

(1976)l _{in the extensive investigation on the free surface condition of the}

linearized wave making resistance theory. He showed that the slender

low speed assumption leads to the classical inhomogeneous linearized free surface condition.

Then it seems better to try again the slender ship

assumption in the low speed theory in order to investigate the diffraction effect and the local nonlinearity at the second approximation of the exact

problem. So far the diffraction as well as the local nonlinearity has been

in actual studied for not full ship, instead for rather slender ship hull

form5.

In this paper the idea of introducing a perturbation scheme based on the slender ship theory will be expanded starting from reviewing the exact formulation of the problem. The problem will be treated by the iteration

procedure similar to Shimomura & Kitazawa's up to the second steps.

The 3-D Neumann-Kelvin problem in the second approximation will be approximated by the method of matched asymptotic expansions. In the process of the analysis, the near field problem will turn out to be the

same one that in the time dependent boundary value problem is defined by

Adachi & Ohmatsu6. The diffraction effect and the local nonlinearity will

be given by the near field potentials which are relatively easier to solve numerically. And the wave resistance formula modified by the diffraction effect and the local nonlinearity will be given by the far field potentials.

2. FORMULATION OF PROBLEM

Let's assume that the fluid is

invisid and imcompressive and the

motion is irrotational. Also the existence of solutions for all the boundary value problems is assumed. Take the cartesian coordinates system, with

z-axis vertically upward and with xy plane on the free surface at rest. The ship is fixed in the space and the uniform flow with velocity U is

directed to x-axis. The origin is to be taken at the bow. Let's denote the disturbance due to a ship by a velocity potential U(x, y, z), then the potential for the flow field can be defined by

ø(x, y, z)= Ux+ U(x, y, z)

Since disturbance potential is harmonic in the fluid domain which extends to infinity, then

[L] P2=O

z<

(1)

where z=(x, y) is the equation of free surface elevation. The boundaries

of fluid are the wetted surface on ship hull, the free surface and the

sur-face surrounding fluid at infinity. On these boundaries the potential must

satisfy the specified conditions. On the wetted hull surface S11, we have

[H] 6x =

an an on S11

(2)

where a/an denotes the normal derivation on S with taking the direction

of the normal vector out of fluid domain. On the free surface S,

kine-matic and dynamic conditions must be satisfied.

(l+x)+xx=O

Ofl S1.

and

X.r +,X V)2 + L) O on S1..

where =g/U, g is the gravitational constant. At the infinity the dis-turbance will die out and the dominant waves will appear only at the

rear of ship. Let's consolidate these conditions as [R]:

[R] Radiation condition, V-+O as R=(x2+y2+z2)2_-*oo

and x=o(xJ112) as x-oo.

The free surface elevation is given by

z=(x, y)== --+--(V)-

11 1

2

(3)

Applying the following relations

-

a

(x+-VX)2)}

a (X+l(VX)2)}

_I{ a

(Xx+1VX2)} ,

ay 2 dz 2

we can combine [A] and [BI conditions into

5

Now let's assume the disturbance potential is decomposed into two

parts, one is the double body flow potential ?V(x, y, z) which gives the basic

flow and the other is the wavy potential Ø(x, y, z) which is thought to

express the wave field over the basic flow.

x(x, y, z)?r(x, y, z)+ç(x, y, z).

₍₅₎

The conditions for the basic flow potential are specified as,

[LI V2=O in D.

[F] _{=O on} S1.

[H] Ç= ax _{on S,}

1°I

VO

asRoo.

(6)

where D,, denotes fluid region in z<O, S0 free surface at z=O and S

wetted surface in z<O. We notice (6) is the equation for the flow around a non-lifting body.

If the velocity of the uniform flow is small, the flow field could be

given by the basic flow potential for the first approximation. This

as-sumption may be fulfilled asymptoticallyas UO. Then (5) can be regarded as an asymptotic expansion with the basic flow potential as leading order

term. _{For the second order velocity potential which must represent wavy}

field over the bathc flow, the conditions are written as, [L] V2çb(x,y, z)=O

_z<ç

[HI çi5,=O on S1 [F] ç

+

= - {2

(V)2 + VT(V)2 + 2Çr + 2]

-

_{[4VlTçb. + 2{1(V?1V». + VT)ç5 + V?1V)ç5}]}

-

I

_{[4VVçb + 2{(VV)?JÇ + (VVW +} + Vç5V(V)i [2 a (VØ)7+ 2{f rO'ç75V)çb + Ç(Vç5V)ç& +

qrvpvj

+ V?l/V(Vç75)2 + 2{ç&r(VçW)?I/.r + ç(VçbV) + 'VçbV(Vç5)2 _{on z=} [R] Radiation condition.

This boundary value problem is nonlinear because of the _{condition at} the unknown free surface z=.

Since the basic flow potential is the leading order term in the

ex-pansion (5), the uniform flow velocity will be small quantity, so that the waves generated over the basic flow also must be small both in amplitude

and length. And the longest wave length can be specified as,

r _{= 0(U2).}

Let's introduce a small parameter representing slowness:

ö=i=0(U2),

(7)

so that the wavy potential will be governed by this parameter. It may

not entirely be wrong to assume that the

differentiations of the wavy

potential with respect to the space variables change the order of

magni-tude, while the basic flow potential does not.

These assumptions lead to the following changes of the order:

(a

Sx 6ya 6z

a )(x,y,z)=o()

(8)

(

a a _{a )Ø(x, y, z) =}

ax' ay' az

with respect to slowness parameter ô.

Moreover, introduce an implicit assumption that the slope of waves

due to the wavy potential should be asymptotically small as 5-+0. Let's

specify, at this moment, the order of magnitude of with respect to ô, from (8), as fr=O(1). The surface elevation (3) becomes.

(x, y)= l(v95)2j

-

0(ö) (10)

[1] [1] [.3-Ip]

The smallness of the wave slope suggests ç=o(ô), since wave length is of

order 0(3). Therefore we can assume çi= 0(52) and this will give for the

first time the meaningfull reduction of the problem.

Expanding the quantities given by around z==0. and defining wave

height,

&(x,y) = - -t- ?V.r + v?r)2} = 0(5),

we have the wave elevation,

C(x, y) = O -

i

_{F()Vc(x, y,} ) +ç(x, y, )

+ + '(vr)2o}]₊₀₍₅₎ (12)

[F]

{(1+)_--

}2ç+ø=D(xy)

on z=8 (16) dx [F] {ør+çb2}z= D(x, y)+E(x, y) (17) D(x, y)= --_{2?v+{2 a a 2 dx dx + + lÇ0)(2(Ç + +

E(xY)=_{{(1+)

_dx

+'Ç

_dy6

-

JO(.2,y)0

dz--(

₊

.

where the basic flow potential is expanded around z=O. Further ex-panding Ø around z=O, we have

dz z6 + O(o4).

42] _EÖ1

Therefore the second term in the right hand side of (12) is written as,

(x, y) = - + + }z= o + O(), (13)

where qç, are evaluated on z=O and the following relations, =O,

xr=o and 1Ç=O on z=O are used. The wave elevation can be decom-posed into two parts,

(14)

is thought to express the waves over slowly varing free surface eleva-tion 8.

As has been clear in the above reduction, the derivatives of can

be evaluated on z=O, while those of (5 on z=8. Therefore, neglecting the

higher order terms, we can have finally the following free surface

condi-tion. [F] çb + çl =

--

+ V1'T(V)2 +

21j

-[1] [1] [1] - [2VVØ + {?JÇ(VV)çb + 'Ç(VV)Ø.0}} on z =8. (15) Ei i Or equivalently, and where and z-O

The free surface conditions in the above are originally given by Newman4,

Baba7 and Maruo respectively. However (17) is slightly different from

Maruo's.

In deriving the free surface condition for the wavy potential, we

translated free surface onto 6, consequently the fluid region should be defined as z<O, thus the wetted surface has been slightly changed. If the

gap arised from replacing free surface would cause only higher order effect

which can be neglected in the problem, the boundary value problem for 6 is defined as follows, [L] V2ç5(x,y,z)=0

in z<8

[F]

{(1+)

a

+8

a

}2+=D(xy)

on z=O ax ay [H] ç!=O on S8 z<8 [Rl Radiation condition. (20)

Some attemps exist to solve this problem. We will make use of the Green's theorem to treat the problem as Maruo did.

If we neglect further the effect of the wetted surface enclosed between z=O and z=0, and continue the functions into D0, we have the conditions

for ç slightly modified from (20). As for the effect of wetted surface neglected in this paper, we can refer to Bessho's work8. Then we have,

[L] V2çb=0 in D. z<0 [F] ç +

= -

i/r(X,y) on z =0 [HI ç,=O on S80 [Rl Radiation condition, (21) where

= DE.

Since E includes the derivatives of çb, the problem is not identical with the Neumann-Kelvin problem with inhomogeneous free surface condition.

However to this problem the Green's theorem can be applied with the

Green's function defined by,

[L] F2G(P, Q)=53(P-Q) in D,,

[F]

G+iG0=0

on z=0

[R] Radiation condition. (22)

The potential at a point P is given by,

ç(P)= -ft

{G(P Q)aÇ$Q Ç5(Q)aG(P Q) jdS(Q)

SBO _aflQ aflQ

-JJ {G(P, Q)

aç&(Q) 6G(P, Q)

Deforming the integral on S, and using [F] condition in (21),

ç(P)= -5f

ç

_dS!

f V-

dr SBO fl0 j L CO +

1

55 i,)G(P, Q)

ddrj

where L is a line at which the wetted surface crosses with free surface

z=O. Moreover, it is possible to express (23) by source distribution on S0, defining appropriately the interior potential in the region enclosed by

S. and free surface z=O inside of hull. The result is,

(P)=JJ

dS(Q)ff(Q)G(P, Q)+1 J d(Q) cos (, n)G(P, Q)

---

ddD(, )G(P, Q)--- 55 ddE(, ,)G(P, Q).

(24)

In this expression unknown functions are source density functions both

on the wetted surface and on the free surface, that is, a(Q) and E(x, y). As it will be shown later, this potential can be obtained by the iterative

procedure. Let the j th approximation be

(P)=

-155

ddD(e, )G(P, Q)+JJ

dS(Q)a,(Q)G(P, Q)

+--- f dc1(Q) cos (, n)G(P, Q)

SL ddE1(, ,2)G(P, Q)

(25)

where E, is evaluated by the j-1 th potential. We can assume E)=O.

The j th potential satisfies the following conditions:

[L] V2ç1,=O in D.

[F] Ç5jxx + = D(x, y) + E1 (x, y) on SF.

[H] ØO

on Se,,

[R] Radiation condition. (26)

This is the Neumann-Kelvin problem with inhomogeneous free surface

condition. It is possible to decompose ç into two potentials,

(27)

çÇ represents the potential due to source distribution on z= O and satisfies the conditions:

[L] Vç57=O in Dr,

9

[F]

_{Ø+iØÇ=D+E,}

on St,,

[R] Radiation condition. (28)

Then, çbÇ must include the reflection effect due to ' on the hull surface, so that it satisfies the conditions:

[L] 17241)1)_O in D,,

[F]

Ø.+.iØ=o

on SFO

[H] çl5=_çbÇ

[R] Radiation condition. (29)

Now it becomes apparent that this problem defines the Neumann-Kelvin

problem.

From the above consideration, the most troublesome part of the low

speed theory takes place in solving ç5 problem, while ç5' can be calculated from the given basic flow pötential and the lower order potentials evaluated on z= O. Substituting (25) into [H] condition in (29), we have an integral equation for o,(Q) with a kernel function which is constructed from the Havelock wave source potential.

If the iteration will have a convergent series, we can construct an

exact solution for the 3-D Neumann-Kelvon problem (21). In that problem, the nonlinear behavior is included in the inhomogeneous term ik(x, y) in the free surface condition. The 'k function has informations about the

propagation of waves over the basic flow field as well as the diffraction effect. Especially E(x, y) in fr may be responsible to such effects. In

order to include the nonlinear effects in the solution, we have to solve

the problem up to the second approximation at least.

3. LOW SPEED SLENDER SHIP THEORY

Although the initial formulation for the low speed theory has been set up for an arbitrary ship hull shape, the boundary value problem for

wave field at the second approximation turns out to be time comsuming in the stage of numerical calculation. And this fact seems to make the

investigation on the local nonlinearity included in the second

approxima-tion almost impossible. For the remedy to this situation, it will be worth

introducing the slender body theory into the low speed theory in the

second approximation. In doing so, of course, ship should be slender in

some sense.

It has been known well that for the steady motion problem the bow

near field has special importance on the wave field in the rear of this

region. The introduction of bow near field was credicted to Ogilvie (1972)

who analyzed the wave making problem by the slender body theory with uniform flow velocity U= 0(1) and showed the importance of this region.

[3/2J [,2]

= D(x, y) + E(x, y) on z = 0. (31)

This implies çí=O(&+). And the wave heights are

1(2+2)

O(012o2)

= -

+ çb) =O(ô312).

Therefore the boundary value problem in the bow near field becomes:

in D,,

on z=0 on z=h(x,y)

(30)

II

For the case of low speed, the importance of the bow near field will not

be changed. Then we will try to analyze the low speed theory by the

slender body theory in the bow near field.

3.1. Bow near field problem

Let's take the slenderness of ship as a parameter . The classical

slender body theory indicates that the orders of magnitude of the basic

flow potential in the bow near field are given by:

= O(e), l, =O(3/2), = O(e),

where the bow near field is defined as region in

x=O(E'2),

y=O(), z=O().

For the wavy potential, we will make one more assumption that when the differentiations with respect to space variables are operated on the

wavy potential in the bow near field, the orders of magnitude change as

follows:

o

= O( /51) a a = O(e'ô').

ax ay dz

From this, the free surface condition (17) writes

[F] _{çL +}

_{= - [xx +}

+ ?1xyy+ =0

[J [3t2J [8/2]

- [2'Çq5+ c',L=0

[M] Matching with the far field solution,

where z=h(x,y) is the equation of ship surface and h=O(), h-=O(E) and h,,=O(1) in the bow near field as assumed by Ogilvie. Immediately the

first approximation is obtained as, [L] Ç'200_{+ Çzz + çb} = O [1] [1] [] [F]

+

= D(x, y) + E(x, .y) L,.] t'] [H]

çh(x, y)+çbhçiS0O

[f512,3] [f0] [ö.]

[Ti

jD_

J ?iyy V '"iz:

[F] çbri+1)çS;'=:O

FT-fl D

till 'j'iN 'J-'IN

[II

z<O z=O

on z=h(x,y)

x<O. (36)

The [I] condition comes out because at x<O there is no hull boundary. The potentials satisfying the above conditions are expressed by the

[L] 0+Ø=O

z<O [F]

ç,+ç=D(x,y)

z=O [H] th,h,,ç$16=O on z==h(x,y) (32) or [Hl Ø =O on z=h(x, y) and '=O(E2ô2).

For the second approximation, 02 can be written

as a sum of two

potentials with different orders,

02 = O(E2ô2)+ O("ô')

with conditions,

[LI Ø2Y+Ø2ZZ=O z<O

[FI

02+i-Ø=D+E

z=O

[H] 02N=O on z=h(x,y). (33)

At least up to the second approximation, we can use the above boundary value problems which have 2-D Laplace equation as the governing field equation and satisfy the free surface condition which includes 3-D effect

through x-derivative. This type of problem is defined as the time depend-ent boundary value problem6' because the free surface condition coincides

with well known time dependent condition if x is replaced by time variable

t through the relation x= Ut.

Let's decompose the potential 0 into two parts as previously done

in (27),

0i=0+0

i=1,2

(34)

with conditions, respectively:

[L]

ó+q=O

_z<O

[F]

0!+ç5=*i(x,y)

z=O

[oc] «-o

as (35)

and

i(x, P)+ÇdeJ d1(Q), Q)G(x_iP Q)

2 0 c() aN

'=Jd

J dl(Q)cr1(e, Q)Ö(xOE; y, z, , ) (40)

where c(x) denotes the contour of hull section at x. The unknown source

density function af(x, Q) will be determined by an integral equation which can be derived by substituting (40) into [H] condition in (36). We have

P on c(x)

la

Green's function which is the potential for a point impulsive source and

satisfies the following conditions,

[L] G+G=ô(yrj)ö(z)ô(x)

[F]

z=0

[I]

&=Ono

x<0. (37)

The Green's function is given by Wehausen & Laitone (1960)'°,

ö(x)

_in

r(y,z--)

y, z, q,

22r r'(y, z-+-)

__H(x_)J

Id dk

cos k(y) sin

(x)

(38)

Using the Green's theorem, we have

c=--

JI.

d J

dr1!r1(e, )O(x; y,

z,rb 8) (39)

(41)

With these equations, the inner problem up to the second

approxima-tion by which the local nonlinearity would be studied turns out to be

solvable numerically more easily than the 3-D Neumann-Kelvin problem. 3.2. Far field problem

In the far field the problems for the pressure potential and the

dif-fraction potential have been given by (28) and (29) without [H] conditions.

When we look the disturbances caused by the pressure distribution and its interaction with hull from the far field, the disturbance due to pressure distribution may appear as a slender pressure distribution near ship, and that due to diffraction effect as a line source distribution on ship center

line. Therefore, the far field potentials will be written as follows,

=

-

E

ø=JI. dEr(E)G(x, y, z; , 0,0). (43)

Now let the Green's function be specified as'°

G(x,y, z; E, ,

)=

-

4JRR')

+-- f

dOsec0P. y

fl dkez) cos[k(xE)cosO] cos{k(yi)sinO]

it2 Jo

kisec2O

--f-

I2o

sec2 0e 8ec2O

sin ['(xE) sec O] cos [(y) sinO sec2 O]

Jr Jo

(44)

where R=[(xE)2+(yi)2+(z)2]'2 and R'=[(xE)2+(y'»2+(z+)2]"2,

and P V means the principal part of the integral to be taken. In order

to take the inner expansion of the far field potential, it suffices to

con-sider the expansion of the following expression,

I(xE, yii, z)

=1

Ç'2 do sec2 OP

v$

dke°0s

[k(xE) cos Ojeos [k(yij) sin O]

ir °

k sec2

sec2Oec2o sin [(xE) seco] cos [(-) sinO sec2 01.

Now let's follow the technique adopted by Baba (1975)11) to seek the inner

expansion of similar equation. Changing variables, we have,

Jr

_{p. vf dw f d}

e/,2+ß2z_{cos (t)(xE) cos}

ir o O w2s/a2+j92

IJ

dß - e sin ß(xE) cos Introducing new variables,

X="°x, E=o1'2E,

Y=,s 'y,

H=o'i, Z=oz,

m=°°w, n=oß

then I is written, in terms of X, Y, Z, as

I=IP V"° Ç dm f

dn

ir Jo Jo

m2_/nb+m

11/2f

eZ _sin

n(X)cos /n(nXYH)

2

and when letting o-0, the leading order term is approximated as, writing

1= IP. V

dw J dß e cos w(xC) cos 8(yi) I, O

-

J: dß

cos ß(yi1) sin 1i(xC).

Now we can use the following relation,

P V I dw cpsw(xC)

sin /(xC)

sgn (xe).

Jo w2-'8 2

Finally the inner expansion of the lowest order is obtained,

I(xC, yv, z)=J

cos k(y) sin '(xC)

x>C

=0 x<C.

Therefore the inner expansions of the far field potentials are defined as,

Ø'(x, y, z)

-

I

Ç dC J

d/ro(C, f

dk _{e cos k(y - ) sin /k(x C)}

- -(45) and (x, y, z)

--- J' der(e) J' dk _{e cos ky sin /k(xC). (46)} ir 0

From (45) it is immediately proved that the inner expansion of the far

field pressure potential matches with near field potential (39).

The matching of the diffraction potcntial needs a little calculation. Since the outer expansion of the near field potential as yO(l) can be

obtained as,

ç[ Re

_--

_{f dC f}

e'°> sin /k(xC) $

dsc(C, Q)e' cos k?2(s).

ir O o -../tk c(f)

This should match with (46), then we have

fdCr(C) sin s/k(xC)=Re f d sin /k(x) J: dsa(C, Q)ei'2>.

(47)

Multiplying e/%/i to both sides of (47) and integrating with respect

to k in the interval [0, cx), where a is an arbitrary small positive constant

which is later let be 0, we have,

and using the relation

15

C' dk

4 e2

CxV'

I- e"4

.

sin 4,/i.kx=-_ ----. i dte'

/i.k 's1a

um

=/ô(x) and

_{$ e'"dt=} we finally have,

fdk

sin /,jk(x_)=

2r

ô(xe)

as a-+0.

Ji

/k

Therefore the unknown source density function r(x) is now determined

from (47),

r(x)= _{J' d}

j' dsa(, Q) j' cos ko(s) sin i,/k(xe). (48)

u o

Now we can complete the matching. The inner expansion is obtained by

letting only the slenderness parameter tend to O asymptotically. This

may be valid because we assumed that small parameter ô pertaining to

the slowness of uniform flow is independent of the ship hull geometry.

If we assume ô is a function of o, the procedure in the above must be

modified.

4. WAVE MAKING RESISTANCE FORMULA

Wave making resistance will be given by the momentum flow analysis

in the infinity downstream, where outward propagating waves are seen

and asymptotic evaluation for the potential is possible. There we can

use the Kochin function for expressing the far field flow field. The Kochin

function is defined for the source disturbance potential as,

K(k, O)

_{= if}

dS(Q)X(Q)e1° Cö 0+1'; SIS (49)

where .Y(Q) is the sum of source distributions on boundaries. In the low

speed theory, the far field potential is composed of two source

distribu-tions, one is on the free surface near ship hull and other one is on the

ship center line. Therefore the Kochin function is defined by,

K(k, 0)==I s: df diJr(e, )e0se81e)

eosO ₍₅₁₎

The wave resistance formula will be given by 2 ¡'r/2

Cw=

J dO K( sec2 O, O) sec3 O.

4 ,r/2

For the first approximation, potential can be written as

17

1(x,y, z)=çj"+q (53)

and the Kochin function becomes

K1(k, O)= --t- j: d j

dD(, )e'

CS + SICO)

100

+1_J-

d1()e'"'''.

(54)

The line source density function is given by

--

jdj dsa1(e, Q)$

dk_ek

cos k7) sin/(x) (55)

22r O dO) °

and the source density function on hull is given by the integral equation,

-L71(x, P)+jCdej

dsi1(e, Q) P on c(x). (56)

2 i _)e) aIV

The second approximation will be given by replacing suffix i to 2 in the above, but slightly modifying the pressure potential,

c'=

_---

_J d J-,, d{D(, )+E1(e, )}G(x, y, z; , ,0). (57)

The source density functions on the free surface D, and E1, are given by,

D(x, y)=(l8)

- (.. +

+ OV)2:O in the near field (58)

E,] [,3/2] [,3/2] and a

+y

}2j

+J° ax 8y - [21SV+Ø1VY],O. (59) [3/2[ [,21

In order to calculate the effect of local nonlinearity, E term should be

included. The formula (54) apparently includes the correction for

diffrac-tion effect. And the second order Kochin function may include not only

diffraction effect but also local nonlinear effect that is thought to come

from the fact that waves propagate over the basic flow field. 5. CONCLUSION

A theory which is based on the low speed theory is developed under the slender ship assumption. The exact boundary value problem for the

velocity potential in the near field is

set up and treated by iterative

the negligence of gap in the wetted surface that is translated from the original free surface to plane at z= O, other important factors such as

diffraction and local nonlinearity effects are considered.

Expanding the exact problem in the bow near field, we have the same

problem as that appeared in the time dependent boundary value problem. The solution for this problem is relatively easier in numerical calculation

than the 3-D Neumann-Kelvin problem. And it is shown that the near

field solution can match with the far field solution by which wave

re-sistance is calculated.

And the first approximation in the far

field

potential without diffraction effect becomes identical with the result of

the low speed theory by Baba and Maruo.

In this paper it becomes clear that for a slender ship the iteration

will converge asymptotically as a ship becomes more and more slender. Then the second approximation will explain most of the nonlinear phe-nomenon related with wave propagation over the basic flow.

Introduction of large high speed computer seems to exclude fine

theoretical analysis from numerical calculation. However, the epact

pro-blem is sometimes beyond numerical capacity and needs tremendous

com-puting time as has been said in the symposium on numerical hydrodynamics1.

Therefore, it can be sometimes better to rely on the approximated method such as the slender body theory. In fact in this paper a several important points have been clarified by the introduction of slender ship theory.

REFERENCES

Maruo, H., Wave resistance of a ship with finite beam at low Froude numbers, Bulletin of the Faculty of Engineering, Yokohama National University, Vol. 28,

pp.59-75, 1975.

Shimomura, Y., Kitazawn, T., Inui, T., and Kajitani, H., The low speed wave resistance theory imposing accurate hull surface condition, Journ. Soc. Naval

Arch. Japan, Vol. 146, pp. 27-34, 1979.

Brard, R., The representation of a given ship form by singularity distributions when the boundary condition on the free surface is linearized, Journ. Ship

Re-search, Vol. 16, No. 1, pp. 79-92, 1972.

Newman, J. N., Linearized wave resistance theory, Proc. International Seminar

on Wave Resistance, Tokyo, pp. 31-43, 1976.

Inui, T., and Kajitani, H., Hull form design, its practice and theoretical back-ground, Proc. International Seminar on Wave Resistance, Tokyo pp. 159-183, 1976. Adachi, H., and Ohmatsu, S., On the influence of irregular frequencies in the integral equation solutions of the time-dependent free surface problems, Journ.

Soc. Naval Arch. Japan, Vol. 146, pp. 119-128, 1979.

Baba, E., and Takekuma, K., A study on free surface flow around the bow of slowly moving full forms, Journ. Soc. Naval Arch. Japan, Vol. 137, pp. 1-10, 1975. Bessho, M., Line integral, uniqueness and diffraction of wave in the linearized theory, Proc. International Seminar on Wave Resistance, Tokyo, pp. 45-55, 1976. Ogilvie, T. F., The wave generated by a fine ship bow, Proc. Symp. Naval Hydro-dyn., 9th, ACR-203, Office of Naval Research, Washington D.C., pp. 1483-1525, 1972.

19

Wehausen, J. V., and Laitone, E. V., Surface waves ,Handbuch der Physik, Vol. 9, Springer-Verlag, Berlin, pp. 446-778, 1960.

Baba, E., Analysis of bow near field of flat ships, Mitsubishi Technical Bulletin, No. 97, Mitsubishi Heavy Industries, Ltd., 1975.

Proc. Second International Conference on Numerical Ship Hydrodynamics,

PAPERS OF SHIP RESEARCH INSTITUTE

No. i Model Teses on Four-Bladed Controllable-Pitch Propellers, by Atsuo Yazaki,

March 1964.

No. 2 Experimental Research on the Application of High Tensile Steel to Ship

Struc-tures, by Hitoshi Nagasawa, Noritaka Ando and Yoshio Akita, March 1964. No. 3 Increase of Sliding Resistance of Gravity Walls by Use of Projecting Keys under

the Bases, by Matsuhei Ichihara and Reisaku moue, June 1964.

No. 4 An Expression for the Neutron Blackness of a Fuel Rod after Long Irradiation,

by Hisao Yamakoshi, August 1964.

No. 5 On the Winds and Waves on the Nothern North Pacific Ocean and South

Ad-jacent Seas of Japan as the Environmental Condition for the Ship, by Yasufumi

Yamanouchi, Sanae Unoki and Taro Kanda, March 1965.

No. 6 A code and Some Results of a Numerical Integration Method of the Photon

Transport Equation is Slab Geometry, by Iwao Kataoka and Kiyoshi Takeuchi,

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No. 7 On the Fast Fission Factor for a Lattice System, by Hisao Yamakoshi, June

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No. 8 The Nondestructive Testing of Brazed Joints, by Akira Kannö, November 1965.

No. 9 Brittle Fracture Strength of Thick Steel Plates for Reactor Pressure Vessels, by

Hiroshi Kihara and Kazuo Ikeda, January 1966.

No. 10 Studies and Considerations on the Effects of Heaving and Listing upon Thermo-Hydraulic Performance and Critical Heat Flux of Water Cooled Marine Reactors, by Naotsugu Isshiki, March 1966.

No. 11 An Experimental Investigation into the Unsteady Cavitation of Marine

Propel-lers, by Tatsuo Ito, March 1966.

No. 12 Cavitation Tests in Non-Uniform Flow on Screw Propellers of the

Atomic-Power-ed Oceanographic and Tender ShipComparison Tests on Screw Propellers De-signed by Theoretical and Conventional Methods, by Tatsuo Ito, Hajime Takahashi and Hiroyuki Kadoi, March 1966.

No. 13 A Study on Tanker Life Boats, by Takeshi Eto, Fukutaro Yamazaki and Osamu Nagata, March 1966.

No. 14 A Proposal on Evaluation of Brittle Clack Initiation and Arresting Temperatures

and Their Application to Design of Welded Structures, by Hiroshi Kihara and

Kazuo Ikeda. April 1966.

No. 15 Ultrasonic Absorption and Relaxation Times in Water Vapor and Heavy Water

Vapor, by Yahei Fujii, June 1966.

No. 16 Further Model Tests on Four-Bladed Controllable-Pitch Propellers, by Atsuo

Yazaki and Nobuo Sugai. August 1966. Supplement No. 1

Design Charts for the Propulsive Performances of High Speed Cargo Liners with CB=

0.575, by Koichi Yokoo. Yoshio Ichihara, Kiyoshi Tsuchida and Isamu Saito, August

1966.

No. 17 Roughness of Hull Surface and Its Effect on Skin Friction, by Koichi Yokoo.

Akihiro Ogawa, Hideo Sasajima, Teiichi Terao and Michio Nakato, September

1966.

No. 18 Experimets on a Series 60, CB= 0.70 Ship Model in Oblique Regular Waves,

by Yasufumi Yamanouchi and Sadao Ando. October 1966.

No. 19 Measurement of Dead Load in Steel Structure by Magnetostriction Effect, by Junji Iwayanagi, Akio Yoshinaga and Tokuharu Yoshii, May 1967.

No. 20 Acoustic Response of a Rectangular Receiver to a Rectangular Source, by

22

No. 21 Linearized Theory of Cavity Flow Past a Hydrofoil of Arbitrary Shape, by Tatsuro Hanaoka, June 1967.

No. 22 Investigation into a Nove Gas-Turbine Cycle with an Equi-Pressure Air Heater, by Kôsa Miwa, September 1967.

No. 23 Measuring Method for the Spray Characteristics of a Fuel Atomizer at Various Conditions of the Ambient Gas, by Kiyoshi Neya, September 1967.

No. 24 A Proposal on Criteria for Prevention of Welded Structures from Brittle Frac-ture, by Kazuo Ikeda and Hiroshi Kihara. December 1967.

No. 25 The Deep Notch Test and Brittle Fracture Initiation, by Kazuo Ikeda, Yoshio Akita and Hiroshi Kihara, December 1967.

No. 26 Collected Papers Contributed to the 11th International Towing Tank Conference, January 1968.

No. 27 Effect of Ambient Air Pressure on the Spray Characteristics of Swirl Atomizers,

by Kiyoshi Neya and Seishirô Satö, February 1968.

No. 28 Open Water Test Series of Modified AU-Type Four. and Five-Bladed Propeller

Models of Large Area Ratio, by Atsuo Yazaki. Hiroshi Sugano, Michio Takahashi and Junzo Minakata, March 1968.

No. 29 _{The MENE Neutron Transport Code, by Kiyoshi Takeuchi, November 1968.} No. 3D Brittle Fracture Strength of Welded Joint, by Kazuo Ikeda and Hiroshi Kihara,

March 1969.

No. 31 Some Aspects of the Correlations between the Wire Type Penetrameter Sensi-tivity, by Akira Kanno, July 1969.

No. 32 Experimental Studies on and Considerations of the Supercharged Once-through

Marine Boiler, by Naotsugu Isshiki and Hiroya Tamaki, January 1970.

Supplement No. 2

Statistical Diagrams on the Wind and Waves on the North Pacific Ocean, by Yasufumi Yamanouchi and Akihiro Ogawa, March 1970.

No. 33 Collected Papers Contributed to the 12th International Towing Tank Conference,

March 1970.

No. 34 Heat Transfer through a Horizontal Water Layer, by Shinobu Tokuda, February

1971.

No. 35 A New Method of COD. MeasurementBrittle Fracture Initiation Character-istics of Deep Notch Test by Means of Electrostatic Capacitance Method, by Kazuo Ikeda, Shigeru Kitamura and Hiroshi Maenaka, March 1971.

No. 36 Elasto-Plastic Stress Analysis of Discs (The ist Report: in Steady State of Thermal and Centrifugal Loathngs), by Shigeyasu Amada, July 1971.

No. 37 Multigroup Neutron Transport with Anisotropic Scattering, by Tornio Yoshimura,

August 1971.

No. 38 Primary Neutron Damage State in Ferritic Steels and Correlation of V-Notch Transition Temperature Increase with Frenkel Defect Density with Neutron

Ir-radiation, by Michiyoshi Nomaguchi, March 1972.

No. 39 Further Studies of Cracking Behavior in Multipass Fillet Weld, by Takuya Kobayashi, Kazumi Nishikawa and Hiroshi Tamura. March 1972.

No. 40 A Magnetic Method for the Determination of Residual Stress, by Seiichi Abuku,

May 1972.

No, 41 An Investigation of Effect of Surface Roughness on Forced-Convection Surface Boiling Heat Transfer, by Masanobu Nomura and Herman Merte, Jr., December

1972.

NO. 42 PALLAS-PL, SP A One Dimensional Transport Code, by Kiyoshi Takeuchi,

February 1973.

No. 43 Unsteady Heat Transfer from a Cylinder, by Shinobu Tokuda, March 1973.

No. 44 On Propeller Vibratory Foces of the Container shipCorrelation between Ship and Model, and the Effect of Flow Control Fin on Vibratory Foces, by Hajine

p

Takahashi. March 1973.

No. 45 Life Distribution and Design Curve in Low Cycle Fatigue, by Kunihiro lida and Hajime moue, July 1973.

No. 46 Elasto-Plastic Stress Analysis of Rotating Discs (2nd Report: Discs subjected to Transient Thermal and Constant Centrifugal Loading), by Shigeyasu Amada and Akimasa Machida. July 1973.

No. 47 PALLAS-2DCY, A Two-Dimensional Transport Code, by Kiyoshi Takeuchi,

November 1973.

No. 48 On the Irregular Frequencies in the Theory of Oscillating Bodies in a Free

Surface, by Shigeo Ohmatsu. January 1975.

No. 49 Fast Neutron Streaming through a Cylindrical Air Duct in Water, by To: himasa

Miura, Akio Yamaji, Kiyoshi Takeuchi and Takayoshi Fuse, Septembe' 1976.

No. 50 A Consideration on the Extraordinary Response of the Automatic Steer ing

Sys-tern for Ship Model in Quartering Seas by Takeshi Fuwa, November 1976.

No. 51 On the Effect of the Forward Velocity on the Roll Damping Moment, by Iwao

Watanabe, February 1977.

No. 52 The Added Mass Coefficient of a Cylinder Oscillating in Shallow Water in the Limit K-.0 and K-.co, by Makoto Kan, May 1977.

No. 53 Wave Generation and Absorption by Means of Completely Submerged Horizontal

Circular Cylinder Moving in a Circular OrbitFundamental Study on Wave Energy Extraction, by Takeshi Fuwa, October 1978.

No. 54 Wave-power Absorption by Asymmetric Bodies. by Makoto Kan, February 1979.

No. 55 Measurement of Pressures on a Blade of a Propeller Model, by Yukio Takei,

Koichi Koyama and Yuzo Kurobe, March 1979.

No. 56 Experimental Studies on the Stability of Inflatable Life Raft, by Osamu Nagata, Masayimi Tsuchiya and Osamu Miyata. March 1979.

No. 57 PALLAS-2DCY-FC, A Calculational Method and Radiation Transport Code in Two-Dimensional (R, Z) Geometry, by Kiyoshi Takeuchi, July 1979.

No. 58 Transverse Pressure Difference between Adjacent Subchannels in a Square Pitch Nuclear Fuel Rod Bundle, by Köki Okumura, November 1979.

No. 59 Propeller Erosion Test by Soft Surface Methodusing Stencil Ink proposed by

the Cavitation Committee of the 14th ITTC, by Yuzo Kurobe and Yukio Takei,

March 1980.

No. 60 Plastic Deformation Energy and Fracture Toughness of Plastic Materials, by

L. I. Maslov. March 1980.

No. 61 Performance of Fireproof Lifeboats of Reinforced Plastics, by Osamu Nagata and Kazuhiko Ohnaga, March 1980.

Supplement No. 3

Winds and Waves of The North Pacific Ocean, by Yoshifumi Takaishi, Tsugio Matsu. moto and Shigeo Obmatsu, March 1980.

No. 62 Elasto-Plastic Stress Analysis of Rotating Discs (The 3rd Report: Application

of Perturbation Method), by Shigeyasu Amada, August 1980.

No. 63 On the Fatigue Damage of Standing Wire Ropes Multiple Step Testing Loading,

Takahisa Otsuru. Hisao Hayashi, Shoju Okada, Yoshihisa Tanaka and Isao

Ueno. December 1980.

In addition to the above-mentioned reports, the Ship Research Institute has another

series of reports, entitled "Report of Ship Research Institute", The "Report" is published in Japanese with English abstracts and issued six times a year.