On the harmonic oscillations of a rigid body on a free surface

__________ *

Lab.

v

Scheepsbouwkunk

J. Fluid Mech. (1965), vol. 21, partS,pp. 427-451

_{'Technische 'HogechooI}

Printed in Great Britain

DeRt

On the harmonic oscillations of a rigid body

on a free surface

By W. D. KIM

Boeing Scientific Research Laboratories,_{Seattle, Washington} (Received 12 May 1964 and in revised form 27 _{July 1964)}

The present paper deals with the practical and rigorous solution of the potential

problem associated with the harmonic oscillation of a rigid body on a free surface.

The body is assumed to have the form of either an elliptical cylinder or an

ellipsoid. The use of Green's function_{reduces the determination of the potential} to the solution of an integral equation. The integral equation is solved numerically

and the dependency of the hydrodynamic quantities such as added mass, added

moment of inertia and damping coefficients _{of the rigid body on the frequency} of the oscillation is established.

1. Introduction

'

A fluid motion eaused by small prescribed oscillations of a rigid body on the

free surface of an incompressible inviscid fluid is studied in this paper. The fluid is assumed to occupy a space bounded by the surface of the body and by the free surface extending in all directions. The induced motion of the fluid in this space

interacts with the oscillating body and exerts the dynamic pressure on the

immersed surface. A point of interest_{here is to find the effects of suchpressure}

on the variation of the inertial and damping _{characteristics of the body}

per-forming a steady oscillation with a certain frequency.

In the formulation of the present problem, the boundary conditions will _be

linearized by neglecting high-order terms in view of the smallness of the motions

involved. It is well known then that the poténtial which satisfies the _linear

conditions on the boundaries in the_{undisturbed position and the proper physical} condition at infinity can be determined_{uniquely. Nevertheless, such a potential}

depends on the mode and frequency of the oscillation as well as the form. of the body.

For a body of general form Kotchin (1940) and John (1950) showed that_the solution of the problem can be represented as a potential corresponding

_{to a}

surface distribution of point sources. The strength of the sources is to be

deter-mined from a Fredhoim integral equation which satisfiesa prescribed kinematic condition (appropriate to a specific oscillation) for the potential _{on the body}

surface.

We present a procedure for the approximate solution of these integral

equa-tions. In two-dimensional problems the kernel of the integral equation is

428 W.D.Kim

of the singular term is one of the major problems of the present work. This

procedure is valid for any shape which can be described analytically.

Once the equations are solved, the dynamic forces Fd and moments Gd onthe

rigid body can be obtained by numerical quadrature. For instance, in three-dimensional problems resolving these into a component in phase with the

acceleration and the other component in phase with the velocity, we write

Fd=_pa3MXJpNZ (j= 1,2,3)

for the linear oscillations X(t)

Re [I(t) e1, and

Gd = _pIûpo4HO5 (j = 4,5,6)

for the angular oscillations

O(t) = Re [O(t) e°'J,

where p represents the density of the fluid, is the half-length of the body, and o. is the frequency of oscillation. The dimensionless quantities M and N are called the added mass and linear damping coefficient, and similar quantities

I and H are called the added moment of inertia and angular damping coefficient,

respectively. For the case of two-dimensional problems the dynamic forces F and moments per unit length can be written in the same form with and a4 replaced by a2 and respectively.

These hydrodynamic quantities are functions of frequency or, more precisely,

functions of the parameter ao.2/g = a, g being the acceleration of gravity. In this paper we present results for simple shapes, ellipses in two dimensions and ellipsoids in three dimensions. In order to gain some insight into the validity of

the present method, the results are compared with the previous results obtained

by a quite different method which was originally developed by Ursell (1 949 a, b). Added mass and the damping coefficient of heaving ellipses presented in figures 15 and 16 showed good agreement with the results of tJrsell (1949 a) and Porter

(1960). The same agreement is noted between the present results of surging ellipses in figures 13 and 14 and the work of Tasai (1961) who used Ursell's method.

The only results in three dimensions which are known to us are due to Havelock

(1955) and Barakat (1962) and pertain only to a heaving sphere. These are

compared with the corresponding curves in figures 3 and 4. It was noted that the present results are in complete agreement with those of Havelock. It would seem, on the basis of the limited evidence available, that the present procedure yields accurate results.

Nevertheless, there is a fundamental limitation to the present numerical

method. The kernels of the integral equations oscillate rapidly as the parameter

a increases. Therefore, unless many subdivisions are used, the numerical quadratures occurring in the present procedure become inaccurate. In this

paper, we have limited the computations to values of a less than four, which covers the range of practical interest. For instance, in the three dimensions, it means that we have waves which are not much shorter than the length of the rigid body. Higher frequency results can be obtained with more computational

labour. The final adopted fox such as the ellipsoids h two-dimern ellipsoids is approximal

2. Gener

Consider of gravity e system Ox and we tak oscillations equilibriuix the surface all directio: The posi at any inst of gravity: linear corn angular co Assuming motion di may be in requires t where V is As the a wave moti the lineari free surfac where 7 re Then the i with k de length of

i _{Oscillatjon8 of a}

rigid body _{on a/ree snrface}

429

The final objective of the_{present study is to}

assess the merit of various models adopted for the investigation of ship form. At first,_{the hydrodynamic quantities} such as the added mass, added moment of inertia, _{and damping coecients}

of the ellipsoids having different

axis ratios are estimated by the strip theory, using the two-dimensional data of_{a long cylinder. Next, the effect of the draft on the}

ellipsoids is examined in _{order to evaluate the applicability of the shallow-draft} approximation (see Kim 1963).

2. General formulation

Consider a rigid bodyimmersed in an inviscid,_{incompressible fluid with its} centre

of gravity on the origin O, and its_{axes on the (, )-plane of}

a space co-ordinate

system Offi. The ()-plane

_{here coincides with the undisturbed free surface,} and we take the i-axis positive upwards. If the_{body is given linearand angular}

oscillations of small _{amplitudes X° and 00 with a certain frequency}

o about its

equilibrium position, VIZ.

(t) = Re [0 e1°"],

1).

(2.1) 0(t) = Re[00e-ioIJ,J

the surface disturbance created by these motions_{travel outwards}

as waves in

all directions.

The position of the_{immersed surface S(, ) relative to the space}

co-ordinates

at any instance

can now be expressed by specifyinga position vector of the_centre of gravity 5 _{= ¡X +112 + k13, and the Eulerian}

angle O _{= ¡04 + 105 + ko6. The}

linear components

_{X, I and Ï}

are called surge, heave and sway, while the

angular components 04, _{05 and 6 are named roll, yaw and pitch,}

respectively. Assuming the fluid to_{attain a time-periodic irrotational motion when}

transient

motion dissipates, _{a velocity potential}

= Re[V(,y,)e-íuiJ

_(2.2) may be introduced to describe the state of the fluid._{The incompressibility}

then requires that V is a solution of the potential equation

V2V(,,)

_{= O in < O,}

(2.3)

where V is complex-valued

As the amplitude of_{oscillation is considered}

small, the amplitude _{of induced} wave motion will also be small in comparison with _{the wavelength.}

Therefore,

the linearized dynamic condition for the velocity_{potential at the undisturbed}

free surface becomes

0, _(2.4)

where

represents the free surface elevation, and g the acceleration of gravity. Then the linearized kinematic condition O, _{; t) = ; t) and (2.4) yields}

V(, 0, _{) - k V(, O, ) = O on}

0, _(2.5)

with k dénoting the _{wave-number which is equal}

to o2/g _{= 2n/X, X being the} length of free wave.

430 W.D.Kim

The kinematic conditiön which states that in the absence of viscosity the

normal velocity across the immersed surface of the body is continuous takes the form

(2.6)

where 1 Ñpresents the position vector of amaterial point on the body and n, the

unit normal of the immersed surface, i.e.

j:

_{= îi+1+fc}

and n

îflx+JAfly+kflg.

Note that as the consequence of the linearization, the kinematic condition is to

be satisfied on the surface in the undisturbed position. Thus, we find

a

-- V(,) =

= _j0.[XO.fl+ØO.(xfl)], (2.7)

an jTh

for six degrees of freedom in the problem.

Finally, a disturbance in the finite region should produc only an outgoing

progressive wave at large distance,

V(?, O,) _A(0)?_ehi_0. as ?-.,

(2.8)

where? = (2+2), e = tair' (z/x).

In order to show clearly the dependence of the solution on the frequency

parameter a= k, being a typisai length of the body (such as a half-length of

the ellipsoid or a half-beam of the cylinder), we shall first make the space variables

and the amplitudes dimensionless, i.e.

x= x/a, i =y/a, z ₌

and X

=

then introduce the pressure function u1 by

= au5(x,,z) (j= 1,2,3),

io171(x, , )/ga05°= au5(x, y, z) (j= 4,5,6).

It follows that the dynamic pressure 11 can now be expressed as

]TI(, t)

= P[j(

7, ; t)]1=pg Re [Xau1(x, y, z)] (j = 1,2, 3),

fl5(,,;t)

=

_p[(D1(,,;t)]1

=pgRe[O'au1(x,y,z)] (j = 4,5,6).

The boundary value problem which arises in the study of small harmonic

oscillations of a body on the free surface is to

find a potential u5(x, y, z),

j

= 1, 2, 3, ..., 6, continuous in the fluid spacesuch that

(A) V2u1(x,y,z)=0

in y<0,

(B) u1(x,0,z)au1(Z,Q,Z)= O outside S(x,z),

(C) u5(x, y, z) = h5(x, y, z) on S(x, z),

(D) u1(r,O, y) - A1(0) r4euU+iar - O as r

.

,

(2.11) (2.9) (2.10) where S(x, z position S(a depends oni are given, r h,(x, y, z) h4(x,y,z) In the ea problem to normal to ti to this piar where C(x) position; C Here we re now corre problem,

3. Repre

The so the sets of in the fo G(x,y G(x,y;g where and S0(a functions and Cias

O8cillation8 of a rigid body on. a/ree 8urface 431 where S(x, z) represents the immersed surface of the body in. the undisturbed position S(x, y) = ay), and h5 denotes the prescribed function which

depends on the mode of oscillation. For six degrees of freedom, the functions h,.

are given, respectively, by

h1(x, y, z) = n, h2(z, y, z) = n, h3(x, y, z) =

h4(x, y, z) = yn. - zn,, h5(x, y, z) = zn - xn, h6(x, y, z) = xn, - yn.

In the case where a rigid body is a cylinder, we expect the solution of the

problem to be two-dimensional; in effect, a solution for an (, )-p1ane which is normal to the axis of the cylinder is valid for all planes cutting the axis parallel

to this plane. Therefore, the boundary value problem is to find a potential

u1(x, y) j = 1, 2, 3, continuous in the fluid space such that

V2u,.(x,y)=O

in y<O,

O) - a'u1(x, O) = O outside C(x),

(2.13)

= h,.(x,y) on

u5(x,y)_BjeaV+i_.O

as x-+±,

where C(x) represents the immersed periphery of the cylinder in the undisturbed position; C(x) = 'O(x) and h,. are given by

h1(x,y)

_{= n,}

h2(x,y) = ni,,

h8(cc,y) = xn-yz.

(2.14) Here we remark that due to the renaming of the co-ordinate axes j = 1,2, and 3 now correspond to the case of sway,' heave and roll in the three-dimensional

problem, respectively.

3.

Representation of the potential

The source potentials G of unit strength in. the lower half-space which satisfy the sets of boundary conditions (2.11 A, B, D) and (2.13 a, b, d) can be expressed

in the form

G(x, y, z; , _, ) = R-' + R'' - iTa e'7> [S0(aw) + Y0(av) - i2J0(avj)]

- 2aea(1'+)f

eius + )4 dia,

(3.1)

=

+sinaJx-J Sia

lxI

-In x-j + 1Tsinax-t -i7rcosa(x-)]

ro

2a ea(y+'12)

e" in [(z

)2 +u2)d/L. (3.2)

J y+v

where

= [(x_)2+(y_)2]I

=

R = [2 + (z )2], R' [F2_{+ (z}

and S0(atu) is the Struve function of order O, .4(aw) and Y0(aw) are the Bessel

functions of the first and second kind of order zero, respectively, and Si a fx

-and Ci a ¡z

_{- ]}

denote the integral sine and cosine functions. It should be noted

432 W.V.Kim

that the source potentialsG are more tractable in thepresent form than in other

expressions using Cauchy's principal-value integrals (see,for example, Wehausen & Laitone 1960).

We now seek the solutionof the boundary valueproblems (2.11) and (2.13) in the following form:

=

_ff

_{f,?7, )}

G(x,y,z; )dS, (3.3)

u.(x, y)

=

_ffí

)G(x, y; , ) dG,

(3.4)

where f represents the strengthof distributed sources over the immersed surface

S(x, z) or C(x), and is a continuous complex function.

According to potential theory, the normal derivatives ofthe potential u on

S(x, z) and C(x) are given by

u(x,y, z)

= -

f5(x, y,

z)+-ff

n ) -G(x,i,z; ,n )¿S,

(3.5)

and 'u(x, y)

= -

f5(x, y) +

_ffi(

) - G(x, y; , ) dG. (3.6)

Therefore, if f is determined from the integral equation

i tC

0G

f5(x,y,z)+-- fl f,n,C)dS

= 2h5(x,y,z), (3.7)

1Tjj

it

0G

or

_f5(xy)+Jfi(C,n)dC

= 2h5(x,y), (3.8)

ti5will satisfy the boundary condition (2.11 C) or (2.13 c). Accordingly, u is the solution of the given problem. In order that these integralequations be soluble, the homogeneous equations must possess only the trivial solution. John (1950)

proved that the homogeneous equation cannot have a non-trivial solution for

sufficiently large wavelength.

We shall examine here the behaviour of the source potentialand its normal derivative at the proximityof a point source: in (3.1) as a variable point (, 17, ) tends to the point source at (x,y,z) on the immersed surface S(x,z), i.e. R-*0,_ro and ui-0, in addition to R-1 being singular, Y0(av) and e#(ji2 w2)_lda

become logarithmically singular since .'

" um [Y0(au7) - In (cur)] = 0,

lT

(3.9)

and

lim{f°

a

2+u2)_1du_lflRY17]

=

wl.0 JlJ+'l 7.

However, because of the opposite signs these logarithmic terms cancel out and we obtain _{G(x,y,z;,n,C) = R_1+G*(x,y,z;,ll,C),} (3.10)

with

G*(x,_y,z; 17, )

= R'-' -

2aea4i)[ln [a(R' - y -n)]

f0

+

ir[S0(a)

+ N0(a) - i2J(a)] +

(e

1)

_p2 w2)4df

11+'?

4

where N1,(a Next in (3 i.e. V7-+O,

CiaIxt

where 'y again can with Let u (3.10) an and In (3. terms i become distanc present

Ix_LIt

of curv (3.15). Thus integra

4. Fo

The imme in pha of the of me coeffic We small 28

_..,

-esciUations of a rigid body on a free eurfacé 433.

where 14(aw) is a regular function defined by

N,(aw) = Y0(avi) - 7T in (aw). (3.11) Next in (3.2) as (, 1) approaches the source at (x, y) along the periphery C(x),

i.e. w-+0, and ¡x-LI

in addition to mw and lnjx-I being

singular,

Cia!x-j behaves as

um [Ciajx_I_(y+h1aIX-Ifl = 0,

(3.12) where y is Euler's constant. Here the opposite signs on the logarithmic terms

again cancel singularities so that we have

G(x,y;,i7) intrjG*(x,y;g,17), (3.13) with

Q*(x,y;,17) =inwl+2eaQ/+cosa(x_)CiaIx_I+siflaIx_iSiaIx_I

ljxg +sinax-j _i1Tcosa(x-g)+a(y+17)lnw'.

X

(eP-1)in[(x_)2+2Jd}.

Let us now turn to the normal derivatives of the source functions. From

(3.10) and (3.13) we find that

G(x,y,z;,17,C) =R_1+G*(X,y,Z;,17,),

(3.14)

and G(x.y;,17) =

I-lnw+LG*(x,y;,?i).

(3.15)

In (3.14) as R and w tend to zero (a/an) R' becomes indeterminate and the

terms in (a/an) G* which contain powers of reciprocal distanc such as vr' or vr2

becomes singular. However, when w = O the multiple factors of the reciprocal distances vanish simultaneously so that these reciprocal distance terms do not present the problem here (Kim 1964, Appendix). Finally in (3.15), as w and

¡z

-

j tend to zero, (a/an) in tu takes the value of (2R0)-', where R0 is the radius

of curvature of the periphery at = z. Therefore, no singular term appears in (3.15).

Thus, the integrals containing the terms R', (a/an) R' and

in w in their

integrands have to be evaluated using the special scheme.

4. Forces and moments

The forces and moments caused by the dynamic fluid pressure acting upon the immersed surface of an oscillating rigid body may be resolved into components

in phase with the acceleration and other components in phase with the velocity

of the rigid body. The former quantities are called added mass or added moment

of inertia while the latter quantities are called linear or angular damping

coefficients.

We shall determine the forces and moments when an ellipsoid is excited into small harmonic oscillations about it equilibrium position on a free surface of

434 W.D.Kim

a fluid. Since theform of an ellipsoid is symmetric about the space axeslying on

the free surface, fromthe result presented by John (1949), the forces F and moments G can be deduced as

F( = F(t) -pgI2(t) i

(j = 1,2,3), 'I

and - (4.1)

G(t) = Gd(t) - îpgO4(t) (2 + 2) - fpgO6(t) + ) (j = 4,5, 6),J where the dynamic forces and moments are given by

Fd(t)

=

fJ11,

t)ndS,

Gd(t) =

ff ll, ,

; t)(r xn)dS,

and S represents the surfacegiven by

= (1_/ä2_2/b2),

< 0.

By (2.1) and (2.10), Fd and Gd can be rewritten in termsof the dimensionless space variables and amplitudes as

Fd(t) = pg Re _{5f au5(x,} ,z) n dSe_10f] (j'= 1,2,3),

Now, expressing the dynamic forces and moments with a component inphase

with the acceleration and that in phase with the velocity of theellipsoid, we

obtain Fd(t) = _MX,«)_NX,(t)

(j= 1,2,3),)

fl1(t)

(j = 4,5,6). ₅ (4.3)

and

G(t) = -

¡5(t)

-Then, the comparison of (4.2) with (4.3) yields

M=-i=Re[55ui(XYZ)fldS]

N=__==Irn[ff1L(X,Y,Z)fldS]

( = 1,2,3),

I =- =

Re{5Ju1(x,z)(rXfl)d5]

H=.!'==Im[f5Ui(X,Y,Z)(rx9)d8]

(j=4,5,6),'

where M and N denote the dimensionless added mass and linear damping

coefficient, and I and H denote the dimensionless added moment of inertia and

angular damping coefficient, respectively. In passing, we note that the quantities

M and N in (4.3) can be related to the added mass coefficient i and damping

coefficient 2h, which are employed by Ursell (1949a) and Havelock (1955) as

M = mii, and N = rno(2Fi), (4.5)

where m represents the actual massof fluid displaced by a rigid body.

Ifa rigid body is a cylinder, the forceF2 and moments G(2) arising from small

forced oscillations are given by

F(2)(t) = FTÎPUX2(t) 2 (j = 1,2), and

G2(t) = Gpg03(t)(ã2+b2),

(4.2) (4.4) (4.6) where

F(t)

and C 112,j forces and and i veloc and we O coe I 1mo d sa res pre coe ate Ap

I I M2

_1f

s A

i

M2) = = _{u(x, i) (in2 +jn) dCj.} N(2)_{= 2 =}

_{Im[fui(xY)(mn2+În)dC]}

_{(j = 1,2),} j(2)

rr

_i =_pa = Re [j u3(x, y) (xn - yn2) dC] = _{= im ¡f u3(x, y) (xn yn2)dC].}

It is often necessary to estimate the _{three-dimensional physical quantities}

knowing only the two-dimensional results. _{For instance, using the}

hydro-dynamic quantities of the cylinder (4.9),_{an attempt can be made to evaluate the} same quantities of the ellipsoid (4.4). A method of relating the two-dimensional

results to the three-dimensional estimates is called

_{the 'strip method'. In the}

present problem, the added mass, Mf,, added moment of_{inertia, I, and damping}

coefficients, N, and H, of_{an ellipsoid for modes of heave and roll can be} evalu-ated by the following strip method formulae_{(for the derivation, see Kim 1963,}

Appendix 2): 1 =

2f

b2( i z2) {ab( 1 x2)J dx, = _{2f1b2(1_x2)Ni2)[ab(i_z2)Jdx,} (4.10) =

2f

b4(1

x2)2I[ab(1 _x2)Jdx,

H = 2f'b4(1_x2)2H)[ab(1_x2)iJd,

28.2 (4.9) F

Oecillation. of a rigid body on a free aurface 435

where the dynamic forces and moments are given by

F)(t)

=

f 11)(,

; t) (In2 +n) dc, and G)(t)

₌

f

; t) (n n2) dG,

and C represents the periphery of the cylinder,

_{= ( 1 x/a),}

_{< O, and}

11),J - 1, 2, 3, is the two-dimensional dynamic_{pressure. Rewriting the dynamic}

forces and moments in the form

= pgRe [Xfaui(xY)(înx+În)dCe_iui]

and = pga3 Re{O _{f au3(x, y) (xn,}

-

_{yn2) dC}e_101],

and identifying the component either in phase with the acceleration or the

velocity from

FÇ)(g)

_{= _M2X(t)_N2,(t)}

_{(j = 1, 2)}

and

_{G(t) = _Ï2)3(t) 23(t),}

j

(4.8)

we obtain the dimensionless added mass, added moment of inertia and_damping coefficients for a two-dimensional problem

as

(j= 12)}

436 W.D.Kim

where MJ2, I and. H denote the data of the cylinder. It will be shown

presently how the estimates obtainedby (4.10) compare with the corresponding three-dimensional data.,

5. Numerical procedure

Wo are ultimately concerned with the solution of equations (3.7)and (3.8) for

arbitrary values of a. Here (3.7) deals with an ellipsoidx2/ä+y_2/C_2+2/2 = 1, < 0, which has length, beam, and draft of2, 2 and while (3.8) deals with

a cylinder having a cross-section

/2+r/2 = 1,

< 0, in which 2 .and j

denote beam and draft. Suitable co-ordinates which conform to these body

forms are the ellipsoidal polar co-ordinates

z = coscxsinç5, y =ccosç5, z = bsinasinçi5,

_i

= cosfisinfr,' = ccosfr, = bsinfisin1r, (5. )

and the cylindrical polar co-ordinates

x=coso,

= Cose,

I =

b sinO,

1. (5.2)

= bsme,

where z, y, z and , i, are dimensionless space variables, and b, c are

dimension-less lengths resulting from dividing each physical length by a typicallength of the body .

By use of the new co-ordinates we can express equations (3.7) and (3.8) in. the form

bc a

(a, ç5)+

_i-f

5 .F(fi, fr)G(a, ç5;fi,7fr)T(fi, fr)sin%frdkdfi =

211(a,ç5),

and

_F,(0)+fFj(6)LG(0e)R(e)d = 214(0),

(5.4)

where T(a, çS) = [(cos2 a + sin2a/b2) sin2 ç5 + cos2

and E(0) = (sjnse+b2cos2O)k.

(5.5)

We restrict ourselves to thethree-dimensional case since the two-dimensional

equation is solved in an analogous manner. The method is based on Fredhoim's

procedure of replacing the integral equationwith a finite set of linear equations. Suppose the fi- and fr-axes which bound the region of integration S(fi, sfr) are.

divided, respectively, into24and 6 equal divisions. Then, a lattice can be formed

on S(fi, %fr) by connectingthe points of divisions with straight lines parallel to the

fi- and fr-axes. The element of such a lattice is a square having the side h

= kir.

Now, choosing the centroids of the elements as pivotal points, equation (5.3)is

to be solved on these discrete points. Since .l, the source strength, is a complex function, equation (5.3) can be separated into a pair of equations for the real and

imaginary parts: - be ± .!. L. ....,.,. , .. . raGi

FT

- Ljm{fi1m)ImL

x T(fi,*,)sinfr, = 2H(a,ç5), (5.6) f be ! ! 1.-. ,

S,.

Í01

.

m[1(fi,11)]Re[}

x T(fi1 1r,) sin fr,, = O i (5.3) where, as h distance bel integral. N We turn and n(a, and (fi, in its pro. where It follo Thus, o possesse part wh where with with where r it can be with and E ==

Oscillation8 of a rigid bod1j on a/ree e'urface 437 where, as has been noted earlier, the integral involving (8/On) R-1, R being the

distance between a point source and a variable point, in Re [00/On] is an improper

integral. Nevertheless, this integral can be shown to exist. We turn to the evaluation of the following integral

2i IT

I=bcf

_L

fl,fr)X(a,ç5)

al

where ₌

with X(c, ç» = Ix(a, ç» +y(a, ç1) + kz(a, ç»,

fr) = î(fi, *) +(fi, ?fr) + L(fi, sfr),

and n(a, ç» being a unit normal on the surface S(ft, 1r). Let us suppose X(a, ç»

and !(fl, sfr) represent the position vectors of a point source and a variable point

in its proximity. Then, by Taylor's expansion, we may write

i

(fl,1r) = X(cz,ç»+ (5.8)

n=1 fl.

with = 8cosr, çb = 8sinr, and 8 = [(ß_a)2l(ç5_fr)2]l, (5.9)

where r = tan-1 (1fr - Ø)/(,8 - a). By the use of (5.8) and the binomial expansion, it can be shown that (Kim 1964, Appendix)

(5.10)

81

1

(2a + j2ç» 112a sin2 çb + ¿2ç5

where A1 =

2

(E+c22sin2ç»F

B1 3(i2a sin2 + ¿2ç» - i2aiçb sin 2çi

4(E+c22qSsin2çS)

with

(&xX + zqX) (12aX +

2&cAÇ5X* +

i2ç5X)

MzX + 2açLX

.

Xq + 22

and E = E(&,tç)

= It follows then that

(5.7)

r81

hrn

LOflR

T(a,)('')] = 0.

(5.1Ï)

Thus, on the transformed plane as (fi, 11e) tends to X(a, ç», the function (8/On) R' possesses a part which involves the reciprocal distance, and an indeterminate part which depends upon the angle of approach.

438 W.D.Kim

By (5.7) and (5.9), the part of the integral

1, taken overthe neighbourhood

element , ear be approximatedas

r2n r4(r)IA bcsin$F(a,ç!) I i (-1+B1)d8ßdr

Jo Jo

\

/

'+l/'

+h1

J + bcF(z, )J _j T(a, ) +B1)] T(/3, sfr) sin 1fr dfi h2 f2 r «

(eos2rsin2+sin2r)dr

- -

bcF(a, ) sin _{Li_} (E + e2 sin27Sifl2 ç»

11r lcI

(cos2rsin2+sin2r)d7] +1 .)1t

(E+c2sin2rsin2)t

33

+

i1j1

(fi.a)2 sin2 5+(fr5ç5)2] T(fi'çfr1)

[E1 + c2(1 )2 3j2

j

Sfl

where the8-integration is carriedout explicitlywith h(r) given by

h(r)

=

h!seer! for

CSC

7T; 7T $ 7 ir; tir,

iii;

iT

and the second integral is evaluated using Simpson's

rule since its integrand

vanishes at (fi, le') = (a, ç». Furthermore, it should be noted

that

r2ir I'h(r)_{B1dô6dr = O, since}

B1(ir+r) = B1(r).

Jo Jo

The remainingpart of the integral I taken over

5 - can be

approximated by

use of Simpson's rule.

Application of theformula (5.12)and Simpson's rule enables us

to write out the doublesummations in (5.6) as alinear combination

of Re [] and

Im [F so that we can obtain a finite setof linear equations for each mode

of oscillation.

By (5.1), the right-hand membersfor the six

degrees of freedom areobtained

from (.12) as

H A cos a sin çiS

H

cos çb H A

sin a sin çf.

- T(a, ç)

'

2 _{cT(a, ç»'} 3(a, bT(a, ç5) H c - b2

sin a sin ç6 cos

_H

b2 1 sin a cos a

ç5

5 13

-

_T(a,çi5) ' b T(a,çS)

)

1c2coscxsinç5cosç5

and ll(a,ç5)=

T(a,çL)

Here the strengthof source .1 in the problem dcpends not only

UOfl the mode and frequency ofthe oscillation,but also upon the form of the

body. Since the

immersed surface of an ellipsoid possessessymmetry about

the and axes, we

expect J to

exhibit thecorresponding symmetry. However, duo to

the trans-(5.12) formation ol the variable

p

I

On the bas equations more, relat produced matrix fo Having pressure

f

where a in Re[Ü with th where It foil [U wher Agni by u facil of o

j

Oscillations of a rigid body on a free surface ₄₃₉

formation of co-ordinates by (5.1) this symmetry is to be expressed in terms of the variable a. For the specific modes of oscillation, we note that

=

= l(a,ç) =

= F2(a,ç) = F2(na,ç,) = F2(r+a,ç!),

F34(,ç) = 4(--a,çb)

= F(n.-a,q!) = F34(n+a,ç),

= (a,q) = (ira,ç) =

On the basis of such symmetric properties, it is sufficient to consider the linear equations (5.6) only at 36 pivotal points _{contained in one quadrant.}

Further-more, relating the values of _{in other quadrants by (5.14),a rectangular matrix}

produced by the double summation of_{(5.6) should be folded into a square-form} matrix for the solution of Re_{[J and Im [] from 72 linear equations.}

Having found the real and imaginary parts of the source strength F,, the

pressure function u can be determined from

bc24 6

[Re[FJ(fllfrnj)JRe[Q]_Im[flj,fm)]Im[G]}

77i=i ni1

+ i{Re [1(fit?frm)] Im [G] + Im [(flit'm)J Re [G]}] T(/31?Ifm) _frm' (5.15) where an approximation formula for the improper integral

r2irrr

₁

'2 = bc

_{F(fl,r)T(fl,fr)sinfrd/îdß}

_(5.16)

.10 JO

in Re [G] may be derived repeating the _{same reasoning employed in connexion} with the improper integral_{4. Here we find that}

Jim

_

(A2+B2]

= 0, (5.17)

(2a+2çf

_i

where A2 = , B2 -

-(E + c2Içb ç)1 2 (E + c2A2ç sin2 çb)l

It follows then that

If+'T _lsecrldr h2 bcF(rz, Ø) sin T(a, ç»

-

(E + e2 sin2 T sin2 q))

¡cscrdr

±1_J (E+c2sin2rsjn2ç»lj where (5.14)

+D,{'

:=i j=i

R,

[E1, + c2(fr5 ç»2 sin2Ø]J T(fi1 fr,) sin (5.18)

B2d86'dr = 0, _{since- B2(n'+r)}

_{= B2(r).}

- ¡(r)

r22T r'

JO Jo

Again, the remaining part of the integral_{4 taken over S - can be approximated}

by use of Simpson's rule. We remark here that the use of the symmetry (5.14) facilitates the evaluation of the pressure function u by (5.15) for a given mode

30

=

a

FICuBE 2. Dampingcoefficiont for surging (or

swaying) spheroids N =ÑJpo as a functionof a = 4. parameter more poin the def1niti a corresp011 by for ellips kerne the o mere requ tatid 440 W.D.Kim

6. ResultS

and discUSSiOn

In this paper, the threedimefl5i01' problem was solved for a

half-ellipsoid having beam tolength ratio = ;and its special case, ahalfspher0id

= 1.

Consideration was given to the effectof draft by varying the half-length to

draft

ratio as H = 4,2,1 and } for the spheroid,

H = 4 and 2 for the ellipsoid. Furthermore, the twodimeflsi0n5 problemwas solvedfor a halfcylinder

of an elliptic cross-section having half-beam to draft ratios

a/ , H = 4, 2, 1 and . The computation was generally performed with the values of the frequency

H 1/2. _Spheroid.

A

half-length

H--j

M o

4 o

PIoB 1. Added massfor surging (orswaying) spheroids M as a function of a = o2/g.

-Osciflation. of a riid body on a/ree snrface 441

parameter a = 0, 0.10, 0.25, 050, 0.75, 1.0, F5, 20, 25, 3.0 and 35 (except when more points were necessary due to a rapid change of curvature). According to

the definition of the frequency parameter the minimum and maximum values of a correspond, respectively, to the cases in which the length of the wave generated

25 20 M1, 15 10 O5 o

i

a

Fxom 3. Added mass for heaving spheroids M1,= as a function of a = o2/g.

2

a

FXGiJRE 4. Damping coefficient for heaving spheroids N,, =

as a function of a = o2/g.

by forced oscillation is approximately equal to 30 and to i times the length of the

ellipsoid (or the beam of the cylinder). From the asymptotic behaviour of the

kernel of the integral equation, it can be seen that as the value of a becomes large

the oscillation of the kernel grows rapidly. For this reason, if the value of a is increased beyond the present range, a quite large number of pivotal points are

required. Therefore, higher frequency results could be obtained with more

compu-tational labour, but it would be profitable to devise a different tecimique in line

3 4

À

-

Spheroid half-length draft

It

\=co

(1961) 4/I

\',

2/1

-- :,

442 W.D.Kim

with Ursell's approach (1953). He obtained the higher frequency

asymptOtiCs

from the solution of an integral equation in which the boundary values of a wavepotefltiaI occurs asunknown.

'z

a

FIGV 5. Added moment of inertia for pitching

(or rolling) splieroids

I, = I:IP4 as afunction of a =

FlousE 6. Dampingcoefficient for pitching (or rolling) spheroids Hz = H/p7a4 as afunction of a = ao2/g.

For each combinationof the parameters H and a oran ellipsoid (or H and a of a cylinder) several sets of linear equationsdescribing specific modes

of

oscilla-tion were solved. Then, from the soluoscilla-tions of the linear equations, the pivotal values of the pressure were determined. Subsequently, summation of the

real

and imaginary parts of these pressures over the immersed surfaceby Simpson's

rule yielded thehydrodynamic quantities. In figures 1-6, the

dependence of the

hydrodynamic quantities on the frequency parameter afor the spheroids having various drafts arepresented. M and N represent the normalized added mass

and associa added mass The having 13-18) paraine added represe dampi

0-06 004 0-02 0-03 O-02 002 O O a

FIGuRE 7. Added mass for surging ellipsoids M = as a function of a = ão2/g.

FIGURE 8. Damping coefficients for surging ellipsoids .N = Ñ/poa as a function of a = cr2/g.

The dependence on the parameter a of the same quantities for the I ellipsoid

having various drafts is presented in figures 7-12. The last set of figures (figures

13-18) elucidate how two-dimensional hydrodynamic quantities vary with the

parameter a and with the draft. Here and N> represent the normalized

added mass and associated damping coefficient for sway, and I?> and B-' represent the normalized added moment of inertia and associated angular

damping coefficient for roll.

E11isoid Beam/length = 1/4 haif-length H_ Ellipsoid

BIlen:ls%s%%

Oscillation8 of a rigid body on a free sirface 443

and associated damping coefficient for surge, 4 and I'Z, represent the normalized added mass and associated damping coefficient for heave, and 1 and represent

the normalized added moment of inertia and associated angular damping

coefficient for pitch, respectively.

4 2 a 3 4 3 2 O

444 W.D.Kim

In figures 3 and 4, and figures 15 and 16, variation of M and N, forspheroids

and that of

and for cylinders are shown. The problem of a heaving

cylinder was first worked out by Ursell (1949 a). He made use of a wave potential which consists of a set of non-orthogonal polynomials and a suitable point source

03 02 o 015 0 10 00 t

i

The

and iß

previol a

FIGURE 10. Damping coefficient for heaving ellipsoids N,, = Ñ,,/paa

as a function of a = o2/g.

at the origin on the physical ground that the forced oscillation of a rigidbody

produces a standing wave in its vicinity and a progressive wave at a large distance

from the body. tJrsell's method has been employed by Porter (1960) for the study of heaving cylinders having an elliptic cross-section. The extension of

Ursell's method to the three-dimensional problem of a heaving spheroid canalso be seen in papers of MacCamy (1954), Havelock (1955) and Barakat (1962).

label orde smal a he 'S Ellipsoid Beam/length = 1/4 haif-length

H

draft

.----

4/1-2/l Ellipsoid Beam/length = 1/4 o 2 3. 4

FIGURE 9. Added mass for heavingellipsoids M,, = as a function of a = aTh2/g.

a

-Oscillaion8 o/a rigid bod7/ on a/ree 8ur/ace 445

The broken curves in the two sets of drawings, figures 3 and 4, and figures15

and 16, indicate the present results, which indicate good agreement with the

previous results obtained respectively by Havelock, Ursell and Porter as so

OE06 0 if 0-010 2/1 O-005 0 o .4 a

FIGuaE 11. Added moment of inertia for pitching ellipsoids 15 = as a function of a = ao2/g.

Ellisoid

Beam/length = 1/4

H=r

n(1963)

Fxou 12. Damping coefficient for pitching ellipsoids H5 =

as a function of a = o2/g.

labelled. The curves attributed to MacCamy in both sets correspond to the first-order solutions of the shallow-draft approximation applied to a rigid body with small draft (see MacCamy 1961). In effect, those represent exact solutions of

a heaving circular disk in figures 3 and 4, and of a heaving plate in figures15_and

Ellipsoid Beam/length = 1/4 half-length 'io2 draft ,-963)

.

2 a i 3 4 004 15 002 o-020 0015

446 W.D.Kim

16. We note that .M,, iV, and increase systematicaily as the draft of the body

decreases. In a limiting case H = when the draft is zero, the wave-making effect attains its maximum.

30 20 10 i

rr

11

1\

_foo+JJfoo+)dS = 2h,

3 4 a

FIGURE 14. Damping coefficient for swaying cylinders = as a function of a = aTh/g.

The low-frequency asymptotics of three-dimensional potentials can be shown (Kim 1964) as

lirnu(x,y, z) =

_{ff f00(+) dS+iO}

(6.1)

where is the solution of

Hence, the li becomes a no nary part of frequency as t Prof. Ut (6.2) and (6. malization u

4

Cylinder half-beam

H

Draft

7%.

'2

\

Ordinate X 10

1/24

4/ H= 1/1 0 1 2 3 4. a

FIGURE 13. Added mass for swaying cylinders M = as a function of a = 15 Cylinder 10 ¡ 1/2 05 _{Ordinate x 10} 2/1 4/1

OscillaiOfl3 of a rigid body on a free 8urface 447

I

_'t

.0 '1 tt

,

MacCamy(1961) Porter (1960) t _{Porter 1960)}

\

Ursell 1949)

t t'

_t 3 2 Cy1nder haif-Iength H _draft H= 2/1 ou o a

Fiou 15. Added mass for heaving cylinders

M =

as a function of a =ao2Jg.

2

a

3

FIGURE 16. Damping coefficient for heaving cylinders = Ñ/poä5

as a function ofa = o2/g.

Hence, the limiting value of M which was evaluated using the real part of u)(0) becomes a non-zero constant, while the limiting value of which uses the

imagi-nary part of 'u(3)(0)vanishes. For a heaving sphere Ursellt found that the high-frequency asymptotics depend upon the potential

_j3a_1e_iar_e' as r-+,

(6.2)

t Prof. Ursell provided the author with his high-frequency results. Here, empressions

(6.2) and (6.3) were obtained by converting Prof. Ursell's results according to the nor-inalization used in the present paper.

., 4

448

and it follows that

We remark that as the present computation ranges up to the wavelength equal to the length of the sphere 2, the high-frequency asymptotics (6.3) cannot be

0'3 02 0 015 0 10 flS) 005 W.D.Kim 97T

and hmN,--.

_a O a

Fmm 18. Damping coefficient for rolling cylinders H = H/po as a function of a = o2/g.

joined to the present results. However, we find that in the range of the parameter

a = 3.0 4.0 the result given by (6.3) lies slightly above the present M, (II = 1) curve. (6.3) I In figur potential i

Notetha'

Next, and 14, associate of the pa i By the u mapping (1961) o ratio agreeme

H = i ii

a = 075, shown b that as t

N and i

magnit.0 i the body be create 29

f

H = 4/i CyIi'nder

H half length_draft

r-1/2ordinatex10

- H=4/1

Cylinder ordinate o i 2 a

FIGuRE 17. Added moment of inertia for rolling cylinders 4 = as a function of a = ao2/g. 2 3 4 where the limit' becomes to the fac the two- s behavioii exhibited estimatifl dimensio i frequenc and it fol

Oscillagion8 of a rigid body_{on. a free .urface 449} In figures 15 and 16, it should be _{observed that due to the property of the}

potential u(2)(0), which also can be shown as

ffoo(

u + In m') dC+ iO if _{is an odd function of z,} u(2)(0) = limu(x,y) _{= (6.4)} a'O

O(lna)_iffdC

if f is an even function ofz,

where f is the solution of the_{two-dimensional equation}

foo+ffoo(+int')dc=:

2h,

the limiting value of _{becomes logarithmically infinite}

while that of .2Vj2) becomes a non-zero constant. Havelock has attributed the infinite valueof M,2>

to the fact that when a_{= O, the condition at the free surface, i.e. u, = O, makes} the two-dimensional problem indeterminate. We emphasize that the different

behaviour of the lower asymptotics in the two- and three-dimensional problems exhibited by (6.1) and (6.4) deprives_{the 'strip method' of beinga useful way of}

estimating the three-dimensional hydrodynamic quantities from the two-dimensional data. For _{a circular cylinder IJrsell (1953) obtained the}

high-frequency asymptotics as

u21(co)

i4a2 eia

eay+iax _as

_{¡xJ -*,}

(6.5)

2

2\

. 16

lim.Ll4)...

---;, and hmN--.

(6.6)

a-+ - a a+ a

Note that the M>(H_{= 1) curve in figure 15 lies slightly below that given by (6.6).} Next, let us look at the two sets of drawings, figures 1 and 2, and _{figures 13}

and 14, which present, for the sway mode, the normalized added mass and

associated damping coefficient of spheroids and those of cylinders_{as a function}

of the parameter a. In the_{case of a spheroid, sway and surge are the same mode.} By the use of Ursell's method, which can be used conveniently whenever the

mapping of the cross-section of a cylinder on a circle is known explicitly, Tasai (1961) obtained the added mass coefficient k2 =_{[2H/iî]M2) and the amplitude}

ratio _{= a[Nfl1 up to the value of a = 15. The present results}

are in complete

agreement with Tasai's results. In figure 1 we observe that _{for the sphere}

H = 1 increases from the initial value 106 to the peak value

_{137 at about}

a = 0.75, falls to O32 at a = 350, then gradually _{rises to the final value 0.57 as}

shown by Macagno & Landweber _{(1960). From (6.1) and (6.4) it can be noted}

that as the frequency tends to_{zero, M and M1 become non-zero constants while}

.& and N2 vanish. For_{sway, the hydrodynamic quantities decrease}

in their

magnitudes with the.reduction of draft. Thus, in the limiting case H = when the body form is a disk, as it is clear from a physical reason, no disturbance will

be created by the mode of_{sway on the free surface.}

29

and it follows that

Fluid Mech. 21

450 W. D. Kirn

The variations of added moment of inertiaand angular damping

coefficient of

spheroids and cylinders with rolling frequency are seen in figures5 and 6, and figures 17 and 18. Since roll and.pitch are the same mode for a spheroid, the

broken curves in figures 5 and 6 indicateadded moment of inertia and associated damping coefficient of a rolling or pitchingcircular disk. We note that either a fiat

or a thinbody form produces a large wave-makingeffect in the mode of roll.

TsBLE 1. Strip methodapproximatiO1

It is clear from (6.1) and (6.4) that the hydrodyflamic quantities associated with roll behave in the same fashion asthose associated with sway in a lower

frequency range. Making use of thecylinder data, the values ofthe normalized

added mass Mt,, damping coefficient N, for heave, and the normalized added moment of inertia I and damping coefficient H for roll, are computed by the strip method formula(4.10). The estimatedresults are shown togetherwith the actual values in table i - The stripmethod does not give a satisfactoryestimate for the threedimeflsi0flaI problem. Beyond the value of a = 2-00, estimated

values of I and Ll approach actual values. Forellipsoids, estimated values of

M7, and N, also become close to theactual value after passing the value a = 2-00.

The autb study, andt for valuabl encouraged performed I B.&s.A.XAT, R 13, 540-HAVELOCK, Proc. B

Jom,F. li

JoB:N,F. 1 W. D. KIM, W. D Scienc KOTCmN, free s TIO. 4( MACAGO horizo McCAMz, Seriee McC&MY 5, 3 PoRTER, cylin Berk TA5AI, F oscill 9, 91 URsELL, Qua URSELI-, Mec URSELL, ,Soc. WEB:AtJS a M5 For spheroid H = M; 2 N,, N; I: H 0-012 H 0-072 0-50 1-65 i.00 1-22 1-50 104 2-00 0-96 2-50 0-93 3-00 091 1-96 1-48 1-41 1-41 1-42 1-45 0-88 0-79 0-63 0-49 0-38 OE24 2-28 1-51 1-10 0-83 0-01 O-46 0-141 0-137 0-112 0-093 O-082 O-075 0-206 0147 0-113 O-094 \ O-084 O-077 O-044 -o58 0055 O-048 O-040 O-150 0-096 O079 0065 O-053 For ellipsoid 1-00 o-128 1-50 0-098 2-00 0-080 2-50 0-072 3-00 0-070 = , H 0-090 0-075 0-073 0-073 0076 2 0-094 0-084 0-068 0-052 0-039 0-138 0-099 O-072 0-052 0-038 X 10' 0-120, 0-115 0-088 0064 0-052 X 10' 0-137 0-090 0-071 0-063 0-058 X 10_1 0-019 0-049 0-066 0-060 0-049 X 10 0-074 0-088 0-077 0-060 0-049 For spheroid H = i 0-50 1-25 1-00 0-91 1-50 0-82 2-00 0-81 2-50 0-83 3-00 0-87 1-40 1-28 1-31 1-41 1-52 1-63 0-72 0-53 0-35 0-24 0-15 0-12 1-89 1-00 0-60 0-34 0-20 0-14 0 0 0 0 For ellipsoid 1-00 0-174 1.50 0-138 2-00 0-113 2-50 0-097 3-00 0-087

= ,H =

0-121 .i00 0-091 0-085 0-080 4 0-116 0-115 0-106 0-093 0-080 0-173 0-142 0-118 0-099 0-084 0 0 0 0

Oscillation8 of a rigid bod1, on. a free eurface 451

The author is grateful to Prof. R. C. MacCamy whose suggestion initiated this study, and to Prof. H. Ashley, Prof. P. Ursell, Dr M. E. Graham and Dr J. F. Price

for valuable discussions. The author is also indebted to Dr T. E. Turner who encouraged the present research, and to Mr W. Cook and Mr D. Peterson who performed the numerical work on the digital computer.

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13, 540-56.

HAvocx, T. 1955 Waves due to a floating sphere making periodic heaving oscillations.

Proc. Roy. Soc. A, 231, 1-7.

Jo, F. 1949 On the motion of floating bodies. I. Comm. Pure Appl. Math. 2, 13-57.

Jomî, F. 1950 Ori the motion of floating bodies, U. Comm. Pure Appi. Math. 3,45-101.

KIM, W. D. 1963 On the forced oscillations of shallow draft ships. J. Ship Rea. 7, 7-18.

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5, 34-43.

PORTER, W. R. 1960 Pressure distributions, added mass and damping coefficients for

cylinders oscillating in a free surface. Tech. Rep., Series 82, Issue 16, Inst. Enng Res., Berkeley, California.

TASAr, F. 1961 Hydrodynamic force and moment produced by swaying and roIling

oscillation of cylinders on the free surface. Rep. Res. Inst. Appl. Mech., Kyuhu Univ., 9, 91-119.

URSELL, F. 1949a On the heaving motion of a circular cylinder on the surface of a fluid.

Quart. J. Mech. AppZ. Math. 2, 218-31.

URSELL, F. 19496 On the rolling motion of cylinders in the surface of a fluid. Quart. J.

Mech. AppI. Math. 2, 335-53.

URSELL, F. 1953 Short surface waves due to an oscillating immersed body. Proc. Roy.

Soc. A, 220, 90-103.

WEHAUSEN, J. V. & LAIToNE, E. V. 1960 Surface waves. Encycl. Phys. 9, 446-778.