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ARCHIEF

THE WAVE RESISTANCE OF

A SURFACE PRESSURE DISTRIBUTION

IN UNSTEADY MOTION

V. K. Djachenko

Translated by:

M. Aleksandrov

1. J. Doctors TP o, 1811

TIlE

DEP4RTM

i4i

C

THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Lab.

v.

Scheepsbouwkundè

Technische Hogeschool

No. 044

Deift

January 1970

0

M/IRINE ENGINEERING

(2)

PROCEEDINGS OF THE

LENINGRAD SHIPBUILDING INSTITUTE

Volume 52

HYDROCHANICS AND THEORY OF SHIPS

1966

The Wave Resistance of a

Surface Pressure Distribution in Unsteady Notion

by.:

V.K. Djachenko

Translated by:

M. Aleksaridrov

L.J. Doctors

The University of Michigan

Department of Naval Architecture and Marine Engineering

(3)

TRANSLATORS'

NOTE:-In this paper the symbols used for the hyperbolic

functions have the following meanings:

sh sinh

ch cosh

(4)

1

We consider the problem of non-steady motion of a surface pressure distribution Of arbitrary form p(, y) over an area .

A coordinate system (0, x, y, z) is fixed on the undisturbed surface, where is in the direction of motion and z is vert-ically upwards.

The disturbance potential which is caused by this pressure distribution is defined by the. soution of Laplace'sEquation,

= 0 with the followthg boundary conditions: On the free surface:

dD'Oq ôtz

5

ox2 2z .dt Ox Ot Ox .atz 0. (1) the bottOm: Liz at z = -H. (2)

In expressions (1) and (2)

all

the variables are

non-dimensionalized with respect to 1, the half-length of the pressure distribution (see Fig. 1), po, the characteristic

pressure, and g, the acceleration of gravity. The disturbance potential can be represented by the following integral form:

2

(5)

and

Making the substitutions

o=gcos8,

flk

Stn9 we get f(x,y, Z) , Sr

PJJkdk(de _!fI

àx

J

J

4r2JJ Ox.0

1

/' ./

/ / /

//

/

// /

/ / / / / Z/ / /

Fig. 1

Substituting (3.) and (4) in (1) and (2), and equating

the integrands (because the

k and

e),

then expressing B and substituting in (1),. we

I .

-vz i(xcose+sinBJ k

?T.[wdwfde fA(f)e

B(t)e

]e

ejh'f(x-.z0)cos9 (W-D)5"idx0cJy,

(3)

I

equality must hold for all

in terms of A from equation (2) obtain

2

We consider the case when Let us assume

(6)

-f--t-F't/7kH t- AL

ikco56)2224

(iKco5B)A -J'jg co 6)

2kH..1(f np,.,

-iK(;cos8t0sins)

=

(1e

)

JJdPii-y--e

dx0d4q0

Now let us introduce the function

-tkCO56f7dt

p

To determine A1 we note that it satisfies the differential equation

(7)

where

_ei

oc0se+o5Lfle?i10 (8)

and

5JL7clt

(8a)

Solving (7) with zero initial conditions,

d41

dt

(equivalent to

/

b and a ), by the method of variation of the integration constant, we get

A4=(jrfçcoi 8tdt) sin

8t

(-f(strz

Itdt)

co,$t

+

(- -f(cotdt)sin$t+(ff.5i7iddt)co58t

(9)

t

where

Lr=

(7)

Therefore the disturbance potential is

Y(x,(,2,t)=

ifledN[cJe.A1.e"

x

[e_e2K]e

iiqrcoiepin

8)

The wave resistance is obtained from

R=-7(X,y)-- dxdi,

assuming that R is equal to the horizontal component of the pressure acting on the area . In formula (11) all variables

are dimensional. In non-dimensional form it would be

ffPu x. )

::i dxdy;

Q

8 aPu LT0 82!f

-

fxPu ft dxd

y

-i;

dxcly

-a'

dxd, =

In the last equation the first term is the horizontal component of the pressure, the second is equal to the

resist-ance during steady motion, and the third term occurs only

because of the unsteadiness. We can obtain

Y?JJ3fdwJde.A. (,

etMM)(jgcogof 8) eINC0S8S

i,qxco$+ jsin8)

&(k,e)=Jfie

dxdy (12)

,

(13)

(14) 4

(10)

(8)

R

p(Z

j-dxd4,-

d'iwcoJ9)1(M,8)Ofr,e) '(g,e,

t)

de,;

(23)

and = if kdA'f do (i t.p

)

(icoso)Ofro)t 2

(15)

The wave resistance can be evaluated from

fJXPu

dxdi-

e2)

(16)

13('k', 8)ii'8

On the basis of (9)

= c,0 8coi8t -c20 8s,78t

cos8tffcoi8tdt sin8tJ(sin6tdt.

(17)

And from (8)

fccoi8tcit=fxe

LW C0g0S

8t(i+

e2K)ff&L ei0&*JL

'dy0dt =

=x(11-e2NN -) C (*'o)Jve

cos8tdt;

(18)

where fr

9)ff.

dxd

(19)

Identically

8tdt

z

(i+e2')'

(A,8)JOe ' 0gB S.

8tdt

(20)

If the motion started from rest, i.e. v-O, tO, then,

using (8) , (9) , (18) and (20)

= x

(ii.e2VH)t L*(A,,9) e,

t),

(21)

where

r

-iico8S

-i,cos8S

(9)

and

1Ncos8S LI.%cOS8

8, t)= e (cosoltfire to,8tdt +

+sLndJ

-j$coS8.

8tdr]

.

ire Sw

Let us now use this solution to calculate the wave re-sistance for the case of a constant pressure distribution

acting on a rectangular area with b/i moving at a constant

speed over deep water. Let us assume that the speed changes according to Fig. 2 below.

Fig. 2

For t + , the required functions become:

k L7&iNC0S8)

(

)

- o"-

cog2 8 O

4LPIin(*'COS8)Sin(A?1sLfl8) /"siao

l?('A',8)

=

sia(*'P sin 0)sin (*' cog8)

'stn 9

s cog8

t i 32 U rz(Au sin 9) Sin'fr cosO)

N2s1n19.*'cosO

(10)

R

n Pt

--

1Ndk1A'cos8). 32p5ir,A'u31r79)Sifl'(A' c0S8)

4,yJ

J

k"SLrfO NcoI9

iY'(-icoS6)

9

'f!ktco329. Ll

Substituting czl'=Acos&, we obtain 8PX

5I

tisjnh&2)sjnA

&dd -

(tl)ft)

V&'-cx'

The integrand has four poles on the real. axis, at. a and a = ±v'T/v (Fig. 3, below). To calculate the inner in-tegral we use the theorem of residues. For the contour of integration we take the semicircle Cr of infinitely large

radius in the upper half of the complex plane a, going under-neath the poles as shown. Substituting A = /v2 in (26),

we obtain finally, as a particular case of (23), the

estab-lished formula of B.P. Bolshakov.

Fig. 3

7

(11)

1öp C

f

R=

Jfin

where 1(M,t)= 11-!fpuksin kudu; XL L

ofsJ.0

dx

J 8u 0

-2 eJA'tVF':7

dt

T2 2

(tz_!).j

co'q'u-x)du =

=fffr

t)cof kxdn'

f (ii',t)5r

A'.x di1',

?LfpNcoiN udu.

8

(27)

After substituting (28) into (2) we obtain the following

system of equations:

where A' = 2g1

TWO - DIMENSIONAL

PROBLEM:-In this case the velocity potential can be expressed as

Y(JC,2,t)f[(ACOIMX+ CSLfl

x)ch

*'z

(Rco *'x + Dsin *'x)s/i it'Z] dw (28)

where A, B, C, rare unknown functions of t and

k.

For the solution we employ (1) and (2). Assuming ap/at = 0, the

(12)

where

AShkW=BthNH

Csh NH = Dch A'H

B =Ath NH D=Cth I(HJ

Sibstituting (28) in (1) and equating the coefficients

of cos kx and sin kx we obtain the following system.:

4_Ns_gbo2A_2v.K-!_w C =

!4

gD_g272C2k k*A

(N,tj

Eliminating B and D from (31)

and (30),

we have

--at

dt

Multiplying the second equation by I and adding to the

first, and introducing

itS

= ce

jg5

f(M,t)=e [(Mt)L/2(Nt)]. 5=fUdt

we obtain the following differential equation for W1 which is represented as

+I

+ =.

(13)

is

10

The solution of this equation with zero initial conditions

8tdt -

COS8tf(stn

tdt

--+

(f(sin8tdt)coJ6t.+(( f(coi8tdt)sin

St

(33)

If the motion starts from rest, the two last terms

be-come zero. Then

fi co, 8tdt

8t dt The functions A, C are

ARe(e

-INS

);

and C

The velocity potential is then

Y(x,z,t)=J{Re (%e5) fchkz+th1H.shM2J

COf MX +

tJrn(e')(chIthkH. .hw] S'iaNx)dk,

where W i is given by (34) and a = Iv dt.

The wave resistance is

R I g 2

Pu

dZzfPu

Pu dx+ fPu

dx--f

U dxöt

dx

at

(14)

where

Substituting (37) in (38) and reversing the order of integration, we obtain po'

dx

!JAi2

(.c)dN_(k

(Jm()]dN,

(39) 1,0 0,5 -INS I -jI(X

G(N)Pe

dx.

a0

-- q = a = 11 (40)

Using these solutions for the case of constant speed,

we get Lamb's formula for the resistance:

I

L7.2

(41)

The result.s of the investigation of the wave resistance for the case of a constant acceleration, using (39) and the

approximate method of Stationary Phase, are presented in

Fig. 4 below. The hump speed for this motion is

consider-ably less than for the case of steady motion.

I!

4p2i;

15

0,5

(15)

FormulaS (23) and (39) are recommended for the de-termination of wave resistance for non-steady motion.

REFRENCES :

-Bolshakov, B.P. "The Wave Resistance of a Surface Pressure Distribution." Proceedings of the

Scien-tific Society of Shipbuilding, Vol. 49, 1963.

Sretensky, L.N. "On the Motion of a Cylinder under the Free Surface." Proceedings of the Central Aero-Hydrodynamics Institute, Vol. 346, 1938.

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