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Calculation of Nonlinear Water Waves around a 2-Dimensional Bully in Uniform Flow by Means of Boundary Element Method K. Suzuki Yokohama National University Yokohama, Japan Abstract In this paper, nonlinear wave making phenomena a- round a two dimensional body is studied. The analysis uses the boundary element method based on Cauchy's in- tegral theorem and the wave profile is calculated by time marching integration based on the semi-Lagrangian ap- proach. Steepening or Breaking waves can be simulated by this scheme. Two problems are discussed; a semi-circular mound in shallow water, and a floating body with semi-infinite length. In cases including a uniform flow component, nu- merical treatments have some difficulties. In the semi- circular mound problem, nodal points on the free surface move out of the calculation region, and in the floating body problem, nodal points concentrate in front of the bow. In the present work, numerical difficulties caused by these problems are settled and numerical examples are given for several cases. 1. Introduction Many numerical schemes hare beets presented fur the purpose of the analysis of free surface flu: jar receipt ~ C 1' S Some of the schemes, however, steed bigly perf~r,~, e computers with very large memory storage. T~ tl~:>se cases some difficulties remain in practical applications. In Flee problems of nonlinear water waves with steepening or breaking, many complicated numerical procedures and considerable CPU time are needed. If we neglect viscos- ity and assume irrotational motion, a numerical scheme based on the less complex boundary integral equation cart be employed, which does not need high performance com- puters. For the problem of two dimensional free surface flow, Longuet-Higgins and Cokelet t1] introduced the boundary element method ~ abbreviated as BEM ~ based on Green's integral theorem, in which they used the mixed Eulerian- LagraIIgiall method in order to follow nodal points on the free surface by means of time marching integration. This scheme was modified by Vine and Greenhow t2], who employed BEM based on Caucl~y's integral theorem and highly accurate scl~e~e of time marching integra- 1 1; 225 tion. Similar schen~es were developped in last decade and iIIvestigatioIls of two dimensional nonlinear water wav including steepening or breaking have been carried out t3] t4] t5] t6] . In most cases, however, wave-making pheno~ll- ena around a body ir1 uniform flow A] have slot been tried. The main difficulty is that nodal points on the free surface can move according to the uniform flow compoIlerlt. In the present study, two problems are described; a semi-circular mound in shallow water, and a rectangu- lar floating body with semi-i~lfinite length. In tile for- mer problems, nodal points on the free surface will pass through the downstream boundary, and in the latter prob- lem, nodal points will concentrate at the stagnation region in front of the bow, if the suitable way cannot be found. In this paper, numerical treatments for these problems alla related ones are discussed, alla several numerical ex- amples are given. Some checks about the validity of the present numerical method are also given by comparing the results to the other theories and experilnellts, a',`:l ail\ varying the number of nodal points, tile size otcalcuJ ,ti -all region and tile time interval. 2. Basic Equations Two examples of nonlinear wave making pllenolllella around a two dimensional body in uniform flow are dis- cussed in this paper. I) Semi-circular mound in shallow water as in Fig. 1. 2) Rectangular floating body with sen~i-infinite length as in Fig. 2. IY N ~ 1 2 3 Free Surface | _ Ct C=C+uC~ Bottom so . h Cal , ~ Cal ~ J Fig. 1 Sell~i-circular mound ill shallow water. x

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The downstream boundary is open in tile first problem, and a stagnation region exists in front of the bow in the second problem. When using the numerical method ex- plained below, various numerical difficulties are arisen be- cause of these features of the problem. Numerical treat- ments for each of these problems are explained in later sections. ~ (y N ~ .2 3 ~ C ~ ,NF Ax _, '/'''`~//C//~"~/"' . Ci I I /////l/////f/Cp/i////////~`ti/////~/  Fig. 2 Rectangular floating body with semi-infinite length. The coordinate system is taken as shown in Fig. 1 or Fig. 2. For the sake of convenience, all equations in this paper are normalized by a characteristic length 1, and a uniform flow U. For example, the normalised time t is expressed as U/l x real time. In the first problem, the radius a of semi-circular mound can be chosen as the char- acteristic length, and in the second problem, the draught d can be chosen as the characteristic length. If we neglect the fluid viscosity and also assume irrotational motion, the governing equation of the flow field is J~aplace's e.~.~- tion. The complex velocity potential call be i,~trod''~d as follows, win; t) = Aid, y; t) + idle, y; t) (,1) z = ~ + is (2) where ~ is the velocity potential and ~ is the stream func- tion. If the contour C of clockwise direction is chosen and the singular point zo is considered on C as shown in Fig. 1 or Fig. 2, the following equation can be introduced by Cauchy's integral theorem, icxw~zo; t) + / (; Adz = 0 (3) c ZTo where cat = or, if the contour is smooth at zO. If ~ is given at z0 ~ C, part of C ), the real part of eq. (3) can be taken. On the contrary, if ~ is given at zo ~ C,~, part of C ), the imaginary part of eq. (3) call be taken. Taking the real and imaginary parts of eq.~3), following equations are obtained. -Ct?,~(Zo; t) + Re,/ ~ Adz = 0 on Cal, (4) (matzo; t) + Im,| ~ ; Adz = 0 on Cal (5) c ZTo These formulae are integral equations of Fredholm type of else second kind, in which ~ is unknown on C, (eq.~4~) and is unknown on Cat (eq.~5~) respectively. To solve these integral equations, nodal points are distributed along the contour C as shown in Fig. 1 or Fig. 2. We 1lave NF nodal points on the free surface which are movable and a total of N1D nodal points on the contour C. On each boundary element divided by these nodal points, it is as- sumed that the complex velocity potential w varies lin- early in z. Using the well known procedures by means of these linear boundary elements, the following discrete expressions are obtained with respect to eq. (4) and (5), where -NP ~ Re ~ Ek,nWn = 0 on ~ n=! -NP Im ~ Ek,nWn = n= ~1 '` (Gil _ O (~t C'7/~ ~ ~ ~ Zk Zn1 Zn Zk Zk Zn+1 Zn+-l [k,n = In + In Zn Zn~ Zn~ Zk Zn Zn+ -1 Zn ZkZk2 Zk1 Zk [k,k~ = ID Zk~ Zk - 2 Zk - 2 Zk 1 Zk+i Zk [k,k = no Zk1 Zk rk,k+1 = k Zk+2 n1 Zk+2 Zk Zk+1 Zk+2 Zk+1 Zk Zk Zk (8) (9) (10) (11) Since terms including known ~ and ~ remain in deft hand side of eq. (6) and (7), these terns must be transposed to the right hand side. A set of sin~ultaneous equations is thus obtained. Since the coefficient matrix of this equa- tion system has a property of diagonal superiority, Gauss- Seidel iterative method can be used to solve the sin~ulta- neous equation. In order to follow nodal points and velocity potential values on the free surface, the niixed Eulerian-Lagrallgian formulation is employed. Namely the dynamical free sur- face boundary condition is expressed as follows by using the material derivative dig = 1 {A + (~1~) }-COY+ 2' (12) where TO = gl/U2. In eq. (12), a uniform flow component is taken into consideration. The kinematical free suface boundary conditions are also expressed as follows. dx = 0+ = Re d- (13) Y = - =-Im (14) dt By dz Integrating the ordinary differential equaticlls (12)-~( l1) numerically with time increment, the `~-~-e profile CR.11 be obtained at each time step. Hanll~illg s predict.,r- corrector method is used as the time marching integral loll technique. This method needs the values at the first three time steps, which are calculated by means of Runge-Kutta method. Wave making drag acting on the mound or the rect- angular floating body can be estimated in each time step. Since the pressure coefficient is expressed as 226

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C P - Po = 1 - (ok) - (my) - 2^yoY - 2 bf (15) by Berno~lli's theorem, the coefficient of wave making drag is obtained by the following pressure integral, Cw= --lu2 =-2,iCpn~ds, (~16) where s is the girth of the mound or the wetted bow part of the floating body, and no is the direction cosine of outward normal on s to ~-axis. 3._Semi-circular Mound in S_allow Water 3.1 Treatments for Numerical Computations General descriptions of the numerical method used in this work are given in the preceding section. In practi- cal free surface computations, however, some difficulties caused by the uniform flow component must be settled. First the initial condition locust be given. In general, the initial condition is given by U = 0 at t = 0 and the flow is increased gradually to uniform to maintain the stability of the numerical computations. In the present study, however, the uniform flow is give at t = 0 as ~ = :e on C+, ~ = 0 on Cat,, (17) in order to eliminate the accelerations effect on to Fly--. This initial condition gives no influence Ott the stability- of the numerical computations. This procedure is effective also for saving CPU time. Since the nodal point NF shown in Fig. 1 can Gove according to the uniform flow component, the treatment of the downstream boundary is more difficult. If this boundary is fixed, nodal points on the downstream free surface will move out of the calculation region. In order to avoid this difficulty, a new nodal point is introduced on the upstream surface, and a nodal point which passed through the downstream boundary is deleted. The de- tailed numerical process can be described as follows: 1) As shown in Fig. 3, a fixed upstream boundary and an initial downstream boundary are given. On the free surface, NF nodal points which are movable with time marching are given. 2) The downstream boundary can Gove corresponding to the nodal point NF. Unless the nodal point NF- 1 pass through the initial downstream boundary, the computa- tion is continued without changing numerical procedures as shown in Fig.3 (1~. 3) If the nodal point NF1 passed through the initial boundary, the downstream boundary is changed to new position of the nodal point NF- 1 at next time step. At the same time, as in Fig. 3, the nodal point NF is deleted, the nodal point slumbers are replaced, and the new nodal point 1 is added between nodal point NP and 2. The position and the velocity potential of the new nodal point 1 is given simply like x~ = 2(\XNP+X2), Y! = 2(YNP+Y2), 1 = 2(NP+2~. (18) This replacement technique is convenient, because the cal- culation region can be kept almost the sense size. NP 1 2 3 1xF-3 I\F-2 1xF-1 NF ( 1 ) a= ,.-_ I I A ,: B C NP 1 2 3 NF-3 NF-2 NF-l NF del. It ~ t t t ~ t t NP 1 2 Nf-4 FF-3 FF-2 NF-l NF (2) 1' 1 A BD C A: Upstream Boundary ( f i x ) B: I ni t ial Downstream Boundary C: Downstream Boundary ( movable ) D: New Downstream Boundary Fig. 3 Treatment of nodal points on the free surface. For the movable downstream boundary-, the ~onditi.:'n for ~ must be given, because this boundary- is C. as in Fig. 1. In the finite difference method, the zero-extrapolation technique is ordinally used for an open boundary of this kind. In this analysis, however, BIDM is employed, and an- other way must be found. When the disturbance velocity potential ~ is introduced as ~ = X + A, IF = XNF + (PNF, (~19) the downstream conditions for ~ are given below, which utilize the solution form based on the linearized free sur- face condition. 1) If YNF > 0, for y ~ o, for y < 0, 2) andifyNF < 0, where by sinh kh + cosh kh IF kYNF sinh kh + cosh kh Oslo key + h) IF kYNF Sigh kh + cash kh (20) (21) = IF hl~k~y + h) (22) k - so tanh kh = 0. (23) These equations satisfy the condition of ~ = IF at y = YNF and the continuity conditioll, and eq. (20) is the linear extrapolation of eq. (21~. As described in the previous section, in order to follow the wave profile in each tinge step, the numerical integra- tions of differential equations (13), (14) are needed. For 227

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this purpose, dw/dz On tile free surface must be evalu- 2.D ated. Several methods are known, however, the following two simplest methods for approximations of dw/dz were used for the present problem. 1) Upstream difference 2) Downstream difference 3.2 Numerical Examples _ . .. . _ dw wnWn-1 (124) dz ZnZn-l dw we'Wn+1 (25, dz ZnZn+l Numerical examples are shown for the semi-circ~lar mound of radius a = 0.1m in the uniform flow of U = 0.5m/sec and the water depth ht = depth/a) = 0.25. Some examples are also given for the other speeds of urti- form flow or the other water depths. As the first example, else schemes for dw/dz on the free surface have to be examined numerically. The computed wave profiles based on the schemes of upstream difference arid downstream difference are shown in Fig. 4. Ilk this example, numerical conditions are chosen as; the position of downstream boundary Amid,, = -7.5' the position of up- stream boundary :Uma~ = 10.0' the number of nodal points on the free surface NF = 100 ~ length of all elements are equivalent ), and the time interval At = 0.05. Fig. 5 shows Glee wave drag coe~cier~ts for the sense cases. In Fig. 4 and 5, the scheme of downstream difference seems to be a suitable one for this problems. However, if ex- tending the wave height to vertical direction as in Fig.6, the reflected wave front the downstream boundary can be observed. For this reason, the upstream difference which can simulate the steep wave as in Fig. 4 is employed as the scheme for dw/dz. 1=3.50 ( 0.70 see ) DOWN-STREAM DIFFERENCE t=2.55 [ 0.51 see ) UP-STREAM DIFFERENCE ~ ~ 4 ,~ Fig. 4 Wave profiles based On tile scllellle of upstream difference and downstream difference. According to the numerical conditions, soluble examples are computed. Fig. 7 shows the cvu~p~ted wave profiles for U = 0.3, 0.4, 0.5m/sec, where h = 2.5 at Teal time = 228 Cw=Rw/[poU2) _ ~.6 '.2 o. 8 n ~ UP-STREAM D I FFERENCE / DO`N-STRERM D I FFEREN~ u.u- I.0 2.0 3.0 4.0 Fig. 5 Wave drag coefficients based on the schemes of upstream difference and downstream difference. 1=3.5 ~ 0.7 see ) Fig. 6 Wave profile based on tlte scheme of downstream difference. Go 4.O 0.5 m/s 0.4 m/s 0.3 m/s - ~ . O Fig. 7 Wave profiles for U = 0.3, 0.4, 0.5m/sec. ( h = 2.5, real time = 0.51sec ~ 0.51sec in all cases. The difference of wave steepening points can be simulated well ill this example. Fig. 8 shows the computed wave profiles for h = 1.25,2.5,5.0, where U = 0.5m/sec and t = 0.5. Fig. 9 shows flee wave drag coefficients for the same cases. The case of h = 1.25 is the limit of what call be computed by the present technique. As shown in Fig. 9, wave breaking occurs immediately after the wave profile ill Fig. 8. As is well known, numerical techniques based on BEM cannot simulate the wave after breaking.

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~ n -.51 -1. 0_ Fig. 8 Wave profiles for h = 1.25, 2.5, 5.0. ~ U = 0.5m/sec, t = 0.5 ~ Cw=Rw/ [po.U2 ) z . c 1.6 1 ~ D. 3 DEPTHS: I .25 1 0. 125 r ) DEPTHS: 2.50 ~ 0. 250 m ) f DEPTH/a: 5.00 1 0.-500 m ) 1 . 0 2. 0 3. 0 4. 0 Fig. 9 Wave drag coefficients for h = 1.25, 2.5, 5.0. ( U = 0.5m/sec ) 2. 2.0 1.61 1.2 . n R ~ _ . _ 0.4 ./ 0.0 _ Cw=Rw/ [poU2 ) or DECO. 05 / I\\ Dl=O. 07 ~ \~ Fig. 10 Wave drag coefficients for the cases of At = 0.03, 0.05, 0.07. 37t h st op t=2. 59 1 O. 518 see ) Dt=O. 07 1 ~~ 1 DEPTH/e : 5. 00 DEPTH/e : 2. SO _.._.._.._. DEPTH/e : 1. 25 Fig. 11 Wave profile for the case of At = 0.07. For the first example, ejects of the time interval are exemplified. The wave drag coefficients for the cases of lit = 0.03, 0.05, 0.07 do not show serious differences caused by the time interval ~ Fig. 10 ). In Fig. 11, however, the wave breaks unusual in case of lit = 0.07. Since the wave profiles for At = 0.03 and 0.05 are almost saline in this problem, At = 0.07 is considered as a rough time step for the present problem. 0.5~ n n -0.5 t.0_ 0.5 - 9. 0 -7.0 XMIN : -lO.0 XMIN : -7.S Fig. 12 Upstream free surfaces for the cases of xmin = -7.5 and-10.0. 2;~, T-0.40 T=t.20 r=2. Do T=2. 80 T=3. 1 2 Fig. 13 Time history of wave profile. ~ U = 0.5m/sec, Cumin = - 10.0, NF = 157, At = 0.04 ~ In some of tile examples, small undesirable ocillatioIIs can be seen on the upstream surface. Whell the numerical conditions are changed to Cumin = -10.0 alla NF = 115, those undesirable ocillations are suppressed ~ Fig. 12 ). In the final three examples, Cumin = -10.0, NF = 157, alla /\t = 0.04 are used ~ Fig. 13, 14 and 15 ). The com- puted time history of the wave evolution is given as in Fig. 13. At the final time step, the wave becomes very steep. In Fig. 14, this steep wave profile just before breaking is conspired with Else result based Ott the finite difference method by Miyata et. al.~84~9] Both results show a fairy good agreen~ellt. The wave profile before steepness pre- dicted by the present method is also compared with the experi~nelltal result by Miyata et.al.~8~9] in I?ig. 15. Since the experimental wave profile is replotted front the pub- lished photograph, small errors are probably included, anal 229

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the time step is not equivalent ifs both cases, because ini- tial conditions are different with each other. Though clear conclusions cannot be described for the above reasons, the present method can be regarded as one of the powerful simulation tools for real phenomena of steep wave. To NF= 157 DT=~. 04 CRL. BY H. ~ I YRTR .ol Fig. 14 Wave profiles by the present method and the finite difference method. ~ U = 0.5m/sec ~ ~ n NF= 157 DT=O. 04 EXP. BY H. MITRTR 1.01 Fig. 15 Wave profiles by Else present method and else experiment. ~ U = 0.5m/sec ~ 4. Rectangular Floating Body with Semi-infinite Length 4.1 Treatment for Numerical Computations As described in section 3.1, tile means of solving some numerical difficulties have to be given also for this prob- lem. If the deep water deftly is assumed, the boundary conditions of upstream and downstream are simply writ- ten as = x on the upstream boundary, (26) ~ = kit on the downstream boundary (27) respectively, because there is no free surface at tile down- stream boundary in this case. However tile condition (27) is not applied to the top and bottom nodal point at the downstream boundary, wl~ich are regarded as points of Cal on the body and the bottom in the present calculation. If flee water depth become shallower, the problem becomes more complicated and other considerations will be needed for both boundary conditions. In order to start the computation, an initial condition is needed. Though the uniform flow condition is employed on all C, region as in eq. (17), it is not able to be applied to the present problem, because of existence of stagna- tion point on the bow. For this type of tile flow field, the numerical solution of the double model flow can be intro- duced as the initial condition. The double model solution can be obtained by tile boundary conditions of = 0 on the body and tile free surface at rest, (28) ~ = -ah on the bottom (29) with eq. (26) and (27~. The obtained values of ~ on the free surface are employed as tl~e initial condition with the other boundary conditions. The most serious problem is caused by the existence of the stagnation region. Because of the uniform flow com- ponent, the nodal points will concentrate in front of the bow and the distance between nodal points NP and 1 will become larger and larger. This causes unusual wave profile around Else nodal point 1, which is shown in the subsequent section by a numerical example. In this case, the replacement technique of nodal points as explained in section 3.1 cannot be introduced, because the verti- cal boundary wills the nodal point NF is not movable and nodal points in front of the bow cannot be deleted in order to simulate the wave breaking. To counter this difficulty, long elements are introduced on y = 0 before the nodal point 1 as shown in Fig. 16. Following bound- ary conditions are imposed on nodal points on these long elements. fib = ;z on the long element region (30) These long elements act as wave suppression plates. ALONG ELEMENTS: Fig. 16 Long elements on tile upstream surface. In order to evaluate dw/dz on tile free surface, the following three methods are used. 1) Upstream difference ~ same as eq. (24) 2) Downstream difference ~ same as eq. (25) 3) Centered difference dw z7~_zW76_~} ~ZnAn- + Wz,,_zW7,+ - Inan+ = R] dz ~ZnAn-. + ~ZnAn+ -. Eq. (31) corresponds to tile weighted mean of the up- stream difference and the downstream difference. Nodal points on the free suface can be followed by the above numerical procedures. For the nodal point NF, how- ever, the horizontal velocity component Re~dw/dz) is ne- glected, because this nodal point must be restricted to move along tile bow. 4.2 Numerical Examples Before allowing several numerical examples, tile nu- merical accuracy of tile double model solution lllUSt be 230

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studied, because it is employed as the initial condition for the present problem. Two numerical solutions of Oman = 20.0 and 80.0, where Cumin = -10.07 are compared with the analytical solution based on Schwarz-Christvffel transfor- mation t10] in Fig. 17. The results are slightly different, though the same tendency of ~ is obtained. The extension of the upstream boundary does not improve the numeri- cal solution. Since the numerical solution is employed as the initial condition, it cannot be avoided that ~ includes small errors initially. However, initial flow velocities on the free surface are accurate, because the velocities are obtained from derivatives of . ''\ "amp "amp . -to As me, " "A '% ~ -- NUMERICaL SOLUTION: SHOD so.n -- aNaLrtlC SOLUTION ~ it SOLUT l ON : XHqX 20.0 ~ Fig. 17 Double model solutions. As discussed in section 3.2, several numerical treat- ments and conditions are studied. These are carried out as d = 0.1m and U = l.Om/sec ~ Fig. 18 ~ Fig. 24 ). First, the electiveness of long elements on the upstream free surface is verified. Long elements are arranged before :r = -10.0, :e = -10.0 ~ -8.0 is divided by 10 normal el- -8 -7 -6 WITH LONG ELEMENTS DO,'N-STRERM DIFFERENCE I ~ t18 -5 - 4 UP-STREAM D I FFERENCE t ~ 1.6 -2 _ 7 / ~ r I I I I I I I I ~-l -9 -8 -7 -6 -5 -4 -3 -2 - ~ x CENTERED D I FFERENCE I = Z6 / \ r I 1 4 1 1 ~ I I me HI > -9 -8 -7 -6 -5 -4 -3 -2 - ~ x Fig. 19 Wave profiles based on tile schemes of upstream, downstream and centered difference. ements, and x = -8.0 ~ 0.0 is divided also by 120 normal elements. By employment of these long elements, unnat- ural waves induced on the upstream surface, shown in the upper example of Fig. 18, are suppressed as in the lower example. In this example, the centered difference scheme is used to calculate dw/dz oft the free surface. The pre- dicted wave profiles at final time step by three schemes of downstream difference, upstream difference and centered difference are shown in Fig. 19. Iior the present prob- le~n, both the downstream difference and the upstream difference are not suitable, because computations failed 1' r WITHOUT LONG ELEMENTS i' -4 -3 -2 -9 -8 -7 -6 -5 -4 -3 -2 ~_- Fig. 18 Wave profiles for the cases without and with long elements on the upstream surface. ( t = 2.4, At = 0.04 ) 231 t

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without sufficient wave overturning. In the case of the centered difference, however, the plunging breaker can be simulated. The centered difference includes both informa- tion from upstream and downstream. In order to express the flow field in the stagnation region, both are needed. Differences in tlte wave drag coefficient are also observed with respect to the upstream difference and the centered difference as in Fig. 20. In all following examples, the scheme of centered difference is employed. Cw=Rw/ (Ed UZ ) 1.8 1.5 1.2 . 0.9 0-~:: 0.3; it/ UP-STREAM D I FFERENCE / ~ CENTERED D I FFERENCE Fig. 20 Wave drag coefficients based on the schemes of upstream and centered difference. Effects of the size of calculation region and the tilde interval are examined. Fig. 21 shows the wave profiles at t = 2.4 for the cases of Oman = 10, 20, 30, and Fig. 22 slows the wave drag coefficients for tl~e stance cases. The wave overturning point is closer to the bow of floating 1 ,.~ 1~ -2 - 1 XM9X- 3G. O - x ~ q x - 2 ~ . 0 Xerox- 1 a. . . 1. O n ~ x Fig. 21 Wave profiles for the cases of Oman = 10, 20, 30. 1VF = 130, t = 2.4, i\t = 0.04 ~ 232 body as ~ma: increases. On the contrary, the extension of the upstream boundary has no effect for the numerical solution. Cw=Rw/ Spa Us ~ . . _ 1.0 . 0.~- . // 0.6. / ~ 4 . 0.2 . 0.0- ~ XMqX- I 0. 0 XM4X=20. 0 - xmQx- 30. 0 , , , , , t 1.0 2.0 3.0 4.0 Fig. 22 Wave drag coefficients for the cases of xmac = 10, 20, 30. Fig. 23 shows the wave drag coefficients for the cases of At = 0.02, 0.04, 0.08, Slid 0.12, where Oman = 20. At At = 0.02 and 0.04, almost the sense results are obtained. As in Fig. 24, the detailed simulation of wave breaking is Flown, where t = 0~ 2.6 and /\t = ().02. Cw=Rw/ (pU U2 ~ ,~,f-# - DT=O. 02 i' ~ DT=O. 04 - DT=O. 08 DT=O. 12 0.2 Fig. 23 Wave drag coefficients for the cases of /\t = 0.02, 0.04, 0.08, 0.12.

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/ \ r 1 x -1 Fig. 24 Wave breaking simulation. U = l.Om/sec,xma~ = 20, t = 0 ~ 2.6, /` t = 0.02 ~ According to the above studies about numerical treat- ments and conditions, simulations of mave making phe- nomena are carried out for Fa = U/~/~d = 0.5, 0.8 and 1.0 as in Fig. 25, 26 and 27 respectively. In the case of Fa = 0.5 a spilling breaker is obtained, though the other cases show plunging breakers. Since reliable experimental data is not available, these solutions cannot be compared with experiments. However, under the nurr~erical treat- ment that the numerical solution of double model flow is employed as the initial condition, these solutions must be considered as accurate ones, if the above mentioned studies are acceptable. \ r 1 _ ~ x .. -1 Fig. 25 Wave breaking simulation for F.' = 0.5. ~ t = 0 ~ 0.9, fit = 0.02 ~ I r 1 A/ -2 x . -1 Fig. 26 Wave breaking simulation for F,l = 0.8. ~ t = 0 ~ 1.92, At = 0.032 ~ ~~ Y 1 /,'\/: y -1 Fig. 27 Wave breaking simulation for Ed = 1.0. ~ t = 0 ~ 2.4, At = 0.04 ~ The present results are compared with the other the- oretical and numerical results. Dagan and Tulin t11] ob- tained the wave profile ill front of the rectangular body by a perturbation method based on small Froude number expansion. In Fig. 28, wave height at the bow ~7 based on the present method is plotted for If, with their re- sult. The present result is not equivalent to ~7 = 0.5F,' by Dagan alla Tulin. Wave steepening is related to their sec- ond order solution, but wave breaking phenomena cannot be explained by their analytical approaches. Finally as shown in Fig. 29, wave profile for F`' = 1.0 is compared with the result based on the similar method by (;rosen- baugl~ and Yeung t73. Fairy good agreement is observed except the sharpness of overturning waves. 233

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1.0~ - 0.5 PRESENT CAL. PAGAN &TULIN . / / / , , 0 0.5 Ed 1.0 Fig. 28 Wave height at the bow by the present method and by Dagan and l'ulin. A\ r - 1 /. - : PRSENT CAL. GROSENBAUGH & YEUNG 1 Fig. 29 Wave profiles for F`` = 1.0 by the present method t4] and by Grosenbaugl~ and Yeung. 5.Conclusiot1 In the present study, flee simulation method of two dimensional nonlinear water waves based on BEM and the mixed Eulerian-Lagrangian approach is applied to two wave making problems around a body ifs uniform flow; the sen~i-circular mound in shallow water and the rectangular floating body with semi-infinite length. For numerical difficulties caused by respective prob- lems, some treatments are given and those effectiveness are confirmed numerically by several examples. The pre- sent method can simulate the nonlinear wave snaking phe- nomena including steepening or breaking, but cannot sim- ulate the wave after breaking. In the case of the floating body problem, overturnig waves ~ plunging breaker ~ in front of the bow can be simulated. Numerical validations 234 of the present method are shown by examples for several cases with numbers of nodal points, sizes of calculation region, and time intervals. For the detailed experimental verifications, reliable data is needed. The present method, however, can be regarded as one of the powerful simula- tion tools for the nonlinear wave making phenomena. The present method can be extended to general problems of two dimensional wave making phenomena. Acknowledgement The author wishes to express his deep appreciation to Prof. M. Ikehata and Emeritus Prof. H. Maruo of Yoko- han~a National University for their useful suggestions and encouragements. Allis paper was written flails at flee Uni- versity of British Columbia under the financial support of the Government of Canada Award. The author would also like to express his deep gratitude to Prof. SKI. Calisal of U.B.C. and the World University Service of Canada. He thanks also Mr. D. McGreer and Dr. J.L.K. Chan of U.B.C., and Mr. D. Jimbo, Mr. N. Nakajilua, Mr. S. Masuda and Mr. K. Furusawa of Y.N.U. for their kind cooperations. References [1] Longuet-Higgins, M.S. and Cokelet, E.D.: "The de- forn~ation of Steep Surface Waves on Water, I. A Nu- r~lerical Method of Computation", Proc. R. Soc. (A), 350 (1976) [2] Vinje, T. and Brevig, P.: "Nonlinear Ship Motions", Proc. 3rd Illt. Conf. Numerical Ship Hydrodynan~cs (1981). [3] Greenhow, M. and villje, T.: " Extreme Wave Forces on Submerged Wave Energy Devices", Ap- plied Ocean Res., Vol. 4, No. 4 (1982). Lin, W.-N., Newman, J.N. and Yue, D.K.: "Nonlin- ear Forced Motions of Floating Bodies", 15th Syrup. On Naval Hydrodynamics ( 1984). t5] Takagi, K., Naito, S. and Nakal~'ura, S.: "Co~npu- tatiorl of Nonlinear Hydrodynamic Forces on Two- Dimensional Body by Boundary Element Method", Journal of Kansai Soc. of Naval Archtects of Japan, Vol. 197 (1985). ~ in Japanese ) t6] Schultz, W.W., Ra~berg, S.E. and Gri~n, O.M. : "Steep and Breaking Deep Water Waves", 16th Symp. on Naval Hydrodynamics ( 1986 ). A] Grosenbough, M.A. and Yeung, R.W.: "Nonlinear Bow Flows - An Experimental and Theoretical I~- vestigation", 17th Sync. on Naval Hydrodyrlarnics (1988). t8] Miyata, M., Matsukawa, C. and Kajitalli, H.: "A Separating Flow near the Free Surface", Osaka Int. Colloquim on Ship Viscous Flow ( 1985).

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[9] Miyata, M., Matsukawa, C. and Kajitani, H. : 6th Ed., p287. "Shallow Water f low with Separation and Breaking Wave", Jounal of Soc. of Naval Archtects of Japan, Vol. 158 ~ 1985 j. t10] Milrte-Thotnso~n " Theoretical Hydrodynamics", t11] Pagan, G. and Tulin, M.P.: "Nonlinear Free-Surface Effects in the Vicinity of BluIlt Ship Bows", 8th Snap. on Naval Hydrodynamics (1970~. 235

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DISCUSSION by R.C. Ertekin I think your paper lacks quite important references on the upstream waves that can be seen in your figures. It is well known by now (see the three papers by Bai et al.; Choi and Mei; and Ertekin & Qian) that when a disturbance moves in finite depth, then upstream waves (solitons) will be generated if the blockage coefficient is significantly high (like yours) and the depth Froude number is not very small (>0.2). So the upstream waves that you obtain are not necessarily "undesirable oscillations" but a gift of nature. By the way, you are solving Laplace's equation and there is no difference between the body moving (steady) in an otherwise calm water and the fixed body placed in an uniform oncoming flow. With regard to the "open-boundary" conditions you can very well calculate the phase speed at these boundaries and use Orlanski's scheme coupled with the Sommerfeld's radiation condition. Your results show that your "open-boundary" conditions are reflective. Author's Reply In the case of solitons, the waves propagate from the body to the upstream. In my case, however, the upstream waves appear around the nodal point 1 and propagate to the downstream direction. It is caused by the numerical technique of the addition of new nodal point 1 and can be avoided by the extension of the calculation region. In the present case, the wave breaking occurs before the generation of solitons. In near future, I would like to simulate the soliton by the present technique. Exactly speaking, your opinion about the radiation condition is right. For the practical use, however, we usually need the simple and numerical radiation condition. For example, in the research field of Rankine source method, several numerical radiation conditions are employed. In the present method, the combined technique of the upstream difference approximation of dw/dz, the replacement of nodal points on the free surface and the employment of the linear solution form at the downstream boundary can be expected as the numerical radiation condition. DISCUSSION by C.G. Kang Usually there is singular behavior at the intersection point between the body and the free surface. Even if the potential and the stream function are not singular at the point, the velocity is singular when the intersection angle is not 90 degrees. Could you show us how to remove the singular behavior? Greenhow showed that the solution using fine grids is poorer than that using coarse grids. Did you check the convergence of the velocity at the intersection point? Author's Reply As described in my paper, the intersection point NF is treated as the free surface nodal point, and its horizontal velocity component calculated by Re(dw/dz) is ignored. Along the bow, only this intersection point is movable, that is, the other nodal points on the bow under the free surface are fixed. In this approximation, the velocity at the intersection can be obtained without difficulty. DISCUSSION by J.H. Hwang I congratulate on your fine presentation. Your calculation is seemed to be basically based on Vinje-Brevig method. Could you give some comments on major advantages of your calculation in the numerical scheme including the treatment of the intersection point between the free surface and the body. Author's Reply As described in my paper, the intersection point NF is treated as the free surface nodal point, and its horizontal velocity component calculated by Re(dw/dz) is ignored. Along the bow, only this intersection point is movable, that is, the other nodal points on the bow under the free surface are fixed. In this approximation, the velocity at the intersection can be obtained without difficulty. Discussion by J.W. Kim We would like to comment on your treatment of the downstream condition and your finite difference schemes. 236

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The downstream condition given in Eqs.(20)-(22) is based on the steady linear solution. But your calculation is made on an unsteady problem. In a transient stage many components of waves with different wave lengths are evolved and eventually hit the downstream boundary. The wave components which do not satisfy the dispersion relation in (23) will be reflected back to the computational domain. Even for the wave components satisfying the equation (23), this equation cannot distinguish incoming or out-going waves with respect to the computation domain. You have tried various difference schemes in your paper and the final choice was made from computational results. We do not understand how one can choose a specific finite difference scheme if we don't know the correct result in advance. We strongly believe that one should decide a certain numerical scheme for a given problem based on rational mathematical analysis, not after comparing with the known result. Author's Reply Strictly speaking, your comments are true. For numerical treatments in my paper, however, it is not suitable to discuss separately the downstream condition and the finite differnce schemes of dw/dz. These treatments connect with each other through eqs.(12)-(14), that is, ~NF in eqs.(20)-(22), which is time dependent variable, is determined from egs.(12) and (19). In this treatment, the position of down stream boundary is not fixed. If we find a suitable way to estimate the wave number as a time dependent variable, these numerical treatments will be improved more precisely. We should not pursue an ideal, but find more convenient way. 237

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