__________ *

Lab. v Scheepsbouwkunk

J. Fluid Mech. (1965), vol. 21, partS, pp. 427-451 'Technische 'HogechooIPrinted in Great Britain

DeRtOn the harmonic oscillations of a rigid body

on a free surface

By W. D. KIMBoeing 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 potentialproblem 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 anellipsoid. The use of Green's function reduces the determination of the potentialto the solution of an integral equation. The integral equation is solved numericallyand the dependency of the hydrodynamic quantities such as added mass, addedmoment of inertia and damping coefficients of the rigid body on the frequencyof the oscillation is established.

1. Introduction '

A fluid motion eaused by small prescribed oscillations of a rigid body on thefree surface of an incompressible inviscid fluid is studied in this paper. The fluidis assumed to occupy a space bounded by the surface of the body and by the freesurface extending in all directions. The induced motion of the fluid in this spaceinteracts with the oscillating body and exerts the dynamic pressure on theimmersed surface. A point of interest here is to find the effects of such pressureon 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 belinearized by neglecting high-order terms in view of the smallness of the motionsinvolved. It is well known then that the poténtial which satisfies the linearconditions on the boundaries in theundisturbed position and the proper physicalcondition at infinity can be determined uniquely. Nevertheless, such a potentialdepends on the mode and frequency of the oscillation as well as the form. ofthe body.

For a body of general form Kotchin (1940) and John (1950) showed that thesolution of the problem can be represented as a potential corresponding to asurface distribution of point sources. The strength of the sources is to be deter-mined from a Fredhoim integral equation which satisfies a prescribed kinematiccondition (appropriate to a specific oscillation) for the potential on the bodysurface.

We present a procedure for the approximate solution of these integral equa-tions. In two-dimensional problems the kernel of the integral equation iscontinuous, but in three-dimensional problems it becomes singular; the handling

428 W.D.Kim

of the singular term is one of the major problems of the present work. Thisprocedure is valid for any shape which can be described analytically.

Once the equations are solved, the dynamic forces Fd and moments Gd on therigid body can be obtained by numerical quadrature. For instance, in three-dimensional problems resolving these into a component in phase with theacceleration 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 oscillationsO(t) = Re [O(t) e°'J,

where p represents the density of the fluid, is the half-length of the body, ando. is the frequency of oscillation. The dimensionless quantities M and N arecalled the added mass and linear damping coefficient, and similar quantitiesI and H are called the added moment of inertia and angular damping coefficient,respectively. For the case of two-dimensional problems the dynamic forces Fand moments per unit length can be written in the same form with and a4replaced 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. Inthis paper we present results for simple shapes, ellipses in two dimensions andellipsoids in three dimensions. In order to gain some insight into the validity ofthe present method, the results are compared with the previous results obtainedby 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 figures15 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 surgingellipses 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 arecompared with the corresponding curves in figures 3 and 4. It was noted thatthe present results are in complete agreement with those of Havelock. It wouldseem, on the basis of the limited evidence available, that the present procedureyields accurate results.

Nevertheless, there is a fundamental limitation to the present numericalmethod. The kernels of the integral equations oscillate rapidly as the parametera increases. Therefore, unless many subdivisions are used, the numericalquadratures occurring in the present procedure become inaccurate. In thispaper, we have limited the computations to values of a less than four, whichcovers the range of practical interest. For instance, in the three dimensions, itmeans that we have waves which are not much shorter than the length of therigid body. Higher frequency results can be obtained with more computationallabour.

The finaladopted foxsuch as theellipsoids htwo-dimernellipsoids isapproximal

2. GenerConsider

of gravity esystem Oxand we takoscillationsequilibriuix

the surfaceall directio:

The posiat any instof gravity:linear cornangular coAssumingmotion di

may be inrequires t

where V isAs the a

wave motithe linearifree surfac

where 7 reThen the i

with k delength of

i Oscillatjon8 of a rigid body on a/ree snrface 429The final objective of the present study is to assess the merit of various modelsadopted for the investigation of ship form. At first, the hydrodynamic quantitiessuch as the addedmass, added moment of inertia, and damping coecients of theellipsoids having different axis ratios are estimated by the strip theory, using thetwo-dimensional data of a long cylinder. Next, the effect of the draft on theellipsoids is examined in order to evaluate the applicability of the shallow-draft

approximation (see Kim 1963).

2. General formulationConsider a rigid bodyimmersed in an inviscid, incompressible fluid with itscentreof gravity on the origin O, and its axes on the (, )-plane of a space co-ordinatesystem Offi. The ()-plane here coincides with the undisturbed free surface,and we take the i-axis positive upwards. If the body is given linear and angularoscillations of small amplitudes X° and 00 with a certain frequency o about itsequilibrium position, VIZ.

(t) = Re [0 e1°"],1).

(2.1)0(t) = Re[00e-ioIJ,Jthe surface disturbance created by these motions travel outwards as waves inall directions.

The position of the immersed surface S(, ) relative to the space co-ordinatesat any instance can now be expressed by specifyinga position vector of the centreof gravity 5 = ¡X +112 + k13, and the Eulerian angle O = ¡04 + 105 + ko6. Thelinear components X, I and Ï are called surge, heave and sway, while theangular components 04, 05 and 6 are named roll, yaw and pitch, respectively.Assuming the fluid to attain a time-periodic irrotational motion when transientmotion dissipates, a velocity potential

= Re[V(,y,)e-íuiJ (2.2)may be introduced to describe the state of the fluid. The incompressibility thenrequires 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 inducedwave motion will also be small in comparison with the wavelength. Therefore,the linearized dynamic condition for the velocity potential at the undisturbedfree surface becomes0, (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 thelength of free wave.

430 W.D.Kim

The kinematic conditiön which states that in the absence of viscosity thenormal velocity across the immersed surface of the body is continuous takes the

form(2.6)

where 1 Ñpresents the position vector of a material point on the body and n, theunit 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 harmonicoscillations 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 space such 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) r4 euU+iar - O as r. ,

(2.11)

(2.9)

(2.10)

where S(x, zposition S(adepends oniare given, r

h,(x, y, z)

h4(x,y,z)

In the eaproblem tonormal to tito this piar

where C(x)position; C

Here we renow correproblem,

3. RepreThe so

the sets ofin the fo

G(x,y

G(x,y;g

where

and S0(afunctionsand Cias

O8cillation8 of a rigid body on. a/ree 8urface 431

where S(x, z) represents the immersed surface of the body in. the undisturbedposition S(x, y) = ay), and h5 denotes the prescribed function whichdepends 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 theproblem to be two-dimensional; in effect, a solution for an (, )-p1ane which isnormal to the axis of the cylinder is valid for all planes cutting the axis parallelto this plane. Therefore, the boundary value problem is to find a potentialu1(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 undisturbedposition; 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 3now correspond to the case of sway,' heave and roll in the three-dimensionalproblem, respectively.

3. Representation of the potentialThe 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 expressedin 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-)]

ro2a 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 Besselfunctions 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 potentials G are more tractable in the present form than in other

expressions using Cauchy's principal-value integrals (see, for example, Wehausen

& Laitone 1960).We now seek the solution of the boundary value problems (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 strength of 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 of the 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 equationi tC 0Gf5(x,y,z)+-- fl f,n,C)dS = 2h5(x,y,z), (3.7)1Tjjit 0G

or _f5(xy)+Jfi(C,n)dC = 2h5(x,y), (3.8)

ti5 will satisfy the boundary condition (2.11 C) or (2.13 c). Accordingly, u is the

solution of the given problem. In order that these integral equations 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 potential and its normal

derivative at the proximity of 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'-' - 2a ea4i)[ln [a(R' - y - n)]

f0+ ir[S0(a) + N0(a) - i2J(a)] + (e 1) p2 w2)4 df11+'?

4

where N1,(a

Next in (3

i.e. V7-+O,

CiaIxt

where 'yagain can

with

Let u(3.10) an

and

In (3.

terms ibecomedistancpresentIx_LItof curv(3.15).

Thusintegra

4. FoThe

immein phaof theof mecoeffic

Wesmall

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 asum [Ciajx_I_(y+h1aIX-Ifl = 0, (3.12)

where y is Euler's constant. Here the opposite signs on the logarithmic termsagain cancel singularities so that we have

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

withQ*(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 theterms in (a/an) G* which contain powers of reciprocal distanc such as vr' or vr2becomes singular. However, when w = O the multiple factors of the reciprocaldistances vanish simultaneously so that these reciprocal distance terms do notpresent 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 radiusof 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 theirintegrands have to be evaluated using the special scheme.

4. Forces and momentsThe forces and moments caused by the dynamic fluid pressure acting upon the

immersed surface of an oscillating rigid body may be resolved into componentsin phase with the acceleration and other components in phase with the velocityof the rigid body. The former quantities are called added mass or added momentof inertia while the latter quantities are called linear or angular dampingcoefficients.

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

28 Fluid Moch. 21

434 W.D.Kim

a fluid. Since the form of an ellipsoid is symmetric about the space axes lying on

the free surface, from the result presented by John (1949), the forces F and

moments G can be deduced asF( = 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 x n)dS,

and S represents the surface given by= (1_/ä2_2/b2), < 0.

By (2.1) and (2.10), Fd and Gd can be rewritten in terms of the dimensionless

space variables and amplitudes as

Fd(t) = pg Re 5f au5(x, , z) n dS e_10f] (j'= 1,2,3),

Now, expressing the dynamic forces and moments with a component in phase

with the acceleration and that in phase with the velocity of the ellipsoid, 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 dampingcoefficient, 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 dampingcoefficient 2h, which are employed by Ursell (1949a) and Havelock (1955) as

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

where m represents the actual mass of fluid displaced by a rigid body.Ifa rigid body is a cylinder, the force F2 and moments G(2) arising from small

forced oscillations are given byF(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 C112,jforces

and

and iveloc

and

we Ocoe

I1modsaresprecoeateAp

and Ge(e) = pfa4 Re [OJO5J auj(x, y, z) (r x n) dSet] (j = 4,5,6).

I

I

M2 1f s A iM2) = = u(x, i) (in2 +jn) dCj.

N(2) = 2 = Im[fui(xY)(mn2+În)dC] (j = 1,2),j(2) rr i

= = Re [j u3(x, y) (xn - yn2) dC]pa

= = im ¡f u3(x, y) (xn yn2) dC].

It is often necessary to estimate the three-dimensional physical quantitiesknowing only the two-dimensional results. For instance, using the hydro-dynamic quantities of the cylinder (4.9), an attempt can be made to evaluate thesame quantities of the ellipsoid (4.4). A method of relating the two-dimensionalresults to the three-dimensional estimates is called the 'strip method'. In thepresent problem, the added mass, Mf,, added moment of inertia, I, and dampingcoefficients, N, and H, ofan 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, and11),J - 1, 2, 3, is the two-dimensional dynamicpressure. Rewriting the dynamicforces and moments in the form

= pgRe [Xfaui(xY)(înx+În)dCe_iui]

and = pga3 Re {O f au3(x, y) (xn, - yn2) dCe_101],

and identifying the component either in phase with the acceleration or thevelocity 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 dampingcoefficients for a two-dimensional problem as

(j= 12)}(4.7)

436 W.D.Kim

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

presently how the estimates obtained by (4.10) compare with the corresponding

three-dimensional data.,

5. Numerical procedureWo are ultimately concerned with the solution of equations (3.7) and (3.8) for

arbitrary values of a. Here (3.7) deals with an ellipsoid x2/ä+y_2/C_2+2/2 = 1,

< 0, which has length, beam, and draft of 2, 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-ordinatesx=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 typical length of

the body .

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

formbc 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 + sin2 a/b2) sin2 ç5 + cos2

and E(0) = (sjnse+b2cos2O)k.(5.5)

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

equation is solved in an analogous manner. The method is based on Fredhoim'sprocedure of replacing the integral equation with a finite set of linear equations.

Suppose the fi- and fr-axes which bound the region of integration S(fi, sfr) are.

divided, respectively, into 24 and 6 equal divisions. Then, a lattice can be formed

on S(fi, %fr) by connecting the 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 complexfunction, equation (5.3) can be separated into a pair of equations for the real and

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

. raGi FT- Ljm{fi1m)ImLx 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 hdistance belintegral. N

We turn

and n(a,and (fi,in its pro.

where

It follo

Thus, opossessepart wh

where

with

with

where rit 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 thedistance between a point source and a variable point, in Re [00/On] is an improperintegral. Nevertheless, this integral can be shown to exist.

We turn to the evaluation of the following integral

2i ITI=bcf Lfl,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 pointin its proximity. Then, by Taylor's expansion, we may writei

(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 + ¿2ç5where A1 =2 (E+c22sin2ç»F

B13(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)

r81hrnLOflR

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 indeterminatepart which depends upon the angle of approach.

438W.D.Kim

By (5.7) and (5.9), the part of the integral 1, taken over the neighbourhood

element , ear be approximated as

r2n r4(r) IAbcsin$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 sin2 7 Sifl2 ç»

11r lcI (cos2rsin2+sin2r)d7]+1

.)1t (E+c2sin2rsin2)t33

+ i1j1(fi.a)2 sin2 5+(fr5ç5)2] T(fi'çfr1)

[E1 + c2(1 )2 3j2 jSfl

where the 8-integration is carried out 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 remaining part of the integral I taken over 5 - can be approximated by

use of Simpson's rule.Application of the formula (5.12) and Simpson's rule enables us to write out

the double summations in (5.6) as a linear combination of Re [] and Im [F so

that we canobtain a finite set of linear equations for each mode of oscillation.

By (5.1), the right-hand members for the six degrees of freedom are obtained

from (.12) as

H Acos a sin çiS H

cos çb H Asin a sin çf.

- T(a, ç) ' 2 cT(a, ç»'3(a, bT(a, ç5)

Hc - b2 sin a sin ç6 cos H

b2 1 sin a cos a ç5 5 13

- T(a,çi5)' b T(a,çS)

)

1c2coscxsinç5cosç5and ll(a,ç5)= T(a,çL)

Here the strength of source .1 in the problem dcpends not only UOfl the mode

and frequency of the 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 the corresponding symmetry.However, duo to the trans-

(5.12)

formation olthe variable

p

I

On the basequationsmore, relatproducedmatrix fo

Havingpressure f

where a

in Re[Üwith th

where

It foil

[U

wher

Agniby ufacilof 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 ofthe 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 linearequations (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 matrixproduced by the double summation of (5.6) should be folded into a square-formmatrix for the solution of Re [J and Im [] from 72 linear equations.

Having found the real and imaginary parts of the source strength F,, thepressure 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 integralr2irrr 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 connexionwith the improper integral 4. Here we find that

Jim_ (A2+B2] = 0, (5.17)

(2a+2çf iwhere A2 = , B2 - -(E + c2Içb ç)1 2 (E + c2A2ç sin2 çb)l

It follows then thatIf+'T lsecrldrh2 bcF(rz, Ø) sin T(a, ç»

- (E + e2 sin2 T sin2 q))

¡cscrdr±1J (E+c2sin2rsjn2ç»lj

where

(5.14)

+D,{':=i j=i R, [E1, + c2(fr5 ç»2 sin2 Ø]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 approximatedby 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 modeof oscillation.

30

=

a

FICuBE 2. Damping coefficiont for surging (or swaying) spheroids N = ÑJpo

as a function of a =

4.

parametermore pointhe def1nitia corresp011

by forellipskernethe omererequtatid

440W.D.Kim

6. ResultS and discUSSiOn

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

having beam to length ratio = ; and its special case, a halfspher0id = 1.

Consideration was given to the effect of 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 problem was solved for 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-lengthH--j

M o

4o

PIoB 1. Added mass for surging (or swaying) 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 whenmore points were necessary due to a rapid change of curvature). According tothe definition of the frequency parameter the minimum and maximum values ofa correspond, respectively, to the cases in which the length of the wave generated

25

20

M1, 15

10

O5o 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 theellipsoid (or the beam of the cylinder). From the asymptotic behaviour of thekernel of the integral equation, it can be seen that as the value of a becomes largethe oscillation of the kernel grows rapidly. For this reason, if the value of a isincreased beyond the present range, a quite large number of pivotal points arerequired. 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- Spheroidhalf-length

draft

It\=co

(1961)4/I

\', 2/1

-- :,

442W.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 as unknown.

'z

a

FIGV 5. Added moment of inertia for pitching (or rolling) splieroids

I, = I:IP4 as a function of a =

FlousE 6. Damping coefficient for pitching (or rolling) spheroids

Hz = H/p7a4 as a function 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 equations describing specific modes of oscilla-

tion were solved. Then, from the solutions 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 surface by Simpson's

rule yielded the hydrodynamic quantities. In figures 1-6, the dependence of the

hydrodynamic quantities on the frequency parameter a for the spheroids having

various drafts are presented. M and N represent the normalized added mass

and associaadded mass

Thehaving13-18)paraineaddedrepresedampi

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/poaas a function of a = cr2/g.

The dependence on the parameter a of the same quantities for the I ellipsoidhaving various drafts is presented in figures 7-12. The last set of figures (figures13-18) elucidate how two-dimensional hydrodynamic quantities vary with theparameter a and with the draft. Here and N> represent the normalizedadded mass and associated damping coefficient for sway, and I?> and B-'represent the normalized added moment of inertia and associated angulardamping coefficient for roll.

E11isoidBeam/length = 1/4

haif-lengthH_

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 normalizedadded mass and associated damping coefficient for heave, and 1 and representthe normalized added moment of inertia and associated angular dampingcoefficient for pitch, respectively.

42

a3

432O

444 W.D.Kim

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

and that of and for cylinders are shown. The problem of a heavingcylinder was first worked out by Ursell (1949a). 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

Theand ißpreviol

a

FIGURE 10. Damping coefficient for heaving ellipsoids N,, = Ñ,,/paaas a function of a = o2/g.

at the origin on the physical ground that the forced oscillation of a rigid body

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 can also

be seen in papers of MacCamy (1954), Havelock (1955) and Barakat (1962).

labelordesmala he

'S EllipsoidBeam/length = 1/4

haif-lengthH draft

.----4/1-

2/l

EllipsoidBeam/length = 1/4

o 2 3. 4

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

a

o 1 2

-

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 figures 15and 16, indicate the present results, which indicate good agreement with theprevious results obtained respectively by Havelock, Ursell and Porter as so

OE06

0

if 0-010

2/1

O-005

0o

.4a

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

EllisoidBeam/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 withsmall draft (see MacCamy 1961). In effect, those represent exact solutions ofa heaving circular disk in figures 3 and 4, and of a heaving plate in figures 15 and

EllipsoidBeam/length = 1/4

half-length

'io2

draft

,-963) .

2

ai 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 bodydecreases. In a limiting case H = when the draft is zero, the wave-makingeffect 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 libecomes a nonary part offrequency as

t Prof. Ut(6.2) and (6.malization u

4

Cylinderhalf-beamH 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 =

15Cylinder

10

¡ 1/2

05 Ordinate x 102/1

4/1

OscillaiOfl3 of a rigid body on a free 8urface 447

I't.0 '1tt , MacCamy(1961)

Porter (1960)t Porter 1960)\ Ursell 1949)t t't

3

2

Cy1nderhaif-Iength

H draft

H=

2/1

ouo

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/poä5as a function of a = 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 equalto 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

97Tand hmN,--.a

O

a

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

joined to the present results. However, we find that in the range of the parametera = 3.0 4.0 the result given by (6.3) lies slightly above the present M, (II = 1)curve.

(6.3)

I

In figurpotential i

Notetha'Next,

and 14,associateof the pa iBy the umapping(1961) oratioagreemeH = i iia = 075,shown bthat as tN and imagnit.0 ithe bodybe create

29

f

H = 4/i

CyIi'nder

Hhalf length

draft

r-1/2ordinatex10

- H=4/1Cylinder

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'becomesto the facthe two- sbehavioiiexhibitedestimatifldimensio ifrequenc

and it fol

Oscillagion8 of a rigid body on. a free .urface 449In 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 of z,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 value of M,2>to the fact that when a = O, the condition at the free surface, i.e. u, = O, makesthe two-dimensional problem indeterminate. We emphasize that the differentbehaviour of the lower asymptotics in the two- and three-dimensional problemsexhibited by (6.1) and (6.4) deprives the 'strip method' of being a useful way ofestimating 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\ . 16lim.Ll4)... ---;, and hmN--. (6.6)a-+ - a a+ aNote 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 13and 14, which present, for the sway mode, the normalized added mass andassociated damping coefficient of spheroids and those of cylinders as a functionof 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 themapping 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 amplituderatio = a[Nfl1 up to the value of a = 15. The present results are in completeagreement with Tasai's results. In figure 1 we observe that for the sphereH = 1 increases from the initial value 106 to the peak value 137 at abouta = 0.75, falls to O32 at a = 350, then gradually rises to the final value 0.57 asshown by Macagno & Landweber (1960). From (6.1) and (6.4) it can be notedthat as the frequency tends to zero, M and M1 become non-zero constants while.& and N2 vanish. For sway, the hydrodynamic quantities decrease in theirmagnitudes with the.reduction of draft. Thus, in the limiting case H = whenthe body form is a disk, as it is clear from a physical reason, no disturbance willbe created by the mode of sway on the free surface.

29

and it follows that

Fluid Mech. 21

t

450W. D. Kirn

The variations of added moment of inertia and angular damping coefficient of

spheroids and cylinders with rolling frequency are seen in figures 5 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 indicate added moment of inertia and associated

damping coefficient of a rolling or pitching circular disk. We note that either a fiat

or a thin body form produces a large wave-making effect in the mode of roll.

TsBLE 1. Strip method approximatiO1

It is clear from (6.1) and (6.4) that the hydrodyflamic quantities associated

with roll behave in the same fashion as those associated with sway in a lower

frequency range. Making use of the cylinder data, the values of the 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 estimated results are shown together with the

actual values in table i - The strip method does not give a satisfactory estimate

for the threedimeflsi0flaI problem. Beyond the value of a = 2-00, estimated

values of I and Ll approach actual values. For ellipsoids, estimated values of

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

The autbstudy, andtfor valuablencouragedperformed I

B.&s.A.XAT, R13, 540-

HAVELOCK,Proc. B

Jom,F. liJoB:N,F. 1

W. D.KIM, W. D

SciencKOTCmN,

free sTIO. 4(

MACAGOhorizo

McCAMz,Seriee

McC&MY5, 3

PoRTER,cylinBerk

TA5AI, Foscill9, 91

URsELL,Qua

URSELI-,Mec

URSELL,,Soc.

WEB:AtJS

a M5

For spheroid H =

M;

2

N,, N; I: H

0-012

H

0-0720-50 1-65i.00 1-221-50 1042-00 0-962-50 0-933-00 091

1-961-481-411-411-421-45

0-880-790-630-490-38OE24

2-281-511-100-830-01O-46

0-1410-1370-1120-093O-082O-075

0-20601470-113O-094 \O-084O-077

O-044-o58

0055O-048O-040

O-1500-096O0790065O-053

For ellipsoid1-00 o-1281-50 0-0982-00 0-0802-50 0-0723-00 0-070

= , H0-0900-0750-0730-0730076

2

0-0940-0840-0680-0520-039

0-1380-099O-0720-0520-038

X 10'0-120,0-1150-08800640-052

X 10'0-1370-0900-0710-0630-058

X 10_1

0-0190-0490-0660-0600-049

X 10

0-0740-0880-0770-0600-049

For spheroid H = i0-50 1-251-00 0-911-50 0-822-00 0-812-50 0-833-00 0-87

1-401-281-311-411-521-63

0-720-530-350-240-150-12

1-891-000-600-340-200-14

0 0 0 0

For ellipsoid1-00 0-1741.50 0-1382-00 0-1132-50 0-0973-00 0-087

= ,H =0-121.i00

0-0910-0850-080

40-1160-1150-1060-0930-080

0-1730-1420-1180-0990-084

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 thisstudy, and to Prof. H. Ashley, Prof. P. Ursell, Dr M. E. Graham and Dr J. F. Pricefor valuable discussions. The author is also indebted to Dr T. E. Turner whoencouraged the present research, and to Mr W. Cook and Mr D. Peterson whoperformed the numerical work on the digital computer.

REFERENCESBnsxr, R. 1962 Vertical motion of a floating sphere in a sine-wave sea. J. Fluid Mech.

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.Knr, W. D. 1964 On oscillating ships in waves. Tech. Memorandum, no. 25. Flight

Sciences Laboratory, Boeing Scientific Research Laboratories.KOTCHIN, N. E. 1940 The theory of waves generated by oscillations of a body under the

free surface of n heavy incompressible fluid. Uchenye Zapiski Moskov. Gos. Univ.no. 46.

MACAGNO, E. O. & LANDWEnER, L. 1960 Amass of a rigid prolate spheroid oscillatinghorizontally in a free surface. J. Ship Rea. 3, 30-6.

MAcCAMY, R. C. 1954 The motion of a floating sphere in surface waves. Tech. Rep.,Series 61, Issue 4, Inst. Engng Res., Berkeley, Califo-r-nia.

M&cCAMY, R. C. 1961 On the heaving motion of cylinders of shallow draft. J. Ship Rea.5, 34-43.

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

29-2