On the reverse flow theorem concerning wave making theory

ARCHIEF

Lab.

y.

Scheepskouwkuride

Technische Hogeschool

Deift

Introduc tian

The presant per presents a general form of the relatirnships between the characteristics of f lova around a ship among waves in direct and reverse f lova, within the limita

of linearized hydralynanitca.

Theorem (I)may offers a systematic

moans of obtaining approximate

solu-tIons for the problems in the

anti-symtric motions of the ship as *n

wing theory Theorem (It)is nevly

introduced in this per and it is applicable to the calculation of over-all forces acting on the ship. SOEne

interesting results are obtained by virtue of ita application, which are likely to be of practical values List of Symbols

Cartesian coordinates vith the origin o in the free surface of deep water, and oz vertically upwards, and

ox in free stream

direc-tion of direct flow free stream velocity length of the ship

time and circular frequen-. cy of oscillation,

respec-tively

Y linear displacements of each point of the ship

Z,., Z,.,- surface elevation of

regil-lar waves and its

ampli-ti.ide

j)L)ÇL,,.21%.vave nunber rate and

wave length of regular ve, respectively

X

heading angle

surge linenr displace-.

monts of the ship'o

e'

sway centre of gravity along the Z,,and heave z-axes

52l/L,

2S/L ,

te't angle of roll (angular

-

displace-e = -8displace-e" angldisplace-e of pitch _monts about the e"

angle of yaw L, Y, and ¿ -axes) - 22 -ßA.,P4, f3., /3g. I3, /

4=

40eÌJ t

0e"

1,

'

Q

X,.

_Wv'

phase lag of surge, away, hsave,roll, pitch, and yaw respectively, behind regular waves at origin acceleratjon cauaecl by

potbntial xeaanee of velocity

f the 3hip

potential J

density and pressure change respectively, associated with «

'acceleratioxi of gravity

projection of the vertical-median plane intoX.E-planee § 1. Bc*indary Condjtjn and

P&d.ro-dynamic Forces Acting on a Shj Amon _'kLvO5

When a ship is slender, we may

asse that boundary condition la

satisfied, instead of on the hu.U.

surface itself, on the projection of this surface

into

the x.pi,ane,

within tho frame of the linearized theory When the ship's form is

given by y z,i), the boundary

condi-tion _{written as}

-$

-&

-ere

+j

x.- X,, = -

7 et,

_Y=

-i

-ii' tyv.x

_(1,4)

and X.,, y.,,, _{z1,,denote the}

Instan-taneous linear displacements of fluid particle due to regular waves Hence,

the complex amplitudes of relative displacements are given by

.Z,'

"x

e' -e'v

-.,,

When the, boundary condition is

symetric with respect to -plane,

the corresponding velocity and

acce-leration fields are symiiOtrio

Smi-lar].y, when the boundary condition is

anti-symmetric with respect to

X-plane, the corresponding velocity and

Proceedings of the 9th Japan National Congress for Appi. Mech., 1959

III-9. On the Reverse Flow Theorem Concerning

ve-?kthg Theory Thtsuro Banao z

V

L

t,

,

acceleration fields are anti-symmetric. ibsequently, symmetric motions do not cause the ship to be acted on by any

anti-syisne trie force Reciprocally, the similar relation is settled in case

of anti-sy!mnetric motions. Therefore,

we see that hylrodynanic coupling forces are negligible. Hence, we classify the ship motions among ves in tuo kinds of motions: one isa longitudinal motion (symmetric motion), that is surging, heaving and pitching

oscillations, axiçi the other is a

later-al motion (anti-iytnmetric motion), that is swaying, yawing and rolling ose

tUa-tiens.

We show the expressions of vari-aus kinds of the hydredynamic forces an& moments acting on a ship among

waves ., These are needed for the

cal-culatton of the motion of a ship among waves.

The acceleration potential is available for the calculation of the flu.id f orc We can divide the accel-eration potential into the respective

part of the ship motions - the

dis-turbances due to surging, swaying, heaving, rolling, pitching yawing, and distortion of ocean waves caused by presence of the ship. If these are designated by ,.

+,'J+,,QA

and+,

the whole acceleration potential is given by

(1.6)

where 4. and $ denote the symmetric and anti-symmetric parts cf + with respect to r -plane respectively.

The hydrodynainic forces are divided into the respective rts of the ship motions by refering to (1.6). We

classify them in two groupa to contri-bute to the longitudinal and lateral motions.

(i Longitudinal motion. If we

express the hydrodynamic forces and moments by the symbols,

Ç- r

r

4jr

force

A-ÀrA,4Or.,

,j heaving force

pitching_f

moment -

0Fe ¡F

_(& where

-;1e'-'s

_

e"

-C-: _ei0t_)F-Af

(1.7)

and F,... - - - are complex

nbera and the real parts of the

right-hand side of the aboye expree-aion are to be taken, they are given

(ji lAteral motions We ex-press the hydrodynaiuie forces and moments by the symbols,

F

force rolling

moment r-Çr 'cF, UFqg (iou) yuwl.ng

moment

-- _e'' where

Nov we introduce the pressure differ-ence ir. between right and left sides of the ship's surface (right side corresponds to the positive -axis).. If ,and t,. denote the values of 4'. on

the port and the starboard, the

pres-sure differee between both sides is

written as

ir. _?(+.,-t,) (1,12)

We can also divide ir0 into the respec-tive part of the ship motion.

The expressions

of

the forces are written as: -' -1T,. 224

-t

- ¿5

t;'

te dzd

(1.8)

s---;

F - F ¿ -

;i.

ta

t.

da

(1.9)

Fo,

-'ë

F8 -&S F, _2'jJ

+.

t,

dxÀr

_(iio)

-

_F,.

- F.. - F,,

- ir.,-ir

_'P

(1.13)

--'ç-.

-i F,.

1f

r.

(1.14)

- F, )J Tt,,

fT

- i/c.

ir0

t.

2

Rever.e Flow Theoreznc

There i. the relation

4'.=.- 1'/ -.

at

between acceleration totemtial

and

f luJ.d pressure p

. whre

is the

elevaiori of free surfFxce due to

d.ts-turbance.

If we take the novtng

coord.nate sytes advarcing with the

shIp, we have the cese of' e current

flow with the general velocity V or4

past the shIp along x -axis according

as the ship advances along the

nega-tive or onsinega-tive x-axis0

We

ay say

the former direct flow and the latter

reversa flcw

If we Introduce an

artificial force

oportional to the

fluid vel.city into the fluid field

acooring to Rayleigh, the relatIc

between the veloctty and acceleration

potentiaì

occse

+.=

:: tp- V

_L

within the 'eatr l.ction of linearized.

theory, where ¿t denotes the coeff

i-cent. cf the artificial resisting

force0

Both

and

satisfy Laplace

equation and the condition& to be

satisfied at the free surface are

face and the ship' s surface and its

locus in both flows and the surface

to close the fluid at infinity, the

l.15)

volume-integral vaniahes, because

I.

and 4.

satisfy laplace eqLkttion.

If

be fi-ite, 4. and 4 vanisi

at

infinity.

Renca. we have

(2.2)

(2.3)

at

= O

where

and

denote the

correspond-ing quantities for the reverse flows

The reverse flow theorema are

based upon Green's theorem,

(2/.)

Theorem(I

-We write

, in place of ' and 4. in

pl.ce of i

The vçj].ujne- integral is

taken throuehout the whole region

occupied by the fluid and the

surface-integral

ver Its hounder.

In

g6ner-al, the potential ftnctIn

is

dis-continuous on tre locue of the ship's

surface.

Therefore, if the

surface-th.egrel is taker, over the free

sur-- 225

(2.5)

lo.

where ., is the ship's surface and

its locus in both flows.

On substitut ing (2,3)

and

per-forcing partial integration

in x

the surface integral vanishes over

the free surface,

When we wabgtitute (2,2)

for 'P. in the first

term of (2.5)

and perform a partial

integra-tion in

, wo baie

(2/,)

(i

Longitudinal motion,

. and

are synmetric with respect to

xi-plane.

Then we have

IIt

''4 o j'J

(2,7)

(iii Lateral notIoi

4,

and

I/n are anti-syonetria with

re-epect to x -plane, and il. and L.

disappear on the locus,

Then we have

*

(2,8)

This is the samd one with the

re-ciprocity relation in wing theory,

Theorem tII,

If t=10 and

in (2,4), we

bave

(2.9)

In +.he following anályaea, we must

notice that the frictional

coeffi-cient ¿k'

is taken to bs infinitecii.1

in final aoluttcns.

i. LongitntTh1 motion.

As t.

and-4. are sysetric with respect

to i? -plane

and diaaçeare

on

the locus, we obtain, fron (2.9)

(.1o)

Substituting (1.1), and performing

partial integrations in x , we obtain

a reciprocity re].atiou as

(2,11)

ji. lateral motion.

As

and

ara anti-syzisietric with

respect to x

-plane, we have

= of

When we subatitute°''î.l) and then

7oj

(2.1)

Let

Let

and

Q'

Then we have, from (2.11)

226

-¿ f( el'_dtt4. 1 If

-ijijt*".,d1GtZ

Using (3.1) and (i,e), we get

F,.z

By 'following the same procedure,

we obtain the relations between the varions kinds of the forces and nonante as shown in Table I.

Fron the above ana]ysia, we see that, if we have the pressure distri-butions due to swaying, rolling and yawing in still water, we can obtain

the exciting forces by means of a closed-form integral over the pres-sure distributions.

B,b1tcrathy

i) Flax, AFI., General Reverse Flow and Variational Theoreme in

Lift-ing-Surface Theory, Journal of the Aeronautical Sciences, Vol, 19,

No 6, pp361-3?4, June, 1952.

2) Hariaoka, T,, Theoretical Investi-gation Concerning aip Motion in Regular Waves, Netherlands ip Model sin, Symposium 1957, PP.266-235

(Member of Zosen Kiokai.

Transportation Technical Researc

Institute.

/

Coupling forces

F = -

L

_F

_F

- '

_F

-¿Ø

L 8&

- L

8k

-surging

rIJ$;t»0'X'

f)e'd.z

heaving

Llfj[('

*1(

pitching

O

Coupling forces

Ea- =

Ç

-

F«

-n

swaying

F

_.-r

-

_-'

_J)'

rre

rolling

LxiJ.ure1ti

1F

e"'°

=

-JJ

perform partial integrations in x. , we

obtain

JJ

w2zJf

(2q12)

Ç 3. App..i.cations of the Reverse F1v Theorems

In the following analysis it is assumed that the ship is symmetrical

fore and oft. en the ship is maidng oscillations, there are the relations, between the acceleration potentials on the ship' s surface in direct and

re-verse f lows.

(x?)

_t6tz,a)

W. -.

(3")

f,.t*x)11.i- xi)

iii

i',

xë and

_{_ &}

-Then ve have, fron (2,11) and (3.1)

k

«g

Using (1,9) and (1,lO),we get