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A Numerical Research of Nonlinear Body-Wave Interactions Z. Zhou, M. Gu (China Ship Scientific Research Center, China) ABSTRACT This paper presents numerical research results of nonlinear body-wave interactions. The body-wave interaction is treated as a transient problem with known initial condi- tions. The development of the flow can be obtained by a time-stepping procedure, in which the velocity potential of the flow at any instant is obtained by utilizing a source density distribution on all boundary surfaces. The Orlanski's method is used to implement the boundary condition at the open boundary. The position of intersection points are determined by a direct method. The contents of this paper are: (1) A study of the nonlinear radiation problem of a floating body; (2) A study of nonlinear wave diffraction problem around large structures; (3) Interaction of nonlinear waves with a free floating body; (4) An attempt in generating a numerical wave tank. Pretty good agreement is met between the numerical results and the analytical solution. NO, lEt~CLATUR`E a cylinder radius d still water depth F. six components of force/moment acting ~ on the body. 9 the acceleration due H incident wave heisht , Id, Iz moment of inertia about x,y and z K wave number K wave number of the m mass of the body t] number of elements on Sb+Sc+Sd+Sf ~I~,Nc number of elements on So, Sc Nd,Nf number of elements on Sd,Sf SO wetted body surface Sc outer boundary or SO bottom surface Sf free surface Zig vertical position wave elevation p fluid density ~ total wave potential Zhenquan thou and flaoxiang Gu, China Ship Scientific Research Center, P.O. Box 115, Audi 214082, Jiangsu, China to gravity axes velocity potential of incident waves (s scattered wave potential I . INTRODUCTION The interactions between large floating structures and the sea wave are generally predicted on the basis of linear ship motion theories, which are formally valid for small-- amplitude sinusoidal forces, the methods which have been applied so successfully to the linear predictions are no longer available. Therefore much attention has been paid to perturbations involving second order potential in the past few years. However, when the wave is very steep and the nonlinear effects more serious, it may not be appropriate to consider only second order forces. waves. For nonlinear wave wave envelope control surface of the body centre An alternative approach to the nonlinear body-wave interaction problem has been adopted by Isaacson [1], in his model, the nonlinear wave-body interaction is solved numerically by a time stepping procedure. In this field, Lin, Newman and Yue [2] have presented a method of matching the finite computational domain to a linear outer solution for the transient heaving motion of an axisymmetric cylinder. Liu and Yangt3] have presented the study results for nonlinear three-dimensional but axisymmetric free-surface problems using a mixed Eulerian-- Lagrangian scheme. The main difficulties in simulating fully three-dimensional nonlinear interactions between a free-surface and a body are as follows: (1) The treatment of the body and free-surface interface. A confluence of boundary conditions exists at the line of intersection of the free-surface and the body surface. As a result, the panel solution exhibits a singularity at points on that line, which is the one of the main causes of diver- gence. (2) The selection of suitable outer boundary condition or radiation condition. The outer boundary in a numerical model is a control surface surrounding the body and waves which reflect the requirement at the far field in guaranteeing for the uniqueness of solution. To accomplish this the outer boundary condition 03

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is made to fulfil the function of allowing all the outgoing radiated and scattered waves to pass through without any reflections into the inner wave field. (3) Stability of the free surface. In the case of nonlinear waves the wave slope is large, which is conducive to unsteadiness, and hence disrupt the computa- tional effort to reach a steadystate solution. (4) The interaction of the body motion and wave motion. For each time step, it is necessary to re-determine the relative positions of the body and the wave, which in turn needs high computa- tional accuracy. It is realized from the very beginning that in dealing with nonlinear computations of body-wave problems, the method adopted at the present state-of-the-art depends on the target aimed for investigation, if it were the local phenomenon such as the nature of singu- larity at the body-wave intersection and of wave breaking and/or spray formation were aimed at, then full attention should be paid to the treatment of the local singularity. However, if it were the global forces and moments acting on the body that is aimed at, then the method should adopt a pragmatic treatment of local singularities, which paying attention to the rest of the problems to keep the computational treatment robust and effic- ient. The method outlined in the present paper follows the latter philosophy. In the present method' the body wave interaction is treated as a transient problem. The development of the flow is obtained by a time stepping procedure, the velocity potential is obtained by utilizing a panel method, in which simple source are distributed over all boundary surfaces, including the free surface, the immersed surface of the body and the bottom surface. Plane quadrilateral surface elements are used to approximate those surfaces, and the physical variables are assumed to be constant at each element. The center of each element is used as the pivotal point. The integral equation for the source density is replaced by a set of algebraic equations. When this set of equations are solved, velocities at and off the surfaces are obtained. The contents of this paper are: (l) A study of the nonlinear radiation problem of a floating body. (2) A study of the non- linear wave diffraction problem around large structures. (3) Interactions of nonlinear waves with a free floating body. (4) An attempt in generating a 3-D numerical wave tank. 2. MATHEMATICAL FORMULATION As in Fig.l, an arbitrary body is shown floating at the free surface and moving in large amplitude in waves. Let x,y,z from a Cartesian coordinate system as indicated in Fig.l, with x measured in the direction of incident wave propagation and z measured upwards from the still water level. The fluid motion is described by a velocity potential ~ which satisfies the lap ,;~/'/ BY 5! I' ~/~7 Scan ~ I (T ' 1 - t- -- if' 1 ~ ~1 Fig.l Coordinates systems lace's equation within the fluid domain, V2~(Xt ye it t) = ~ the boundary conditions are as follow: a(x,y,z,t) = van on sb (2) it = at ~ at 30 + ii 30 on Sf (3) 3$ + go + 1(v$.v) =o on Sf (4) 3 (x,y,z,t) = 0 at z = -d (5) also satisfies a suitable radiation condi- tion to be discussed later and a proper initial condition. In order to solve nonlinear interaction of the Body and wave, The development of the flow is obtained by a time stepping procedure, with the velocity potential field at any one instant obtained by a panel source method. In this method, ~ is represented as the velocity potential of the field point which is evaluated by (x,y,z) = i:~(q~r(l rids am where rid p , q) is the distance between the points p and q. The normal derivative of the integral in (6) at p of the surface 2Q is Vn = -~2~(p)+n~p)~(q) -,qyds] am where the unit vector n is normal to the body surface and pointing outwards into the fluid. In order to make the computer program extensible to arbitrary bodies of 3 dimensions, the surface So, Sc, Sf are divided into finite number of elements. Assuming that the source 104

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where density is taken as constant over each element the potential at p of Eq. (6) may be written in a discretized form as: N hi = i. tjjcj i=1 2 rat (8) ~ i j = | J s j [ ( x-~; ~ Z+( yen ) 2+z2 ] 2 The normal induced velocity is then N where ni j-1 ii i i=1,2, ,~1 (-2n j= i ~Jsjr3 ilk In conducting the time stepping procedure the boundary conditions at each time interval t+6t are established in explicit forms containning quantities at times At and t- At which are known from previous iterations e.g: (3n~t+At F ($t ~t-At~3n~t' On~t-At) on fit ~ FC(t, ~t-At ad t al~t-At) on Calf At = Fort (t-At 3 t 3n~t-At) onS f 3.1 Formulation (an~t+At = F\t ~t-At An t an~t-At) on (12) Takinginto account of (8) and (10) above and the boundary conditions (12) a set of surface integrals on source density cite At) may be up-dated to the advanced time interval t+At thus: N j-1 id ~ Y Z t+At) = Gj .t+~t i 1 (13) in which the matrix of coefficients are and K.. id Mjj tij K.. id r. ~j = ~d+1~-~'Nd+Nc+Nf (14) i Nd+~c+Nf+1' ~ 'N ~d an i t+At . c Fuji t+At (~) tat (3n) t+At i=Nd+NC+Nf+1 N the velocity of a point in the field may be expressed as i=Nd+1, . . ., Nd+NC =Nd+NC+1, , Nd+NC+~'lf j = Whenever the velocity potential ~ and the velocity v as well as the body motions are known at time t the quantities of these variables at the advanced time interval t+At may be evaluated according to the following process. Update the wave elevation over the free surface Sf at time t+ At; according to the motion of the body determining the new position of the body' which modifies the immersed body surface at interval t+6t under the new free surface Sf~t+At); in the new fluid domain Q(t+ At) re-calculate the coefficient matrix Mij' and update the matrix of boundary conditions Gj t+At;finally find the solution of source density ott+At) at the interval t+At from (13~. The pressure p over the body surface is obtained by the unsteady Bernoulli s (11) equation: p=_p tgz+$t+2V.V~ ~ (17) The force components Fk may be calculated as appropriate integrations of pressure ok =-J:PnkdS (18) ,. where subscript KC1 2 ... 6 and Fk correspond to fores in and moments about x y z direction respectively. 3. THE NONLINEAR RADIATION PROBLEM OF A FLOATI NO BODY As indicated in Fig.1 consider the forced heaving motion of an arbitrary floating body on the free surface a Lagrangian s description is used the kinematic and dynamic boundary condi- tions (3) (4) are rewritten as: Ox _ a`. Dy an Dz _ arm. Dt ~ ax Dt ~ ay Dt Liz `19' Dt = 2V$.V~ _ 9D The above free-surface conditions are discre- tized by an explicit time-step scheme as follows: xp~t+At) = xp~t)+26t{3(at~t-(D8~)t-At}P yp~t+At) = yp~t)+2At{ 3`a't~`aa't-~t} P zp~t+At) = zp~t)+26t{3(a$)t~(a$)t-At}P On the pivotal point the potential is: (20) . ~ t+lit )=j ~ t )+2At{ 3( 2V.V~-go it _( 27~9rl~t-~t} ~ 21 j~l~ciixel+cijye2+0idze3~i (16) Let the body be surrounded by a verti Cal control surface Sc . Ili thi n the immedi ate vicinity of Sc . the scattered wave element may be approxi mated as a pi ane wave, propagati ng outwards with celerity along a direction 1. Assumi ng that the scattered wave near the outer 105

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boundary satisfies the Qrlanski's [4] condi- tion, one have an + cR`ty3 = 0 (22) where CR is the phase velocity for potential function. Let R denote the intersection point of the free surface and the outer boundary, and point R-1 a neighbouring point AR distant from point R along the radial direction, the phase velocity CR is then obtained from ~ of the free-surface as: t -tR l+tR 1 RC = 4~_1 26t(Vr)~-1 (23) Considering that the scattered wave should be outgoing waves, Cp > O. In practical computa- tions, the limits on OR may be set as [~] fat if tR-At C~(t) = c(t) if O-CR At (24) lo if CR<0 where CR(t) is obtained from Eq. (23). The solution of the radiation problem may be obtained from (13), together with the boundary conditions listed in (19)-(24). 7.2 The Panelling of The Instantaneous '1etted Surface Near The Free Surface. At intersection line of the body and the free-surface a confluence of boundary condition exists. As a result, the solution exhibits a singularity at the line. To accommodate this, we determine the position of intersection points by a direct method. From a physical point of view, the fluid particle may not penetrate into the body , it may only slide along the body surface. Therefore, in the numerical model, the vertical position of the intersection point is obtained by interpolation with the body draft along the radius form wave elevations. The horizontal positions (x,y) of the intersection point may be obtained by setting their absolute increments as zero. Another advantage of this method is in using the centre point of the panel as the pivotal point, this avoiding singularity at the intersection point of the body and the free surface. At the intersection of the freesurface and the outer boundary, the vertical position of the intersection point is determined by allowing it to be the same as the wave eleva tion calculated at the outer boundary surface. For each time step, the immersed body surface changes in accordance to the motion of the body and of the wave surface. There are several configurations which the panel on the body may be intersected by the free surface Anti. Fig.2. ? Y:,5 Fig. 2: Nodal points on the intersected body panel. Let (x1 AYE ~Zl ~ and (x2 ~Y2 ~Z2 ~ be two nodes on a panel line, which are located under and above the wave surface acts respectively, thus, the new node on the wave surface nuts is: z0 = Anti z -anti Yo = Y1 + (Y2-Y1 ~ ~ z~-~(t) X0 = X1 + (X2-X1) (25) 3.3 ('umerical Examples of Large Amplitude Radiation Problems and Discussion In this section, results are presented for the case of forced heaving motion of a floating truncated vertical circular cylinder of radius a and mean draft a/2 . The length, time and mass units are so chosen that the radius a, gravity 9 and density p all equal unity. The vertical velocity of the body is prescribed to be V(t) = Hmsin~t (?6) in(t) = -2 - Scout (27) The total number of panels is 232, with tid=26, '''calls' 'Jf=84' "!b=74 Let the radius of outer boundary be P=5.0, amplitude be ',4 =0.0S, frequency w=~n, time increment be ~t=O.l, and water depth be d=4, T=24, i.e. a total of 6 periods. The comparison between Lin's resultst2] and our results for the above problem is shown in Fig. 3, it seems agreement is good. 3.3.1 The Effect of Different Radiation Co-nditions Adopted for The "onlinear Radiation Solution. Two groups of computations are carried out to investigate the effect of different radia 106

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.~ gin i.? 'i't 9-\ . '. ~ (- c, 4 / Fig.3 Comparison of 'neave force 'nistGry - present result; *8 Lin's result r24. tiGn conditions. flare, the time interval T=l6.G . In group hi, radiation condition as proposed by Isaacsontl] is adopted. ~ = 0, am = ~ In group A, the radiation condition of A. (2?) i s adopted. 278 /39 -739 -27~ ~.~- Fig. 4: Co~,parison of 'neave fGrce for different radiation conditions. ~ v v group A; - group B. to comparison of vertical force acting on the cylinder in both cases is shown in F j g. ~ . In the first one and half period, the result of group ~ is consistent ,,~it'n that of group B. But as the time i ncreases, there begins a difference between them; the period of group A becomes gradually less than that of group .'9. Thi s phenomenon ari ses from the ref 1 ecti or of scattered saves in group Hi. fit the beginning, the scat-tared Haves d i d not reach the outer boundary Sc , therefore results of t'ne two groups are consistent. 'hen tine scattered -eves reach the boundary Sc, in the case of group ,$ the condition of (~) induces a ingoing reflection of the scattered slaves. It is the interference of the scattered waves that causes the chage in the period of the wave forces. 3.3.2 The Effect of Changing The Distance of The ~u or ~ ~,~ ,~7~ = t~,~ Ike ()rigin. A comparison of three groups o, calcula- tion using different radii of the outer boundary surface are carried out to investigate the effect of changing distances of outer boundary surface. 'here (a) 0=5.O, (b) R=3.0, (c) R=lO.~. Fig. 5 shows the time history of wave elevation at an intersection point of the body and free surface. here, the result of R=Q.Q is consistent with that of R-lO.O, but the result for P=5.0 is different. For the first cycle, (a) is consistent with (~) and (c), but difference of amplitude appear when time interval increases. (as) Am/ 107 Fig. 5 History of an intersection point. --- group a; -** group b; 0 oo group c. The free surface profiles in the radial direction corresponding to groups (a), (b), (c), for t=6.0, S.0, 14~6r are shown in Fig. 6-~. For t=5.0' groups (a), (bland (c) are consistent. For t=3.O, a slight reflection causes the difference between group (a) and group (b), (c). For t=14. 4 . inconsistencies Bevel oped even between c)~=0, and 9=1(3. Lear the body surface, the wave prof i 1 es are consi stent, but near the boundary S c t',ey are different. Fi g. ~ i s t'ne pressure on the Cottons center panel . Fi g. 10 i s the verti Cal forces acti ng on the cyl i nder of group ( a), ( ~ ), (c ) . It i s shown that by changi ng the di stance of Sc form, ha origin little effect is introduced as far as the hydrodynami c forces are concerned. Fro,-.n the above co~-`,pari sons a concl usi ve rears may be - drawn: For the Orlanski's condition, chant the radius of the outer boundary causes 1 i ttl ~ i n f 1 uence to the wave force but may have a l arge i nf l fiance to the wave profiles. For coruputatiolial stablity it is advi sable that t'ne outer boundary surface 'ce place'! at a distance sufficient!: far from t'ne 'cocky.

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~E-3 E-2 /440+ 933 467 00t A -4~1 _~ -/4lX1 ~F,_~ Mp,''- ~~l~v~A~' At, R 9.33 4.67 000 -4.67 -933 20.~ . fly 667 0.00 -6.47 ,3 _ Fig. 7 Free surface profi les (tag. 0~. _*_group a;-O_grou? b;_ ^~- group c. F_3 ~-_ -~ ~00 -ado too , E-l Fig.8 Free surface profiles (t=14.4) - *- group a; - 0 - group b; -6 - group c. 108 Future _/~ - ~, R /0~# E-J 00 ''28.~ 4~ ~00 92~0 in' ,/ ;,! Fig. 6 Free surface profi les (t=6) _ *_group a; _~- group b; -~- group c. 32-~ Ad/ 49 4 47~9 .02 A f4 i.76 /3& 0.00 is too' SOLO 6~00 0200 NATO -f38 [-1 -2.76 _~4 i t' 3.,/' 6.~i 9~ I`?.72 Of.. Fig. 9 Pressure on the central element *** group a; ~ 0 Q group b, Err, ~,.'rc~ Fig.10 Heave forces for different radii. *~*** group a; ~ 0 0 group b; - group c. 3.3.3 The Effect of Chanqinq the Amplitude of Body's Oscillation. Generally, large-amplitude motions result in nonlinear phenomena. Four cases of large amplitude forced heaving motions are investi- gated in which R=8.0, w-ln, d=4.0, T=16.0, t=Q.1 and in case (a) H =0.05, da=H'=lQZ, h is the mean draft. (b) t] =0.10, ~ b=20/o' (c) H=0.15, dc-30% (d) H=0.25, Gd=50%.Fig.11-14 show the radial free surface profiles for the above four cases. With increasing amplitude, the wave profiles change drastically as steeper waves appear and greater amount of wave -energy are transported outwards. Fig.15 shows wave surface profiles of four cases of I] (at t= 9.0 ~ respectively. In each figure, configuration and phase of wave profiles for these four cases are the same, but the greater the oscillation amplitude, the steeper the wave and the stronger the non- linearity.

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Fig.16 is the time history of a body wave surface intersection point which is the same as the wave elevation on the body surface. It is shown that a larger wave motion appears for a l arger body osci l l ati on ampl i tude, but the peri od and trend are the same. I n the case of the l arger ampl i tudes, there appears more asymmetry in the curves which easily results in i nstabi l i ty of the numeri Cal process. Fi 9. 17 shoals the pressure acti ng on the bottom centre panel. A distortion at the peak of the pressure 'history as a result of strong nonlinearity appears when the osc i 1 1 ati on arr~pl i tude becomes too 1 arge, whi 1 e a drop i n pressure appears i n the very begi nni ng. Si nce the body starts to rise from rest and form its initially displaced pos i ti on at the botoom of the stroke. The compari son of the verti Cal hydrodynami c Force acti ng on the body i n Fi 9. 18 shows l i ttl e change for the four cases consi dered. It i s shown that the vari ati ons i n the radial free surface profi les and in the local pressures are much more serf ous than i n i ts vertical forces for different ampl itudes of oscillation. Thus the nonlinear effect should be considered when deal ing with a local strength probl em i nvol vi ng i ntegrati on of l ocal pressures. 300 Loot zoo J.oo / ^~00 - ~ _ 2.40 3.~0 42 5/ 3605 70~ I,, 7' die ~ Fig. 15 Wave surface profiles for different amplitudes (t=9.03. -~ - case a; 33 - - case b; -~- case c; case d. 165 4. NONLINEAR WAVE DIFFRACTION AROUND LARGE STRUCTURES l6S 4. 1 Formu l at i on In approach to the diffraction problem, the wave diffraction is treated as a transient probl em wi th known i n i ti al cond i ti ons corre- spondi ng to sti l l water i n the immedi ate vi ci ni ty of the structure, and `'i th a pre- scribed incident wave form approaching froin x=~ and propagati ng past the structure, Fi g. 19. 685 437 ?2t Lid a -20 2~4 -451 6~ 5542 238~ ~ 9 S2 iIdb 4.7~Il4 ~It9 Fig. 1& History of an intersection point for different arr,plitud.~s. E_ ~ ~ rat ~, . 00 2.3' 476 7/4 9~' Fi 9. 17 Pressure on the bottom central panel. -3.' ~:_~\t~^~ t-0rGe Time 1/ 2~\g, S.,S :8.93 V~' Fi g. 18 Heave Forces for di fferent ampl i tudes. -case a; 0 0 0 case b; 666 case c; ~ ** case d. 109

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Fig. 19 An isometric view of body-wave i nteracti on. H 2 /d=1 /3, t=1 . 3. then incident waves exist, Lagrangian free surface conditions of the form (10) clay easily l ead to i nterpenetrati on of the slave surface panel and the body surface. I n order to avoi d the difficulty of shaping and rearranging, the free surface panel, a set of Euleri an free-- surface cond i ti ons i s proposed here. at ~ n = 0 ~ (29) at + ire + 2`v~2 Q: For the outer boundary condition provided that i t i s lyi ng at a sui tabl e di stance from the bod:', then i n the begi nni ng ,oeri od when the scatterer] waves are sti 11 in the inner domain, the outer boundary condition may be written as the Isaacson ' s versi on, i . e. an aO an On ~ = to (30' however, when T > Tc, where Tc is the tine when the scattered waves travel l i no at i ts group vel oci ty Yogi n to meet the outer boundary, the Qrlansl i's condition is adopted: at ~ C,p(KrarS + K~r9) = 0 K = 06.a~s /,/(a~s)2+(a~s)2 Kin = <~.a~s /l/~2~(3'~s)2 (32) Here ec J1 if apse O an t-1 if a s ~ 0 I t i s assumed here that the scattered wave po- tenti al As = ~ - It must be mentioned that a note of n~athemati- cal i nconsi stency occurs here as for nom i near wave dynamics, the principle of superposition i s not stri ctl y appl i cabl e. For the initial condition, as there is i n i ti a l l y no i no i dent Slave moti on i mmed i atel y ad j acent to the body, SO the scattered poten- tial is initially zero. I n our numeri Cal procedure, the corner point q0 of the wave surface panel is deter- mi ned by the area average of i ts four surroun~- ing panels, let rl, s represent wave elevation and area respect) vel y, then ~-~+~eDo+~+~Oa No= (3~) iSA4SB+~SD For each tirre step, the adjusted wave surface wi 11 truncate a part of the body panel s near the free-surface. If the re~,ai ni ng part of the body panel becomes seal ler then , a predeter- mi ned number, then i t i s del eted from the body surface to avoid divergence. 4.2 humeri Cal ~xampl es ~ corn,outer program which incorporates the method descri bed above has been used to generate results for a few specific situations i n order to establ i sh the practi Cal vi anti l i ty of the method used. Suitable comparisons with avai fable results may be made only for relatively few restricted cases for which known di ffracti on sol uti ons are avai l abl e. [ 1 ~ . d.2.1 T'ne Diffraction of Small A.molitu de 'flames Past A Verti Cal Ci rcu l ar Cyl i nder A surface-pi erci ng verti Cal cyl i nder i s sub jected to a 1 i near _ i nci dent .lav=. here, d/a=2, Fi/a=n. 1, Ka=1. 5, k/k=~. S. Fig. 2~) shoes a) the incident wave elevation at x-O, b) the time history of the hori zontal force acti ng on the cyl i nder. The resul ts obtai net! front our program i s shown i n dotted l i nes, the pre;di ctec force vari ati on agrees wi th that pi ven i n ~ 1]. The program has al so been tested for the case of z surface piercing vertical cylinder sub jected to a ,arescri bed sol i tary cave tGr`~` (31) given by ~ = Hsech2 [R(xs-Ct) ~ (36) w'nere T= /3~/4d~, Xs=X+a+3.~/?, and d/a=~.5, H/d=~. 1. Comparison of results with closed-forr~ solutions giver, by Isaacson t1 ] is presented in Fig. 21. The solid line indicates the closed-- form solution given by Isaacson, the broken line is t'ne numerical results calculated By I saacson ' s model ~ 11 , and the dotted l i no i s the numerical result calculated by the present (33) model. Results evaluated by this report serves to highl ight the 1 act that the outer boundary condition presented in ~ hiis section is capabl ~ of al 1 owi ng the outgo) ng scattered `.'aves to pass through efrectiv-~ly wit'.. 1 iti;le (34! reflection inwards. It also enables the 110

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computation of wave forces to run on for a sufficient duration of time so that the wave motion is fully established at the vicinity of the body. 4.2.2 The Diffreaction of Large Amplitude Shallow Water leaves ~ Results are presented for a circular dock fixed at the free surface subjected to a lrage amplitude shallow water wave. let d/a-1.5, -I.| / ~t/T to F/pd~ad W~ 00 - ~ -/.0 Fig. 20: a) The incident wave elevation at X=O. b) History of horizontal wave force. --- closed-form solution [13; o o ~ present results. ~2 F . -/ fit 1` 1 4 ~6 ~\ ~em. --2 quit ~' Fig. 21: Horizontal wave force. - closed-form solutiont13; --a. -- numerical resul ts[13; ~ ~ , present results. h/a=~. 5, d/gT -0.018, and Hi /d=1/4, H2/d-1/S, where h is the draft of the dock, T is the wave period, ill, H2 are wave heights. Fig. 22 shows a) the incident wave elevation at x=O, b) comparison of horizontal forces for cases of two wave heights. The dot-chain line represents the result predicted by liner wave theory [53. The broken line and the solid line represent results given by the present theory for H/d=1/4 and 1/3 respec- tively. There is a 00 phase difference between the wave force and the incident wave form. The maximum force coefficient occurring at t/T=Q.488 are respectively 15% and 217 greater than that of the linear-theory predic- tion. The total wave profiles occurring for diffraction around the fixed dock are presented in Fig. 23. It is noticeable that the steepness is larege. In Fig. 23, it is seen that the incident waves diffract around the body, and propagate past the body, comparing the wave crests at two sides of the body, a little delay of the phase may be noticed found. It shows that the wave behind the body may come from diffraction of the wave in front of the body. Since the wave is obstructed by the body, the wave elevation in front of the body swells up to a val ue, much hi gher than that behi nd the body. As a result, a pressure difference is formed and a periodic wave force is generated. Fi g. 19 i s an i sometri c vi ew of the body-wave interaction showing the mechanism of di ffracti on wi th Li /d=1/3, t=1 . 3. E -! Jon s.oo o,oo a -moo -10 -1 / sot ALL moo -1000 F/p~la2 , ~ +/r ~ . . ~ . . no too moo 600 ~E ~ Fig. 22: a) Incident slave elevation at X=O. b) Hori zontal wave forces. --. - linear diffraction solutiont53; --- H1/d=1/4; tl2/d=1/3. 5. IFITER;ACTIONS OF 51Q~ILIi'5~AR SURFACE '`lAVE WITH A FREELY FLOATI FIG BODY 5.1 Formulation In order to consider the effects of three dimensional nonlinear interactions between slave and body, the boundary conditions on the immersed body surface are expressed in terms of velocity Uk (k=1,2,...6) which are the six velocity components in the moving co-ordinate system fixed to the body. Thus 3$ 6 (37) The six values of Uk may be determined from the dynamical equations of motion:

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F1 6= or siniply, F1 = m(Ul+U5U3-U6U2) F2 = m(U2+U6Ul-U4U3) F3 = m(u3+u4u2-u5u1) F4 = Ixu4-( Iy~Iz)u5u6 F5 = Iyu5~( Iz-Ix)u6u4 Izu6 (Ix Iy)u4u5 w,,=. ~ (3S) may be expressed as jj Q-~Q;kzimknkQnkj/`s' (49) which was neglected in his num,erical computati ons. Final ly, substituting Uk in (45) into the body boundary cond i ti on ( 37 ), the body surface equati on may be rewri tten as: N 6 i2-1 (Ki j-Aj j )0i = k1nk; Hk/mk ( 38) where i=1,2, ,N (48) Fk mkuk + fk llere the external force F k may be grouped into the fol 1 owi ng subgroups Fk FAk+FBk+FCk (4Q) where FAk denotes the fluid dynamic forces and moments of potent) al nature, FBki s that due to gravi ty of the body and FCk that due to external moori ng forces or vi scous darnpi ny forces. The f 1 u i d dynam,i c ~ orce components are: FAk -P j_3 n k j ~S j [ 9Z~ - vev+2 ( V~ ) 2 ] substi~utin~q (,)~1 ), into (39), we have + p Zb a~ ~s - p Z~ { nkjASj [9Z-Ve.V+2(V)2 ] j} fk + F3k + FCk (42) I n the coord i nate system f i xed to the body center G. nkj6S do not vary v.'ith time for the tin~e step consi~ered, thus, (42) may be written as aHk at (t) = ink(t) and for the next time ste,~ t+At. where Hk(t+At) = Hk(t)+-At{ 3hk(t)-hk(t-At)} ) H k ( t ) = m kU k+ P j _ 1 d~j n k; [S; hk = - PjZb{0kj [Sj [gZ-Ve-V+2(V)2 ~ jI fk + FBk + FCk (46) Ve is the velocity at the point considered on the body. V = V + ~xr e 9 (47) Vg i s the vel oci ty of the center C, r i s the posi ti on vector rel ati ve to the ori si n 0. The ter,n -n`tS j~e.v OCR for page 103
Fig. 24: a) Incident wave elevation at X=O. b) Heave vari ati on for a f 1 oati ng doc k . --- H 1 /d=1 /4; H 2/d=1 /3 . -/o. 00 E-1 A0,~ 1/d .o 000 G -500 ,/~OOIF~/~'0 i .~o o.oo ~, -7oo 00 y 4.00 6.00 &. -/~oo . E-l a fixed dock would easily satisfy the initial condition as reguired by the present paper' vi z. at t-O there i s no i nci dent wave immedi- ately adjacent to the fixed dock. The hi story of hori zontal wave force due to diffraction by the fixed dock is shown in Fig. 29. Here, d/a=1. 5. A maximun~ horizontal force coefficient of F/ p g`1a2 =Q.65 seems reasonabl e compared to Fi 9. 22. ~~[~ ~-Y~ 1 ~ ~ 1 ~ 1 ~I ~ I ' ~ '' i : R/a-8.- Fig. 25: a) Incid~nt wave elevation at X=Q. b) -' Hori zontal wave , orce. ',1 /d=1/~. '/0` 6. A~] ATTE`VlPT TO SIM~JLATE A t:!UMEnICAL '~'A'JE TANK A numeri cal ~nodel i s constructed for a cyl i ndri cal wavernaker operati ng i n a seakeepi ny tank of finite depth but of infinite dimen- s i ons. Attenti on i s focussed on numeri cal 1 y simulating the nonlinear waves with a wave front and acti ng on another body i n a numeri- cal seakeepi ng tank. The fi rst step i s to generate numerically the waves simulating 3-D nonlinear waves generated by a wavemaker. The second step i s to s i mu l ate the tan k experdr,ient by requiring these numerical `~aves wi th a wave front to act on cornputati onal model s of l arge offshore structures or shi ps. The attempt to justify the prediction of nom i near ~`ave force on an offshore structure nur`~eri cal l y i s a real i sti c proposi ti on. For illustration, the results of simulat- i ng the di ffracti on ~x,oeriment of a 1 arge floating dock , ixed on the free surface and subjected to an approaching nonlinear wave with a wave front generated by a cyl i ndri cal wave maker numeri cal l y are presented. The computi onal model i s as Fi g. 27. The cylindrical wavemaker is a circular floating dock operati ng i n a forced heave mode, the i nstantaneous draft of whi ch i s ex,urnssed as h~t) - -DF+Hsi n~t ~ 50) where the vel oci ty of i ts centre ~ i s accord- i ngly: Vg~t) ~H - . cos~t (51 ) l~lere, OF/a=l.O, t] /a=0.2, d/~=~.O. m=~ The generated wave time history (t) and wave potent) al hi story 0(t) at ~ poi nt ,~ ( x/a=-O. 8, y/a=~) are shown in Fiy. 2~. The nun~erical ly yenerated wave al ong x di recti on i s a propagat- i ng wave with a front, which xYhen acted on Fig. 27: Computional model of a numerical tank. E -` ~ :~6.00 -Ao.oO -fs,oo , 10' i~-640 2.~0 0.00 -2.gO -500 Fi 9. 28: a) The generated numeri cal wave; b) The generated wave potent) al . tE -1 F/egHa~ -5o ~,, C' ~ ~ i0. ~20 30 \40 / 50. E~] ~ ~ 5 0 _ _10 \J Fi g. 29: F1ori zon ual wave force.

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Fig. 11 Nonlinear radial free surface profiles, case a, amplitude H=0.05, 6=107 (l OCR for page 103
Fig. 13 Nonlinear radial free surface profiles~case c, amplitude H=0.15, 6=30% (1<~<8, t=0.1, 0.3,...) Fig. 14 Nonlinear radial free surface profiles' eases d@ amplitude H=0.25, 6=507 (l OCR for page 103
Fig. 23 The wave profiles occurring for diffraction around a fixed dock. [/d-1/4 (-9.0 OCR for page 103
7. CONCLU S ~ 01~ A numerical study of body-wave interaction as an approach to 3-D nonlinear sea!~eeping problems has been attempted. For illustration, several numerical examples are presented to demonstrate the influences of the change of different ,iJara,~.aeters, several i nteresti ng remarks flay be dra``n. ~1 ) Regarding the outer bounder: condition, ecuat.i on i s adopted as treatment of the al though ncl anski ' s , an open boundary condition, the position of the outer boundary sti 1 1 counts. It resul ts i n a more serf Gus i no 1 thence on the wave surface prof i 1 es than on the forces. Therefore for di fferent engi r`eer- i ng probl ems' ~ di fferent radi us of outer boundary may be adopted. (2) General ly there is ~ difference between the resul ts predi cted from nom i near theory and that from 1 i near theory. So,i,cti,,ies al though thi s di fference i s not 1 urge from a gloha1 point of reviewer, it flay exhibit sir~nifi- cant di f~erences at 1 cca1 areas. (3) The blockage due to a large fixed body to the wave wi 11 cause local swell up. The pri nc i pal zone of i nf l uence relay be restri cted to a sn!al 1 zone ad jack the boa>. If the zones are ~ 1 ittle further from;, the body, the cleave tori ons are rnai n 1 y i ncoIiii ng waves. Thi s phenomena may al 1 ow us to treat the bcdy-~'ave interaction by matc!-'ing tale nom inear sol uti on i n an i nner dori,ai n near the body \~'i th 1 i near sol uti on i n the outer domai n. (4) An unfavorabl e phasi ng of wave and body ~;~oti on i s demonstrated i n whi ch shi ppi ng of green water or sl ammi ng nay occur, especi al 1 y i n the case of 1 arge all i tube moti ons. tan k of novel idea o, ~ heaving cylinder is attempted, and validated by cor~iputa' ion t'nat such waves `~i th a wave front ;.~aiy be brought upon an offshore structure wi th proper i ni ti ail condi- tions and nu~.r,erical ly treated to Field non- 1 i near wave forces of the ri ght order. (5) Difficulties associated with the; i ntersecti on poi nts of the body ant! the wave-surface has been ,;:ragmatical ly resolved by the i nterpol ati on on the free-surface and avoiding the singularity at the intersection point. This may not be justified for the local phenomena, but may be appl icabli~ for global force eva1 uati ons. An effort to sinful ate a nur~eri Cal wave infinite horizontal diriiensio,ns by a; .FFEPAl!C~ ~1 ~ Isaacson, i;] de St. O., "~Ionlir'car Slave Forces on Large Offshore Structures", Coastal /Ocean Ens i petri no Report, ten i vers i ty of r ri ti sh Col ui`,bi a, 1 Q,'31 . t2] Lin, '-J. -'i,. . t'ewman, 3.'3. and Yue, O. it., Nonlinear Forced rlocior~s of Floating Bodies", Proc. 1 5th O~/S,'t1, I-Ja,~bury, Siej<:t. 2-7, 1~!. -l ~ ~ Li u, Y. Z. and Yang, A,., "owl i near Radi ati on Probl eta or An Axi symi~etri c Cyl i nuder", Advances i n T~ydrodynar,~ics, China. 'iol.Z, r'iJ.l, -1~ [4] Orlanski, L., ''A Simple Boundary Condition For Unbounded Hyperbol i c F1 ows", J. Computa ti onal Physi cs 21. 197~. ~] : saacson, i,. de St. O., "nonlinear Eve Effects on sir; xed and F 1 oati ng !~`odi es'', J. F 1 u i c' tech., `101. 2Q: pa ~G7-281 ~ 90?. 117

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DISCUSSION T. Francis Ogilvie Massachusetts Institute of Technology, USA You use the Orlanski condition for closure of the computational domain. This is a condition for hyperbolic systems. Please explain how you are able to use it in the water-wave problem, which is elliptical. AUTHORS' REPLY Orlanski's condition is a numerical condition on the open boundary. So far, the availability of Orlanski's condition cannot be proved in mathematics, however, it is useful in numerical computation. DISCUSSION Choung M. Lee Pohang Institute of Science and Technology, Korea We always find difficulty in determining the point of intersection of the free surface on a body under motion. It is mentioned in your paper the free-surface intersection was obtained by interpolation of the free-surface to the body. Would it not violate the kinematic body boundary condition at the intersection point? Since your attempt is to obtain an exact solution for nonlinear body-wave interaction, the problem of finding the intersecting point should be addressed more clearly. AUTHORS' REPLY As mentioned in the first section of this paper, the singularity of intersection point is a local problem. In our paper, we adopt a pragmatic treatment to local singularity. We have tested many methods for treating the intersection points in 3-D problem, the direct interpolation method may be the best one in these methods from our numerical test; it keeps the computation robust and efficient. 118