Theory of ship wave resistance for unsteady motion in still water

(1)

THEORY OF SHIP WAVE RESISTANCE FOR UNSTEADY MOTION IN STILL WATER

by A. N. Shebalov Translated by Michail Aleksandrov and Geoffrey Gardner

The Department of Naval Architecture and Marine Engineering The University of Michigan

College of Engineering

(2)

The linear problem of forces acting upon a slender ship in unsteady motion was studied in References [1] through [5] The following is an approximate solution to the non-linear

problem of wave resistance and hydrodynamic forces for unsteady motion, starting from a static position.

Let us assume that the ship is moving in an ideal, un-bounded, incompressible fluid, with a variable speed c(t)

and starting from a static position. We can place a coordinate system on the surface of the still water with the x-axis in the direction of ship motion, the z-axis vertically up and

We can assume that the wetted surface of the ship is given by

i: = 1(X,

) ,

Ç

The velocity component normal to the ship's surface is then assumed to be zero,

i-7i 8W

JUi

dti 8z

(4)

the y-axis pointing to the right. Let us denote the potential of the absolute velocities of the fluid particles as: (x,

y, z, t) . This potential can be determined using the LaPlace

equation

(1)

with the following boundary and initial conditions. On the free surface described by z = (x, y, t) the pressure must be constant as given by the Lagrange integral

2

r2

¿

-2

-r

L'1 ('

th

2L(c2rI (2)

The boundary condition for the same surface is

(3)

In infinity

?wdcp--Q x oci

ÇO-O X)

(5)

Initial conditions: when t O the ship is in a static position with o(o) O. Furthermore, o) = O. Then as

the initial conditions we may have:

dt t=c

(6)

Expressions (1) - (6) completely define the problem

of finding the velocity potential. The difficulty is that equations (2) - (4) are non-linear partial differential

equations, and the surface equations z = (x, y, t) for conditions (2) and (3) are unknown. The following is an approximate solution to the problem using a series expan-sion of the ship motion parameters relative to some small factor (e). We will use the method developed by Sysov, Reference [61.

Let us assume that the dynamic and kinematic parameters of the moving fluid depend on the small factor (e) and that they may be expanded in a converging series according to (e) Here we can use as this small parameter either the ship's "fineness" or the small amplitude of waves arising from ship motions. Without losing much generality, we can assume that when e = O,

.t,

E) = (xy,

t,O) ¡(X, X, E) o

(7)

The series expansion of the velocity potential is:

ç!x,i,t,e) =E9(x,/,,t)+E'ÇP2('X,y,,/)+ EÇç(x,y,,t)+

(8)

Identically the equations of free surface and the ship's wetted surface can be expressed as

(4)

z

=

e1 _{(X, Z) +}

The velocity components v, v, v

and vt can be expanded as:

(X.,Z,E,)=E

Eço

Ir

(.r,í,o,t) +

[,

çp _(x,i,o,t) + ç ('rlo,t)] + -

(II)

t

_{(x,y,Z,/,) .Ç(re,o,L)}

_{(Of) -+- Ç(X,L,Û,t)J +}

- (12)

(xy.Í,e) _{(X,e,4t) + ¿ [ço}

(x,I,a,/) + _ç ₊

(X,y, Z,

=

e (X,y, +

[,

42 (x,, ( L) f ç (x,, L,,L)]± -

-Taking into account (9) - (14) we can write the non-linear

differential equation in the following form: dynamic boundary condition (2)

Comparing coefficients of E on the right and left hand sides of this equation, we have

-yZ,±

1c(t)ço

=0

,p.r 11 i 2 21 c(t»o -t-

ÇPc('t)0_

12-[4ç

f! j

±40 ±40 O 21 2x 2 3

_,+E+ ¿31

2 ¿

- C(t) [e+

e

o,+

r)]

[cç

-4-40 +40)-4-,

Equations (15) are justified for z 0. From (15) it follows that the free surface equation, in linear form, is:

cipo

=o.

(16)

]=Q.

J (10) (15) ('3) (14)

(5)

Let us write, in identical form, the kinematic _boundary conditions (3)

-(E2 +E2

_Ix _2x

-f-

.) 2 +(E

+êì

-f-.

2 + + E + reco

+E(ç

_+ç

+ L

',x

fx

2x?

E+E('ço

1i

+çi

)+

'

y 2 --

Io

±2(ç

+ L 1 _f2' ..2/

-

+ C(t) ? ₌

-

ZÏ

c(t)?=

çp2+

f

9',,-

'Ia

for

+z.

7 Pt =0 -j

Again, comparing the coefficients of E Ofl both sides of the

equation we have the system of equations which _{are justified}

whenz=O

-Solving simultaneously the equations of the first, second and following lines from equations (15) and (17) we obtain

-

2c(t)'

_{-f C"t)(p} -f _{- C'(t)ÇD = - (z,I,o,t)}

'ritt

'ttXX f7 FiX where 2 2 ₂₁ (7, t) _c(/) ço

-

_çp

_{y +}

Lix

rj

Let us now transform the boundary condition on the ship's surface. We substitute in for the corresponding series with respect to E

íIr

- c(!)

_{j «f+ E}

+

_{-) ±}

2

+ -- -) =o

1f 2

Again, comparing the coefficient of E, we have the system:

I!=o217

(19)

(17)

(6)

which is justified for y = O, where:

= c,/t)4 (z, z);

c(t)/ (X, ) -

Çi/

- çI/

The condition,

allows us to assume that sources with the intensity g (x, z)

are distributed along the ship's center plane. Let us denote:

00

=

a,

QAx»yE) =

This function Q(x, z, E) represents the density of the sources. Thus, we can represent the body flow as the interaction of the outside flow with the sources. For defining the velocity potential i(x, y, z, t, E) we have a boundary problem A = O

'X

-

2c(t+ C(L)0+

yp -

= r(x,i, e)

}

=IQ(z,,)

(z,)C5

_cl

_o;

_çol

o

Jo

o

(x,2)5

'

1t0

_{¡t_o}

with a single solution.

On the basis of (21) we can see that the non-linear boundary problem defined by (1) - (5) with the help of (8)

-(14), becomes linear, (21). The velocity potential (x, y, z, t, E)

5

where P and P are the unity Green functions defined as AP = O

1 2 1

-

_{2c(1ì q1 . Ct)P}

_{+ QP}

-'rx

q f

(7)

p'

(x-,-)

s'

=0;

P

=0

t-o _t-o

1=0

_1-0

- 2C(t)/+

C?(t)P ₊ _{- C'E»}

.!3(x-& ,-7)

viherz = o 2,rx when z = O P

=0;

p

2L

where Q is the half-plane z O, y > O. Expanding the integrals of the right hand side of (22) to a series with respect to E we get the following expressions for the com-ponents of the potential:

ÇÇ

+fl2('7)J

-

(25)

()9y [, ?)]J[x,y.

d +

where: S0 is the suerged part of the center plane bounded by the center plane contour

t(x,E)=?,+E!,(x)+.-.

and line

L is the x interval for the points on S . Formulas

o o

(25) and (26) allow us to obtain an expression for the full potential. The problem of defining the Green function

=0;

t-0

1-0

=0

t-o

1-0

(23) (24) (26)

(8)

=

V

2

72

2 =

To obtain the Green unity function P we have AP = O

2 2

82r, P.

=0;

?

PZ,DU y=Q t-o t=O

To solve this system, we may use a Fourier Transformation. Let us introduce the function

oc f

¡

(K,I,,7,o,t)=

d.t

Then on the basis of the inverse Fourier Transformation we have

/(x,y,,7,o,/)=

dx

Substituting (30) into the equation AP2 = O we get:

2

210

(31)

Following the substitution of (30) into (18) when z = O we can write

is described by the system of equations

(23). The function P must give the velocity potential for

the motion of a single source placed at the point

(E,

, Ç)

of the lower half-plane.

Based on Reference

[lii ,

P must be written as

t

j

110()

where

ith

(t- )ep

?

+ ± t(X -

)w38

-

7»zns}}

d6xd,

(9)

¿1'2Pq

ü

_{2 2}

dc(L)J

----2ic(t)A'---[KC(t)

+LK-_-jJ -±

?-JLAX

X-'-7).

(32)

After applying the Fourier Transformation to the expression in the brackets in (32) we have

/

o

-3

will be sought in the form:

20t)=/Jf

(*,z,t)aii tzydn

Substituting the last expression in (31) , we obtain equations

with

respect to A(k, z, t)

u,f

-- (X +

'z )J

If P 0, when z - co1 which is a condition of still water,

3

then

Á(A-,,t) =a(A-,L)e

The meaning of the function a(k, t) may be found from equation (33)

i 22f

êa

_2Éc(t)th

_{Kc() +}

zxcf) -

T/7+1]

a = = (38)

-y-)e

¿v'zdxdn

The other boundary condition, (28), with the help of be written as: (30) can -=0. =0; t 0

tQ

?zpU y=Q; o

Therefore, the function P as defined by (31) , (32) , and (34)

-2 zkC(t)-Ç _{K2C1t

_ikfl

_{1((x_}

(10)

In the theory of S-functions, Reference [811 , it is shown that

r r

JJd(x-,y--7)e

cv6rLa'xdn=e

cvz7.

u

-Substituting (39) into (38) and using

t

fr aí'r,t) =b(k,t)e o we find that _t

vy+n2

='k[ß'co5fz7

i+fC(!dt)

(40)

Applying the Lagrange method to the solution of (40) we can write: t t t'fr t)

lfTcrjjrz7

-tk( r

=

e

j

exp[zx/c(î)dz]

zt4/'

(t - ï) dt o

We can now refer back to the function a(k, t)

t t ir'cosrt' c1(K,t)z e exp

[t/mat]

in

dt.

o P may be introduced as j/k2+fl2 e -LK e coin zzrz7a'ttx o t t

¡exp [tr/c(fl ali] iiP7J/'

(t-The function P can be written as

/

LJJ

e'X)

X

- o

7/[z/c(t)dh]

itn(t-i)dndî.

From expressions (44) we can see that the obtained Green function satisfies the initial conditions. When t = 0, the inner integral is equal to zero, which makes P = O also. This shows that the liquid is in a static position.

Let us count the forces and moments acting on a ship

(11)

lo

moving with changing speed. The main hydrodynamic force is

-J(p-p.) ñc/s,

(45) where a hydrodynamic pressure is defined by the Lagrange

integral ¡C2(I) 2 z ₍₄₆₎ After substitution -

ff

f2

i

Ç R

jo]

_1--C(t)j

_ffd.s

-4-J

ñt-J5

The second integral in (47) is equal to

/ñds

where: V = the volume bounded by the surface S. Applying the Kraejn Method, Reference [7] to the rest of (47) we have - V

_{fds fQ(x.e)wdv(/e)ds,}

where

'(x,j,Z,t,5)

fO(.7E)

L)th

-Q '.-o'. E t -__L-1

r r K

-2J

j j ==czo

[lKco8/c()cft]

Oln

J700

z exp { K [z ± +

_{- ws +}

_{- 7)irn&] } &}

The projection of (48) on the x-axis gives us the ship's resistance for unsteady motion

(51)

(12)

Substituting (51) into (49) and (50) we get: ¿.ffr / f

-

)y-

7)4(+JQ

where

fil

exp(LK8fcö.i

_J -i7o o x exp

{K

[i +

±

_{+ 4-7»th6] }}

t)d6dd!. (53)

The first member in (51) represents the added masses of the ship moving along a rigid plane which represents the free

surface--

_{IJff(_4)Q4..Ew3rtrcJsds}

55

The two following members give the ship wave resistance. Then, taking into account (26) and

= ¿Ra-4- E3RÊ ± .. -.

we find out that

ReetÇ/(

(ST,

vdxth/f

_{(, ¿)G} _d

x/c(z)fjexp

_±

4x-)

CO3G +

icaiefcmdrj

} x

-ffa(.,) [

L/d5

--f/r(7.E)

If we substitute: 11 (52)

(53)'

(54) X (55)

(13)

t

' zK[(x-)+f

_{c(r)cf r] c038}

{K(x-)+fc(r)drJ

21/ -JI

then (55) will be transformed into the form _previously developed by Sretensky, Reference [1]

ji 'X, t R Reel JL7

f Í'91'2r

f )ü'ia'i

[[[° +l-VJ(tW]

S2

2 !i

50S0

Kexp[IKCV36 JC(r)dtj

_{a?3l/7E- Z)d&a'ka'î}

₊

_{Reef7/f((2_}

_{qilÇ )didi x}

S.S.

1/f k2c()coi8

_{K [z-t +f'X-)Wj6+ ii'y-7j}

e

exp[z

iS (c(v)a't]

t

+

ReeL'4//(

i

p(+) +X(x-)

-ooOO X CO3/7y

Identically we can write the expressions for

Rt

(1=3,4,.... ti)

Let us now consider the lifting force which is the projection of (48) on the vertical axis

R=

3'V_p/[tcv(,z,z) ±Q(X,Z.E)4D0]d3 s Using (49), (50) and

= E2R+

_'2

-we can obtain

=

ìc'eet'p/f[,(

_{kt 92J di d5}

se So 12 X (60)

(14)

13

R =

- 2qqKJ did ±

5050 (61)

djct, (kf

55Z

If we take only the first members of the series expansions for RB and R we have a linear solution to the problem. The following members, however, contain sorne additional elements which satisfy the boundary conditions better. Thus, we can obtain the solution with any degree of accuracy, depending upon the number of members taken into account.

(15)

REFE RENCE S

[1] Z. N. Sretensky, "Formula for Wave Resistance Cal-culation," Proceedings of C.A.H.I., 1937

[21 T. Havelock, T. Quart, Mechanical and Applied Math,

Vol. 2, 1949

T. Lunde, Transactions of the Society of Naval Ar-chitects and Marine Engineers, 59, 1951

H. Maryo, The Society of Naval Architects of Japan 60 T. H. Anniversary Series, Tokyo, 1957

A. N. Shebalov, "Upon the Forces Acting on a Body in Unsteady Motion under the Free Surface," Proceed-ings of U.S.S.R. No. 2, 1969

[61 W. G. Sisov, "To the Theory of Ship Wave Resistance

in Still Water," Proceedings of A.S. of U.S.S.R. No. 1, 1961

A. A. Kostukov, "Theory of Ship Waves and Wave Resis-tance," Sudpromgys, 1959

A. A. Sokolov, "Classical Theory of Fields," G.I.T.T.L., 1951