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Time-Domain Calculation of the Nonlinear Hydrodynamics of Wave-Body Interaction* C. Yang, Y. Z. Liu Shanghai Jiao Tong University, Shanghai, China N. Takagi Nihon University Chiba, Japan Abstract S source point The boundary element method coupled with time-marchthg finite difference is adopted and improved to calculate the nonlinear hy- drodynamics of wave-body interaction. The radiation condition and initial condition have been studied through specially chosen examples such as the cylinder undergoing forced heave motion or forced sway motion and the body floating or standing in periodic waves with wave front, and steady solution of practical interest has been obtaind in a de- finite calculation domain by less computer time. A few comparisons are made with avail- able solution and model test results. It is concluded that the method is capable of pre- dicting forces due to nonlinear wave quite accurately with requirement of median com- puter. XG ZG H C T a DF d Nomenclature ~ horizontal velocity of the body vertical velocity of the body wave height wave velocity wave frequency or oscillating fre- quency of the body undergoing forced motion wave period oscillating amplitude of the body undergoing forced motion radius of circular cylinder mean water draft of the body still water depth radius of the outer open boundary mass density of fluid oxyz frame of reference, with z pointing 1. Introduction upward and z=0 the still water ~~~ surface In many design cases, the application of linear diffraction theory is not entirely appropriate for the prediction of wave forces on large offshore structures of general form. For 3-D nonlinear free surface problem, basi- cally there are two approaches commonly used in literature. One is based on finite dif- ference method, in which the solution of Navier-Stokes equations by MAC and its various modifications 5MAC,SUMMAC,ABMAC,IMP and re- cently TUMMAC (13~2) 0 ) are relatively po- pular. This aproach appears to have the capa- city tocope with large amplitude nonlinear waves and even breaking, but considerable further development will be necessary to be realistically used due to its high cost and need of supercomputer. The other approach is based on boundary element method coupled with finite difference time-marching method, first introduced by Longuet-Higgins and Cokelet (43 and then followed by Faltinsen (51, Vinje et al. (6), Isaacson (7), Lin et al. (8), and others. This approach is more suitable to deal with the wave diffraction of large off- shore structures for engineering use and is sb Sf n V n t At velocity potential immersed body surface free surface outer open boundary surface unit normal vector directed outward from the fluid region n=(n~,n,,nz) normal velocity of the body surface acceleration of gravity time variable time increment (x,y,t) elevation of the 3-D case (x,t) elevation of the 2-D case field point the Project Supported Partly by National Natural Science Foundation of China. 341

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adopted in this paper. The work of this paper is to solve 2-D and 3-D nonlinear problems with free surface. The wave-body interaction is treated as a tran- sient problem with known initial condition and is solved by integral-equation method based on Green's theorem. It is emphasized to deal with the following aspects through dif- ferent model problems in the paper: (a) the radiation condition. (b) the point at the junction of the body and the free surface, (c) the initial condition for a body floating or standing in periodic wave.Only after des- cribing these three aspects correctly is the solution stable and practical. Numerical calculation should be truncated at finite distance and the smaller the domain the better, while the physical domain is in- finite. Therefore a numerical radiation con- dition should be posed so as no reflected waves from the truncated surface, ie. outer open boundary. For an axisymmetrical cylinder heaving in still water, several approaches for the formulation of the radiation condi- tion have been tested, and it is found that the usual one-dimensional Sommer-feld condi- tion is the simplest and can give reasonable results both for the wave pattern and wave force. For the cylinder sway in still water, the Som~er-feld condition is extended to 2-D case and the wave direction is determined only by wave itself. The numerical results show good agreement with the tested ones done in Nihon University, Japan. For the diffrac- tion of a solitary wave upon a fixed vertical circular cylinder, the Sommer-feld condition is further extended as a radiation condition by assuming that the scattering of the outer- going wave is small at the outer open boun- dary. The numerical results is appropriate compared with Isaacson's t73 analytical solu- tion. The same works have been done for the diffraction of a periodic wave upon a fixed cylinder. In the numerical calculation, Lagrangian free surface condition is used. The point at the junction of the body and the free surface is determined by extrapolation for axisym- metrical flow and by satisfying both the con- ditions on the free surface and on the body for general 3-D flow,that is, according to the body condition we can obtain the velocity of the intersection point, by which the new position of intersection point con be deter- mined through satisfying the free surface condition. In order to shorten initial transient pro- cess, appropriate initial boundary condition should be posed. Numerical tank is good for this purpose, and it can make the vicinity of the body be still water.The another advantage of the numerical tank is capable to produce waves with any water depth and any water bottom condition. The approach of numerical tank is described in detail in the doctoral thesis by C.Yang t93. A 2-D nonlinear Stokes wave with wave front is also suitable for the initial condition. The details are described in subsequent section. 342 2. Formulation of the Problem 2.1 Basic Equations . The basic assumptions are that the fluid is inviscid, incompressible and the flow is irrotational. Select the coordinate system and computation domain as shown in Fig.1, the velocity potential ~x,y,z,t) satisfies V20(x,y,z?t)=0 in fluid domain (1) 00=0 on S (2) an d a0=v On n Do 30 Dt~a~ I Dy 30 ~ Dt by j Dz 30 1 Dt- Liz ) on Sb (3) t on Sf (4) Dt ~ 0.570 V0 on S' (5) Here the conditions on the free surface are expressed in Lagrangian form. These equations are solved with suitable initial and radia- tion conditions. 2.2 Time-marching Procedure The free surface and the velocity poten- tial on it are calculated by finite dif- ference time-marching method and they can be expressed in the form xn+1=xn+O.56t(3(3a0~) ~(~0x) ) yn+1 yn+o 5^t(3(0a0)n~(~0y) ) n+1=zn+O 5At(3(0a0)n-(~0) ) 0 =0 +O.5At(3(- ~+O.5V0 90) -(- ~+O.5V0 V0) ) (7) } (6) Where superscript n denotes the value at taunt. Eq. ( 7) can be rewritten as follows 1=F1(n,~n-1,(30)n (~)n-1) on Sf ( 8 ) Where F. is a known function. According to the mode in which body moves we can obtain ( ~ 0 ) 1 = F2 ( An, An- 1 , ( 00 ) n ( ~ ) n- 1 And 1 ) on :+1

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Where F is a known function for forced moron and it c2an be obtained by solving Newton's Law of momentum simultaneously with the solu- tion of the velocity potential 0 for freely motion, because implicit scheme is usually used in the Bernoulli's equation t9] for this case. With suitable radiation condition the velocity potential on the outer open boundary can be described as 0 =F3(0 ,0 '( ~ ~ ,ta0) ) c ( 1 0 ) Where F. is also a known function and can be obtained in section 3. Once the free surface Sf, immersed body surface Sb, outer open boundary Sc and the velocity potential On, normal velocity (30/~n) on them are known (of course before time t=n~t all of these quantities are also known), Sf+1 and 0 1 on Sf+1 can be obtained from Eqe.(6)~(7). ~ 1 and (30/3n~n+1 on <~1 can be determined in terms of the new points at the junction of the body and the free sur- face and Eq.(9) separately. SC+1 and 0 1 on Sc 1 can be determined in terms of the radia- tion condition Eq.(10). From Green's third fomula, we have (~=~ i(G(x,9 8~0(~) ~ ~ 0(~))ds (11) Where S=Sf+Sb+Sc. Both points ~ and ~ are on the boundary S. ~(~)=2~ for a smooth surface at point x. Rankine source with suitable images is used as the Green function G(x,8), which is so chosen that the bottom condition are satisfied automatically. Now the integral equation (11) can be solved numerically at time t=(~+1)At, and we can obtain (30/~n)n+1 on Sf 1 and SO ~ 0 on ~ 1. In this way we can go further to time t=(n+2) at and calculations can be ad- vanced over a sufficient duration. Once the values of the velocity potetial on the body surface are known, the pressure on the body surface can be obtained from Bernoulli equation represented in a frame of reference fixed on the body. P=-p(-~-Ve V0+0.5V0 V0+gz) (12) where ( a ~ 0/~ t)n+1=(0n+1_0n)/^t 0n+1 and On are the values of the velocity potential on the same point of the body at different time steps and ve is the velocity of the body at that point. The hydrodynamic forces on the body are calculated from the formula 3.1 Forced Motion 3. Radiation Condition (13) Assuming that ~ cylinder starts to move in still water, initial condition could be~atis- fied easily, so we specially choose the e~am- ples of forced heave motion and forced sway motion as the calculation models to find out the propriate radiation condition. In numerical calculations we use a cylin- drical coordinate system (ores) on the outer open boundary S and make r=R=const represent S . First let a circular cylinder undergo a forced heave motion in still water, here the flow is axisymmetric. Before the wave reaches the S we have c 0=0 on so (14) With such radiation condition we have tried different radius R. i.e. expanding the open boundary gradually to find out how the scale of the outer open boundary affects the nume- rical solution. Then we assume that the wave near the outer boundary satisfies Sommer-feld radiation condition t10) 3~0t+C03~0-0 on SO (15) The phase velocity C0 varies only with time on So, and it can be determined by 0 1 and (30/3n) on Sf near Sc. Then 0 1 on SO can be obtained according to C0. Eq.(15) can make the wave on SO be propagating outerwerd along r direction. Next let a ci rcular cylinder undergo forced sway mo ti on. Now the flow i ~ non-ax) symm e t ri c and Sommer-feld condition is described as at+C03l- on So (16) Here 1 is an unknown direction of outergoing wave on the outer boundary, phase velo ci ty Cp varies with time t and angle (3 and is equal at the same vertical line. From Eq.( 16) we have where 343 0(kr ~ ~ R~a6)=0 (17) kr=cos(l,]) ~ =cos( 1,6) (18)

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The solution of Eq.(17) can be expressed as 0=0(krr+\ RD-Cpt) (19) where kr=d ~~(~0r)2+(~0) )% ~ =dR80oo/t taaO' 2+( R83Oo)2 )% J1 arts )-1 if Trio (20) (21) such ~ can make the wave at outer boundary be outergoing propagating wave. Substituting Eq.(20) into Eq.~17), we have ~ dC(( ~~2+(R~0)2~%=0 (22) Up to now by Eq.~22) we can determine Cp in teems of 0~+' and (30/an~n on Sf near So, and obtain 0 +1 on So according to Cp. 3.2 Wave Diffraction Problem For the diffraction of a solitary wave or periodic wave upon a fixed or floating verti- cal cylinder, the Sommer-feld condition is further extended as a radiation condition by assuming that the scattering of the outergo- ing wave is mall on the outer open boundary. Replacing O in Eq.(22) by Is, we can also obtain 0n+1 on So, here 0~3=0-0W and 0 is total velocity potential and Ow is velocity potential of incoming wave. 4. Dencription of the Initial Condition . _ . 4.1 Radiation Problem . (23) For forced heave motion mentioned above we have Where 0(x,y,z,0)=0 on Sf It 0 (24) or, n(~tY,z~o)=(zGnz)|t=O+ on ~ It=o (25) 0~x,y,z,O)=O on Sc|t=o (26) For forced sway motion, the formulas of initial conditions on Sf and Sc are similar with Bqe.(24),(26) and that on ~ becomes an( ~ MY. Z'O)=( Sign,,) It o+ on ~ |t=o 4.2 Body Standing or Floating in Waves If the incoming wave is a solitary wave, (27) 344 as used by Isaacson t7 ~ because it decays rapidly away from its crest' the flow near the body can be taken zero as initial condi- tion. If the incident wave is periodic Stokes or Conoidal waves and the body is assumed to be stationary at certain instant (as initial time), there must be a transient period before a steady state is approached. Sometimes as reported by Vinje, Xie and Brevig 611], even numerical troubles occur. In order to formu- late the initial condition properly, a numerical tank is set up using the same pro- cedure. A 2-D cylinder with different cross- section heaving can produce the required wave profile, see Fig.2 and Fig.3. The details are in doctoral thesis by C.Yang :9). With such 2 incident wave, the solution can approach steady rapidly and initial condition can be described as 0It=o=w It=o 0 lt=o=~w IT on Sflt=0 (28) on So |t=o (29) The numerical tank can be applied to any water depth and any type of water bottom boundary. Another expresion of the incident wave is a 2-D nonlinear Stokes wave with wave front, it can be described as (x,y,t)=A(x)(~ osO 16 sh3(kd)(2+ch(2kd))cos26) (30) (x,y,t)=A(~)(2~ ch~k9d0~sinS + '42 ~ sin29) 6=k~-~, s=z+d, x_~(x-Cgt) (kick) I Cg_> 1+sh(2kd)), C=(k (kd)) ~ (32) (31) 1-exp(x+ka) if x+ka'O A(x)= ~ O if x+Ea~O (33a) 1 if x+Ea`- A(x)=~0.5(1-cos(x+ka)) if -~ OCR for page 341
for 3-D case or linearly distributed for 2-D case, as in typical boundary method, the above Green's fomula Eq.(11) becomes ~ ~j0~+ ~ Bii(~6n0)j+ ~ Ci;( ~ )~Di where i=1,2, ,NN A iaG(x ~ d Bij= ~ G(xi,~ ds Cij=Bij NB NF NC D. =-~ Bij(an)~- ~ Fiji ~ Aij0; r1 i=.i (34) > (35) i j No in j ) ~G=hOsinc~t and NN=NB+NF+NC is total number of elements (for 3-D) or nodes (for 2-D) on boundary S=~+Sf+SC, and NB on ~ ,NF on Sf, NC on SO respectively. These algebric equations can be solved either by direct or iterative method. The junction point of the body and free surface is determined in the paper by extra- polation for axisymmetric flow and by satis- fying both the conditions on the body and on the free surface for general 3-D flow, that is, according to the body condition we can obtain the velocity of the intersection point, by which the new position of intersection point can be determined through satisfying thefree surface condition. 6. Numerical Examples and Conclusion 6.1 Forced Heave Motion As an example, we consider the forced heave motion of a floating truncated vertical gy~hder of radius a and mean draft a/2. The vertical velocity of the body is prescribed to be ZG= housing with body draft H(t)= 2 hocos~t (36) (37) In order to make our computation com- parable to Lin's (8) results, we also choose that a=1, p=1 and g=1, the other initial input data for calculation is ~ =~/2, h =0.05, d=8, at=0.1.Besides the radius of the Outer boundary and the radiation condition for calculation are divided into following three gI~Up8 345 (1) {Red. Cond. Eq.(14) JR--4 7 Grad. Cond. Eq.(14) (3) {Red Cond. Eq.(15) Fig.4, Fig.5 and Fig.6 show respectively the time history of free surface profiles con- sisting with above three cases. From Fig.6 we can observe the wave reflecting from the outer boundary, Fig.7 shows the comparison of heave force of case(3) with Lin's (8], and Fig.8 shows the effect of water depth on heave forces in which the calculation method is similar to case(3). 6.2 Forced Sway Motion Let a floating trancted cylinder undergo forced sway motion. The radius of the cylin- der is a, the mean draft of it is a/2 and the water depth is a. The gravity center of the body can be described as (38) here h =0.05a and 4(a/g)%=0.8028. Comparison has been made between the calculations and experiments by Dr. N.Takaki in Nihon Univer- sity,Japan. Fig.9 shows good agreement of the results of the free surface elevations at the fixed point. Fig.10 gives the free surface profiles at some fixed time. 6.3 Diffraction Problem The diffraction problems of a vertical circular cylinder standing on the seabed and piercing the free surface by a solitary wave has been calculated. Fig.11 shows the com- parison of hydrodynamic coefficients among present results, Li's t12) difference solu- tion and Isaacson's (7] closed-form solution. Fig.12 gives numerical calculation model of diffraction problem, in which the 2-D in- coming wave is obtained by the forced heave motion of a 2-D cylinder, i.e. numerical tank. Fig.13 is the time history of incoming wave elevation Ed velocity potential of incoming wave at point ~-R, y=0 ( see Fig. 1 2 point A ) . Fig. 14 is the time history of horizontal wave fo roe. The interaction of the 2-D floating rectan- gular cylinder and the wave has been calcu- lated, where the cylinder is only with one degree of freedom in z direction and the wave is also produced by the numerical tank whi ch ensure no wave in the vicinity of the cylind- er. Fig. 15 is the free surface elevations at point ~-R. Fig. 16 is the variations of hori- zontal wave force and vertical wave to roe . Fig. 17 shows the variations of the body cen- ter aid the body velocity in z direction. Fig. 18 arid Fig. t9 show the results of a 2-D periodic Stokes wave (described by Eq.( I)) and Eq. ( 31 ) ) upon a fixed circular cylinder and a truncated fixed circular cylinder at

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the free surface. 6.4 Conclusions From above examples the following conclu- sions are obtained: (a) The boundary element method coupled with time-marching finite difference shows good prospect for practical use with rea- sonable cost and requirement of median com- puter. (b) Sommer-feld condition with varying wave speed used approximately as radiation condition for radiation and diffraction po- tential in nonlinear case seems to be accep- table, at least for the case we have deft with. (c) Numerical tank is good for establish- ment of initial boundary condition with shorter transient process, and it can be applied to any water depth and any water bottom condition. A 2-D nonlinear Stokes wave with wave front is also suitable for initial condition. (d) The determination of the location of the intersection points at the free surface and the body is serious problem, the approach we used is succeded in our cases. References 1. Bourianoff, G.I., Penumalli, B.R., "Nume- rical simulation of ship motion by DGerian hydrodynamic techniques", Proc. of Second Int'l Conf. on Numerical Ship Hydrodynamics (1977). 2. Miyata, H. and Nishimura, S., "Finite- difference simulation of nonlinear ship waves, J.Fluid Mech. Vol.157, pp.327-357 (1985). 3. Nishimura, S., Miyata, H. and Kajitani,~., "Finite-difference simulation of ship waves by the TUMMAC-IV method and its application to hull-form design", J. Soc. Nav. Archit. Japan, Vol.157, pp.1-14 (1985). 4. Longuet_Higgins, M.S. and Cokelet, E.D., t'The defo Ration of steep surface waves on water't, Proc. Roy. Soc. Series A, Vol.350 <1976). ,. Faltinsen, 0., ''Numerical solutions of transient nonlinear free surface motion outside or inside moving bodies"', Proc. Second Int'1 Conf. on Numerical Ship Hydro- dynamics (1977). 6. Vinje, T. and Brevig, P., ''Nonlinear ship motion't, Proc. Third Int'l Conf. on Numeri- cal Ship Hydrodynamics (1981?. 7. Isaacson, M. de st Q., "'Nonlinear-wave effects on fixed and floating bodies", J. Squid Tech. Vo].120, pp.~67-281 (t982). 346 8. Lin, V.M., Newman, J.N. and Yue, D.~., "Nonlinear forced motion of floating body", Proc. of 15th Symp. on Naval Hydrodynamics (1984). 9. Yang, C., "Time domain calculation of three dimensional nonlinear wave forces", Doctoral thesis, Shanghai Jiao Tong Univer- sity, China (1987~. 10. Olanski, L., "A Simple boundary condition for unbounded hyperbolic flows", J. Comp. Phys. Vol.21 (1976). 11. Vinje, T., lie, M.G. and Brevig, T.,"A numerical approach to nonlinear ship motion ,Proc. of 14th Symp. on Naval Hydrodynamics (1982). 12. Lin, B.Y. and Lu, Y.L., "A numerical model for nonlinear wave diffraction around large offshore structure", Proc. Second Asian Congress on Fluid Mech. (1986). SD i Am it An= -d - Fig.1 Frames of referenc" and integration surface ( show with ~=0 ) o)~-O. 3}45 DF/a-1. O h,,/a.O. ~ d-/a-1 . 5 T.2~/u) _ ~ ~ ~ 0 20 40 60 80 ~ 00 X/e Fig.2 Free surface elevation at various times a~fi77~0.50163 DF^-~.0 hO/a-O.1 d/a.~.5 'r-2~) Fig.3 Free surface elevation at various times

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Fig.d, .Noclicear free surface profiles for case (1) (l~r<6.5. t. 0. Cat. 0.1, ) Fig. 6 .Noalisear free surface profiles for c&se (3) <~544.,, t~n, o.~, . . . _51 , , ~ ~ 4 8 1~, TIKIE Fig. 7 Heave fo rce va riationthO=0 .0 53 case( 3) Lin' s resul ts ~1 Fig.8 Heave force variation for different water depths --- d= 1 +++ d= 2 d=8 0. 250 1 7(/hO ( x=4a, Y-O) 0.12` . 0.000 -0.1 25 ~0. 250 0.250 0.125 O. .000 -0. 1 25 -0. 250 347 | ~k~ 4;, ~ ~ t/T ' i/ho ( ~-" ' 00 ) t/T Fig.9 Free surface elevations the fixed point model test results --- num e ri cal resul ts t= 1. 5T Fig. 10 Free surface profiles

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n -1 . -2 . Fig. 12 Numerical calculation model 0.1' 0.1] 0.osl 0.00f -o.oS~ -0.10 -0.15 Fig. 1 3a Incoming wave elevation for 3-D fixed ci Ocular cylinder at point A ( =--R, y=O) 0.15~0/ o. 10. - o.os o An _0.05 -0.10 . -O. IS ~ A ~ ~ ~ oo a:, \ / 1;0 ~ ~ 5 ~ 2 0 x lot Fig. 1 3b Velocity potential of incoming wave to r AD fixed ci Ocular cylinder at point ~ (xR,y=O) lo~1 t 0.6~ 0.41 '1 n n . _ _ -0.2 - .4 -0.6 -o.a 2! f~Haa Boy /.5, H/~=o ~ 0.44 ; ~ ~ Liz;\; 6~8oo'0~ 0-~7 no \ ' try Fig. 11 Horizontal wave force variation coo present results Li ' s results --- Isaacson' s results 0.6 0.4 _ /y 0.2 / X o.o 2a -0.4 , -0.6 . . - t/ 1~2o~o - om 1 -0.16 ~ 0.22 0.22 0.44 . ,/Dr I_._ \ ~ / \ ,1 ~'5 ' Fig. 15 Incoming wave elevation at ~-R for 2-D Moating rectangular cylinder ( d/DE 4, a/DF= 5. 6, R/DE~26 ) FX/p6D~d --_ r /f~D?d ~ , ~ __ ~ \ \ 1 \J~- Fig. 16 Horizontal and vertical wave- force variations for 2-D floating rectagular cylinder ( d/DF=4, a/DF=5. 6, R/DF=26) s0Io. sag so/D, , \~ ''a An\ ' / '\ \ `~t -~ 2.'5 " it \\ \ 7.S- ~ _ ~ \ Fig.17 Variations of the body center and the body velocity for 2-D floating rectangular cylinder in z direction ( d/DF= 4, a/DF= 5. 6, R/DF= 26 ) Id/ - d _ 4/~2. /0.65, s/~.4 i/. r' ~ r I -0.1t -owl ~~/: MAIL* offs go gay lo ~szo Fig. 18 Incoming wave elevation at ~0 and horizontal wave-force for AD fixed circular cylinder rx/~~ \~//\ /, 2.0 x 10 Fig. 14 Horizontal wave-force variation for 3-D fixed circular cylinder 0.6 ! , o.. - 0.2 - o.o -0.2 _0.4 348 /~' 4/~1.S. /0.65, 8/_0.4, Dr/~-o.s I''--` d'" ~ \ , {K Fig. 19 Incoming wave elevation at ~0 and horizontal wave-force for 3-D truncated circular cylinder at the free surface

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DISCUSSION by K.J. Bai First of all, the authors should be congratulated on the impressive numerical work reported in the present paper. This paper is a most welcome addition to the literature on numerical computations for the nonlinear free surface flow problem. In the following, I would like to make three comments: (l)the radiation condition given in Eq.(15) is true only for linear (or nonlinear) hyperbolic-type wave. However, in the water wave problem there exists local disturbance term besides the propagating waves. Therefore this radiation condition should be imposed at a sufficient distance away from the heaving vertical cylinder. Specifically, for this heave motion, the local term in the potential behaves like a pulsating free-space (Rankine) source. In some cases, at the sufficient distance away from the heaving cylinder where the local disturbance term is negligible, the propagating waves may be treated as linear. This is because the nonlinear three- dimensional wave will be linear as it spreads out. Recently we have made some numerical tests on the matching of the Kelvin source distribution and the local nonlinear numerical scheme along the numerical radiation boundary, which replaces the radiation condition in Eq.(15). This matching procedure worked very well in our numerical test. I wonder if you have ever tested this scheme. I would like to know how far one should take the radiation boundary in order to use the radiation condition Eq.(15). (2) Similarly to the above question, I do not understand the radiation condition given in Eq.(22) for the sway motion of a vertical cylinder. I think that for this asymmetric motion for the swaying cylinder, the wave number vector should be radial vector. It may be seen from the fact that the potential for the swaying vertical cylinder can be expressed in a Fourier-Bessel series in a sufficient distance away from the cylinder. (3)I do not understand the validity of the equation in (23)' which is entirely based on the linearity. Even though the diffracted waves become small at the radiation boundary, I do not see the logic behind the linear superposition of the local nonlinear part(i.e., the incoming wave, I guess) and the linear part. Author's Reply (1) Although Olanski condition is just only an approximation as radiation condition for radiation and diffraction problems in nonlinear case, the numerical results have shown that this radiation condition can absorb the reflected wave on the open boundary. Besides because of the Sommerfeld radiation condition with varying phase velocity every time step, this condition is acceptable in nonlinear case so long as the distance between open boundary and body is large enough. Of course that nonlinear solution matching with linear Green's function on the open boundary is also usable as an approximation, but Sommerfeld condition is easier. (2) If the vertical circular cylinder undergoes forced sway motion, the flow would be non-axisymmetric in the vicinity of the body, and we should solve this problem in 3-D flow, and it is different from the heave motion of a circular cylinder. Because the distance between the open boundary and body is just only large enough, the direction of the reflected wave on open boundary and body is just only large enough, the direction of the reflected wave on open boundary is an unknown quantity which can be determined by Eq.(20). If the open boundary is very far away from the body, the direction of the reflected wave will be along r direction, and it can be obtained from Eq.(20). (3) in the wave diffraction problem we use ~S=~~OW just only on the open boundary. On the free surface we let ~ satisfy nonlinear free surface condition, and Eq.(23) only used as satisfying radiation condition on open boundary. DISCUSSION by R.C. Ertekin The authors should be commended for their paper which initiates one of the first steps in solving the exact nonlinear diffraction/radiation problems governed by the potential theory. I have a few questions on the formulation and results. 1) What is the form of the Green function which satisfy a~/an=0 on the sea floor? I know of a way of placing image singularities if the sea floor is horizontal so that 3~/3Z=0 there. But not the form of Green function if it is arbitrary so that aO/3n=0 349

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2) How did you deal with the 3-D problem of neighboring panel compatibility, i.e. that the surface elevation is continuous in passing from one panel to another and having a common boundary (without holes) between adjacent elements? If the problem is axisymmetric, one can use cubic spline interpolation, for instance, to deal with this problem, as it was done by Brown University researchers to solve the problem of rain drop collision with a free surface using the BEM (Symp. on Fluid Dynamics, California Institute of Technology, Pasadena, 19~39). I can see the resolution of this difficulty in the case of a heaving cylinder, but not so easily, in swaying cylinder case in 3-D. 3) Wang, Wu and Yates solved the solitary wave diffraction by a vertical cylinder using Boussinesq equations (17th ONR symp., The Hague, 1988) and compared their results with Isaacson's that you cited. They found considerable discrepancy especially behind the cylinder. Whereas your results agree well with Isaacson's. Can you explain what you think might be the reason for this? 4) Could you comment on the accuracy efficiency of your time stepping method? Author's Reply and 1) Here Green Function is so chosen that it can satisfy a9/an=0 on the sea floor when sea floor is flat, that is G=1/r+1/r'. If it is not flat, Green function can be described as G=1/r. We must place singularities along the sea floor. At this time SQ/3ni0 , and the sea floor will become one part of the integration surface. 2) In the numerical calculation we can follow the fluid particles at every time step by use of Lagrangian free surface condition and time stepping scheme, then we can determine the free surface elevations and the shape of panel. If the shape of panel is very different from the initial shape, we can redivide the free surface element in terms of some regulations. 3) The diffraction problem of a vertical circular cylinder standing on the seabed and piercing the free surface by a soliton wave has been calculated. There are only small differences between our results and Isaacson's. I think that the different methods by which the radiation condition and the intersection points of the body and the free surface can be described will affect the numerical results. 4) The accuracy of time stepping method will depend on the discretion of the elements, the quantities of the time step and element size, the scheme of finite difference and the method of integration. If the radiation condition and the determination of intersection point are not appropriate, the calculation will diverge after a few time steps. If the free surface has very serious nonlinear, for examples, the body enter into the water suddenly the usual method to deal with the free surface elevation could not be very efficient. 350